http://2014.igem.org/wiki/index.php?title=Special:Contributions/Jan_glx&feed=atom&limit=50&target=Jan_glx&year=&month=2014.igem.org - User contributions [en]2024-03-29T08:33:15ZFrom 2014.igem.orgMediaWiki 1.16.5http://2014.igem.org/File:Team_Heidelberg_RFC_draft.pdfFile:Team Heidelberg RFC draft.pdf2015-01-09T20:02:11Z<p>Jan glx: uploaded a new version of &quot;File:Team Heidelberg RFC draft.pdf&quot;: now for real</p>
<hr />
<div>RFC [i] as submitted to the BioBrick Foundation.</div>Jan glxhttp://2014.igem.org/File:Team_Heidelberg_RFC_draft.pdfFile:Team Heidelberg RFC draft.pdf2015-01-09T19:58:56Z<p>Jan glx: uploaded a new version of &quot;File:Team Heidelberg RFC draft.pdf&quot;: Added RFC number
Fixed some spellings</p>
<hr />
<div>RFC [i] as submitted to the BioBrick Foundation.</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Linker-Software_DocuTeam:Heidelberg/pages/Linker-Software Docu2014-10-18T03:58:24Z<p>Jan glx: /* Bundeled version */</p>
<hr />
<div>=General=<br />
<br />
During the iGEM competition we have written a software, that can predict the best linker to circularize a protein. At first, connections between the ends of proteins are found, these are weighted on their quality for the linker and then these paths are retranslated to biological sequences.<br />
The software is mainly made possible by python's numpy package for easily handling and processing huge amount of data. Numpy is one of the most used python packages in scientific computing, providing a powerful N-dimensional array object and fast C/C++ written functions to process them. Thus we were able to handle the huge amount of different linker paths (in the scale of 10^9 paths). <br />
Python, as a high level programming language with its various packages, enabled us to write such a powerfull software within the short time period of this years iGEM competition. Being an interpreter language, python's runtime of course is much higher. As python is able to integrate fast C-Code natively, the runtime of Numpy calculations is not that much higher than compared to classical precompiled C-code. This is achieved by using the same type of entries in an array, so that the whole array can be processed at once. On the other hand this consumes much more memory, since the whole array has to be loaded in the RAM in every processing step, causing one of the major problems for us. Therefore, the software requires at least 2 GB of free RAM.<br />
<br />
==Usage==<br />
<br />
Two different versions can be downloaded. One is a bundeled version with reduced features, but enhanced usability. The other is the source code, where in the beginning definitions need to be made in the code, giving the full usability at hand of the user. If not all features are needed, we recommend using the reduced version. The bundeled version can be used only on linux PCs.<br />
Both versions require minimum 2GB of free RAM. <br />
<br />
===Bundeled version===<br />
<br />
Please download CRAUT from [https://github.com/igemsoftware/Heidelberg_2014 here ]here. You only need to add the protein structure file in PDB format and an instructions file. For detailed information please see the README in the folder.<br />
<br />
The bundeled version is nicer accessible more user friendly, butdoes not provide full usability. Mainly, it is missing adjustability. Some functions that can not be used are:<br />
<br />
Weighting of different aminoacids: A function, which allows you to define aminoacids and regions that should be omitted by the linker.<br />
<br />
Checkpointing: The program has to run in total at once, it can not be stopped during the calculation, because no files are stored during computation. Calculations can take up to days.<br />
<br />
Size of linkers: As no checkpointing is enabled in this version, also the maximum angles are reduced to two angles in the linker. This shold be sufficient for most proteins. but If this does not produce results, please refer to the python version from the website.<br />
<br />
Circularizing only a part of the protein: The complete protein, provided by the PDB file is circularized, in contrast to the the complete version, where the user can define the amino acid site where the protein circularization should occure.<br />
<br />
Ignoring different parts of proteins: It is not possible to ignore additional parts of the PDB. Only the defined subunit will be processed, other subunis are ignored.<br />
<br />
Attachment-Sequences: The exchange of attachment-sequences (normally GG in the beginning of the linker) has to be done manually.<br />
<br />
===The python file===<br />
<br />
For this version you need a running python 2.7 environment, please see below on the used packages. Please download the .py file from [[###here###]] and insert the data in the header of the code to the different variables.<br />
<br />
Two folders have to be creaded in the same directory as the python program is running, one named "files", in which checkpoints are saved, and one called "protein", which contains the PDB file.<br />
<br />
There can be made several choices on which parts of the protein should be taken into account for circularization and with which sequences this should be done. The extein sequence or the sortase scar should be inserted at ScarsAtEnd variable.<br />
Also a weighting of certain amino acids can be defined in the beginning.<br />
<br />
The software will store temporary data after reaching a processing checkpoint so that calculation can be resumed afterwards again. The code is split in different procedures, that can be run in IDEs like [https://pythonhosted.org/spyder/overview.html spyder ] separately. Each of these sections defines a piece of code, <br />
<br />
One <br />
Afterwards the calculations can just be resumed at one of the various pickle.load checkpoints.<br />
<br />
For most proteins, not too many paths need to be checked, which results to a runtime of so it runs in several hours on a 2.6GHz intel i5. The software might calculate several days for the linkers. For big proteins and long linkers, about 8 GB of RAM are needed to run completely. But for<br />
<br />
In the end a single results file is saved on the disk, containing the sequnces of the linkers and the weightings of them.<br />
<br />
=Biggest problems we had=<br />
<br />
We encountered many bigger or smaller problems while programming. Some are quite serious issues and are mainly due to the brute force approach ansatz we made, but were mainly solved to an acceptable extent. <br />
<br />
==RAM usage==<br />
<br />
For example w The calculation of the distance from the connection in a path to the protein is calculated, encompasses always the distance of all of the atoms of the protein to this connection. This easily results in arrays of 100 000 000 * 3 * 6000 * 3 shape can occur, which is just too much for normal RAM sized Computer. <br />
<br />
On the other hand, using numpy arrays, the bigger the processed arrays are , the faster the program is in total, because every functions only have to be loaded once . Therefore in each step the arrays are sliced in a way, that the following procedure can take place in the RAM of the computer, see [[#make_small_generator_offset(listofarraysinRAM, PointArray, repetition, RAM, tobesplitlength, ProteinArray = None):|functions]] part.<br />
<br />
===Array Size===<br />
<br />
Arrays with about 300 000 000 float 16 entries, like they occur for large proteins, while [[software-representation#Point generation|point generation]] are too large in size for the RAM. Thus we had to manually store them on the harddrive for processing. Of course also this slowed the procedure down. But fortunately we could use python's [http://www.h5py.org/ |h5py] package, which allowed us, just taking out certain lines from the array stored on the disk.<br />
<br />
==Runtime==<br />
<br />
The longest calculation took about 11days on a 2.6 GHz intel i5 CPU with 8GB RAM and an SSD harddrive. Therefore, results are frequently stored on the harddisk using python's [[#cpickle|cpickle]] package, which allows a fast storage of complete numpy objects. Thus calculations can be stopped and restarted after certain points.<br />
On the other hand, calculation time increases with the number of points for the connections and for the protein. Reducing the points of the protein to the atoms on the surface would be the next step to take, which could reduce calculation time to one sixth of the time now.<br />
Due to lack of time, we often had to make the tradeoff between fast programming and fast calculation. Having the huge resources of [[iGEM@home|###i@h###]] we clearly decided for fast programming and not optimizing the code yet for velocity.<br />
<br />
==Flexible ends==<br />
<br />
Long flexible (non helical) regions might cause problems until now are kind of an issue, but we have implemented several functions that should handle those. The problem is, that tThe number of possible conformations of two or three OF WHAT? exceedes our capabilities and could not be easily handeled in the building-block system we choose to implement. Therefore also helical regions are handeled as straight connections, but with varying length. After the flexible regions there are no angular restraints given to the attached helical block.<br />
<br />
==Path storage==<br />
<br />
The possible paths are always stored as the angle points of the paths under following variables: firstpointsflexible, secondpointsflexible, thirdpointsflexible, firstpointstriangle, secondpointstriangle, thirdpointstriangle, erstepunkte, zweitepunkte and drittepunkte. As each point in 3d has three coordinates, all of these variables are n*3 arrays. The first index identifies the path then. This creates the possibility it possible, that a path is just deleted, to erase a path by deleting the line of all the arrays. This way also the arrays can be easily sliced, making it possible to process parts of an array in a fast way.<br />
<br />
==Differences in proteins==<br />
<br />
Proteins can differ widely in their size and sometimes even in their definition in the PDB files. It was a big issue we encountered in setting up to set up a stable version running with all these different PDBs varying in size by the factor of 20.<br />
<br />
=Different definitions=<br />
<br />
==Protein data==<br />
<br />
In the PDB file, all coordinates of non-hydrogen atoms are stored. These are then loaded into arrays x, y and z, just containing one coordinate of each a point. these are arrays of length n. Since this kind of array is complicated to handle, the information is restored in different point arrays of shape n*3.<br />
<br />
*PointsOfAllSubunits: These are all points from the PDB file, that should not be ignored. The user has to decide, which parts of the protein should not be taken into account.<br />
<br />
*pkte: These are all the points of part ???, that should be circularized, so this are all the atom-coordinates between N- and C-terminus<br />
<br />
*OtherPoints: These are all points of PointsOfAllSubunits that are not in pkte.<br />
<br />
<br />
==List of angles and rods==<br />
<br />
Rods: "AEAAAK", "AEAAAKA", "AEAAAKAA", "AEAAAKEAAAK", "AEAAAKEAAAKA", "AEAAAKEAAAKEAAAKA", "AEAAAKEAAAKEAAAKEAAAKA", "AEAAAKEAAAKEAAAKEAAAKEAAAKA"<br />
<br />
Angles, alwys mean, then std and the pattern: [(29.7, 8.5, "NVL"), (38.7, 30., "KTA"), (60., 12., "AADGTL"), (74.5, 27., "VNLTA"), (117., 12., "AAAHPEA"), (140., 15., "ASLPAA"), (160., 5., "ATGDLA")]<br />
<br />
==General definitions==<br />
<br />
*minabstand: the radius of an alpha-helix, also the minimal distance an atom needs from a connection<br />
<br />
*LengthOfAngle: The distance between the end of an angle pattern and the geometrical turning point. <br />
<br />
*LengthOfFlexibleAA: 3.5 &Aring;<br />
<br />
*FlexAtStartSeq and FlexAtEndSeq: This is not only the regions missing from the PDB on both sides, but are also the flexible endsof the linker and the extein on the other side.<br />
<br />
<br />
=Functions=<br />
<br />
==imported python modules==<br />
<br />
===necessary===<br />
numpy: the basic module for numerical calculations in large scale<br />
h5py: Storing arrays on the harddisk, allowes slicing of the arrays on the disk.<br />
os: Used for reading and writing files<br />
sys: module used for exiting the program at certain points.<br />
<br />
<br />
===recommended===<br />
matplotlib.pyplot: can be used for what<br />
from mpl_toolkits.mplot3d, Axes3D: for what<br />
time: is nedd to see the progression of the calculations and observe the calculation times.<br />
fnmatch: allows wildcard search in strings, important for finding specific linker patterns<br />
cPickle: used for intermediate storing of the arrays, so that calculations coul be continued after restarting the program.<br />
<br />
<br />
==selfwritten functions==<br />
<br />
===angle_between_connections_array(startingarray, middlearray, endingarray):===<br />
<br />
<br />
:: calculates the angles between the vectors from startingarray to middlearray and middlearray to endingarray. If there is no displacement between the arrays it returns zero as angle. startingarray and endingarray can be only one single point, middlearray should always be an array of points in 3d space.<br />
:: returns values between [0,pi] in an array of size Startarray.<br />
<br />
----<br />
<br />
===angle_between_vectors(vect1, vect2):===<br />
<br />
:: calculates the angle between two arrays of vectors. If one of the vectors is 0, the angle is set to 0. The result is based on arccos.<br />
:: returns the angles between two vectors.<br />
----<br />
<br />
<br />
===distance_from_connection(Startarray, Endarray, Points):===<br />
<br />
::takes a connection from Startarray to Endarray and calculates the perpendicular distance of the points from the connection. Startarray or Endarray can also be single points.<br />
::returns an array of size (Startarray * points) with all perpendicular distances or distances of the endpoints.<br />
----<br />
<br />
===punktebeigerade(minabstand, pkte, gerade, aufpunkt, laenge):===<br />
<br />
::checks whether there are points too close to a straight line coming from aufpunkt with in direction of gerade with length laenge.<br />
::returns True if no point of pkte is closer to the straight line than minabstand<br />
----<br />
<br />
===test_accessible_angles(winkelarray, length, anfangspunkt, proteinpoints, gerade=np.array([0, 0, 1])):===<br />
<br />
:: winkelarray is an array of angles that should be checked, whether they are accessible from anfangspunkt. Accessible means that no point of proteinpoints is too close to the straight line, which is produced by rotating gerade with the angles of winkelarray. Gerade always starts at anfangspunkt angles are measured from z-axis, if gerade is not defined else.<br />
::returns a boolean array with which winkelarray can be sliced.<br />
<br />
----<br />
<br />
===reduce_angles_from_redundancies(winkelarray):===<br />
::takes an array of angles in the format [phi, theta] and looks which angles produce the same result in the vector.<br />
::returns an array with all indices, that can be deleted along the 0 axis of winkelarray.<br />
----<br />
===make_displacements(lengtharray, displacementarray):===<br />
<br />
::generates all possible displacements from displacementarray (an array of vectors) and lengtharray (array of different lengths)<br />
::returns an array with displacementvectors in different lengths<br />
<br />
----<br />
<br />
===sort_out_by_protein(startingarray, endingarray, proteinpoints, mindist, beforearray = None):===<br />
<br />
::sorts out the connections between startingarray and endingarray with proteinpoints. A connection is sorted out, if one point of the proteinpoints is nearer to the connection, than mindist.<br />
::returns only the points for the connections, that are good. If beforearray is set, returns also beforearray<br />
<br />
----<br />
<br />
===naechstepunkte(anfangsarray, verschiebungsarray):===<br />
<br />
::generates for each point of anfangsarray, all points that are made by displacements of that point with verschiebungsarray.<br />
::returns two arrays of equal size, the enlarged anfangsarray and the array resulting from verschiebungsarray.<br />
<br />
----<br />
<br />
===aussortierennachpunken(punktearray, proteinpunkte, minabstand, maxabstand):===<br />
<br />
::sorts all the points of punktearray out, that are nearer than minabstand to one of the points from proteinpunkte, or farther away than maxabstand.<br />
<br />
::returns a boolean array, with which one can slice punktearray.<br />
<br />
----<br />
<br />
===angle_weighing(anglearray, angletosequence=angletosequence):===<br />
<br />
::weighting of the angles form anglearray. The better an angle fits to the angles provided by angletosequence, the lower the value is. The best angle gets a weighing of 1, the worst angle of 2. <br />
::returns a weighingarray for the angles of anglearray. Each weighing is in the range between 1 and 2. The weighing is based on gaussian distributions.<br />
<br />
----<br />
<br />
===angle_function(StartingArray, MiddleArray, EndingArray):===<br />
<br />
::makes a weighing of the connection from Startingarray, over Middlearray to Endingarray based on the weighing of the angles.<br />
::returns an angle weighting for each connection.<br />
<br />
----<br />
<br />
===unpreferable_places(Start, End, ProteinPoints, AminoacidNumberArray, ToBeWeighedAAInput, WeighingofAA, substratelist):===<br />
<br />
::Calculates a weighting for the connection from the points of Start to the points of End based on the distance from regions that should be omitted. These aminoacids should be defined in the ToBeWeighedAAInput array and the WeighingofAA array defines how important this region is. If one wants whole substrates to be omitted, they should be added in the substratelist. The total returned number is normalized, so the weighting of the regions is independent of the number of places, that should be omitted.<br />
<br />
::Parameters:<br />
::::Start: The points where the rod starts,<br />
::::End: The points where the rod ends<br />
::::ProteinPoints: The points of the protein, that should be taken into account.<br />
::::AminoacidNumberArray: The array, that tells, to which amino acid one atom belongs<br />
::::ToBeWeighedAAInput: One output of make_weighing_arrays<br />
::::substratelist: A list of tuples (amino acid nr, size of substrate). Amino acid nr, is the amino acid, where the substrate binds to.<br />
<br />
::returns the weighing of the connections, because of the regions, where the linker passes through.<br />
<br />
----<br />
<br />
===distance_from_surface(beforearray, testarray, ProteinPoints, Afterpoint = None):===<br />
<br />
::calculates the distances of the testarray points from the surface as just the minimum of the distances to all proteinpoints. It doesn't calculate the points that are equal to the points of the beforearray, so that these are not taken twice. Additionally, it checks that the points don't lie on the endpoint.<br />
<br />
::Returns the weighting of the distance by subtracting mindist, dividing it through mindist for making it unitless and then squaring, so that the values are better distributed. <br />
<br />
----<br />
<br />
===weighing_function_rigids(StartPoint, FirstArray, SecondArray, ThirdArray, EndPoint, ProteinPoints,AminoacidNumberArray, ToBeWeighedAA, WeighingofAA=None, substratelist=None):===<br />
<br />
::makes the weighting of rigid linkers, with angle, distance, length and regions distribution.<br />
::returns a list of 5 arrays: weighedvalue, normed lenghtweighing, Angleweighing, Siteinfluence and the distances<br />
<br />
----<br />
<br />
===weighing_function_flex(StartPoint, FirstArray, SecondArray, ThirdArray, EndPoint, ProteinPoints, AminoacidNumberArray, ToBeWeighedAA, WeighingofAA = None, substratelist = None):===<br />
<br />
::makes the weighting of flexible linkers, with angle, distance, length and regions distribution.<br />
::returns a list of 5 arrays: weighedvalue, normed lenghtweighing, Angleweighing, Siteinfluence and the distances<br />
<br />
----<br />
<br />
===make_weighingarrays(Userstring):===<br />
<br />
::Userstring is of the shape: 273,10 280-290,5 298,7,35.6 etc. (spaces separate entries, "," is for single residues "-" for anges, second "," for the diameter of the substrate)<br />
<br />
::If nothing should be weighted, insert ""<br />
::returns the information in arrayform (Shouldbeweighed and Weighingarray) and a substratelist<br />
<br />
----<br />
<br />
===sort_out_by_angle (startingarray, middlearray, endingarray, angletosequence):===<br />
<br />
::sorts out the paths from startingarray over middlearray to endingarray. A path is sorted out, when the angle it would need is too far away from the possible angles in angletosequence<br />
::returns a boolian array which paths to keep, middle and endingarray must have same dimension, if startingarray is only one point, it returns only middlearray and endingarray, else all three arrays are returned<br />
<br />
----<br />
<br />
===make_small_generator(PointArray, repetition, RAM, tobesplitlength, ProteinArray = None):===<br />
<br />
::calculates how often PointArray needs to be split so that the following calculations still fit into the RAM.<br />
<br />
::RAM in GByte<br />
::repetition means how often is the largest array repeated. Repetition must be manually found and adjusted as the real amount of repetitions is only a hint.<br />
::returns MakeSmall and teiler<br />
<br />
----<br />
<br />
===make_small_generator_offset(listofarraysinRAM, PointArray, repetition, RAM, tobesplitlength, ProteinArray = None):===<br />
<br />
::calculates how often PointArray needs to be split so that the following calculations still fit into the RAM.<br />
::In the listofarraysinRAM can be either just the arrays or the size of the arrays, same for PointArray <br />
<br />
::RAM in GByte,<br />
::repetition means how often is the largest array repeated. Repetition must be manually found and adjusted as the real amount of repetitions is only a hint.<br />
<br />
::returns MakeSmall and teiler<br />
<br />
----<br />
<br />
===sort_out_by_distance(startingpoints, endingpoints, firstpoints, distance, variation):===<br />
<br />
::generates all possible connections from startingpoints to endingpoints, that lie in one of the distances plus minus the variation.<br />
::returns three arrays with all possible paths, made out of all possible combinations startingpoints to endingpoints that are in a certain distance<br />
<br />
----<br />
<br />
===sort_out_by_length (comefrompoints, gotopoints, linkerlaengen):===<br />
<br />
::sorts out the connections between comefrompoints and gotopoints, when they don't fit to the linkerlengths from linkerlaengen.<br />
<br />
::Either comefrompoints or gotopoints can be only one point, but never both of them.<br />
::returns a boolean array, with which you can slice the points, True means the values are kept<br />
<br />
----<br />
<br />
===length_to_sequence(lengtharray, linkerdatenbank, linkerlaengen):===<br />
<br />
::translates the length from lengtharray to sequences according to the different linkerpieces in linkerdatenbank.<br />
::returns an array of the sequences that reproduce the length<br />
<br />
----<br />
<br />
===angle_to_sequence(anglearray, angletosequence, angleseparators):===<br />
<br />
::translates the angles from anglearray to sequences according to the different angletosequence data.<br />
::returns an array of the sequences that reproduce the angles<br />
<br />
----<br />
<br />
===translate_paths_to_sequences(startpoint, firstflex, secondflex, thirdflex, firstrig, secondrig, thirdrig, endpoint, linkerdb, linkerlKO, angletosequence, angleseparators, weightflex, weightrig):===<br />
<br />
::translates all paths to sequences according to the patterns provided in angleosequence and linkerdb<br />
::returns an array with sequences for each path<br />
----</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:54:06Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology [[#References| [5]]]. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions [[#References| [4]]]. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect [[#References| [5]]], that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
Notice that many methods for parameter estimation in these types of models have been developed [[#References| [6]]] [[#References| [7]]].<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. A more detailed description on the model development can be found [[Team:Heidelberg/Modeling/Enzyme_Modeling_detailed|here]]. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T * V^{lysozyme}_{Max, 37.0}$. Mathematically this just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the ord1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 1 min incubation at 44.5 °C and for diminished activity after 10 min incubation at 54 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the ord1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software [[Team:Heidelberg/Project/Linker_Screening|here]].<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_detailedTeam:Heidelberg/pages/Enzyme Modeling detailed2014-10-18T03:53:02Z<p>Jan glx: </p>
<hr />
<div>=The long way to the model=<br />
<br />
For long time we have thought it would be easy to extract the relevant data out of our lysozyme assays. But this though somehow perished one week before wiki freeze. For showing that science is always trying and failing, we wanted to explain how we came up with the upper model in the next part.<br />
<br />
==First try, easy fitting==<br />
<br />
The first thought when looking to the curves was that the reaction was clearly exponentially with some basal substrate decay and a small offset due to the different types of proteinmix added. The relevant parameter for us would be only the exponente which would be equal to some constant k times the enzyme concentration present. We would assume, that the constant doesn't change after heatshock, but the part of enzyme that survived. This assumption is based on a paper by Di Paolo et al., who claim that for pH denaturation of $\lambda$-lysozyme there are only two transition states, folded and unfolded. [[#References|[1]]]<br />
This way curves for the temperature decay can be measured for each kind of lysozyme and finally compared to each other.<br />
<br />
A basic problem of this method was, that it could never be excluded, that the temperature behaviour is not due to some initial concentration effects and that is why we chose to try Michaelis menten fitting, as in a perfect case, one could make an estimation on the amount of enzyme in the sample. As the exponential is just a special case of Michaelis Menten, one can always enlarge the model with this contribution.<br />
<br />
==Fitting Michaelis-Menten kinetics to the concentration data==<br />
<br />
<br />
As we are screening different lysozymes in high-throughput we tried to use the whole data obtained from substrate degredation over time by applying integrated michaelis menten equation [[#References|[2]]] But as there is always an OD shift because of the plate and the cells lysate we need to take this parameter into account while fitting.<br />
<br />
The basic differential equation for Michaelis-Menten kinetics [[#References|[3]]] is:<br />
<br />
\[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]<br />
<br />
Where $\left[S\right]$ means substrate concentration at time 0 or t respectively, $ V _{max}$ is the maximum enzyme reaction velocity, $K_m$ is Michaelis-Menten constant and t is time. This leads to:<br />
<br />
\[ \frac{K_m + \left[S\right]}{\left[S\right]} \frac{d\left[S\right]}{dt} = -V_{max}\]<br />
<br />
Which we can now solve by separation of variables and integration.<br />
<br />
\[ \int_{\left[S\right]_0}^{\left[S\right]_t} \left(\frac{K_m }{\left[S\right]} + 1\right) d\left[S\right] = \int_{0}^{t}-V_{max} dt'\]<br />
<br />
This leads to,<br />
<br />
\[K_m \ln\left( \frac{\left[S\right]_t}{\left[S\right]_0} \right) + \left[S\right]_t - \left[S\right]_0 = -V_{max} t \]<br />
<br />
what we reform to a closed functional behaviour of time.<br />
<br />
<br />
\[t = - \frac{K_m \ln\left( \frac{\left[S\right]_t}{\left[S\right]_0} \right) + \left[S\right]_t - \left[S\right]_0 } {V_{max}} \]<br />
<br />
As the functional behaviour is monotonous we can just fit this function to our data, which should directly provide us with $V_max$ which is the interesting parameter for us.<br />
<br />
As OD in the range where we are measuring still is in the linear scope with some offset due to measurement circumstances and the absorption of the cells lysate we can use $\left[S\right] = m OD + a$, where a is the offset optical density, when all the substrate has been degraded, OD is the optical density at 600 nm and m is some parameter that needs to be calibrated.<br />
<br />
So we can refine the functional behaviour as: <br />
<br />
<br />
\[t = - \frac{K_m \ln\left( \frac{ m ({\mathit{OD}}_t - a)}{ m (\mathit{OD}_0 - a)} \right) + m \mathit{OD}_t - m \mathit{OD}_0 } {V_{max}} \]<br />
<br />
<br />
$OD_0$ is just the first measured OD we get, this parameter is not fitted.<br />
<br />
But these fits did completely not converge so we needed to find another solution. Liao et al. proposed and compared different techniques for fitting integrated Michaelis-Menten kinetics, [[#References|[4]]] which didn't work out for us, because starting substrate concentration was too low for us and thus the fits converged to negative $K_m$ and $V_Max$. Finally we fitted Michaelis-Menten kinetics using the method proposed by Goudar et al. [[#References|[5]]] by directly fit the numerically solved equation, using lambert's $\omega$ function, which is the solution to the equation $ \omega (x) \exp (\omega(x)) = x$. So we fitted <br />
<br />
\[ \left[S\right]_t = K_m \omega \left[ \left( \left[A\right]_0 / K_m \right) \exp \left( \left[A\right]_0 / K_m -V_{max} t / K_m \right) \right] \]<br />
<br />
This worked well, the fits converged reliably, but sometimes produced huge errors for $K_m$ and $V_Max$ of the order of $10^5$ higher than the best fit for these values. This simply meant, that from most data, these parameters could not be identified. On the other hand simple exponential fit reproduced the data nearly perfectly, which made us concluding, that we're just working in the exponential regime, because $K_m$ is just much too high for the substrate concentrations we're working with, so that the differential equation from the beginning would transform into:<br />
<br />
\[\frac{d\left[S\right]}{dt} = - V _{max} \left[S\right] \]<br />
<br />
which is solved by a simple exponential equation<br />
<br />
\[ \left[S\right]_{t} = \left[S\right]_0 e^{\left( - V_{max} t \right)} \]<br />
<br />
As we're measuring OD function fitted to the data results in:<br />
<br />
\[ \mathit{OD}_{t} = \left(\mathit{OD}_0 - a\right) e^{\left( - V_{max} t \right)} + a \]<br />
<br />
with a a parameter for the offset in OD due to the plate and the proteinmix.<br />
<br />
This method seemed to be the method of choice, as it also produced nice results. We have written a python skript that handled all the data, the plotting and the fitting and in the end produced plots with activities normalized to the 37°C activity. These results though had too large errorbars, so we tried to set up a framework to fit multiple datasets in one, with different parameters applied to different datasets. We chose to work with the widely used [https://bitbucket.org/d2d-development d2d arFramework] developed by Andreas Raue [[#References|[6]]] running on MATLAB. As all the datahandling had already happened in python we appended the script with the generation of work for the d2d framework, so that our huge datasets could be fitted at once. The fitting worked out quite well, but some strange results could not be explained yet with that.<br />
<br />
==Modeling product inhibition==<br />
<br />
We observed many different phenomena we could not explain properly. For example when the activity at 37°C started low, it seemed, that the protein doesn't loose it's activity after heatshock. This meant that there was some kind of basal activity, independent of the enzyme concentration. On the other hand activity was not completely linear to enzyme concentration.<br />
But the most inexplicable part was, that some samples even after 1h of degradation stayed constant at an $OD_{600}$ level, nearly as high as the starting $OD_{600}$. This could only be due to the substrate not being degraded, so we checked this by adding fresh lysozyme to the substrate. We observed another decay in $OD_{600}$, which clearly meant, that not the substrate ran out, but the enzyme somehow lost activity during measurement. This meant, that our basic assumption from above was completely wrong and the results completely worthless, as we are only detecting a region of the kinetics, where already some enzyme has been lost due to inhibition. But therefore nothing about initial enzyme concentration in the sample could be said.<br />
We even found this based in a paper from 1961 written in french [[#References|[7]]].<br />
<br />
=Grand model=<br />
<br />
We then tried to always fit one grand model to all the data we have obtained from all the different assays, with curves for different temperatures, biological replicates and technical replicates. In total these were about 100 000 data points we feeded in and up to 500 parameters we fitted. This did not work out, because the variation in starting amounts of enzyme was too large even between the technical replicates from different days. This might be due to the freeze thaw cycles, that the enzyme stock was subdued to. On the other hand this model was just way too complex to be handled easily, as it took hours only for the initial fits. Calculating the profile likelihoods took about one day. Therefore we chose to take another approach, always modeling the data of one single plate, as on that for sure the variations were much less. Of course thus different parameters would not be identifiable, for example the enzyme concentration would not be comparable between the different samples. On the other hand, the only parameters, that are interesting for our purpose, the bahavior after heatshock would still be identifiable.<br />
<br />
===References===<br />
<br />
[1] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[2] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[3] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[4] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[5] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[6] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[7] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:52:48Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology [[#References| [5]]]. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions [[#References| [4]]]. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect [[#References| [5]]], that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
Notice that many methods for parameter estimation in these types of models have been developed [[#References| [6]]] [[#References| [7]]].<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. A more detailed description on the model development can be found [[Team:Heidelberg/Modeling/Enzyme_Modeling_detailed|here]]. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T * V^{lysozyme}_{Max, 37.0}$. Mathematically this just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the ord1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 1 min incubation at 44.5 °C and for diminished activity after 10 min incubation at 54 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the ord1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity. It can be clearly observed, that none of the formerly identifiable activities are identifiable anymore. Therefore this was not a considerable model.<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software [[Team:Heidelberg/Project/Linker_Screening|here]].<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/Modeling/Enzyme_ModelingTeam:Heidelberg/Modeling/Enzyme Modeling2014-10-18T03:50:16Z<p>Jan glx: </p>
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subtitle= Modeling of lysozyme activity with product inhibition<br />
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The theory of enhancing heatstability by circularization suggests, that the more the ends are constrained, the better stabilization works. Based on this theoretical assumption we identified patterns to have the ability to customly build rigid linkers, that fit between the ends of the protein. Based on over 1000 experimental degradation curves from 9 different constructs with biological and technical replicates of lambda lysozyme, we wanted to gain quantitative insight into the effect of different linkers into the functionality after heatshock. Therefore, a comprehensive approach based on quantitative dynamic modeling (ODEs) was made to determine the relevant information. We started with easy fitting of the data and continued by testing different enzyme kinetics models. We evaluated and identified the most relevant and robust parameters by [[#PLE_analysis|profile likelihood analysis]]. Research suggested that a Michaelis-Menten model with substrate inhibition was appropriate for our data. Our final findings show that circularization of lysozyme can have a considerable influence on enzyme thermostability and that suboptimal linker design can decrease thermostability.<br />
These results were extraordinarily important for the calibration of [[Team:Heidelberg/Software/Linker_Software | CRAUT]].<br />
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{{:Team:Heidelberg/templates/mathjax}}</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_detailedTeam:Heidelberg/pages/Enzyme Modeling detailed2014-10-18T03:49:01Z<p>Jan glx: </p>
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<div>=The long way to the model=<br />
<br />
For long time we have thought it would be easy to extract the relevant data out of our lysozyme assays. But this though somehow perished one week before wiki freeze. For showing that science is always trying and failing, we wanted to explain how we came up with the upper model in the next part.<br />
<br />
==First try, easy fitting==<br />
<br />
The first thought when looking to the curves was that the reaction was clearly exponentially with some basal substrate decay and a small offset due to the different types of proteinmix added. The relevant parameter for us would be only the exponente which would be equal to some constant k times the enzyme concentration present. We would assume, that the constant doesn't change after heatshock, but the part of enzyme that survived. This assumption is based on a paper by Di Paolo et al., who claim that for pH denaturation of $\lambda$-lysozyme there are only two transition states, folded and unfolded. [[#References|[1]]]<br />
This way curves for the temperature decay can be measured for each kind of lysozyme and finally compared to each other.<br />
<br />
A basic problem of this method was, that it could never be excluded, that the temperature behaviour is not due to some initial concentration effects and that is why we chose to try Michaelis menten fitting, as in a perfect case, one could make an estimation on the amount of enzyme in the sample. As the exponential is just a special case of Michaelis Menten, one can always enlarge the model with this contribution.<br />
<br />
==Fitting Michaelis-Menten kinetics to the concentration data==<br />
<br />
<br />
As we are screening different lysozymes in high-throughput we tried to use the whole data obtained from substrate degredation over time by applying integrated michaelis menten equation [[#References|[2]]] But as there is always an OD shift because of the plate and the cells lysate we need to take this parameter into account while fitting.<br />
<br />
The basic differential equation for Michaelis-Menten kinetics [[#References|[3]]] is:<br />
<br />
\[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]<br />
<br />
Where $\left[S\right]$ means substrate concentration at time 0 or t respectively, $ V _{max}$ is the maximum enzyme reaction velocity, $K_m$ is Michaelis-Menten constant and t is time. This leads to:<br />
<br />
\[ \frac{K_m + \left[S\right]}{\left[S\right]} \frac{d\left[S\right]}{dt} = -V_{max}\]<br />
<br />
Which we can now solve by separation of variables and integration.<br />
<br />
\[ \int_{\left[S\right]_0}^{\left[S\right]_t} \left(\frac{K_m }{\left[S\right]} + 1\right) d\left[S\right] = \int_{0}^{t}-V_{max} dt'\]<br />
<br />
This leads to,<br />
<br />
\[K_m \ln\left( \frac{\left[S\right]_t}{\left[S\right]_0} \right) + \left[S\right]_t - \left[S\right]_0 = -V_{max} t \]<br />
<br />
what we reform to a closed functional behaviour of time.<br />
<br />
<br />
\[t = - \frac{K_m \ln\left( \frac{\left[S\right]_t}{\left[S\right]_0} \right) + \left[S\right]_t - \left[S\right]_0 } {V_{max}} \]<br />
<br />
As the functional behaviour is monotonous we can just fit this function to our data, which should directly provide us with $V_max$ which is the interesting parameter for us.<br />
<br />
As OD in the range where we are measuring still is in the linear scope with some offset due to measurement circumstances and the absorption of the cells lysate we can use $\left[S\right] = m OD + a$, where a is the offset optical density, when all the substrate has been degraded, OD is the optical density at 600 nm and m is some parameter that needs to be calibrated.<br />
<br />
So we can refine the functional behaviour as: <br />
<br />
<br />
\[t = - \frac{K_m \ln\left( \frac{ m ({\mathit{OD}}_t - a)}{ m (\mathit{OD}_0 - a)} \right) + m \mathit{OD}_t - m \mathit{OD}_0 } {V_{max}} \]<br />
<br />
<br />
$OD_0$ is just the first measured OD we get, this parameter is not fitted.<br />
<br />
But these fits did completely not converge so we needed to find another solution. Liao et al. proposed and compared different techniques for fitting integrated Michaelis-Menten kinetics, [[#References|[4]]] which didn't work out for us, because starting substrate concentration was too low for us and thus the fits converged to negative $K_m$ and $V_Max$. Finally we fitted Michaelis-Menten kinetics using the method proposed by Goudar et al. [[#References|[5]]] by directly fit the numerically solved equation, using lambert's $\omega$ function, which is the solution to the equation $ \omega (x) \exp (\omega(x)) = x$. So we fitted <br />
<br />
\[ \left[S\right]_t = K_m \omega \left[ \left( \left[A\right]_0 / K_m \right) \exp \left( \left[A\right]_0 / K_m -V_{max} t / K_m \right) \right] \]<br />
<br />
This worked well, the fits converged reliably, but sometimes produced huge errors for $K_m$ and $V_Max$ of the order of $10^5$ higher than the best fit for these values. This simply meant, that from most data, these parameters could not be identified. On the other hand simple exponential fit reproduced the data nearly perfectly, which made us concluding, that we're just working in the exponential regime, because $K_m$ is just much too high for the substrate concentrations we're working with, so that the differential equation from the beginning would transform into:<br />
<br />
\[\frac{d\left[S\right]}{dt} = - V _{max} \left[S\right] \]<br />
<br />
which is solved by a simple exponential equation<br />
<br />
\[ \left[S\right]_{t} = \left[S\right]_0 e^{\left( - V_{max} t \right)} \]<br />
<br />
As we're measuring OD function fitted to the data results in:<br />
<br />
\[ \mathit{OD}_{t} = \left(\mathit{OD}_0 - a\right) e^{\left( - V_{max} t \right)} + a \]<br />
<br />
with a a parameter for the offset in OD due to the plate and the proteinmix.<br />
<br />
This method seemed to be the method of choice, as it also produced nice results. We have written a python skript that handled all the data, the plotting and the fitting and in the end produced plots with activities normalized to the 37°C activity. These results though had too large errorbars, so we tried to set up a framework to fit multiple datasets in one, with different parameters applied to different datasets. We chose to work with the widely used [https://bitbucket.org/d2d-development d2d arFramework] developed by Andreas Raue [[#References|[6]]] running on MATLAB. As all the datahandling had already happened in python we appended the script with the generation of work for the d2d framework, so that our huge datasets could be fitted at once. The fitting worked out quite well, but some strange results could not be explained yet with that.<br />
<br />
==Modeling product inhibition==<br />
<br />
We observed many different phenomena we could not explain properly. For example when the activity at 37°C started low, it seemed, that the protein doesn't loose it's activity after heatshock ###figure needed###. This meant that there was some kind of basal activity, independent of the enzyme concentration. On the other hand activity was not completely linear to enzyme concentration.<br />
But the most inexplicable part was, that some samples even after 1h of degradation stayed constant at an $OD_{600}$ level, nearly as high as the starting $OD_{600}$. This could only be due to the substrate not being degraded, so we checked this by adding fresh lysozyme to the substrate. We observed another decay in $OD_{600}$, which clearly meant, that not the substrate ran out, but the enzyme somehow lost activity during measurement. This meant, that our basic assumption from above was completely wrong and the results completely worthless, as we are only detecting a region of the kinetics, where already some enzyme has been lost due to inhibition. But therefore nothing about initial enzyme concentration in the sample could be said.<br />
We even found this based in a paper from 1961 written in french [[#References|[7]]].<br />
<br />
=Grand model=<br />
<br />
We then tried to always fit one grand model to all the data we have obtained from all the different assays, with curves for different temperatures, biological replicates and technical replicates. In total these were about 100 000 data points we feeded in and up to 500 parameters we fitted. This did not work out, because the variation in starting amounts of enzyme was too large even between the technical replicates from different days. This might be due to the freeze thaw cycles, that the enzyme stock was subdued to. On the other hand this model was just way too complex to be handled easily, as it took hours only for the initial fits. Calculating the profile likelihoods took about one day. Therefore we chose to take another approach, always modeling the data of one single plate, as on that for sure the variations were much less. Of course thus different parameters would not be identifiable, for example the enzyme concentration would not be comparable between the different samples. On the other hand, the only parameters, that are interesting for our purpose, the bahavior after heatshock would still be identifiable.<br />
<br />
===References===<br />
<br />
[1] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[2] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[3] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[4] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[5] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[6] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[7] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:45:39Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology [[#References| [5]]]. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions [[#References| [4]]]. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect [[#References| [5]]], that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
Notice that many methods for parameter estimation in these types of models have been developed [[#References| [6]]] [[#References| [7]]].<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. A more detailed description on the model development can be found [[Team:Heidelberg/Modeling/Enzyme_Modeling_detailed|here]]. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T * V^{lysozyme}_{Max, 37.0}$. Mathematically this just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the ord1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 1 min incubation at 44.5 °C and for diminished activity after 10 min incubation at 54 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the ord1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software [[Team:Heidelberg/Project/Linker_Screening|here]].<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_detailedTeam:Heidelberg/pages/Enzyme Modeling detailed2014-10-18T03:45:25Z<p>Jan glx: </p>
<hr />
<div>=The long way to the model=<br />
<br />
For long time we have thought it would be easy to extract the relevant data out of our lysozyme assays. But this though somehow perished one week before wiki freeze. For showing that science is always trying and failing, we wanted to explain how we came up with the upper model in the next part.<br />
<br />
==First try, easy fitting==<br />
<br />
The first thought when looking to the curves was that the reaction was clearly exponentially with some basal substrate decay and a small offset due to the different types of proteinmix added. The relevant parameter for us would be only the exponente which would be equal to some constant k times the enzyme concentration present. We would assume, that the constant doesn't change after heatshock, but the part of enzyme that survived. This assumption is based on a paper by Di Paolo et al., who claim that for pH denaturation of $\lambda$-lysozyme there are only two transition states, folded and unfolded. [[#References|[3]]]<br />
This way curves for the temperature decay can be measured for each kind of lysozyme and finally compared to each other.<br />
<br />
A basic problem of this method was, that it could never be excluded, that the temperature behaviour is not due to some initial concentration effects and that is why we chose to try Michaelis menten fitting, as in a perfect case, one could make an estimation on the amount of enzyme in the sample. As the exponential is just a special case of Michaelis Menten, one can always enlarge the model with this contribution.<br />
<br />
==Fitting Michaelis-Menten kinetics to the concentration data==<br />
<br />
<br />
As we are screening different lysozymes in high-throughput we tried to use the whole data obtained from substrate degredation over time by applying integrated michaelis menten equation [[#References|[4]]] But as there is always an OD shift because of the plate and the cells lysate we need to take this parameter into account while fitting.<br />
<br />
The basic differential equation for Michaelis-Menten kinetics [[#References|[5]]] is:<br />
<br />
\[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]<br />
<br />
Where $\left[S\right]$ means substrate concentration at time 0 or t respectively, $ V _{max}$ is the maximum enzyme reaction velocity, $K_m$ is Michaelis-Menten constant and t is time. This leads to:<br />
<br />
\[ \frac{K_m + \left[S\right]}{\left[S\right]} \frac{d\left[S\right]}{dt} = -V_{max}\]<br />
<br />
Which we can now solve by separation of variables and integration.<br />
<br />
\[ \int_{\left[S\right]_0}^{\left[S\right]_t} \left(\frac{K_m }{\left[S\right]} + 1\right) d\left[S\right] = \int_{0}^{t}-V_{max} dt'\]<br />
<br />
This leads to,<br />
<br />
\[K_m \ln\left( \frac{\left[S\right]_t}{\left[S\right]_0} \right) + \left[S\right]_t - \left[S\right]_0 = -V_{max} t \]<br />
<br />
what we reform to a closed functional behaviour of time.<br />
<br />
<br />
\[t = - \frac{K_m \ln\left( \frac{\left[S\right]_t}{\left[S\right]_0} \right) + \left[S\right]_t - \left[S\right]_0 } {V_{max}} \]<br />
<br />
As the functional behaviour is monotonous we can just fit this function to our data, which should directly provide us with $V_max$ which is the interesting parameter for us.<br />
<br />
As OD in the range where we are measuring still is in the linear scope with some offset due to measurement circumstances and the absorption of the cells lysate we can use $\left[S\right] = m OD + a$, where a is the offset optical density, when all the substrate has been degraded, OD is the optical density at 600 nm and m is some parameter that needs to be calibrated.<br />
<br />
So we can refine the functional behaviour as: <br />
<br />
<br />
\[t = - \frac{K_m \ln\left( \frac{ m ({\mathit{OD}}_t - a)}{ m (\mathit{OD}_0 - a)} \right) + m \mathit{OD}_t - m \mathit{OD}_0 } {V_{max}} \]<br />
<br />
<br />
$OD_0$ is just the first measured OD we get, this parameter is not fitted.<br />
<br />
But these fits did completely not converge so we needed to find another solution. Liao et al. proposed and compared different techniques for fitting integrated Michaelis-Menten kinetics, [[#References|[7]]] which didn't work out for us, because starting substrate concentration was too low for us and thus the fits converged to negative $K_m$ and $V_Max$. Finally we fitted Michaelis-Menten kinetics using the method proposed by Goudar et al. [[#References|[6]]] by directly fit the numerically solved equation, using lambert's $\omega$ function, which is the solution to the equation $ \omega (x) \exp (\omega(x)) = x$. So we fitted <br />
<br />
\[ \left[S\right]_t = K_m \omega \left[ \left( \left[A\right]_0 / K_m \right) \exp \left( \left[A\right]_0 / K_m -V_{max} t / K_m \right) \right] \]<br />
<br />
This worked well, the fits converged reliably, but sometimes produced huge errors for $K_m$ and $V_Max$ of the order of $10^5$ higher than the best fit for these values. This simply meant, that from most data, these parameters could not be identified. On the other hand simple exponential fit reproduced the data nearly perfectly, which made us concluding, that we're just working in the exponential regime, because $K_m$ is just much too high for the substrate concentrations we're working with, so that the differential equation from the beginning would transform into:<br />
<br />
\[\frac{d\left[S\right]}{dt} = - V _{max} \left[S\right] \]<br />
<br />
which is solved by a simple exponential equation<br />
<br />
\[ \left[S\right]_{t} = \left[S\right]_0 e^{\left( - V_{max} t \right)} \]<br />
<br />
As we're measuring OD function fitted to the data results in:<br />
<br />
\[ \mathit{OD}_{t} = \left(\mathit{OD}_0 - a\right) e^{\left( - V_{max} t \right)} + a \]<br />
<br />
with a a parameter for the offset in OD due to the plate and the proteinmix.<br />
<br />
This method seemed to be the method of choice, as it also produced nice results. We have written a python skript that handled all the data, the plotting and the fitting and in the end produced plots with activities normalized to the 37°C activity. These results though had too large errorbars, so we tried to set up a framework to fit multiple datasets in one, with different parameters applied to different datasets. We chose to work with the widely used [https://bitbucket.org/d2d-development d2d arFramework] developed by Andreas Raue [[#References|[8]]] running on MATLAB. As all the datahandling had already happened in python we appended the script with the generation of work for the d2d framework, so that our huge datasets could be fitted at once. The fitting worked out quite well, but some strange results could not be explained yet with that.<br />
<br />
==Modeling product inhibition==<br />
<br />
We observed many different phenomena we could not explain properly. For example when the activity at 37°C started low, it seemed, that the protein doesn't loose it's activity after heatshock ###figure needed###. This meant that there was some kind of basal activity, independent of the enzyme concentration. On the other hand activity was not completely linear to enzyme concentration.<br />
But the most inexplicable part was, that some samples even after 1h of degradation stayed constant at an $OD_{600}$ level, nearly as high as the starting $OD_{600}$. This could only be due to the substrate not being degraded, so we checked this by adding fresh lysozyme to the substrate. We observed another decay in $OD_{600}$, which clearly meant, that not the substrate ran out, but the enzyme somehow lost activity during measurement. This meant, that our basic assumption from above was completely wrong and the results completely worthless, as we are only detecting a region of the kinetics, where already some enzyme has been lost due to inhibition. But therefore nothing about initial enzyme concentration in the sample could be said.<br />
We even found this based in a paper from 1961 written in french [[#References|[2]]].<br />
<br />
=Grand model=<br />
<br />
We then tried to always fit one grand model to all the data we have obtained from all the different assays, with curves for different temperatures, biological replicates and technical replicates. In total these were about 100 000 data points we feeded in and up to 500 parameters we fitted. This did not work out, because the variation in starting amounts of enzyme was too large even between the technical replicates from different days. This might be due to the freeze thaw cycles, that the enzyme stock was subdued to. On the other hand this model was just way too complex to be handled easily, as it took hours only for the initial fits. Calculating the profile likelihoods took about one day. Therefore we chose to take another approach, always modeling the data of one single plate, as on that for sure the variations were much less. Of course thus different parameters would not be identifiable, for example the enzyme concentration would not be comparable between the different samples. On the other hand, the only parameters, that are interesting for our purpose, the bahavior after heatshock would still be identifiable.<br />
<br />
===References===<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).<br />
<br />
[10] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:43:47Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology [[#References| [5]]]. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions [[#References| [4]]]. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect [[#References| [5]]], that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
Notice that many methods for parameter estimation in these types of models have been developed [[#References| [6]]] [[#References| [7]]].<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. A more detailed description on the model development can be found [[Team:Heidelberg/Modeling/Enzyme_Modeling_detailed|here]]. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T * V^{lysozyme}_{Max, 37.0}$. Mathematically this just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 44.5 °C and for diminished activity after 10 min incubation at 54 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software [[Team:Heidelberg/Project/Linker_Screening|here]].<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:42:35Z<p>Jan glx: </p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology [[#References| [5]]]. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions [[#References| [4]]]. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect [[#References| [5]]], that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
Notice that many methods for parameter estimation in these types of models have been developed [[#References| [6]]] [[#References| [7]]].<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. A more detailed description on the model development can be found [[Team:Heidelberg/Modeling/Enzyme_Modeling_detailed|here]]. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T * V^{lysozyme}_{Max, 37.0}$. Mathematically this just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software [[Team:Heidelberg/Project/Linker_Screening|here]].<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/Modeling/Enzyme_Modeling_detailedTeam:Heidelberg/Modeling/Enzyme Modeling detailed2014-10-18T03:42:05Z<p>Jan glx: Created page with "{{:Team:Heidelberg/templates/wikipage_new| title=ENZYME MODELING | white=true | red-logo=true | header-img= | header=background-color:#DE4230 | header-bg=black | subtitle= The de..."</p>
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{{:Team:Heidelberg/templates/mathjax}}</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:39:19Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology [[#References| [5]]]. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions [[#References| [4]]]. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect [[#References| [5]]], that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
Notice that many methods for parameter estimation in these types of models have been developed [[#References| [6]]] [[#References| [7]]].<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T * V^{lysozyme}_{Max, 37.0}$. Mathematically this just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
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|All plates <br />
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<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $V_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software [[Team:Heidelberg/Project/Linker_Screening|here]].<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_detailedTeam:Heidelberg/pages/Enzyme Modeling detailed2014-10-18T03:38:59Z<p>Jan glx: Created page with "=The long way to the model= For long time we have thought it would be easy to extract the relevant data out of our lysozyme assays. But this though somehow perished one week bef..."</p>
<hr />
<div>=The long way to the model=<br />
<br />
For long time we have thought it would be easy to extract the relevant data out of our lysozyme assays. But this though somehow perished one week before wiki freeze. For showing that science is always trying and failing, we wanted to explain how we came up with the upper model in the next part.<br />
<br />
==First try, easy fitting==<br />
<br />
The first thought when looking to the curves was that the reaction was clearly exponentially with some basal substrate decay and a small offset due to the different types of proteinmix added. The relevant parameter for us would be only the exponente which would be equal to some constant k times the enzyme concentration present. We would assume, that the constant doesn't change after heatshock, but the part of enzyme that survived. This assumption is based on a paper by Di Paolo et al., who claim that for pH denaturation of $\lambda$-lysozyme there are only two transition states, folded and unfolded. [[#References|[3]]]<br />
This way curves for the temperature decay can be measured for each kind of lysozyme and finally compared to each other.<br />
<br />
A basic problem of this method was, that it could never be excluded, that the temperature behaviour is not due to some initial concentration effects and that is why we chose to try Michaelis menten fitting, as in a perfect case, one could make an estimation on the amount of enzyme in the sample. As the exponential is just a special case of Michaelis Menten, one can always enlarge the model with this contribution.<br />
<br />
==Fitting Michaelis-Menten kinetics to the concentration data==<br />
<br />
<br />
As we are screening different lysozymes in high-throughput we tried to use the whole data obtained from substrate degredation over time by applying integrated michaelis menten equation [[#References|[4]]] But as there is always an OD shift because of the plate and the cells lysate we need to take this parameter into account while fitting.<br />
<br />
The basic differential equation for Michaelis-Menten kinetics [[#References|[5]]] is:<br />
<br />
\[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]<br />
<br />
Where $\left[S\right]$ means substrate concentration at time 0 or t respectively, $ V _{max}$ is the maximum enzyme reaction velocity, $K_m$ is Michaelis-Menten constant and t is time. This leads to:<br />
<br />
\[ \frac{K_m + \left[S\right]}{\left[S\right]} \frac{d\left[S\right]}{dt} = -V_{max}\]<br />
<br />
Which we can now solve by separation of variables and integration.<br />
<br />
\[ \int_{\left[S\right]_0}^{\left[S\right]_t} \left(\frac{K_m }{\left[S\right]} + 1\right) d\left[S\right] = \int_{0}^{t}-V_{max} dt'\]<br />
<br />
This leads to,<br />
<br />
\[K_m \ln\left( \frac{\left[S\right]_t}{\left[S\right]_0} \right) + \left[S\right]_t - \left[S\right]_0 = -V_{max} t \]<br />
<br />
what we reform to a closed functional behaviour of time.<br />
<br />
<br />
\[t = - \frac{K_m \ln\left( \frac{\left[S\right]_t}{\left[S\right]_0} \right) + \left[S\right]_t - \left[S\right]_0 } {V_{max}} \]<br />
<br />
As the functional behaviour is monotonous we can just fit this function to our data, which should directly provide us with $V_max$ which is the interesting parameter for us.<br />
<br />
As OD in the range where we are measuring still is in the linear scope with some offset due to measurement circumstances and the absorption of the cells lysate we can use $\left[S\right] = m OD + a$, where a is the offset optical density, when all the substrate has been degraded, OD is the optical density at 600 nm and m is some parameter that needs to be calibrated.<br />
<br />
So we can refine the functional behaviour as: <br />
<br />
<br />
\[t = - \frac{K_m \ln\left( \frac{ m ({\mathit{OD}}_t - a)}{ m (\mathit{OD}_0 - a)} \right) + m \mathit{OD}_t - m \mathit{OD}_0 } {V_{max}} \]<br />
<br />
<br />
$OD_0$ is just the first measured OD we get, this parameter is not fitted.<br />
<br />
But these fits did completely not converge so we needed to find another solution. Liao et al. proposed and compared different techniques for fitting integrated Michaelis-Menten kinetics, [[#References|[7]]] which didn't work out for us, because starting substrate concentration was too low for us and thus the fits converged to negative $K_m$ and $V_Max$. Finally we fitted Michaelis-Menten kinetics using the method proposed by Goudar et al. [[#References|[6]]] by directly fit the numerically solved equation, using lambert's $\omega$ function, which is the solution to the equation $ \omega (x) \exp (\omega(x)) = x$. So we fitted <br />
<br />
\[ \left[S\right]_t = K_m \omega \left[ \left( \left[A\right]_0 / K_m \right) \exp \left( \left[A\right]_0 / K_m -V_{max} t / K_m \right) \right] \]<br />
<br />
This worked well, the fits converged reliably, but sometimes produced huge errors for $K_m$ and $V_Max$ of the order of $10^5$ higher than the best fit for these values. This simply meant, that from most data, these parameters could not be identified. On the other hand simple exponential fit reproduced the data nearly perfectly, which made us concluding, that we're just working in the exponential regime, because $K_m$ is just much too high for the substrate concentrations we're working with, so that the differential equation from the beginning ###link### would transform into:<br />
<br />
\[\frac{d\left[S\right]}{dt} = - V _{max} \left[S\right] \]<br />
<br />
which is solved by a simple exponential equation<br />
<br />
\[ \left[S\right]_{t} = \left[S\right]_0 e^{\left( - V_{max} t \right)} \]<br />
<br />
As we're measuring OD function fitted to the data results in:<br />
<br />
\[ \mathit{OD}_{t} = \left(\mathit{OD}_0 - a\right) e^{\left( - V_{max} t \right)} + a \]<br />
<br />
with a a parameter for the offset in OD due to the plate and the proteinmix.<br />
<br />
This method seemed to be the method of choice, as it also produced nice results. We have written a python skript [ ???Link???] that handled all the data, the plotting and the fitting and in the end produced plots with activities normalized to the 37°C activity. These results though had too large errorbars, so we tried to set up a framework to fit multiple datasets in one, with different parameters applied to different datasets. We chose to work with the widely used [https://bitbucket.org/d2d-development d2d arFramework] developed by Andreas Raue [[#References|[8]]] running on MATLAB. As all the datahandling had already happened in python we appended the script with the generation of work for the d2d framework, so that our huge datasets could be fitted at once. The fitting worked out quite well, but some strange results could not be explained yet with that.<br />
<br />
==Modeling product inhibition==<br />
<br />
We observed many different phenomena we could not explain properly. For example when the activity at 37°C started low, it seemed, that the protein doesn't loose it's activity after heatshock ###figure needed###. This meant that there was some kind of basal activity, independent of the enzyme concentration. On the other hand activity was not completely linear to enzyme concentration.<br />
But the most inexplicable part was, that some samples even after 1h of degradation stayed constant at an $OD_{600}$ level, nearly as high as the starting $OD_{600}$. This could only be due to the substrate not being degraded, so we checked this by adding fresh lysozyme to the substrate. We observed another decay in $OD_{600}$, which clearly meant, that not the substrate ran out, but the enzyme somehow lost activity during measurement. This meant, that our basic assumption from above was completely wrong and the results completely worthless, as we are only detecting a region of the kinetics, where already some enzyme has been lost due to inhibition. But therefore nothing about initial enzyme concentration in the sample could be said.<br />
We even found this based in a paper from 1961 written in french [[#References|[2]]].<br />
<br />
=Grand model=<br />
<br />
We then tried to always fit one grand model to all the data we have obtained from all the different assays, with curves for different temperatures, biological replicates and technical replicates. In total these were about 100 000 data points we feeded in and up to 500 parameters we fitted. This did not work out, because the variation in starting amounts of enzyme was too large even between the technical replicates from different days. This might be due to the freeze thaw cycles, that the enzyme stock was subdued to. On the other hand this model was just way too complex to be handled easily, as it took hours only for the initial fits. Calculating the profile likelihoods took about one day. Therefore we chose to take another approach, always modeling the data of one single plate, as on that for sure the variations were much less. Of course thus different parameters would not be identifiable, for example the enzyme concentration would not be comparable between the different samples. On the other hand, the only parameters, that are interesting for our purpose, the bahavior after heatshock would still be identifiable.</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:38:52Z<p>Jan glx: /* Different models tested */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology [[#References| [5]]]. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions [[#References| [4]]]. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect [[#References| [5]]], that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
Notice that many methods for parameter estimation in these types of models have been developed [[#References| [6]]] [[#References| [7]]].<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T * V^{lysozyme}_{Max, 37.0}$. Mathematically this just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software [[Team:Heidelberg/Project/Linker_Screening|here]].<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:37:32Z<p>Jan glx: /* Introduction */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology [[#References| [5]]]. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions [[#References| [4]]]. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect [[#References| [5]]], that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
Notice that many methods for parameter estimation in these types of models have been developed [[#References| [6]]] [[#References| [7]]].<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T * V^{lysozyme}_{Max, 37.0}$. Mathematically this just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software [[Team:Heidelberg/Project/Linker_Screening|here]].<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:37:25Z<p>Jan glx: /* Discussion */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T * V^{lysozyme}_{Max, 37.0}$. Mathematically this just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software [[Team:Heidelberg/Project/Linker_Screening|here]].<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:34:27Z<p>Jan glx: /* Final model */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T * V^{lysozyme}_{Max, 37.0}$. Mathematically this just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:34:04Z<p>Jan glx: /* Final model */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T * V^{lysozyme}_{Max, 37.0}$. Mathematicall this just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:33:27Z<p>Jan glx: /* Final model */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T * V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:32:28Z<p>Jan glx: /* PLE analysis */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies. By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:32:01Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state [[#References|[3]]]. We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:30:13Z<p>Jan glx: /* PLE analysis */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [8]]][[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/PartsTeam:Heidelberg/pages/Parts2014-10-18T03:29:15Z<p>Jan glx: </p>
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<h1 id="Favorite Parts">Favorite Parts.</h1><br />
<p>The iGEM Team Heidelberg 2014 had built a new biological system for the iGEM community integrating split-inteins. <br />
Intein splicing is a natural process that excises one part of a protein and leaves the remaining parts irreversibly attached. This great function allows you to modify your protein in numerous ways.</p><br />
<p>Creating a toolbox including all great functions and possibilities of inteins, we need a new standard for the scientific world of iGEM. This standard, the RFC of the iGEM Team Heidelberg 2014, allows us to easily and modulary work with split inteins.</p><br />
<br />
<p>Our favorite Parts represent the basic constructs of our toolbox – the Assembly and the Circularization construct, which are both tested in many methods and applications. </p><br />
<p>In the following we present you <br />
<a href="http://parts.igem.org/Part:BBa_K1362000">BBa_K1362000</a>, the construct for circularization, <br />
<a href="http://parts.igem.org/Part:BBa_K1362100">BBa_K1362100</a> and <br />
<a href="http://parts.igem.org/BBa_K1362101">BBa_K1362101</a>, the N- and the C-construct for assembly. Take a look and visit the Partsregistry to read the associated documentation.</p><br />
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<h3> Circularization Construct. BBa_K1362000 </h3> <br />
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<h4>BBa_K1362000</h4><br />
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<img src="/wiki/images/7/7c/BBa_K1362000.png" class="img-responsive" alt="Circularization Construct"><br />
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<h3 id="Assembly"> Assembly Constructs. BBa_K1362100 and BBa_K1362101 </h3> <br />
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<h4>BBa_K1362100</h4><br />
<p>This intein assembly construct is part of our strategy for cloning with split inteins. Inteins are naturally occuring peptide sequences that splice out of a precursor protein and attach the remaining ends together to form a new protein. When splitting those intein sequence into an N-terminal and a C-terminal split intein one is left with a powerful tool to post-translationally modify whole proteins on the amino-acid sequence level. This construct was designed to express any protein of interest fused to the Nostoc punctiforme DnaE N-terminal split intein. </p><br />
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<img src="/wiki/images/8/81/BBa_K1362100.png" class="img-responsive" alt="Assembly Constructs"><br />
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<h4>BBa_K1362101</h4><br />
BBa_K1362101 is the corresponding C-terminal construct to BBa_K1362100. Upon coexpression or mixture of the N- and C-constructs protein splicing takes place and the N- and C-terminal proteins of interest are irreversibly assembled via a newly formed peptide bond.</p><p><br />
This mechanism can be applied for a variety of different uses such as the activation of a protein through reconstitution of individually expressed split halves. See our <a href="https://2014.igem.org/Team:Heidelberg/Project/Reconstitution">split sfGFP experiment</a> set a foundation that you guys can build on!</p><br />
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<img src="/wiki/images/3/3f/BBa_K1362101.png" class="img-responsive" alt="Assembly Constructs"><br />
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<h1 id="Sample Data Page">Sample Data Page for our favorite Parts.</h1><br />
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<h3> Circularization Construct. BBa_K1362000 </h3> <br />
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<img src="/wiki/images/9/9b/SampleData_Circularization.png" class="img-responsive" alt="Circularization Construct"><br />
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This part represents an easy way to circularize any protein. In a single step you can clone your protein in the split intein circularization construct. Exteins, RFC [i] standard overhangs and BsaI sites have to be added to the coding sequence of the protein to be circularized without start- and stop codons by PCR. By Golden Gate assembly, the mRFP selection marker has to be replaced with the protein insert.<br />
If the distance of the ends of your protein of interest aren't close enough to connect them you will need a linker. <a href="http://parts.igem.org/Part:BBa_K1362000">BBa_K1362000</a>, the split intein circularization construct, includes a strong T7 RBS (<a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362090">BBa_K1362090</a>), we sent to the parts registry as well, and the split intein Npu DnaE. The T7 RBS derived from the T7 phage gene 10a (major capsid protein). </div> <br />
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The resulting plasmid can be used to express the protein of interest with the obligatory linker and the N- and C-intein.<br />
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In an autocatalytic in vivo reaction, the circular protein is formed. To read more about the trans-splicing reaction visit our <a href="https://2014.igem.org/Team:Heidelberg/Project/Background">Intein Background</a> page. If corresponding split inteins are added to both termini of a protein, the trans-splicing reaction results in a circular backbone. <br />
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<div class="well well-sm"><br />
Circular proteins offers many advantages. While conserving the functionality of their linear counterpart, circular proteins can be superior in terms of thermostability, resistance against chemical denaturation and protection from exopeptidases. Moreover, a circular backbone can improve in vivo stability of therapeutical proteins and peptides.<br />
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<h3> Assembly Construct. BBa_K1362100 and BBa_K1362101 </h3> <br />
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These parts represent an easy way to build fusion constructs of intein parts with any protein or peptide of interest (POI). In a single step you can fuse your protein to a split intein part. Desired extein residues, RFC [i] standard overhangs and BsaI sites generating these overhangs have to be added to the coding sequence of the POI by PCR. Facilitating the highly efficient GoldenGate assembly reaction, the mRFP selection marker can be replaced with the POI insert.</div> <br />
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The resulting parts can be concatenated using standard BioBrick cloning into an expression backbone to coexpress the POIs fused to a N- and a C-intein.<br />
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<div class="well well-sm"><br />
In an auto catalytic in vivo reaction, the inteins will cleave themselves out and ligate the exteins - here your POIs - together. For example we used this mechanism to <a href="https://2014.igem.org/Team:Heidelberg/Project/Reconstitution">reconstitute the fluorescence of split sfGFP</a> . To read more about the trans-splicing reaction visit our <a href="https://2014.igem.org/Team:Heidelberg/Project/Background">Intein Background</a> page. The whole process becomes significant more interesting with <a href="https://2014.igem.org/Team:Heidelberg/Project/LOV">conditional protein trans splicing</a>. <br />
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<div class="well well-sm"><br />
If performing experiments with split inteins make sure you always have a non-splicing negative control. Check out our non splicing assembly constructs (with part names ending with 2 or 3) for example: <a href="http://parts.igem.org/Part:BBa_K1362102">BBa_K1362102</a> and <a href="http://parts.igem.org/Part:BBa_K1362103">BBa_K1362103</a><br />
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<h1 id="Intein Library">Intein Library.</h1><br />
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Inteins are the basic unity of our toolbox. They are integrated as extraneous polypeptide sequences into habitual proteins and do not follow the original protein function. Inteins perform an autocatalytic splicing reaction, where they excite themselves out of the host protein while reconnecting the remaining chains on both end, so called N and C exteins, via a new peptide bond. Read more about it in our [https://2014.igem.org/Team:Heidelberg/Project/Background| project background]!<br />
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To characterize the different types and groups of split-inteins and inteins we collect many details about them to develop a intein library. It gives you a great and clear overview about the most important facts.<br />
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{| class="table table-hover"<br />
|-<br />
!Split intein<br />
!Special features<br />
!Size N [aa]<br />
!Size C [aa]<br />
!Reaction properties<br />
!Origin<br />
!References<br />
|-<br />
| Npu DnaE||fast; robust at high temperature range and high-yielding trans-splicing activity, well characterised requirements||102||36||t1/2 = 63s , 37°C , k=~1x10^-2 (s^-1); activity range 6 to 37°C||S1 natural split intein, Nostoc punctiforme||[[#References|[1]]] [[#References|[2]]] <br />
|-<br />
| Ssp DnaX||cross-reactivity with other N-inteins, transsplicing in vivo and in vitro, high yields||127||6||k=~1.7x10^-4(s^-1); efficiency 96%||engineered from Synechocystis species||[[#References|[3]]] [[#References|[4]]] <br />
|-<br />
| Ssp GyrB|| very short Nint facilitates trans-splicing of synthetic peptides||6||150||k=~1x10^-4(s^-1), efficiency 40-80%||S11 split intein enginered from Synechocystis species, strain PCC6803||[[#References|[4]]] [[#References|[5]]] <br />
|-<br />
| Ter DnaE3||trans-splicing activity with high yields||102||36||k=~2x10^-4(s^-1), efficiency 87%||natural split intein, Trichodesmium erythraeum||[[#References|[4]]] [[#References|[6]]] <br />
|-<br />
| Ssp DnaB||relatively fast||||||t1/2=12min, 25°C, k=~1x10^-3(s^-1)||engineered from Synechocystis species, strain PCC6803||[[#References|[2]]] <br />
|-<br />
| Gp41-1||fastes known reaction ||88||38||t1/2=20-30s, 37°C, k=~1.8x10^-1 (s^-1); activity range 0 to 60°C||natural split intein, Cyanophage||[[#References|[7]]] [[#References|[8]]] <br />
|-<br />
|}<br />
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<h3>References</h3><br />
<p>[1] Iwai, H., Züger, S., Jin, J. & Tam, P.-H. Highly efficient protein trans-splicing by a naturally split DnaE intein from Nostoc punctiforme. FEBS Lett. 580, 1853–8 (2006).</p><br />
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<p>[2] Zettler, J., Schütz, V. & Mootz, H. D. The naturally split Npu DnaE intein exhibits an extraordinarily high rate in the protein trans-splicing reaction. FEBS Lett. 583, 909–14 (2009).</p><br />
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<p>[3] Song, H., Meng, Q. & Liu, X.-Q. Protein trans-splicing of an atypical split intein showing structural flexibility and cross-reactivity. PLoS One 7, e45355 (2012).</p><br />
<br />
<p>[4] Lin, Y. et al. Protein trans-splicing of multiple atypical split inteins engineered from natural inteins. PLoS One 8, e59516 (2013).</p><br />
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<p>[5] Appleby, J. H., Zhou, K., Volkmann, G. & Liu, X.-Q. Novel Split Intein for trans-Splicing Synthetic Peptide onto C Terminus of Protein. J. Biol. Chem. 284, 6194–6199 (2009).</p><br />
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<p>[6] Liu, X.-Q. & Yang, J. Split dnaE genes encoding multiple novel inteins in Trichodesmium erythraeum. J. Biol. Chem. 278, 26315–8 (2003).</p><br />
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<p>[7] Carvajal-Vallejos, P., Pallissé, R., Mootz, H. D. & Schmidt, S. R. Unprecedented rates and efficiencies revealed for new natural split inteins from metagenomic sources. J. Biol. Chem. 287, 28686–96 (2012).</p><br />
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<p>[8] Dassa, B., London, N., Stoddard, B. L., Schueler-Furman, O. & Pietrokovski, S. Fractured genes: a novel genomic arrangement involving new split inteins and a new homing endonuclease family. Nucleic Acids Res. 37, 2560–73 (2009).</p><br />
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<h1 id="Backbones">Our Backbones.</h1><br />
<p>Standard BioBrick cloning is a universal way of putting two BioBrick parts together to build a new BioBrick part. Despite several alternative cloning methods allow the assembly of multiple parts at one its simplicity and the broad availability of compatible parts keep it the 'de facto' standard of the iGEM-community.</p><br />
<p>Using standard BioBrick cloning, the generation of translationally active parts requires often more than one round of cloning. The ability to easily test the functionality of a protein before cloning them into complicated circuits has the potential to prevent many unsuccessful experiments of iGEM teams and may improve the characterization of the parts in the parts registry. However the extra amount of work required to clone such an additional construct may inhibit this behavior. We therefore improved the standard plasmids pSB1X3 and pSB4X5 by inserting a lacI repressible T7 promoter directly upstream to the BioBrick prefix of those plasmids. This promoter is completely inactive in 'E. coli' strains lacking a T7 RNA polymerase such as TOP10 or DH10beta bute inducible in strains carrying the T7 RNA polymerase under a lacI repressible promoter such as DE3 strains. This enables the use of the same backbone for cloning and over expression. Using 3A assembly a translational active part can be cloned from an RBS and a coding part in one step while maintaining the full flexibility of standard BioBrick assembly. These new RFC 10 conform backbones eliminate one cloning step needed for the expression and thus the characterization of a newly BioBricked protein. Version number 30 was claimed for the high copy variants and version number 50 for the low copy variants.</p><br />
<p>High copy BioBrick expression backbone:</p><br />
<ul><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362091">pSB1A30</a>(Part:BBa_K1362091): High copy BioBrick cloning/expression backbone carrying Amp resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362092">pSB1C30</a>(Part:BBa_K1362092): High copy BioBrick cloning/expression backbone carrying Cm resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362093">pSB1CK30</a>(Part:BBa_K1362093): High copy BioBrick cloning/expression backbone carrying Kan resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362094">pSB1CT30</a>(Part:BBa_K1362094): High copy BioBrick cloning/expression backbone carrying Tet resistance</li><br />
</ul><br />
<p>Low copy BioBrick expression backbone:</p><br />
<ul><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362095">pSB4A50</a>(Part:BBa_K1362095): High copy BioBrick cloning/expression backbone carrying Amp resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362096">pSB4C50</a>(Part:BBa_K1362096): High copy BioBrick cloning/expression backbone carrying Cm resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362097">pSB4K50</a>(Part:BBa_K1362097): High copy BioBrick cloning/expression backbone carrying Kan resistance</li><br />
</ul><br />
<p>Because of the great experience we had using our expression vectors, we sent them to the iGEM team Aachen and Tuebingen. <a href="https://2014.igem.org/Team:Heidelberg/Team/Collaborations">We helped them</a> solving their problems with the expression of their products.</p> <br />
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<h1 id="allParts"><span style="font-size:170%;">List of Parts</span style="font-size:170%;"> <!-- – <span style="font-size":50%">Placeholder --></h1><br />
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</html></div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:28:27Z<p>Jan glx: /* OD to concentration calibration */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = ((1.160 \pm 0.004 \frac {\mathrm{ml}} {\mathrm{mg}}) * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:26:38Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 4). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability (Fig. 5). Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:24:41Z<p>Jan glx: /* Data */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells. After that the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:23:54Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b).<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:22:12Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [[Media:Heidelberg_orig_multi_plot.png|here]].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:21:38Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [[File:Heidelberg_orig_multi_plot.png|here]].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/PartsTeam:Heidelberg/pages/Parts2014-10-18T03:20:13Z<p>Jan glx: </p>
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<h1 id="Favorite Parts">Favorite Parts.</h1><br />
<p>The iGEM Team Heidelberg 2014 had built a new biological system for the iGEM community integrating split-inteins. <br />
Intein splicing is a natural process that excises one part of a protein and leaves the remaining parts irreversibly attached. This great function allows you to modify your protein in numerous ways.</p><br />
<p>Creating a toolbox including all great functions and possibilities of inteins, we need a new standard for the scientific world of iGEM. This standard, the RFC of the iGEM Team Heidelberg 2014, allows us to easily and modulary work with split inteins.</p><br />
<br />
<p>Our favorite Parts represent the basic constructs of our toolbox – the Assembly and the Circularization construct, which are both tested in many methods and applications. </p><br />
<p>In the following we present you <br />
<a href="http://parts.igem.org/Part:BBa_K1362000">BBa_K1362000</a>, the construct for circularization, <br />
<a href="http://parts.igem.org/Part:BBa_K1362100">BBa_K1362100</a> and <br />
<a href="http://parts.igem.org/BBa_K1362101">BBa_K1362101</a>, the N- and the C-construct for assembly. Take a look and visit the Partsregistry to read the associated documentation.</p><br />
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<h3> Circularization Construct. BBa_K1362000 </h3> <br />
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<img src="/wiki/images/7/7c/BBa_K1362000.png" class="img-responsive" alt="Circularization Construct"><br />
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<h3 id="Assembly"> Assembly Constructs. BBa_K1362100 and BBa_K1362101 </h3> <br />
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<h4>BBa_K1362100</h4><br />
<p>This intein assembly construct is part of our strategy for cloning with split inteins. Inteins are naturally occuring peptide sequences that splice out of a precursor protein and attach the remaining ends together to form a new protein. When splitting those intein sequence into an N-terminal and a C-terminal split intein one is left with a powerful tool to post-translationally modify whole proteins on the amino-acid sequence level. This construct was designed to express any protein of interest fused to the Nostoc punctiforme DnaE N-terminal split intein. </p><br />
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<img src="/wiki/images/8/81/BBa_K1362100.png" class="img-responsive" alt="Assembly Constructs"><br />
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<h4>BBa_K1362101</h4><br />
BBa_K1362101 is the corresponding C-terminal construct to BBa_K1362100. Upon coexpression or mixture of the N- and C-constructs protein splicing takes place and the N- and C-terminal proteins of interest are irreversibly assembled via a newly formed peptide bond.</p><p><br />
This mechanism can be applied for a variety of different uses such as the activation of a protein through reconstitution of individually expressed split halves. See our split sfGFP experiment and the respective parts in the registry for more information. Protein splicing offers many new possibilities and we hope to have set a foundation that you guys can build on!</p><br />
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<h1 id="Sample Data Page">Sample Data Page for our favorite Parts.</h1><br />
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<h3> Circularization Construct. BBa_K1362000 </h3> <br />
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This part represents an easy way to circularize any protein. In a single step you can clone your protein in the split intein circularization construct. Exteins, RFC [i] standard overhangs and BsaI sites have to be added to the coding sequence of the protein to be circularized without start- and stop codons by PCR. By Golden Gate assembly, the mRFP selection marker has to be replaced with the protein insert.<br />
If the distance of the ends of your protein of interest aren't close enough to connect them you will need a linker. <a href="http://parts.igem.org/Part:BBa_K1362000">BBa_K1362000</a>, the split intein circularization construct, includes a strong T7 RBS (<a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362090">BBa_K1362090</a>), we sent to the parts registry as well, and the split intein Npu DnaE. The T7 RBS derived from the T7 phage gene 10a (major capsid protein). </div> <br />
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The resulting plasmid can be used to express the protein of interest with the obligatory linker and the N- and C-intein.<br />
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In an autocatalytic in vivo reaction, the circular protein is formed. To read more about the trans-splicing reaction visit our <a href="https://2014.igem.org/Team:Heidelberg/Project/Background">Intein Background</a> page. If corresponding split inteins are added to both termini of a protein, the trans-splicing reaction results in a circular backbone. <br />
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Circular proteins offers many advantages. While conserving the functionality of their linear counterpart, circular proteins can be superior in terms of thermostability, resistance against chemical denaturation and protection from exopeptidases. Moreover, a circular backbone can improve in vivo stability of therapeutical proteins and peptides.<br />
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<h3> Assembly Construct. BBa_K1362100 and BBa_K1362101 </h3> <br />
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These parts represent an easy way to build fusion constructs of intein parts with any protein or peptide of interest (POI). In a single step you can fuse your protein to a split intein part. Desired extein residues, RFC [i] standard overhangs and BsaI sites generating these overhangs have to be added to the coding sequence of the POI by PCR. Facilitating the highly efficient GoldenGate assembly reaction, the mRFP selection marker can be replaced with the POI insert.</div> <br />
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The resulting parts can be concatenated using standard BioBrick cloning into an expression backbone to coexpress the POIs fused to a N- and a C-intein.<br />
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In an autocatalytic in vivo reaction, the inteins will cleave themselfes out and ligate the exteins - here your POIs - together. For example we used this mechanism to reconstitute the fluorescence of split sfGFP. To read more about the trans-splicing reaction visit our <a href="https://2014.igem.org/Team:Heidelberg/Project/Background">Intein Background</a> page. If corresponding split inteins are added to both termini of a protein, the trans-splicing reaction results in a circular backbone. <br />
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<div class="well well-sm"><br />
If performing experiments with split inteins make sure you always have a non-splicing negative control. Check out our non splicing assembly constructs (with part names ending with 2 or 3) for example: <a href="http://parts.igem.org/Part:BBa_K1362102">BBa_K1362102</a> and <a href="http://parts.igem.org/Part:BBa_K1362103">BBa_K1362103</a><br />
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<h1 id="Intein Library">Intein Library.</h1><br />
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Inteins are the basic unity of our toolbox. They are integrated as extraneous polypeptide sequences into habitual proteins and do not follow the original protein function. Inteins perform an autocatalytic splicing reaction, where they excite themselves out of the host protein while reconnecting the remaining chains on both end, so called N and C exteins, via a new peptide bond. Read more about it in our [https://2014.igem.org/Team:Heidelberg/Project/Background| project background]!<br />
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To characterize the different types and groups of split-inteins and inteins we collect many details about them to develop a intein library. It gives you a great and clear overview about the most important facts.<br />
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{| class="table table-hover"<br />
|-<br />
!Split intein<br />
!Special features<br />
!Size N [aa]<br />
!Size C [aa]<br />
!Reaction properties<br />
!Origin<br />
!References<br />
|-<br />
| Npu DnaE||fast; robust at high temperature range and high-yielding trans-splicing activity, well characterised requirements||102||36||t1/2 = 63s , 37°C , k=~1x10^-2 (s^-1); activity range 6 to 37°C||S1 natural split intein, Nostoc punctiforme||[[#References|[1]]] [[#References|[2]]] <br />
|-<br />
| Ssp DnaX||cross-reactivity with other N-inteins, transsplicing in vivo and in vitro, high yields||127||6||k=~1.7x10^-4(s^-1); efficiency 96%||engineered from Synechocystis species||[[#References|[3]]] [[#References|[4]]] <br />
|-<br />
| Ssp GyrB|| very short Nint facilitates trans-splicing of synthetic peptides||6||150||k=~1x10^-4(s^-1), efficiency 40-80%||S11 split intein enginered from Synechocystis species, strain PCC6803||[[#References|[4]]] [[#References|[5]]] <br />
|-<br />
| Ter DnaE3||trans-splicing activity with high yields||102||36||k=~2x10^-4(s^-1), efficiency 87%||natural split intein, Trichodesmium erythraeum||[[#References|[4]]] [[#References|[6]]] <br />
|-<br />
| Ssp DnaB||relatively fast||||||t1/2=12min, 25°C, k=~1x10^-3(s^-1)||engineered from Synechocystis species, strain PCC6803||[[#References|[2]]] <br />
|-<br />
| Gp41-1||fastes known reaction ||88||38||t1/2=20-30s, 37°C, k=~1.8x10^-1 (s^-1); activity range 0 to 60°C||natural split intein, Cyanophage||[[#References|[7]]] [[#References|[8]]] <br />
|-<br />
|}<br />
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<h3>References</h3><br />
<p>[1] Iwai, H., Züger, S., Jin, J. & Tam, P.-H. Highly efficient protein trans-splicing by a naturally split DnaE intein from Nostoc punctiforme. FEBS Lett. 580, 1853–8 (2006).</p><br />
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<p>[2] Zettler, J., Schütz, V. & Mootz, H. D. The naturally split Npu DnaE intein exhibits an extraordinarily high rate in the protein trans-splicing reaction. FEBS Lett. 583, 909–14 (2009).</p><br />
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<p>[3] Song, H., Meng, Q. & Liu, X.-Q. Protein trans-splicing of an atypical split intein showing structural flexibility and cross-reactivity. PLoS One 7, e45355 (2012).</p><br />
<br />
<p>[4] Lin, Y. et al. Protein trans-splicing of multiple atypical split inteins engineered from natural inteins. PLoS One 8, e59516 (2013).</p><br />
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<p>[5] Appleby, J. H., Zhou, K., Volkmann, G. & Liu, X.-Q. Novel Split Intein for trans-Splicing Synthetic Peptide onto C Terminus of Protein. J. Biol. Chem. 284, 6194–6199 (2009).</p><br />
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<p>[6] Liu, X.-Q. & Yang, J. Split dnaE genes encoding multiple novel inteins in Trichodesmium erythraeum. J. Biol. Chem. 278, 26315–8 (2003).</p><br />
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<p>[7] Carvajal-Vallejos, P., Pallissé, R., Mootz, H. D. & Schmidt, S. R. Unprecedented rates and efficiencies revealed for new natural split inteins from metagenomic sources. J. Biol. Chem. 287, 28686–96 (2012).</p><br />
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<p>[8] Dassa, B., London, N., Stoddard, B. L., Schueler-Furman, O. & Pietrokovski, S. Fractured genes: a novel genomic arrangement involving new split inteins and a new homing endonuclease family. Nucleic Acids Res. 37, 2560–73 (2009).</p><br />
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<h1 id="Backbones">Our Backbones.</h1><br />
<p>Standard BioBrick cloning is a universal way of putting two BioBrick parts together to build a new BioBrick part. Despite several alternative cloning methods allow the assembly of multiple parts at one its simplicity and the broad availability of compatible parts keep it the 'de facto' standard of the iGEM-community.</p><br />
<p>Using standard BioBrick cloning, the generation of translationally active parts requires often more than one round of cloning. The ability to easily test the functionality of a protein before cloning them into complicated circuits has the potential to prevent many unsuccessful experiments of iGEM teams and may improve the characterization of the parts in the parts registry. However the extra amount of work required to clone such an additional construct may inhibit this behavior. We therefore improved the standard plasmids pSB1X3 and pSB4X5 by inserting a lacI repressible T7 promoter directly upstream to the BioBrick prefix of those plasmids. This promoter is completely inactive in 'E. coli' strains lacking a T7 RNA polymerase such as TOP10 or DH10beta bute inducible in strains carrying the T7 RNA polymerase under a lacI repressible promoter such as DE3 strains. This enables the use of the same backbone for cloning and over expression. Using 3A assembly a translational active part can be cloned from an RBS and a coding part in one step while maintaining the full flexibility of standard BioBrick assembly. These new RFC 10 conform backbones eliminate one cloning step needed for the expression and thus the characterization of a newly BioBricked protein. Version number 30 was claimed for the high copy variants and version number 50 for the low copy variants.</p><br />
<p>High copy BioBrick expression backbone:</p><br />
<ul><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362091">pSB1A30</a>(Part:BBa_K1362091): High copy BioBrick cloning/expression backbone carrying Amp resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362092">pSB1C30</a>(Part:BBa_K1362092): High copy BioBrick cloning/expression backbone carrying Cm resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362093">pSB1CK30</a>(Part:BBa_K1362093): High copy BioBrick cloning/expression backbone carrying Kan resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362094">pSB1CT30</a>(Part:BBa_K1362094): High copy BioBrick cloning/expression backbone carrying Tet resistance</li><br />
</ul><br />
<p>Low copy BioBrick expression backbone:</p><br />
<ul><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362095">pSB4A50</a>(Part:BBa_K1362095): High copy BioBrick cloning/expression backbone carrying Amp resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362096">pSB4C50</a>(Part:BBa_K1362096): High copy BioBrick cloning/expression backbone carrying Cm resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362097">pSB4K50</a>(Part:BBa_K1362097): High copy BioBrick cloning/expression backbone carrying Kan resistance</li><br />
</ul><br />
<p>Because of the great experience we had using our expression vectors, we sent them to the iGEM team Aachen and Tuebingen. <a href="https://2014.igem.org/Team:Heidelberg/Team/Collaborations">We helped them</a> solving their problems with the expression of their products.</p> <br />
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<h1 id="allParts"><span style="font-size:170%;">List of Parts</span style="font-size:170%;"> <!-- – <span style="font-size":50%">Placeholder --></h1><br />
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</html></div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:19:57Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
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As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [Heidelberg_orig_multi_plot.png|here].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:19:18Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [[Heidelberg_orig_multi_plot.png|here]].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C).<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:19:18Z<p>Jan glx: /* Different models tested */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_{max}$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [[File:Heidelberg_orig_multi_plot.png|here]].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C). ''Loss of enzymatic activity was characterized by the average enzymatic activity in this window (Fig. Y).''<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:18:42Z<p>Jan glx: /* Final model */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^{lysozyme}_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided to always fit the data of one plate on its own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_Max$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [[File:Heidelberg_orig_multi_plot.png|here]].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C). ''Loss of enzymatic activity was characterized by the average enzymatic activity in this window (Fig. Y).''<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/Modeling/Enzyme_ModelingTeam:Heidelberg/Modeling/Enzyme Modeling2014-10-18T03:18:33Z<p>Jan glx: </p>
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subtitle= Modeling of lysozyme activity with product inhibition<br />
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titles={{:Team:Heidelberg/templates/title|Introduction}}<br />
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abstract=<br />
The theory of enhancing heatstability by circularization suggests, that the more the ends are constrained, the better stabilization works. Based on this theoretical assumption we identified patterns to have the ability to customly build rigid linkers, that fit between the ends of the protein. Based on over 1000 experimental degradation curves from 9 different constructs with biological and technical replicates of lambda lysozyme, we wanted to gain quantitative insight into the effect of different linkers into the functionality after heatshock. Therefore, a comprehensive approach based on quantitative dynamic modeling (ODEs) was made to determine the relevant information. We started with easy fitting of the data and continued by testing different enzyme kinetics models. We evaluated and identified the most relevant and robust parameters by [[#PLE_analysis|profile likelihood analysis]]. Research suggested that a Michaelis-Menten model with substrate inhibition was appropriate for our data. Our final findings show that circularization of lysozyme can have a considerable influence on enzyme thermostability and that suboptimal linker design can decrease thermostability.<br />
These results were extraordinarily important for the calibration of [[Team:Heidelberg/Software/Linker_Software | CRAUT]].<br />
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{{:Team:Heidelberg/templates/mathjax}}</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:17:22Z<p>Jan glx: /* PLE analysis */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation (PLE) enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. By applying PLE analysis and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d Framework], operating on Matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^lysozyme_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided always to fit the data of one plate on it's own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_Max$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [[File:Heidelberg_orig_multi_plot.png|here]].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C). ''Loss of enzymatic activity was characterized by the average enzymatic activity in this window (Fig. Y).''<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:17:13Z<p>Jan glx: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when, it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. Applying PLE analysis one and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d arFramework], operating on matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^lysozyme_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided always to fit the data of one plate on it's own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_Max$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a denatured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 2). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 3). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 3)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [[File:Heidelberg_orig_multi_plot.png|here]].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C). ''Loss of enzymatic activity was characterized by the average enzymatic activity in this window (Fig. Y).''<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 4)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 5)|<br />
file = ples_bad_model.png|<br />
descr= Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
}}<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:15:46Z<p>Jan glx: /* Michaelis Menten kinetics and Competitive Enzyme Kinetics */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. $V_{max}$ is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when, it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. Applying PLE analysis one and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d arFramework], operating on matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^lysozyme_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided always to fit the data of one plate on it's own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_Max$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a detured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 1). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 2). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 1)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [[File:Heidelberg_orig_multi_plot.png|here]].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C). ''Loss of enzymatic activity was characterized by the average enzymatic activity in this window (Fig. Y).''<br />
<br />
{{:Team:Heidelberg/templates/image-quarter|<br />
caption = Figure 3)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
Figure: Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:14:54Z<p>Jan glx: /* OD to concentration calibration */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. V_{max} is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration})$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when, it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. Applying PLE analysis one and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d arFramework], operating on matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^lysozyme_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided always to fit the data of one plate on it's own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_Max$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a detured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 1). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 2). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 1)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [[File:Heidelberg_orig_multi_plot.png|here]].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C). ''Loss of enzymatic activity was characterized by the average enzymatic activity in this window (Fig. Y).''<br />
<br />
{{:Team:Heidelberg/templates/image-quarter|<br />
caption = Figure 3)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
Figure: Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:14:23Z<p>Jan glx: /* OD to concentration calibration */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. V_{max} is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathrm{concentration}$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when, it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. Applying PLE analysis one and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d arFramework], operating on matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^lysozyme_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided always to fit the data of one plate on it's own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_Max$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a detured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 1). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 2). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 1)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [[File:Heidelberg_orig_multi_plot.png|here]].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C). ''Loss of enzymatic activity was characterized by the average enzymatic activity in this window (Fig. Y).''<br />
<br />
{{:Team:Heidelberg/templates/image-quarter|<br />
caption = Figure 3)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
Figure: Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:13:59Z<p>Jan glx: /* OD to concentration calibration */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. V_{max} is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentration differences resulting in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathrm{ml}} {\mathrm{mg}} * \delta \mathtext{concentration}$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when, it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. Applying PLE analysis one and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d arFramework], operating on matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^lysozyme_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided always to fit the data of one plate on it's own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_Max$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a detured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 1). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 2). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 1)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [[File:Heidelberg_orig_multi_plot.png|here]].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C). ''Loss of enzymatic activity was characterized by the average enzymatic activity in this window (Fig. Y).''<br />
<br />
{{:Team:Heidelberg/templates/image-quarter|<br />
caption = Figure 3)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
Figure: Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Enzyme_Modeling_newTeam:Heidelberg/pages/Enzyme Modeling new2014-10-18T03:13:37Z<p>Jan glx: /* Different models tested */</p>
<hr />
<div>=Introduction=<br />
<br />
Enzyme kinetics is a widely studied field in biology. From the derived kinetic parameters one can make many different predictions about the function of a certain enzyme. A commonly used approach for the determination of the enzyme kinetic parameters, is the measurement of the reaction rate in time-dependent manner and with varying substrate concentrations. As this approach would be too laborious to apply in a high throughput manner, we instead decided to record the degradation curves for each lysozyme.<br />
<br />
==Lysozyme as model enzyme==<br />
Lysozyme of the $\lambda$-phage suits well as model for kinetic enzyme studies as it is a well characterized protein. Able to degredade the procaryotic cell wall composed of peptidoglycans. As already stated we anticipated that the lysozyme of the $\lambda$ bacteriophage could reasonably fulfill the requirements for our linker screen. <br />
<br />
As described in the [[Team:Heidelberg/Project/Linker_Screening|Linker screening project description]], we try to infer the loss of activity of $\lambda$-lysozyme due to heatshock, by observing the kinetic behavior on the degradation of the peptidoglycane outer layer of ''M. lysodeikticus''. This dynamic process, which ultimately leads to a change of turbidity, is very complex and has been widely discussed for more than 40 years now. On the other hand the activity of lysozyme is highly sensitive to outer conditions like salt concentrations in the media [[#References| [-1]]] and the lysozyme concentration itself [[#References| [0]]].<br />
<br />
We have not only observed the non-enzymatic activity maximum of lysozyme described by Düring et al. [[#References | [1]]] but also many observed effects can be explained by applying theory of product inhibition to the kinetics [[#References|[2]]]. On the other hand lysozymes unfolding behavior from 37°C seems to be dominated by a rapid collapse when it is denaturated [[#References|[3]]].<br />
<br />
==Michaelis Menten kinetics and Competitive Enzyme Kinetics==<br />
Michaelis Menten theory describes the catalytical behaviour of enzymes in simple reactions. It's basic reactions are assumed as<br />
\[ E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P \] , with E the enzyme, S substrate, ES the enzyme-substrate complex and P the reaction product. $k_r$, $k_f$ and $k_\mathrm{cat}$ are catalytical constants. This means part of the enzyme is always bound in an enzyme substrate complex. This kinetic behavior can be simplified in the basic differential equation: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]. V_{max} is the maximum reaction velocity, obtained from $V_{max} = k_{cat} * E$ and $K_m$ being the michaelis-menten constant<br />
<br />
Competitive product inhibition has the effect, that part of the Enzyme is also bound in the enzyme-product complex EP. This leads to an apparent increase of $K_m$ as: $K^\text{app}_m=K_m(1+[I]/K_i)$ Thus the differential equation changes as: \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m \left( 1 + \frac{S_0 - S}{k_i} \right) + \left[S\right]} \] where $S_0$ means the substrate concentration at start of the reaction and $k_i$ an inhibitory constant.<br />
<br />
=Methods=<br />
==Data==<br />
Using the [[Team:Heidelberg/Notebook/Methods#Lysozyme_Assay|Lysozyme Assay]] assays we have obtained over 1000 degradation curves for different lysozyme variants. In total, we got more than 100 000 data points from 12 assays performed on 96 well plates. From each well we obtained the degradation curves of M. lysodeiktikus by lysozyme, measured by turbidimetry change at 600 nm. We tested 8 different constructs of circular lysozyme and as reference also linear lysozyme. For all but two constructs, not only technical replicates on one plate were made, but also biological replicates from different growths. On each plate we subjected the lysozymes a heat-shock for one minute at different temperatures. This led to minimally 4 different curves per biological replicate per temperature and per lysozyme.<br />
<br />
Each degradation curve consisted in a measurement of the initial substrate concentration withoud lysozyme added, then there is a gap about 2 minutes, varying because of the sequence in that the plate-reader was measuring the wells, and then the degradation was measured every 100 seconds for 100 minutes. The first gap is due to the pipetting step, when adding the enzyme to the substrate and mixing the wells.<br />
<br />
<br />
Notice, that in regards to conditions used for the measurements, particular care was taken for the following aspects: The reactions always took place at the same temperatures. Also another crucial part was the time after adding the enzyme to the substrate: This was minimized as much as possible and we tried to keep it constant. We always made the dilutions in buffer from the same stock, in order to keep salt concentrations fixed.<br />
<br />
==OD to concentration calibration==<br />
<br />
There was performed a measurement for calibrating the $OD_{600}$ to substrate concentration. We have seen that until a substrate concentration of 0.66 mg/ml in the 300 µl wells the behaviour is linear with an offset due to the protein mix and the well plate. We have concentrationdifferences result in an $OD_{600}$ difference of: $\delta \mathit{OD} = (1.160 +- 0.004 \frac {\mathtext{ml}} {\mathtext{mg}} * \delta \mathtext{concentration}$. With this result one can easily calculate the concentration differences in each assay. <br />
Also the $OD_{600}$ of a well, where all the substrate was completely degraded needed to be measured. We found out, that the influence of the added protein mix on the $OD_{600}$ could be neglected.<br />
<br />
==Assumptions and data-based considerations==<br />
The time between when lysozyme was added to the substrate and the first measurement in the platereader was measured and assumed that it nearly took the same time for each measurement with normally distributed errors. Also, the platereader took about 1s for measuring one well. This delay was also taken into account.<br />
<br />
==PLE analysis==<br />
<br />
Often when fitting large models to the data there one has the problem that parameters are connected functionally. The method of Profile likelihood estimation enables to reveal of such dependencies.[[#References | [10]]] By evaluating the profile likelihood unidentifiable parameters can be grouped into structurally unidentifiable and practically unidentifiable parameters. [[#References | [9]]] A parameter is structurally unidentifiable when, it is in a functional dependence of one or more other parameters from the model. It is only practically unidentifiable if the experimental data is not sufficient to identify the parameter. This can be easily distinguished from the profile likelihood. Applying PLE analysis one and identifying structurally unidentifiable parameters, one is able to reduce the complexity of a given model.<br />
In our analysis we relied on [https://bitbucket.org/d2d-development d2d arFramework], operating on matlab and providing PLE analysis in an easy to use and fast manner.<br />
<br />
==Final model==<br />
<br />
For our model of the degradation we decided to apply product inhibited Michaelis Menten kinetics. As all our data was measured in $OD_{600}$ so at first the substrate concentration had to be calculated. Therefore we include an offset turbidity value, that is due to the turbidity of an empty well and included the OD to substrate calibration. Also the initial substrate concentration was inserted. $V_{Max}$, $K_M$, $K_I$ were the three enzymatical parameters that were fitted. Furthermore the error was fitted automatically too. For temperatures higher than 37.0 °C $V_{Max}$ was replaced by a ratio, called the activity of a temperature. Representing how much activity is left, compared to the activity of 37°C. It was defined by: $V^{lysozyme}_{Max, T} = act^{lysozyme}_T V^lysozyme_{Max, 37.0}$. This just meant exchanging one parameter by another for enhanced readability. On the other hand we assumed $K_M$ and $K_I$ to stay the same for different temperatures, but to vary between different lysozyme types. We decided always to fit the data of one plate on it's own, because we observed variation in functional behavior between the measurements from the different days. In table 1 it is shown which parameters are fixed for which part of the model.<br />
<br />
{|class="table table-hover" style="text-align: center;"<br />
|+'''table 1''': The span of parameters.<br />
!span of a parameter <br />
!$K_M$ <br />
!$K_I$ <br />
!$V_{Max}$ <br />
!$k_{decay}$ <br />
!OD offset <br />
!init_Sub <br />
!Error<br />
|-<br />
| colspan = "8" | '''Lysozymes'''<br />
|-<br />
|All lysozymes on the same plate <br />
| <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
|-<br />
|Same biological replicates of lysozyme on the same plate <br />
| x <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Same biological replicates of lysozyme on the same plate and the same temperature <br />
| <br />
| <br />
| x <br />
| <br />
| <br />
| <br />
| <br />
|-<br />
|colspan = "8" | '''Plate'''<br />
|-<br />
|The same plate <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| x<br />
|-<br />
|All plates <br />
| <br />
| <br />
| <br />
| <br />
| x <br />
| x <br />
| <br />
|}<br />
<br />
==Different models tested==<br />
<br />
During the development of our model, we have tested and compared different models. We tried many models describing the data of all the assays at once. These resulted often in calculations going on for hours. Mainly they were all variations of the final model, always based on product inhibited Michaelis Menten theory. In all the models modeling all the assays, $V_Max$ was split up into $k_{cat} * E$ where k_{cat} would be the same over different biological replicates and different plates, but E could vary.<br />
<br />
In the second model we have fixed $k_{cat}$ arbitrarily to 1 for all the different enzymes. In the third model we have tried $K_M, K_{cat}, K_I$ were fixed for the different temperatures, varying for the different types of lysozymes. In the next model (4) $K_M, K_{cat}, K_I$ were fitted separately for each temperature and each enzyme type.<br />
Substantially different was model 5, where we have inserted ratios for the enzyme concentrations. These ratios were obtained from coomassie gels (Fig. 1). Unfortunately no calibration could be made, so we could not introduce concentrations, but just ratios from the different types. For all the models on the whole dataset, the enzyme concentration was fixed between biological replicates.<br />
<br />
{{:Team:Heidelberg/templates/image-half| align=right| caption=Figure 1) Coomassie Gel of the linker constructs| descr=The expression levels of the linker constructs are different. The lysozyme band is the thick band above the N-intein.| file=62.png}}<br />
<br />
Model 6 was built to model the kinetics of one single plate. In contrast to the final model, here the kinetic parameters $K_{cat}, K_I$ were fitted for each temperature separately.<br />
<br />
=Results=<br />
To analyze the effect of circularization on the thermostability of the lysozyme variants, the heat shock dependent reaction rate parameters $v_{max}$ for all lysozyme variants had to be identified. For this purpose we analyzed the observed substrate degradation dynamics for the different lysozyme variants by ODE modeling. As detailed in the introduction, the enzymatic reaction mechanism of the lambdaphage lysozyme can be described by Michaelis-Menten kinetics with product inhibition. Furthermore, experiments on pH-dependent lysozyme degradation have shown that lysozyme exists in two distinct states when challenged with pH changes: the normal, functional state and a detured, nonfunctional state (REF). We hypothesized that lysozyme deformation under heat shock conditions could be described by a similar shift from a functional conformation to a distinct, denatured state. Consequently, enzymatic activity after heat shock was assumed to be exerted by only one, homogeneous, population of functional lysozymes, differing in size depending on heat shock intensity. Because the structure of the active enzyme species was assumed to be identical independent of the applied heat shock, the kinetic parameters of the enzymatic reactions could be assumed to be independent of heat shock intensity. Therefore, based on this model of enzyme denuration, enzymatic activity after heat shock could be assumed to be only dependent on the remaining fraction of functional lysozymes. <br />
<br />
This model was fitted to all available data, using simultaneous multi-model fitting where appropriate. The model could emulated the substrate degradation dynamics for all lysozyme variants (Fig 1). Profile likelihood-based identifiability analysis was employed to verify practical identifiability of the relevant kinetic parameters. While the kinetic parameters representing enzyme affinity for the substrate and the inhibitors could not be identified in the model, the maximal reaction rate $v_{max}$ where identifiable in all cases (Fig 2). The complete result of the profile likelihood analysis can be found here.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 1)|<br />
file = kinetics.png|<br />
descr= Dynamics of peptidoglycan degradation by the lambdaphage lysozyme can be emulated by a simple model assuming Michaelis-Menten kinetics with competitive product inhibition. The model was implemented with the assumption that lambdaphage lysozyme exists in two distinct states – functional or deformed - after heat shock within the considered range of intensities (citation). Following this assumption, kinetic parameters of the enzymatic reaction can be assumed to be independent of heat shock intensity. Thus, model complexity is considerably reduced, as explained in detail in the text. Exemplary measurements of peptidoglycan degradation by the linear lysozyme (a) and by a circularized lysozyme with the sg1 linker (b) are shown together with model fits. Substrate degradation is shown for basal enzyme activity after 10 min incubation at 37 °C and for diminished activity after 10 min incubation at 42 °C.}}<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
caption = Figure 2)|<br />
file = ple_linear.png|<br />
descr= The ratios of heat shock dependent maximal reaction rates $v_{max}$ are identifiable for all lysozyme variants. Likelihood profiles of $v_{max} after 1 min incubation at 44.5 °C and 54 °C are shown for the linear lysozyme (a) and a circularized lysozyme with the sg1 linker (b). Likelihood profiles for all parameters are documented [[File:Heidelberg_orig_multi_plot.png|here]].<br />
}}<br />
<br />
To compare thermostability of the different lysozyme variants, we analyzed the relationship between heat shock intensity and loss of enzymatic activity. As a measure for enzymatic activity, we used the normalized maximal reaction rate (the ratio of the enzymatic activity after heat shock and the basal enzymatic activity after incubation at 37 °C). Heat-shock dependent loss of enzymatic activity differed considerably between the different lysozyme variants (Fig 3). For a direct comparison of lysozyme variant thermostability we sought a robust statistic characterizing heat-shock resistance. This statistic should incorporate the threshold heat-shock intensity upon which significant loss of activity occurs as well as the steepness of the heat-shock intensity dependent loss of activity. We decided to focus on the heat-shock intensity window where most of the enzymatic activity was lost (45 °C to 57 °C). ''Loss of enzymatic activity was characterized by the average enzymatic activity in this window (Fig. Y).''<br />
<br />
{{:Team:Heidelberg/templates/image-quarter|<br />
caption = Figure 3)|<br />
file = resultsofscreening_new.png|<br />
descr= Heat-shock dependent enzyme activity for the linear lysozyme and 8 circularized lysozyme variants. Enzymatic activity is described here as the normalized maximal reaction rates, computed as the ratio of the maximal reaction rate after heat shock at the respective temperature and the maximal reaction rate after incubation at 37 °C. Two biological replicates were available for 7 of the 9 lysozyme variants and the $v_{max}$ values computed for each replicate are plotted separately. Temperature dependent decrease of the enzyme activity was fitted by splines to provide a better visualization of the relationship of heat shock intensity and enzyme deformation.<br />
}}<br />
<br />
Figure: Introduction of heat shock dependent reaction rates does not significantly improve the model fit. It was tested whether the model fit could be improved by assuming that heat shock induced enzyme deformation occurs gradually and not in distinct stages. In this case, the kinetic parameters of the enzymatic activity are dependent on the heat shock intensity.<br />
Exemplary measurements of peptidoglycan degradation by the linear lysozyme at 37 °C are shown for the simplified model assuming heat shock independent kinetic parameters (a) and the full model with heat shock dependent kinetic parameters (b). These data illustrate the general observation that the model fit was not significantly improved (see text for details).<br />
<br />
Finally, we tested whether the mechanistic assumption of a distinct transition between a single active and inactive state upon heat shock had affected the quality of the model fit. The alternative hypothesis concerning the mechanism of enzyme deformation would allow for continuous changes of the lysozyme structure in response to heat shock intensity. Thus, a gradual shift towards more deconformed structures would be expected for higher heat shock intensities. This would result in different kinetic parameters for the same lysozyme species under differing heat shock treatment. To test the effect of implementing this alternative deconformation mode in the model, model fitting was repeated with independent kinetic parameters for different heat shock intensities. Manual inspection of the fitting results did not show a better fit to the data. However, freeing the kinetic parameters resulted in a loss of parameter identifability. Therefore, the increased number of kinetic parameters was considered to negatively affect the usability of the model and the original, parameter-reduced, model structure was retained for analysis.<br />
<br />
=Discussion=<br />
<br />
Using dynamic ODE modeling, we could extract the heat-shock dependent maximal reaction rates of different lysozyme variants from simple substrate degradation measurements. The $v_{max}$ parameters were identifiable in spite of the complex reaction mechanism of the lysozyme. This allowed us to compute a normalized enzymatic activity for all lysozyme variants after a variety of different heat shock challenges. By comparing these enzymatic activities, thermostability of the different lysozymes variants could be directly compared.<br />
<br />
Our findings show that circularization of the lysozyme can have a considerable influence on enzyme thermostability. Similar findings have been reported for a variety of other proteins (sources). Here, we extend previous findings by demonstrating that the effect of circularization strongly depends on the chosen linker structure. Suboptimal linker design can decrease thermostability. The most evident example in the findings presented here is the sho2 linker which was chosen for testing as an example for linkers too short to bridge the natural distance between the C- and N-terminus of the lysozyme. In silico guided design of optimized linker sequences on the other hand can indeed result in increased thermostability, as demonstrated by the ord1 and ord3 linkers. These linkers where chosen as examples for linkers with a very low likelihood of crossing the active center of the enzyme. The implications of this analysis for the linker design are discussed in more detail in the documentation of the linker design software (here).<br />
<br />
=References=<br />
[-1] Mörsky, P. Turbidimetric determination of lysozyme with Micrococcus lysodeikticus cells: reexamination of reaction conditions. Analytical biochemistry 128, 77-85 (1983).<br />
<br />
[0] Friedberg, I. & Avigad G. High lysozyme concentration and lysis of Micrococcus lysodeikticus, Biochim. Biophys. Acta, 127 (1966) 532-535 <br />
<br />
[1] Düring, K., Porsch, P., Mahn, A., Brinkmann, O. & Gieffers, W. The non-enzymatic microbicidal activity of lysozymes. FEBS Letters 449, 93-100 (1999).<br />
<br />
[2] Colobert, L. & Dirheimer G. Action du lysozyme sur un substrat glycopeptidique isolé du micrococcus lysodeiktikus. B1OCHIMICA ET BIOPHYSICA ACTA, 54, 455-468 (1961)<br />
<br />
[3] Di Paolo, A., Balbeur, D., De Pauw, E., Redfield, C. & Matagne, A. Rapid collapse into a molten globule is followed by simple two-state kinetics in the folding of lysozyme from bacteriophage λ. Biochemistry 49, 8646-8657 (2010).<br />
<br />
[4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31.<br />
<br />
[5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009.<br />
<br />
[6] Liao, Fei, et al. "The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis–Menten rate equations of different predictor variables." Journal of biochemical and biophysical methods 62.1 (2005): 13-24.<br />
<br />
[7] Goudar, Chetan T., Jagadeesh R. Sonnad, and Ronald G. Duggleby. "Parameter estimation using a direct solution of the integrated Michaelis-Menten equation." Biochimica et Biophysica Acta (BBA)-Protein Structure and Molecular Enzymology 1429.2 (1999): 377-383. <br />
<br />
[8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).<br />
<br />
[9] Raue, a et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 19239 (2009).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/Software/Linker_SoftwareTeam:Heidelberg/Software/Linker Software2014-10-18T03:10:39Z<p>Jan glx: </p>
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As already [https://2014.igem.org/Team:Heidelberg/Toolbox/Circularization introduced], artificially circularized proteins may gain some heat stability by restraining the C- and N-terminus from moving around freely. This circularization may be trivial when the protein termini are very close to each other, which seems to be reasonably common [[#References|[1]]]. However, if the ends are too far from each other, a long linker is needed to connect them. This linker should not change the natural conformation of the protein and should constrain the relative position of the ends to restrict the degrees of freedom and thus to stabilize the structure even when heated up. On top, these linkers should not affect any of the protein functions. Consequently it is important to prevent linkers from passing through the active site or from covering binding domains to other molecules for example. Therefore one needs to be able to define the shape of possible linkers. This section describes the software we developed to design such linkers. We would like to stress that this work has been made possible thanks to the feedback between computer modeling and experimental work: We could first design linkers in silico, test them experimentally and use the results to further calibrate the software. To our knowledge, this is the first time that such an approach is used to customly design rigid linkers with angles to connect protein extremities.<br />
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{{:Team:Heidelberg/templates/mathjax}}</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/PartsTeam:Heidelberg/pages/Parts2014-10-18T03:10:26Z<p>Jan glx: </p>
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<h1 id="Favorite Parts">Favorite Parts.</h1><br />
<p>The iGEM Team Heidelberg 2014 had built a new biological system for the iGEM community integrating split-inteins. <br />
Intein splicing is a natural process that excises one part of a protein and leaves the remaining parts irreversibly attached. This great function allows you to modify your protein in numerous ways.</p><br />
<p>Creating a toolbox including all great functions and possibilities of inteins, we need a new standard for the scientific world of iGEM. This standard, the RFC of the iGEM Team Heidelberg 2014, allows us to easily and modulary work with split inteins.</p><br />
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<p>Our favorite Parts represent the basic constructs of our toolbox – the Assembly and the Circularization construct, which are both tested in many methods and applications. </p><br />
<p>In the following we present you <br />
<a href="http://parts.igem.org/Part:BBa_K1362000">BBa_K1362000</a>, the construct for circularization, <br />
<a href="http://parts.igem.org/Part:BBa_K1362100">BBa_K1362100</a> and <br />
<a href="http://parts.igem.org/BBa_K1362101">BBa_K1362101</a>, the N- and the C-construct for assembly. Take a look and visit the Partsregistry to read the associated documentation.</p><br />
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<h3> Circularization Construct. BBa_K1362000 </h3> <br />
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<h4>BBa_K1362000</h4><br />
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<img src="/wiki/images/7/7c/BBa_K1362000.png" class="img-responsive" alt="Circularization Construct"><br />
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<h3 id="Assembly"> Assembly Constructs. BBa_K1362100 and BBa_K1362101 </h3> <br />
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<h4>BBa_K1362100</h4><br />
<p>This intein assembly construct is part of our strategy for cloning with split inteins. Inteins are naturally occuring peptide sequences that splice out of a precursor protein and attach the remaining ends together to form a new protein. When splitting those intein sequence into an N-terminal and a C-terminal split intein one is left with a powerful tool to post-translationally modify whole proteins on the amino-acid sequence level. This construct was designed to express any protein of interest fused to the Nostoc punctiforme DnaE N-terminal split intein. </p><br />
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<img src="/wiki/images/8/81/BBa_K1362100.png" class="img-responsive" alt="Assembly Constructs"><br />
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<h4>BBa_K1362101</h4><br />
BBa_K1362101 is the corresponding C-terminal construct to BBa_K1362100. Upon coexpression or mixture of the N- and C-constructs protein splicing takes place and the N- and C-terminal proteins of interest are irreversibly assembled via a newly formed peptide bond.</p><p><br />
This mechanism can be applied for a variety of different uses such as the activation of a protein through reconstitution of individually expressed split halves. See our split sfGFP experiment and the respective parts in the registry for more information. Protein splicing offers many new possibilities and we hope to have set a foundation that you guys can build on!</p><br />
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<img src="/wiki/images/3/3f/BBa_K1362101.png" class="img-responsive" alt="Assembly Constructs"><br />
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<h1 id="Sample Data Page">Sample Data Page for our favorite Parts.</h1><br />
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<h3> Circularization Construct. BBa_K1362000 </h3> <br />
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<img src="/wiki/images/9/9b/SampleData_Circularization.png" class="img-responsive" alt="Circularization Construct"><br />
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<!-- <div class="margin-top"> --><br />
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This part represents an easy way to circularize any protein. In a single step you can clone your protein in the split intein circularization construct. Exteins, RFC [i] standard overhangs and BsaI sites have to be added to the coding sequence of the protein to be circularized without start- and stop codons by PCR. By Golden Gate assembly, the mRFP selection marker has to be replaced with the protein insert.<br />
If the distance of the ends of your protein of interest aren't close enough to connect them you will need a linker. <a href="http://parts.igem.org/Part:BBa_K1362000">BBa_K1362000</a>, the split intein circularization construct, includes a strong T7 RBS (<a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362090">BBa_K1362090</a>), we sent to the parts registry as well, and the split intein Npu DnaE. The T7 RBS derived from the T7 phage gene 10a (major capsid protein). </div> <br />
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The resulting plasmid can be used to express the protein of interest with the obligatory linker and the N- and C-intein.<br />
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<div class="well well-sm"><br />
In an autocatalytic in vivo reaction, the circular protein is formed. To read more about the trans-splicing reaction visit our <a href="https://2014.igem.org/Team:Heidelberg/Project/Background">Intein Background</a> page. If corresponding split inteins are added to both termini of a protein, the trans-splicing reaction results in a circular backbone. <br />
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<div class="well well-sm"><br />
Circular proteins offers many advantages. While conserving the functionality of their linear counterpart, circular proteins can be superior in terms of thermostability, resistance against chemical denaturation and protection from exopeptidases. Moreover, a circular backbone can improve in vivo stability of therapeutical proteins and peptides.<br />
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<h3> Assembly Construct. BBa_K1362100 and BBa_K1362101 </h3> <br />
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<div class="row"><br />
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<img src="/wiki/images/5/5c/SampleData_Assembly.png" class="img-responsive" alt="Assembly Constructs"><br />
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<div class="col-md-5 col-sm-12 col-xs-12"><br />
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<div class="well well-sm"><br />
These parts represent an easy way to build fusion constructs of intein parts with any protein or peptide of interest (POI). In a single step you can fuse your protein to a split intein part. Desired extein residues, RFC [i] standard overhangs and BsaI sites generating these overhangs have to be added to the coding sequence of the POI by PCR. Facilitating the highly efficient GoldenGate assembly reaction, the mRFP selection marker can be replaced with the POI insert.</div> <br />
<!-- </div> --><br />
<div class="well well-sm"><br />
The resulting plasmid can be used to express the protein of interest with the obligatory linker and the N- and C-intein.<br />
</div><br />
<div class="well well-sm"><br />
In an autocatalytic in vivo reaction, the circular protein is formed. To read more about the trans-splicing reaction visit our <a href="https://2014.igem.org/Team:Heidelberg/Project/Background">Intein Background</a> page. If corresponding split inteins are added to both termini of a protein, the trans-splicing reaction results in a circular backbone. <br />
</div><br />
<div class="well well-sm"><br />
If performing experiments with split inteins make sure you always have a non-splicing negative control. Check out our non splicing assembly constructs (with part names ending with 2 or 3) for example: <a href="http://parts.igem.org/Part:BBa_K1362102">BBa_K1362102</a> and <a href="http://parts.igem.org/Part:BBa_K1362103">BBa_K1362103</a><br />
</div><br />
</div><br />
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<div class="col-md-12 col-sm-12 col-xs-12" style="margin: 100px 0;"><br />
<img src="/wiki/images/9/9a/Heidelberg_dna.png" class="img-responsive" alt="Circularization Construct"><br />
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<h1 id="Intein Library">Intein Library.</h1><br />
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Inteins are the basic unity of our toolbox. They are integrated as extraneous polypeptide sequences into habitual proteins and do not follow the original protein function. Inteins perform an autocatalytic splicing reaction, where they excite themselves out of the host protein while reconnecting the remaining chains on both end, so called N and C exteins, via a new peptide bond. Read more about it in our [https://2014.igem.org/Team:Heidelberg/Project/Background| project background]!<br />
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To characterize the different types and groups of split-inteins and inteins we collect many details about them to develop a intein library. It gives you a great and clear overview about the most important facts.<br />
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{| class="table table-hover"<br />
|-<br />
!Split intein<br />
!Special features<br />
!Size N [aa]<br />
!Size C [aa]<br />
!Reaction properties<br />
!Origin<br />
!References<br />
|-<br />
| Npu DnaE||fast; robust at high temperature range and high-yielding trans-splicing activity, well characterised requirements||102||36||t1/2 = 63s , 37°C , k=~1x10^-2 (s^-1); activity range 6 to 37°C||S1 natural split intein, Nostoc punctiforme||[[#References|[1]]] [[#References|[2]]] <br />
|-<br />
| Ssp DnaX||cross-reactivity with other N-inteins, transsplicing in vivo and in vitro, high yields||127||6||k=~1.7x10^-4(s^-1); efficiency 96%||engineered from Synechocystis species||[[#References|[3]]] [[#References|[4]]] <br />
|-<br />
| Ssp GyrB|| very short Nint facilitates trans-splicing of synthetic peptides||6||150||k=~1x10^-4(s^-1), efficiency 40-80%||S11 split intein enginered from Synechocystis species, strain PCC6803||[[#References|[4]]] [[#References|[5]]] <br />
|-<br />
| Ter DnaE3||trans-splicing activity with high yields||102||36||k=~2x10^-4(s^-1), efficiency 87%||natural split intein, Trichodesmium erythraeum||[[#References|[4]]] [[#References|[6]]] <br />
|-<br />
| Ssp DnaB||relatively fast||||||t1/2=12min, 25°C, k=~1x10^-3(s^-1)||engineered from Synechocystis species, strain PCC6803||[[#References|[2]]] <br />
|-<br />
| Gp41-1||fastes known reaction ||88||38||t1/2=20-30s, 37°C, k=~1.8x10^-1 (s^-1); activity range 0 to 60°C||natural split intein, Cyanophage||[[#References|[7]]] [[#References|[8]]] <br />
|-<br />
|}<br />
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<h3>References</h3><br />
<p>[1] Iwai, H., Züger, S., Jin, J. & Tam, P.-H. Highly efficient protein trans-splicing by a naturally split DnaE intein from Nostoc punctiforme. FEBS Lett. 580, 1853–8 (2006).</p><br />
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<p>[2] Zettler, J., Schütz, V. & Mootz, H. D. The naturally split Npu DnaE intein exhibits an extraordinarily high rate in the protein trans-splicing reaction. FEBS Lett. 583, 909–14 (2009).</p><br />
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<p>[3] Song, H., Meng, Q. & Liu, X.-Q. Protein trans-splicing of an atypical split intein showing structural flexibility and cross-reactivity. PLoS One 7, e45355 (2012).</p><br />
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<p>[4] Lin, Y. et al. Protein trans-splicing of multiple atypical split inteins engineered from natural inteins. PLoS One 8, e59516 (2013).</p><br />
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<p>[5] Appleby, J. H., Zhou, K., Volkmann, G. & Liu, X.-Q. Novel Split Intein for trans-Splicing Synthetic Peptide onto C Terminus of Protein. J. Biol. Chem. 284, 6194–6199 (2009).</p><br />
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<p>[6] Liu, X.-Q. & Yang, J. Split dnaE genes encoding multiple novel inteins in Trichodesmium erythraeum. J. Biol. Chem. 278, 26315–8 (2003).</p><br />
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<p>[7] Carvajal-Vallejos, P., Pallissé, R., Mootz, H. D. & Schmidt, S. R. Unprecedented rates and efficiencies revealed for new natural split inteins from metagenomic sources. J. Biol. Chem. 287, 28686–96 (2012).</p><br />
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<p>[8] Dassa, B., London, N., Stoddard, B. L., Schueler-Furman, O. & Pietrokovski, S. Fractured genes: a novel genomic arrangement involving new split inteins and a new homing endonuclease family. Nucleic Acids Res. 37, 2560–73 (2009).</p><br />
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<h1 id="Backbones">Our Backbones.</h1><br />
<p>Standard BioBrick cloning is a universal way of putting two BioBrick parts together to build a new BioBrick part. Despite several alternative cloning methods allow the assembly of multiple parts at one its simplicity and the broad availability of compatible parts keep it the 'de facto' standard of the iGEM-community.</p><br />
<p>Using standard BioBrick cloning, the generation of translationally active parts requires often more than one round of cloning. The ability to easily test the functionality of a protein before cloning them into complicated circuits has the potential to prevent many unsuccessful experiments of iGEM teams and may improve the characterization of the parts in the parts registry. However the extra amount of work required to clone such an additional construct may inhibit this behavior. We therefore improved the standard plasmids pSB1X3 and pSB4X5 by inserting a lacI repressible T7 promoter directly upstream to the BioBrick prefix of those plasmids. This promoter is completely inactive in 'E. coli' strains lacking a T7 RNA polymerase such as TOP10 or DH10beta bute inducible in strains carrying the T7 RNA polymerase under a lacI repressible promoter such as DE3 strains. This enables the use of the same backbone for cloning and over expression. Using 3A assembly a translational active part can be cloned from an RBS and a coding part in one step while maintaining the full flexibility of standard BioBrick assembly. These new RFC 10 conform backbones eliminate one cloning step needed for the expression and thus the characterization of a newly BioBricked protein. Version number 30 was claimed for the high copy variants and version number 50 for the low copy variants.</p><br />
<p>High copy BioBrick expression backbone:</p><br />
<ul><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362091">pSB1A30</a>(Part:BBa_K1362091): High copy BioBrick cloning/expression backbone carrying Amp resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362092">pSB1C30</a>(Part:BBa_K1362092): High copy BioBrick cloning/expression backbone carrying Cm resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362093">pSB1CK30</a>(Part:BBa_K1362093): High copy BioBrick cloning/expression backbone carrying Kan resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362094">pSB1CT30</a>(Part:BBa_K1362094): High copy BioBrick cloning/expression backbone carrying Tet resistance</li><br />
</ul><br />
<p>Low copy BioBrick expression backbone:</p><br />
<ul><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362095">pSB4A50</a>(Part:BBa_K1362095): High copy BioBrick cloning/expression backbone carrying Amp resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362096">pSB4C50</a>(Part:BBa_K1362096): High copy BioBrick cloning/expression backbone carrying Cm resistance</li><br />
<li><a href="http://parts.igem.org/wiki/index.php?title=Part:BBa_K1362097">pSB4K50</a>(Part:BBa_K1362097): High copy BioBrick cloning/expression backbone carrying Kan resistance</li><br />
</ul><br />
<p>Because of the great experience we had using our expression vectors, we sent them to the iGEM team Aachen and Tuebingen. <a href="https://2014.igem.org/Team:Heidelberg/Team/Collaborations">We helped them</a> solving their problems with the expression of their products.</p> <br />
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<img src="/wiki/images/9/9a/Heidelberg_dna.png" class="img-responsive" alt="Circularization Construct"><br />
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<h1 id="allParts"><span style="font-size:170%;">List of Parts</span style="font-size:170%;"> <!-- – <span style="font-size":50%">Placeholder --></h1><br />
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</html></div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Linker_SoftwareTeam:Heidelberg/pages/Linker Software2014-10-18T03:10:20Z<p>Jan glx: /* Abstract */</p>
<hr />
<div>=General procedure=<br />
In short, the software can provide a weighted list of linkers to circularize any protein of interest with a known structure. Those linkers are made of rigid alpha helices segments connected with defined angles. Contrary to flexible linkers, those rigid linkers were expected to constrain the protein extremities and to confer better heat stability. Such an idea was already developed [[#References|[2]]] but only with alpha helices defining simple rods, and without any possibility to introduce angles. To generate those linkers, we first defined the geometrical paths, with segments and angles, that they should follow. The geometrical paths that are biologically feasible are afterwards translated into amino acid sequences. Both the compatibility of paths with possible structures and the translation were made possible thanks to our [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling approaches]. The first approach consisted in performing a statistical analysis of more than 17000 known non-homologous structures containing alpha helices connected with angles. For the second approach, we modeled the conformation of linkers circularizing proteins of known structure and analyzed them for certain properties. This second approach was run for a large number of proteins thanks to our distributing computing system [https://2014.igem.org/Team:Heidelberg/Software/igemathome igemathome]. The software provides different possible linkers with weights that provide the ranking of the linkers depending on their capacity to maintain protein activity at higher temperatures. They were generated thanks to an extensive [https://2014.igem.org/Team:Heidelberg/Project/Linker_Screening linker screening] on the target protein lambda-lysozyme, using the first modeling approach.<br />
The documentation of our CRAUT software can be found [https://2014.igem.org/Team:Heidelberg/Software/Linker_Software/Documentation here].<br />
{{:Team:Heidelberg/templates/image-half|<br />
align=right|<br />
caption=Figure 0)|<br />
descr= A representation of the general concept of CRAUT. At first the user provides it with a protein structure, by providing it with a PDB file. Then he can add relevant data, like binding sites, that he has found in databases. Then he chooses which parts of the protein the software should circularize and which parts of the protein should be ignored. After the calculations have finished, the user gets a sequence of the best linker, with which he could circularize the target protein. |<br />
file=how_user_should_use.png}}<br />
<br />
=Background=<br />
Classically, protein linkers were designed in three different manners. The easiest way is to define the length that a linker should cover and then simply use a flexible glycine-serine peptide with the right amount of amino acids to match this length. Glycine is used for flexibility, as it has no sidechain and does not produce any steric hindrance, while serine is used for solubility, as it has a small polar side chain. This solubility is important, as the linkers should not pass through the hydrophobic core of the protein, but should be dissolved in the surrounding medium. These flexible linkers were normally used for circularization but also for connecting different proteins, when the main goal is that the different parts are connected, but not how they are connected, or when the flexiblity of the linker was required for specific applications. <br />
<br />
A second strategy consists in using rigid helical linkers to keep proteins or protein domains at a certain distance from each other. This is especially important for signalling proteins and fluorescent proteins. One major property of alpha helices is that they always fold in a defined way with well defined angles and lengths. There are also many different helical patterns that differ in stability and solubility. Although they have been used to design cirularizing linkers [[#References|[2]]]. One big disadvantage of this strategy is that one can only build straight linkers with helices. So in the context of circularization, if an artificial line that would connect protein extremities is crossing the protein, this strategy is not an option.<br />
<br />
The third option, which served as a base to develop our approach and which came from discussions with the group of Rebecca Wade in Heidelberg, Germany, consists in designing customly tailored linkers for each specific application. These linkers can be obtained from protein structure prediction. At first one needs to define the path that the linker should take to connect the protein ends. Afterwards, one designs a possible linker sequence that might fit well. Next one makes a structure prediction of the linker attached to the proteins to validate the prediction. Several different linkers, with slight changes, can be compared. This is repeated several times until the linker effectively follows the expected path. This method requires a strong knowledge on protein folding and protein structure prediction and is computationaly intensive. On the other hand, the benefit can be important as the interaction of the linker with the protein surface can be taken into account and as one can accurately define the path taken by the linker to the resolution of protein structure.<br />
<br />
We have set up a completely new strategy to design rigid linkers. As further detailed in the [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling] part, it is possible to define the shape of a linker, by combining rigid alpha helical rods with well-defined angle patterns. Therefore, by defining, in a geometrical way, the possible paths of the circularizing linkers for a given protein, we can then propose potential linkers. This definition of the geometrical path can be very difficult, especially for large proteins with complex shapes. Moreover, this definition is further constrained by the fact that linkers must avoid hiding active sites of the protein of interest. Finally the paths have rotational degrees of freedom at the extremities of the protein, and depending on their orientation, they may or may not match the geometry of the protein. The tool we present here covers the two steps: defining geometrical paths with some weights and translate them into feasible linkers, also with weights. This tool is universal as it has the capacity to design circularizing linkers for any protein with a known structure. Moreover it is modular as, thanks to our [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling approach], we have designed linkers as exchangeable blocks of rods of different lengths and of angle patterns. The following sections detail the different steps followed by our software to design proper linkers.<br />
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<br />
==PDB analysis==<br />
{{:Team:Heidelberg/templates/image-quarter|<br />
align=right|<br />
caption=Figure 1) Accessible ends|<br />
descr= When checking whether the ends of the protein are covered, at first for all the directions it is checked, whether some of the protein points are in the way. This was done for discrete angles incremented at 5° |<br />
file=ends-covered.png}}<br />
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At first, the PDB file containing the structure of the target protein is parsed and the coordinates of the atoms are stored, in the metric unit. After this, some initial tests are made with the protein structure. First, we checked whether the C- and N-termini lie on the surface of the protein and are accessible to the solvent, which is crucial for circularization. We defined a line originating from an extremity of the protein with the two angles of the spherical coordinates around the z-axis. From that, we could determine the accessible angles by rejecting all the lines that are too close to the protein. As the future linker will be made of alpha helices and will therefore have a radius of 5 &Aring;, we used this length as the minimal allowed distance.<br />
Those allowed angles are stored for the coming linker generation.<br />
<br />
==Generation of geometric paths==<br />
As our strategy consists in building linkers with helical rods and connecting angles, a path is completely defined by the coordinates of the angle points. Advancing one step from an existing point is always done by adding a displacement vector on this point. This vector is defined by the two spherical angles, chosen here in a discrete manner with an increment of 5 degrees, and by a length, also chosen in a discrete manner. This discrete length was used in two different contexts: it may correspond to the length of an alpha helix or to the length of the flexible part that appears at the extremity of the protein. The coordinates that are reached thanks to this vector defines the new coordinates of an angle point, no matter if the vector corresponds to an alpha helix or to a flexible part. <br />
{{:Team:Heidelberg/templates/image-quarter|<br />
align=right|<br />
caption=Figure 2) Linker going around torus|<br />
descr=The worst shape we could think of for circularization was a torus without hole with ends in the middle. Even this shape could be circularized with our linkers. |<br />
file=torus.png}}<br />
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As we screen for all possible angles in a discrete manner, those angle points coordinates are regularly distributed on a sphere. As further detailed in the next sections, those spheres are defined from both ends, either once or in several steps. Then the software checks for possible straight connections of given lengths for each pair of angle points originated from both extremities. <br />
{{:Team:Heidelberg/templates/image-full|<br />
align=right|<br />
caption=Figure 3) The generation of the paths|<br />
descr= Easy representation of the process of points generation. At first all the points from the start are generated, left. Then the points from the end are generated. Then all possible connections between the points are checked for their validity. This is done for every point from the beginning. |<br />
file=figure3.png}}<br />
<br />
The linkers are built in a modular way, with blocks of well-defined size. From the [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling] of potential linkers, we could derive 8 different alpha helical rods, all with different lengths. On top, the length of the two segments inside an angle block was always 8&Aring;, so exchanging angle blocks do not affect the length of the linker. This means that the distance between the angle points is well defined, an essential aspect of our strategy of linker design.<br />
The software proceeds in three steps. First, it checks for the possibility of direct single alpha helix linker. for this, it applies the procedure just mentioned with spheres of radius that reasonably corresponds to the length of the short parts at the extremity of the protein. Second, it tests if a linker containing two alpha helices connected with a right angle allows the circularization. Finally it searches the possible linkers with three angle points. The next parts will explain those three steps in detail.<br />
This method has been chosen, because it could be implemented easily and efficiently in our program. However, this strategy generated paths that crossed the protein. Therefore we put big efforts in the sorting out of the paths.<br />
<br />
===Step 1===<br />
As a simple rigid linker with no angle would be easier to design and likely more thermostable than the ones containing angles, the software first checks if this simple solution is possible.<br />
{{:Team:Heidelberg/templates/image-quarter|<br />
align=right|<br />
caption=Figure 4) Step 1|<br />
descr= Only one single alpha helix connects the flexible ends of the protein. |<br />
file=one_helix_flex_ends.png}}<br />
For this, we took into account the fact that proteins have some flexible amino acids at their extremities. This flexible part may come from the protein itself, but also from the 2 glycines that are included at the N-terminal part and from the extein at the C-terminal part. Those two latter parts comes from our linkers. Those parts have no preferential angles and offers a large amount of possibilities to insert fitting linkers. But this flexibility is also a drawback as we have to include this large amount of possible angles and length to our path search.<br />
In this first step, the software explicitely takes these flexible parts into account to check for the possibility of straight linkers. As the angles and the length of the flexible parts are variable, the software position their extremity on a sphere centered on the last fixed position of the structure as explained above. The radius of this sphere is incremented in a discrete manner, in 4 steps, from 5.25 &Aring; to the maximum length of the flexible part.<br />
Then all possible straight segments between the points and the lastpoints are tested. If they are closer than 5 &Aring; to the protein, of if they cross it, then they are rejected. If they are kept, then the software checks whether the length of the segments is compatible with the feasible alpha helices in terms of length: if the length of a given segment equal one of the 8 alpha helix lengths plus or minus 0.75 &Aring;, then the path is eventually saved.<br />
<br />
===Step 2===<br />
The next possibilty to design more complex rigid linkers while still taking flexible ends into account with a reasonable calculation time was to reduce amount of possible angles. As we originally thought that 90° angle would be practically feasible, the software was designed to generate linkers with flexible ends and one 90° angle. <br />
<br />
{{:Team:Heidelberg/templates/image-quarter|<br />
align=right|<br />
caption=Figure 5) Step 2|<br />
descr= The linker forms an angle of 90°. |<br />
file=figure6.png}}<br />
This choice was notably made because of the simplicity to calculate lengths of right triangle edges. We already saw in Step 1 that the length of an edge can only take 8 different values. As the linkers have to start from the extremities of the protein, and as we impose a right angle, the number of possible paths is therefore low, making them easy to compute. Practically, the extremities of the proteins are positioned in a flexible way as in Step 1. From each of the positions allowed by this flexibility, the software searches for all the allowed right triangles. This was mainly done, as the degrees of freedom needed to be restricted to keep calculations feasible.<br />
<br />
===Step 3===<br />
Finally the software also provides the possibility to find paths with up to 4 edges, meaning 4 alpha helices and 3 angles. Thanks to the modularity of the possible linkers, such paths can offer the possibility to circularize theoretically any kind of protein, see figure 2).<br />
<br />
To keep the calculation feasible in a reasonable time, we design the searching strategy so that the flexible part at the extremity are oriented in the same direction as the consecutive alpha helix. This is obviously restricting the search but as these orientations are allowed for the flexible part, this approach remains fully correct. <br />
First, potential ending points of the first alpha helical rod are calculated from the N-terminal point of the protein. The orientation is chosen in a discrete manner, with an incrementation of 5 degrees for the two angles of the spherical coordinates. The distance from the origin corresponds to the 8 possible lengths allowed by the alpha helices, as already seen in Step 2, plus a length of 0, which mimics a linker with 3 instead of 4 edges. The exact same procedure is repeated to define all the potential ending points of the second alpha helical rod starting from all the possible ending points of the first alpha helical rod. Thanks to the possibility of a length of 0 for the first and the second rods, the software also calculate paths with 2 edges. Then, the same is done only once from the C-terminal point of the protein, defining 1 edge. The final step consists in checking if the points originating from the N- and C-terminal points can be linked by an potential alpha helix, i.e. if they are separated by the appropriate distance. If any of the potential alpha helix length lies within the distance between two points plus or minus 0.75 &Aring;, then the path is eventually saved. In the same way, if two points are directly closer than 0.75 &Aring;, then the path is also saved.<br />
<br />
==Sorting out of paths==<br />
The previous part described the generation of paths that can connect the two extremeties of the protein irrespective of the position of these paths relative to the protein. While this allows a fast computing of the geometrical paths, this also implies that the paths that are not practically feasible need to be sorted out. This is the most time consuming part of the computing as about 1 billion paths are generated. Three criteria are considered for the sorting. The first one is the feasibility of the linker: can the software find angle patterns that correspond to the one defined by the geometrical path? This question was part of the motivation for a large modeling effort (link) to determine the possible angles between consecutive angles. This was achieved by analyzing the [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling distribution of angles] between alpha helices found in the ArchDB database. As nearly any angle could be found between 20 and 170 degrees, only few paths were actually rejected at that step. The next criteria was the position of the angle point: if they appear inside the protein, then the path is rejected. Finally, the software checks if any of the atoms of the protein is less than 5&Aring; away from any of the alpha helices, then the path is also rejected.<br />
<br />
==Shifting paths to the patterns==<br />
<br />
The strategy described in step 3 gives a certain freedom for the rod that connect the last two angle points that were generated from the N- and C-terminal points. As this freedom is actually not permitted by the alpha helix and the angle pattern, but is permitted by the flexible part for example at the C-terminal end, the software slighty refine the path by rotating the segment that originates from the C-terminal point.<br />
<br />
==Weighting of paths==<br />
Before translating the paths into sequences and thus into linkers that can be expressed, each path needs to be evaluated for its potential capacity to enhance heat stability. For this we have identified different contributions that should be combined into one value that defines how good the linker may be and that consequently defines its ranking among all possible linkers. The smaller this value, the more we expect that the linker will enhance thermostability. An important step for all these contributions is the normalization, as explained in the next paragraphs.<br />
The first contribution we considered was the linker length. We assumed that a short linker is better to constrain the protein extremities, and that a long helix might give more flexibility. Because we wanted this value to be independent of the size of the protein, the length of the linker is normalized to the distance between the two termini.<br />
The second contribution relates to the angles used in the linker. We learned from the [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling] that angles formed by a certain angle pattern follow a certain distribution. First, we assumed that the narrower the distribution, the more likely the alpha helices would actually produce this angle. Second, the angles found by the software should be as close as possible to those well-defined angles. In this case, the weight value from this contribution should be low.<br />
Then the distance of the linker to the protein is taken into account. Because the linkers should not disturb the protein in its normal environment, linkers that pass close to the protein surface are considered better linkers. The distance was defined as the minimal distance between the linker and all the atoms of the protein. As already mentioned for the sorting of the paths, a linker cannot come closer than 5 &Aring; and this distance was used for normalization of calculated distances.<br />
After this, the places a linker should avoid are calculated. Each protein can interact with other molecules on some oarts of its surface. The user can specify where and how big those parts are. If a linker passes in front a potential molecule binding domain, the value of the corresponding path goes to infinity, so that the linker is discarded. Conversely the farther a linker is from a potential ligand binding domain, the smaller its weighting value. The user can also specify the importance of certain regions. In the end the total weighting is normalized to the amount of binding domains.<br />
<br />
===Calibrating the weighting function===<br />
Every contribution has its own distribution. You can see an example in figure <br />
{{:Team:Heidelberg/templates/image-quarter|<br />
align=right|<br />
caption=Figure 6) Distribution of length contribution of lysozyme|<br />
descr=Each weight has it's own distribution. As an example here the distribution of the length weighting of lambda lysozyme is shown. One can clearly see the gaps due to the discrete lengthes of the building blocks.|<br />
file=histogram_lengths_lys.png}}<br />
, but all of them have different shapes. The aim is to find the paths that globally minimize all of these distributions. Therefore, for simplicity,in the weighting function the four mentioned contributions were combined in a linear manner:<br />
\[ W(p) = \alpha L(p) + \beta A(p) + \gamma D(p) + u(p) \]<br />
where W is the final weighting, p the path, L the length contribution, A the angle contribution, D the distance contribution and u the contribution from the forbidden regions. $\alpha, \beta, \gamma, \delta$ are the weighting constants that needed to be found. The normalization performed for each of the contribution were made so that each of them is dimensionless and that all have reasonably similar values.<br />
The weighting constants were obtained from the [https://2014.igem.org/Team:Heidelberg/Project/Linker_Screening linker screening] performed with lysozyme and the [https://2014.igem.org/Team:Heidelberg/Modeling/Enzyme_Modeling modeling of the enzyme activity]. Their calculation is presented in the results below.<br />
<br />
==Translating paths to sequence==<br />
As already mentioned before the software is provided with two databases, one for the possible angle patterns and one for the helix patterns. The choice of the patterns was inspired by known crystal structures extracted from databases and described in different papers.<br />
A huge in silico screening for refining the preferences of the patterns was then set up using the [https://2014.igem.org/Team:Heidelberg/Software/igemathome distribution calculation] system. For the complete description of search for suitable patterns, one can read the [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling] page. <br />
All the possible paths are now split up at the angles and compared with the possible patterns in the databases. The most suitable patterns are identified and added together to build the paths sequence. It is important to notice that this is only possible because of the modularity of our linker patterns used as building blocks: each block, being an alpha helix or an angle pattern, is not affected by the other. Thus for each possible path, one sequence is produced.<br />
<br />
==Clustering of paths==<br />
Many different paths are represented by the same sequence [[###Figure that shows, different paths have same properties, already before in the text ###]] and we therefore clustered such paths. The weigths for those clustered paths were then calculated by averaging the weights of the different paths that compose a cluster.<br />
<br />
=Results=<br />
==DNMT1==<br />
A major motivation of our effort to design rigid linkers with angles was the [https://2014.igem.org/Team:Heidelberg/Project/PCR_2.0 circularization of the DNA methyltranferase Dnmt1]. The truncation form used in our project is composed of 900 amino acids and the N- and C-terminal extremities are well separated. To circularize it, two linkers were designed: a flexible one made of glycine and serine, and a rigid one designed by the software. The rigif linkers for DNMT1 were obtained from an early state of the software. At that time the calculation took 11 days on a laptop computer with intel i5 processor and 8GB of RAM, which shows the importance of a distributed computing system for large proteins. But from that state on, the software has still improved a lot, resulting in reduced calculation time to about 1 day for DNMT1.<br />
<br />
==Feedback from wet lab==<br />
The results from the software for lysozyme were tested as described in the [https://2014.igem.org/Team:Heidelberg/Project/Linker_Screening linker-screening] part and evaluated as described in the [https://2014.igem.org/Team:Heidelberg/Modeling/Enzyme_Modeling enzyme modeling] part. We have performed a large linker screening on 10 different lysozymes with different linkers. As the purpose of the lysozyme screen was the calibration of the software,those linkers (Table 1) were designed according to the four contributions previously mentioned. One of them was the shortest possible, one had the best possible angle, and so on.<br />
{| class="table table-hover" style="text-align: center;"<br />
|+ '''Table 1''': Linker and their amino acid sequence. Green: attachment sequences to prevent the flexible regions from being perturbed; Blue: angle; Purple: extein.<br />
! Linker<br />
! Amino acid sequence<br />
! activity<br />
! length- contribution<br />
! angle- contribution<br />
! binding site contribution<br />
! distance from surface<br />
! weightingvalue after calibration<br />
|- <br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|'''Very good linkers'''<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| sgt2<br />
| <span style="color:#3ADF00;">GG</span>AEAAAK<span style="color:#00BFFF;">AAAHPEA</span>AEAAAK<span style="color:#A901DB;">RGTCWE</span><br />
| 0.7477<br />
| 1.9205<br />
| 6.7789<br />
| 0.002259<br />
| 10.525<br />
| 114912<br />
|-<br />
| rigid<br />
| <span style="color:#3ADF00;">GG</span>AEAAAKEAAAKAA<span style="color:#3ADF00;">P</span><span style="color:#A901DB;">RGKCWE</span><br />
| 0.9447<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| '''Average linkers'''<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| may1<br />
| <span style="color:#3ADF00;">GG</span>AEAAAKEAAAKA<span style="color:#00BFFF;">AAAHPEA</span>AEAAAK EAAAKA<span style="color:#00BFFF;">KTA</span>AEAAAKEAAAKA<span style="color:#A901DB;">RGTCWE</span><br />
| 0.7489<br />
| 6.2225<br />
| 13.19<br />
| 0.00384<br />
| 1095.2<br />
| 196414<br />
|-<br />
| ord1<br />
| <span style="color:#3ADF00;">GG</span>AEAAAKEAAAK<span style="color:#00BFFF;">ATGDLA</span>AEAAAKAA<span style="color:#A901DB;">RGTCWE</span><br />
| 0.956<br />
| 4.936<br />
| 4.639<br />
| 0.00055708<br />
| 220.8<br />
| 27985<br />
|-<br />
| ord3<br />
| <span style="color:#3ADF00;">GG</span>AEAAAKEAAAK<span style="color:#00BFFF;">ASLPAA</span>AEAAAKEAAAK<span style="color:#A901DB;">RGTCWE</span><br />
| 1.390<br />
| 4.949<br />
| 7.116<br />
| 0.000545<br />
| 261.2<br />
| 28557<br />
|-<br />
|<br />
| <br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| '''Short linkers'''<br />
| <br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| sho1<br />
| <span style="color:#3ADF00;">GG</span><span style="color:#A901DB;">RGTCWE</span><br />
| 0.7087<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| sho2<br />
| <span style="color:#3ADF00;">GG</span>AEAAAK<span style="color:#A901DB;">RGTCWE</span><br />
| 0.5743<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| flexible linker<br />
| GGSGGGSGRGKCWE<br />
| 0.6851<br />
| <br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| linear lysozyme<br />
| no linker<br />
| 0.7039<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
In figures 7 and 8 one can compare the modeled structure of the linker to the predicted path of the software. The lysozyme is oriented nearly in the same directions.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
align=right|<br />
file =circ_lam_lys_nils.png|<br />
caption = Fig 7) Circular lambdalysozyme structure|<br />
descr= A linker calculated by the software was modeled using modeller.}} <br />
{{:Team:Heidelberg/templates/image-half|<br />
align=left|<br />
file = nice_linker_lysozyme_flexible_ends.png|<br />
caption = Fig 8) Path predicted by software|<br />
descr= A path the software predicted in step 1. The points resemble the turning points, the green cross shows the size of an alpha helix.}}<br />
<br />
<br />
<br />
In the end we obtained a ranking of the in vitro tested linkers from the [https://2014.igem.org/Team:Heidelberg/Project/Linker_Screening linker-screening] and chose the parameters $\alpha, \beta, \gamma, \delta$ of the weighting function so that the ranking from the software represented the ranking from the assays. These values were at first fitted, so that the ranking predicted by the software resembles. But this could not work out perfectly because for example may1 linker was worse in every contribution than sgt2 but was tested better. Therefore the different parameters were adjusted afterwards by hand. The final values, $\alpha = 1.85 * 10 ^{-6}, \beta = 0.57 , \gamma = 50.8 * 10^6$, set up a function that could reproduce the ranking oberved in the wetlab experiments.<br />
<br />
=Discussion=<br />
The software described here allowed us to design rigid linkers with well-defined angles. This represents a major advance compared to previous approaches like [[#References|[2]]] as these linkers can circularize any protein of known structure with any complex geometry.<br />
The feedback between the modeling and the experiment work on lysozyme activity was a crutial step in the development of the software. It allowed the testing of our approach and the calibration of the contribution of different features of the linkers to heat stability. This calibration was performed on one enzyme, and can improve in the future with the testing of more enzymes. This will also be refined thanks to a complete modeling and analysis of protein structures with linkers.<br />
Further on we could refine our assumptions on the different contributions of the weighing function. At first we assumed, that length would dominate, but the data suggests, that the contribution from omitting the substrate would be most important.<br />
<br />
=References=<br />
<br />
[1] Thornton, J.M. & Sibanda, B.L. Amino and carboxy-terminal regions in globular proteins. Journal of molecular biology 167, 443-460 (1983).<br />
<br />
[2] Wang, C.K.L., Kaas, Q., Chiche, L. & Craik, D.J. CyBase: A database of cyclic protein sequences and structures, with applications in protein discovery and engineering. Nucleic Acids Research 36, (2008).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/Software/Linker_SoftwareTeam:Heidelberg/Software/Linker Software2014-10-18T03:09:33Z<p>Jan glx: </p>
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abstract=As already introduced, artificially circularized proteins may gain some heat stability by restraining the C- and N-terminus from moving around freely. This circularization may be trivial when the protein termini are very close to each other, which seems to be reasonably common [1]. However, if the ends are too far from each other, a long linker is needed to connect them. This linker should not change the natural conformation of the protein and should constrain the relative position of the ends to restrict the degrees of freedom and thus to stabilize the structure even when heated up. On top, these linkers should not affect any of the protein functions. Consequently it is important to prevent linkers from passing through the active site or from covering binding domains to other molecules for example. Therefore one needs to be able to define the shape of possible linkers. This section describes the software we developed to design such linkers. We would like to stress that this work has been made possible thanks to the feedback between computer modeling and experimental work: We could first design linkers in silico, test them experimentally and use the results to further calibrate the software. To our knowledge, this is the first time that such an approach is used to customly design rigid linkers with angles to connect protein extremities.<br />
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{{:Team:Heidelberg/templates/mathjax}}</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/pages/Linker_SoftwareTeam:Heidelberg/pages/Linker Software2014-10-18T03:06:26Z<p>Jan glx: /* PDB analysis */</p>
<hr />
<div>=Abstract=<br />
As already [https://2014.igem.org/Team:Heidelberg/Toolbox/Circularization introduced], artificially circularized proteins may gain some heat stability by restraining the C- and N-terminus from moving around freely. This circularization may be trivial when the protein termini are very close to each other, which seems to be reasonably common [[#References|[1]]]. However, if the ends are too far from each other, a long linker is needed to connect them. This linker should not change the natural conformation of the protein and should constrain the relative position of the ends to restrict the degrees of freedom and thus to stabilize the structure even when heated up. On top, these linkers should not affect any of the protein functions. Consequently it is important to prevent linkers from passing through the active site or from covering binding domains to other molecules for example. Therefore one needs to be able to define the shape of possible linkers. This section describes the software we developed to design such linkers. We would like to stress that this work has been made possible thanks to the feedback between computer modeling and experimental work: We could first design linkers in silico, test them experimentally and use the results to further calibrate the software. To our knowledge, this is the first time that such an approach is used to customly design rigid linkers with angles to connect protein extremities.<br />
<br />
=General procedure=<br />
In short, the software can provide a weighted list of linkers to circularize any protein of interest with a known structure. Those linkers are made of rigid alpha helices segments connected with defined angles. Contrary to flexible linkers, those rigid linkers were expected to constrain the protein extremities and to confer better heat stability. Such an idea was already developed [[#References|[2]]] but only with alpha helices defining simple rods, and without any possibility to introduce angles. To generate those linkers, we first defined the geometrical paths, with segments and angles, that they should follow. The geometrical paths that are biologically feasible are afterwards translated into amino acid sequences. Both the compatibility of paths with possible structures and the translation were made possible thanks to our [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling approaches]. The first approach consisted in performing a statistical analysis of more than 17000 known non-homologous structures containing alpha helices connected with angles. For the second approach, we modeled the conformation of linkers circularizing proteins of known structure and analyzed them for certain properties. This second approach was run for a large number of proteins thanks to our distributing computing system [https://2014.igem.org/Team:Heidelberg/Software/igemathome igemathome]. The software provides different possible linkers with weights that provide the ranking of the linkers depending on their capacity to maintain protein activity at higher temperatures. They were generated thanks to an extensive [https://2014.igem.org/Team:Heidelberg/Project/Linker_Screening linker screening] on the target protein lambda-lysozyme, using the first modeling approach.<br />
The documentation of our CRAUT software can be found [https://2014.igem.org/Team:Heidelberg/Software/Linker_Software/Documentation here].<br />
{{:Team:Heidelberg/templates/image-half|<br />
align=right|<br />
caption=Figure 0)|<br />
descr= A representation of the general concept of CRAUT. At first the user provides it with a protein structure, by providing it with a PDB file. Then he can add relevant data, like binding sites, that he has found in databases. Then he chooses which parts of the protein the software should circularize and which parts of the protein should be ignored. After the calculations have finished, the user gets a sequence of the best linker, with which he could circularize the target protein. |<br />
file=how_user_should_use.png}}<br />
<br />
=Background=<br />
Classically, protein linkers were designed in three different manners. The easiest way is to define the length that a linker should cover and then simply use a flexible glycine-serine peptide with the right amount of amino acids to match this length. Glycine is used for flexibility, as it has no sidechain and does not produce any steric hindrance, while serine is used for solubility, as it has a small polar side chain. This solubility is important, as the linkers should not pass through the hydrophobic core of the protein, but should be dissolved in the surrounding medium. These flexible linkers were normally used for circularization but also for connecting different proteins, when the main goal is that the different parts are connected, but not how they are connected, or when the flexiblity of the linker was required for specific applications. <br />
<br />
A second strategy consists in using rigid helical linkers to keep proteins or protein domains at a certain distance from each other. This is especially important for signalling proteins and fluorescent proteins. One major property of alpha helices is that they always fold in a defined way with well defined angles and lengths. There are also many different helical patterns that differ in stability and solubility. Although they have been used to design cirularizing linkers [[#References|[2]]]. One big disadvantage of this strategy is that one can only build straight linkers with helices. So in the context of circularization, if an artificial line that would connect protein extremities is crossing the protein, this strategy is not an option.<br />
<br />
The third option, which served as a base to develop our approach and which came from discussions with the group of Rebecca Wade in Heidelberg, Germany, consists in designing customly tailored linkers for each specific application. These linkers can be obtained from protein structure prediction. At first one needs to define the path that the linker should take to connect the protein ends. Afterwards, one designs a possible linker sequence that might fit well. Next one makes a structure prediction of the linker attached to the proteins to validate the prediction. Several different linkers, with slight changes, can be compared. This is repeated several times until the linker effectively follows the expected path. This method requires a strong knowledge on protein folding and protein structure prediction and is computationaly intensive. On the other hand, the benefit can be important as the interaction of the linker with the protein surface can be taken into account and as one can accurately define the path taken by the linker to the resolution of protein structure.<br />
<br />
We have set up a completely new strategy to design rigid linkers. As further detailed in the [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling] part, it is possible to define the shape of a linker, by combining rigid alpha helical rods with well-defined angle patterns. Therefore, by defining, in a geometrical way, the possible paths of the circularizing linkers for a given protein, we can then propose potential linkers. This definition of the geometrical path can be very difficult, especially for large proteins with complex shapes. Moreover, this definition is further constrained by the fact that linkers must avoid hiding active sites of the protein of interest. Finally the paths have rotational degrees of freedom at the extremities of the protein, and depending on their orientation, they may or may not match the geometry of the protein. The tool we present here covers the two steps: defining geometrical paths with some weights and translate them into feasible linkers, also with weights. This tool is universal as it has the capacity to design circularizing linkers for any protein with a known structure. Moreover it is modular as, thanks to our [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling approach], we have designed linkers as exchangeable blocks of rods of different lengths and of angle patterns. The following sections detail the different steps followed by our software to design proper linkers.<br />
<br />
<br />
==PDB analysis==<br />
{{:Team:Heidelberg/templates/image-quarter|<br />
align=right|<br />
caption=Figure 1) Accessible ends|<br />
descr= When checking whether the ends of the protein are covered, at first for all the directions it is checked, whether some of the protein points are in the way. This was done for discrete angles incremented at 5° |<br />
file=ends-covered.png}}<br />
<br />
At first, the PDB file containing the structure of the target protein is parsed and the coordinates of the atoms are stored, in the metric unit. After this, some initial tests are made with the protein structure. First, we checked whether the C- and N-termini lie on the surface of the protein and are accessible to the solvent, which is crucial for circularization. We defined a line originating from an extremity of the protein with the two angles of the spherical coordinates around the z-axis. From that, we could determine the accessible angles by rejecting all the lines that are too close to the protein. As the future linker will be made of alpha helices and will therefore have a radius of 5 &Aring;, we used this length as the minimal allowed distance.<br />
Those allowed angles are stored for the coming linker generation.<br />
<br />
==Generation of geometric paths==<br />
As our strategy consists in building linkers with helical rods and connecting angles, a path is completely defined by the coordinates of the angle points. Advancing one step from an existing point is always done by adding a displacement vector on this point. This vector is defined by the two spherical angles, chosen here in a discrete manner with an increment of 5 degrees, and by a length, also chosen in a discrete manner. This discrete length was used in two different contexts: it may correspond to the length of an alpha helix or to the length of the flexible part that appears at the extremity of the protein. The coordinates that are reached thanks to this vector defines the new coordinates of an angle point, no matter if the vector corresponds to an alpha helix or to a flexible part. <br />
{{:Team:Heidelberg/templates/image-quarter|<br />
align=right|<br />
caption=Figure 2) Linker going around torus|<br />
descr=The worst shape we could think of for circularization was a torus without hole with ends in the middle. Even this shape could be circularized with our linkers. |<br />
file=torus.png}}<br />
<br />
As we screen for all possible angles in a discrete manner, those angle points coordinates are regularly distributed on a sphere. As further detailed in the next sections, those spheres are defined from both ends, either once or in several steps. Then the software checks for possible straight connections of given lengths for each pair of angle points originated from both extremities. <br />
{{:Team:Heidelberg/templates/image-full|<br />
align=right|<br />
caption=Figure 3) The generation of the paths|<br />
descr= Easy representation of the process of points generation. At first all the points from the start are generated, left. Then the points from the end are generated. Then all possible connections between the points are checked for their validity. This is done for every point from the beginning. |<br />
file=figure3.png}}<br />
<br />
The linkers are built in a modular way, with blocks of well-defined size. From the [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling] of potential linkers, we could derive 8 different alpha helical rods, all with different lengths. On top, the length of the two segments inside an angle block was always 8&Aring;, so exchanging angle blocks do not affect the length of the linker. This means that the distance between the angle points is well defined, an essential aspect of our strategy of linker design.<br />
The software proceeds in three steps. First, it checks for the possibility of direct single alpha helix linker. for this, it applies the procedure just mentioned with spheres of radius that reasonably corresponds to the length of the short parts at the extremity of the protein. Second, it tests if a linker containing two alpha helices connected with a right angle allows the circularization. Finally it searches the possible linkers with three angle points. The next parts will explain those three steps in detail.<br />
This method has been chosen, because it could be implemented easily and efficiently in our program. However, this strategy generated paths that crossed the protein. Therefore we put big efforts in the sorting out of the paths.<br />
<br />
===Step 1===<br />
As a simple rigid linker with no angle would be easier to design and likely more thermostable than the ones containing angles, the software first checks if this simple solution is possible.<br />
{{:Team:Heidelberg/templates/image-quarter|<br />
align=right|<br />
caption=Figure 4) Step 1|<br />
descr= Only one single alpha helix connects the flexible ends of the protein. |<br />
file=one_helix_flex_ends.png}}<br />
For this, we took into account the fact that proteins have some flexible amino acids at their extremities. This flexible part may come from the protein itself, but also from the 2 glycines that are included at the N-terminal part and from the extein at the C-terminal part. Those two latter parts comes from our linkers. Those parts have no preferential angles and offers a large amount of possibilities to insert fitting linkers. But this flexibility is also a drawback as we have to include this large amount of possible angles and length to our path search.<br />
In this first step, the software explicitely takes these flexible parts into account to check for the possibility of straight linkers. As the angles and the length of the flexible parts are variable, the software position their extremity on a sphere centered on the last fixed position of the structure as explained above. The radius of this sphere is incremented in a discrete manner, in 4 steps, from 5.25 &Aring; to the maximum length of the flexible part.<br />
Then all possible straight segments between the points and the lastpoints are tested. If they are closer than 5 &Aring; to the protein, of if they cross it, then they are rejected. If they are kept, then the software checks whether the length of the segments is compatible with the feasible alpha helices in terms of length: if the length of a given segment equal one of the 8 alpha helix lengths plus or minus 0.75 &Aring;, then the path is eventually saved.<br />
<br />
===Step 2===<br />
The next possibilty to design more complex rigid linkers while still taking flexible ends into account with a reasonable calculation time was to reduce amount of possible angles. As we originally thought that 90° angle would be practically feasible, the software was designed to generate linkers with flexible ends and one 90° angle. <br />
<br />
{{:Team:Heidelberg/templates/image-quarter|<br />
align=right|<br />
caption=Figure 5) Step 2|<br />
descr= The linker forms an angle of 90°. |<br />
file=figure6.png}}<br />
This choice was notably made because of the simplicity to calculate lengths of right triangle edges. We already saw in Step 1 that the length of an edge can only take 8 different values. As the linkers have to start from the extremities of the protein, and as we impose a right angle, the number of possible paths is therefore low, making them easy to compute. Practically, the extremities of the proteins are positioned in a flexible way as in Step 1. From each of the positions allowed by this flexibility, the software searches for all the allowed right triangles. This was mainly done, as the degrees of freedom needed to be restricted to keep calculations feasible.<br />
<br />
===Step 3===<br />
Finally the software also provides the possibility to find paths with up to 4 edges, meaning 4 alpha helices and 3 angles. Thanks to the modularity of the possible linkers, such paths can offer the possibility to circularize theoretically any kind of protein, see figure 2).<br />
<br />
To keep the calculation feasible in a reasonable time, we design the searching strategy so that the flexible part at the extremity are oriented in the same direction as the consecutive alpha helix. This is obviously restricting the search but as these orientations are allowed for the flexible part, this approach remains fully correct. <br />
First, potential ending points of the first alpha helical rod are calculated from the N-terminal point of the protein. The orientation is chosen in a discrete manner, with an incrementation of 5 degrees for the two angles of the spherical coordinates. The distance from the origin corresponds to the 8 possible lengths allowed by the alpha helices, as already seen in Step 2, plus a length of 0, which mimics a linker with 3 instead of 4 edges. The exact same procedure is repeated to define all the potential ending points of the second alpha helical rod starting from all the possible ending points of the first alpha helical rod. Thanks to the possibility of a length of 0 for the first and the second rods, the software also calculate paths with 2 edges. Then, the same is done only once from the C-terminal point of the protein, defining 1 edge. The final step consists in checking if the points originating from the N- and C-terminal points can be linked by an potential alpha helix, i.e. if they are separated by the appropriate distance. If any of the potential alpha helix length lies within the distance between two points plus or minus 0.75 &Aring;, then the path is eventually saved. In the same way, if two points are directly closer than 0.75 &Aring;, then the path is also saved.<br />
<br />
==Sorting out of paths==<br />
The previous part described the generation of paths that can connect the two extremeties of the protein irrespective of the position of these paths relative to the protein. While this allows a fast computing of the geometrical paths, this also implies that the paths that are not practically feasible need to be sorted out. This is the most time consuming part of the computing as about 1 billion paths are generated. Three criteria are considered for the sorting. The first one is the feasibility of the linker: can the software find angle patterns that correspond to the one defined by the geometrical path? This question was part of the motivation for a large modeling effort (link) to determine the possible angles between consecutive angles. This was achieved by analyzing the [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling distribution of angles] between alpha helices found in the ArchDB database. As nearly any angle could be found between 20 and 170 degrees, only few paths were actually rejected at that step. The next criteria was the position of the angle point: if they appear inside the protein, then the path is rejected. Finally, the software checks if any of the atoms of the protein is less than 5&Aring; away from any of the alpha helices, then the path is also rejected.<br />
<br />
==Shifting paths to the patterns==<br />
<br />
The strategy described in step 3 gives a certain freedom for the rod that connect the last two angle points that were generated from the N- and C-terminal points. As this freedom is actually not permitted by the alpha helix and the angle pattern, but is permitted by the flexible part for example at the C-terminal end, the software slighty refine the path by rotating the segment that originates from the C-terminal point.<br />
<br />
==Weighting of paths==<br />
Before translating the paths into sequences and thus into linkers that can be expressed, each path needs to be evaluated for its potential capacity to enhance heat stability. For this we have identified different contributions that should be combined into one value that defines how good the linker may be and that consequently defines its ranking among all possible linkers. The smaller this value, the more we expect that the linker will enhance thermostability. An important step for all these contributions is the normalization, as explained in the next paragraphs.<br />
The first contribution we considered was the linker length. We assumed that a short linker is better to constrain the protein extremities, and that a long helix might give more flexibility. Because we wanted this value to be independent of the size of the protein, the length of the linker is normalized to the distance between the two termini.<br />
The second contribution relates to the angles used in the linker. We learned from the [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling] that angles formed by a certain angle pattern follow a certain distribution. First, we assumed that the narrower the distribution, the more likely the alpha helices would actually produce this angle. Second, the angles found by the software should be as close as possible to those well-defined angles. In this case, the weight value from this contribution should be low.<br />
Then the distance of the linker to the protein is taken into account. Because the linkers should not disturb the protein in its normal environment, linkers that pass close to the protein surface are considered better linkers. The distance was defined as the minimal distance between the linker and all the atoms of the protein. As already mentioned for the sorting of the paths, a linker cannot come closer than 5 &Aring; and this distance was used for normalization of calculated distances.<br />
After this, the places a linker should avoid are calculated. Each protein can interact with other molecules on some oarts of its surface. The user can specify where and how big those parts are. If a linker passes in front a potential molecule binding domain, the value of the corresponding path goes to infinity, so that the linker is discarded. Conversely the farther a linker is from a potential ligand binding domain, the smaller its weighting value. The user can also specify the importance of certain regions. In the end the total weighting is normalized to the amount of binding domains.<br />
<br />
===Calibrating the weighting function===<br />
Every contribution has its own distribution. You can see an example in figure <br />
{{:Team:Heidelberg/templates/image-quarter|<br />
align=right|<br />
caption=Figure 6) Distribution of length contribution of lysozyme|<br />
descr=Each weight has it's own distribution. As an example here the distribution of the length weighting of lambda lysozyme is shown. One can clearly see the gaps due to the discrete lengthes of the building blocks.|<br />
file=histogram_lengths_lys.png}}<br />
, but all of them have different shapes. The aim is to find the paths that globally minimize all of these distributions. Therefore, for simplicity,in the weighting function the four mentioned contributions were combined in a linear manner:<br />
\[ W(p) = \alpha L(p) + \beta A(p) + \gamma D(p) + u(p) \]<br />
where W is the final weighting, p the path, L the length contribution, A the angle contribution, D the distance contribution and u the contribution from the forbidden regions. $\alpha, \beta, \gamma, \delta$ are the weighting constants that needed to be found. The normalization performed for each of the contribution were made so that each of them is dimensionless and that all have reasonably similar values.<br />
The weighting constants were obtained from the [https://2014.igem.org/Team:Heidelberg/Project/Linker_Screening linker screening] performed with lysozyme and the [https://2014.igem.org/Team:Heidelberg/Modeling/Enzyme_Modeling modeling of the enzyme activity]. Their calculation is presented in the results below.<br />
<br />
==Translating paths to sequence==<br />
As already mentioned before the software is provided with two databases, one for the possible angle patterns and one for the helix patterns. The choice of the patterns was inspired by known crystal structures extracted from databases and described in different papers.<br />
A huge in silico screening for refining the preferences of the patterns was then set up using the [https://2014.igem.org/Team:Heidelberg/Software/igemathome distribution calculation] system. For the complete description of search for suitable patterns, one can read the [https://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling] page. <br />
All the possible paths are now split up at the angles and compared with the possible patterns in the databases. The most suitable patterns are identified and added together to build the paths sequence. It is important to notice that this is only possible because of the modularity of our linker patterns used as building blocks: each block, being an alpha helix or an angle pattern, is not affected by the other. Thus for each possible path, one sequence is produced.<br />
<br />
==Clustering of paths==<br />
Many different paths are represented by the same sequence [[###Figure that shows, different paths have same properties, already before in the text ###]] and we therefore clustered such paths. The weigths for those clustered paths were then calculated by averaging the weights of the different paths that compose a cluster.<br />
<br />
=Results=<br />
==DNMT1==<br />
A major motivation of our effort to design rigid linkers with angles was the [https://2014.igem.org/Team:Heidelberg/Project/PCR_2.0 circularization of the DNA methyltranferase Dnmt1]. The truncation form used in our project is composed of 900 amino acids and the N- and C-terminal extremities are well separated. To circularize it, two linkers were designed: a flexible one made of glycine and serine, and a rigid one designed by the software. The rigif linkers for DNMT1 were obtained from an early state of the software. At that time the calculation took 11 days on a laptop computer with intel i5 processor and 8GB of RAM, which shows the importance of a distributed computing system for large proteins. But from that state on, the software has still improved a lot, resulting in reduced calculation time to about 1 day for DNMT1.<br />
<br />
==Feedback from wet lab==<br />
The results from the software for lysozyme were tested as described in the [https://2014.igem.org/Team:Heidelberg/Project/Linker_Screening linker-screening] part and evaluated as described in the [https://2014.igem.org/Team:Heidelberg/Modeling/Enzyme_Modeling enzyme modeling] part. We have performed a large linker screening on 10 different lysozymes with different linkers. As the purpose of the lysozyme screen was the calibration of the software,those linkers (Table 1) were designed according to the four contributions previously mentioned. One of them was the shortest possible, one had the best possible angle, and so on.<br />
{| class="table table-hover" style="text-align: center;"<br />
|+ '''Table 1''': Linker and their amino acid sequence. Green: attachment sequences to prevent the flexible regions from being perturbed; Blue: angle; Purple: extein.<br />
! Linker<br />
! Amino acid sequence<br />
! activity<br />
! length- contribution<br />
! angle- contribution<br />
! binding site contribution<br />
! distance from surface<br />
! weightingvalue after calibration<br />
|- <br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|'''Very good linkers'''<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| sgt2<br />
| <span style="color:#3ADF00;">GG</span>AEAAAK<span style="color:#00BFFF;">AAAHPEA</span>AEAAAK<span style="color:#A901DB;">RGTCWE</span><br />
| 0.7477<br />
| 1.9205<br />
| 6.7789<br />
| 0.002259<br />
| 10.525<br />
| 114912<br />
|-<br />
| rigid<br />
| <span style="color:#3ADF00;">GG</span>AEAAAKEAAAKAA<span style="color:#3ADF00;">P</span><span style="color:#A901DB;">RGKCWE</span><br />
| 0.9447<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| '''Average linkers'''<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| may1<br />
| <span style="color:#3ADF00;">GG</span>AEAAAKEAAAKA<span style="color:#00BFFF;">AAAHPEA</span>AEAAAK EAAAKA<span style="color:#00BFFF;">KTA</span>AEAAAKEAAAKA<span style="color:#A901DB;">RGTCWE</span><br />
| 0.7489<br />
| 6.2225<br />
| 13.19<br />
| 0.00384<br />
| 1095.2<br />
| 196414<br />
|-<br />
| ord1<br />
| <span style="color:#3ADF00;">GG</span>AEAAAKEAAAK<span style="color:#00BFFF;">ATGDLA</span>AEAAAKAA<span style="color:#A901DB;">RGTCWE</span><br />
| 0.956<br />
| 4.936<br />
| 4.639<br />
| 0.00055708<br />
| 220.8<br />
| 27985<br />
|-<br />
| ord3<br />
| <span style="color:#3ADF00;">GG</span>AEAAAKEAAAK<span style="color:#00BFFF;">ASLPAA</span>AEAAAKEAAAK<span style="color:#A901DB;">RGTCWE</span><br />
| 1.390<br />
| 4.949<br />
| 7.116<br />
| 0.000545<br />
| 261.2<br />
| 28557<br />
|-<br />
|<br />
| <br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| '''Short linkers'''<br />
| <br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| sho1<br />
| <span style="color:#3ADF00;">GG</span><span style="color:#A901DB;">RGTCWE</span><br />
| 0.7087<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| sho2<br />
| <span style="color:#3ADF00;">GG</span>AEAAAK<span style="color:#A901DB;">RGTCWE</span><br />
| 0.5743<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| flexible linker<br />
| GGSGGGSGRGKCWE<br />
| 0.6851<br />
| <br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| linear lysozyme<br />
| no linker<br />
| 0.7039<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|}<br />
<br />
In figures 7 and 8 one can compare the modeled structure of the linker to the predicted path of the software. The lysozyme is oriented nearly in the same directions.<br />
<br />
{{:Team:Heidelberg/templates/image-half|<br />
align=right|<br />
file =circ_lam_lys_nils.png|<br />
caption = Fig 7) Circular lambdalysozyme structure|<br />
descr= A linker calculated by the software was modeled using modeller.}} <br />
{{:Team:Heidelberg/templates/image-half|<br />
align=left|<br />
file = nice_linker_lysozyme_flexible_ends.png|<br />
caption = Fig 8) Path predicted by software|<br />
descr= A path the software predicted in step 1. The points resemble the turning points, the green cross shows the size of an alpha helix.}}<br />
<br />
<br />
<br />
In the end we obtained a ranking of the in vitro tested linkers from the [https://2014.igem.org/Team:Heidelberg/Project/Linker_Screening linker-screening] and chose the parameters $\alpha, \beta, \gamma, \delta$ of the weighting function so that the ranking from the software represented the ranking from the assays. These values were at first fitted, so that the ranking predicted by the software resembles. But this could not work out perfectly because for example may1 linker was worse in every contribution than sgt2 but was tested better. Therefore the different parameters were adjusted afterwards by hand. The final values, $\alpha = 1.85 * 10 ^{-6}, \beta = 0.57 , \gamma = 50.8 * 10^6$, set up a function that could reproduce the ranking oberved in the wetlab experiments.<br />
<br />
=Discussion=<br />
The software described here allowed us to design rigid linkers with well-defined angles. This represents a major advance compared to previous approaches like [[#References|[2]]] as these linkers can circularize any protein of known structure with any complex geometry.<br />
The feedback between the modeling and the experiment work on lysozyme activity was a crutial step in the development of the software. It allowed the testing of our approach and the calibration of the contribution of different features of the linkers to heat stability. This calibration was performed on one enzyme, and can improve in the future with the testing of more enzymes. This will also be refined thanks to a complete modeling and analysis of protein structures with linkers.<br />
Further on we could refine our assumptions on the different contributions of the weighing function. At first we assumed, that length would dominate, but the data suggests, that the contribution from omitting the substrate would be most important.<br />
<br />
=References=<br />
<br />
[1] Thornton, J.M. & Sibanda, B.L. Amino and carboxy-terminal regions in globular proteins. Journal of molecular biology 167, 443-460 (1983).<br />
<br />
[2] Wang, C.K.L., Kaas, Q., Chiche, L. & Craik, D.J. CyBase: A database of cyclic protein sequences and structures, with applications in protein discovery and engineering. Nucleic Acids Research 36, (2008).</div>Jan glxhttp://2014.igem.org/Team:Heidelberg/ModelingTeam:Heidelberg/Modeling2014-10-18T03:04:06Z<p>Jan glx: </p>
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<h2><span>LINKER MODELING</span></h2><br />
The design of linkers, which are essential for our intein-based circularization and assembly methods, is a complicated task. Here we employ structural modeling and statistical analyses to guide our linker experiments and their design.<br />
<div class="clearfix"></div><br />
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<h2><span>ENZYME KINETICS MODELING</span></h2><br />
Based on over 1000 degradation curves of our circularized lysozyme variants, we tried to interpret the effect of linkers on protein thermostability. This was achieved by use of quantitative dynamical modeling.<br />
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<h2><span>LINKER MODELING</span></h2><br />
The design of linkers, which are essential for our intein-based circularization and assembly methods, is a complicated task. Here we employ structural modeling and statistical analyses to guide our linker experiments and their design.<br />
<div class="clearfix"></div><br />
</a><br />
</div><br />
</br><br />
<div style="width:100%;padding:0 15px 0 15px;"><br />
<a href="/Team:Heidelberg/Modeling/Enzyme_Modeling" class="box" style="display:block; width:100%;"><br />
<h2><span>ENZYME KINETICS MODELING</span></h2><br />
Based on over 1000 degradation curves of our circularized lysozyme variants, we tried to interpret the effect of linkers on protein thermostability. This was achieved by use of quantitative dynamical modeling.<br />
<div class="clearfix"></div> <br />
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