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 Revision as of 22:09, 17 October 2014 (view source)Igemnils (Talk | contribs) (→Background)← Older edit Latest revision as of 03:10, 18 October 2014 (view source)Jan glx (Talk | contribs) (→Abstract) (45 intermediate revisions not shown) Line 1: Line 1: - - =Abstract= - As already [http://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. - =General procedure= =General procedure= 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 [http://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 [http://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 [http://2014.igem.org/Team:Heidelberg/Project/Linker_Screening linker screening] on the target protein lambda-lysozyme, using the first modeling approach. 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 [http://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 [http://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 [http://2014.igem.org/Team:Heidelberg/Project/Linker_Screening linker screening] on the target protein lambda-lysozyme, using the first modeling approach. + The documentation of our CRAUT software can be found [http://2014.igem.org/Team:Heidelberg/Software/Linker_Software/Documentation here]. + {{:Team:Heidelberg/templates/image-half| + align=right| + caption=Figure 0)| + 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. | + file=how_user_should_use.png}} =Background= =Background= - 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. + 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. - 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. + - The third option, which served as a base to develop our approach, 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  two  amino acids. 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. ###TODO:  Reference, WADE paper###  This method is time consuming as it is not  only computation intensive, but also requires a strong knowledge on  protein folding and protein structure prediction. On the other hand, the  benefit  can be important as the interaction of the linker with the  proteins  surface can be taken into account and as one can accurately  define the path taken by the linker to the resolution of  protein structure. + - We  have set up a completely new strategy to design rigid linkers. As further detailed in the  [[###link###|modelling]] 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 constained 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 modeling approach [link] we have design 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. + - At first the proteinstructure is analysed and then all possible paths that connect the two ends are found, that have less than three additional edges. In the end all these paths should be sorted by well it would be, if we circularize the protein using this path. But as the final weighing is quite computation intensive, at first the paths need to be sorted out. A path is only sorted out, if it is breaching any rule for linkers. For example paths should never pass through the protein. After all the paths have been generated, the paths are improved, by shifting the points according to the underlying linkermodel, so that no paths are taken into account, that could not be built with our building blocks. + 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. - After all paths being correctly generated, the paths are weighted by several factors. Afterwards one weighting for a path is calculated, that corresponds to the goodness of this certain path. Then the path is retranslated by usage of our amino acid patterns to produce an amino acid sequence. But as one sequence can follow more than one path,  all the paths built up by this sequence are clustered. In the end the average of the weighting of the pathclusters is calculated and thus it for every linker only one weightingvalue is produced with the contributions of all paths represented by this linker. + - ==PDB parsing== + 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. - At first the PDB file containing the structure of the target protein is parsed and thus the information about the coordinates of the atoms are stored. From this data a calibration of coordinate system of the PDB to metric units is made. For this purpose the distance between certain atoms in all the glycines is measured. This distance is a well known distance and thus a calibration can be made, leading normally to 100 pm per unit in the PDB file. + - After this some first tests with the protein structure are made. At first it is tested, whether the C- and N-terminus lie on the solvent accessible surface of the protein, which is crucial for circularisation. The angles from which the ends are accessible are stored for the linkergeneration afterwards. An end is accessible in a certain angle, if the axis that is rotatet from z-achsis there with the angle, is not too near to any of the atompoints. The minimal distance a connection must have from the protein is set to the radius of an alpha helix 5 Å . ###fig needed### + We  have set up a completely new strategy to design rigid linkers. As further detailed in the [http://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 [http://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. - ==Generation of paths== + - As our model is to build linkers with helical rods and connecting angles, a path is completely defined by the coordinates of the angle points. Thus always just the angle points are generated and then the good ones are sorted out. As making shifts to existing points in our programmingstyle is very efficient, this was easier than only to generate the points, that are representing good connections. + - In the [[###link###|modelling]] part we have already described the patterns we are using for building up our linkers. The modularity was crucial for the success of the software. Therefore the angle patterns were always chosen, so that the end of the anglepattern was ???8??? Å away from the turning point ###figure needed###. Thus only displacements had to be made with a certain length, not depending on the direction in which it is going and not depending on the direction in that the linker will continue. This modularity makes the calculations more efficient than it would be, with just generating points randomly. + ==PDB analysis== - ===Flexible ends=== + {{:Team:Heidelberg/templates/image-quarter| - Most proteins have flexible regions at the ends, that are not pointing in a certain direction. Often these flexible ends are even missing in the structure files but still our software estimates how they could behave. Furtheron due to circularization non-helical sequences remain at the ends of the protein. This gives a huge possibility to insert fitting linkers. But this is also a big problem as the estimation of flexible parts is not easy with our brute-force ansatz. + align=right| - For generation of linkers taking into account the flexibility of the ends until now have been included two functions. + caption=Figure 1) Accessible ends| - ====One helix at flexible ends==== + 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° | - The only variability we have in the customly tailored linker there is the length of the helix. Most likely we no suitable linker can be found, because if there is some obstacle between the ends, the linker can't bend around it. But even though if there is the possibility of such a linker, it should be found, because this will be one of the best linkers predicted. ###figure needed### + file=ends-covered.png}} - So for all accessible angles that were calculated before, we calculate all points from the N-terminus and from the C-terminus that lie in certain distances from the terminus. The points are spread over varying distances  from 4.5 Å to the maximum length of the flexible part. ###figure### + - Then all possible connections between the firstpoints and the lastpoints are tested It is tested whether they are too close at the protein or even pass through the protein. This is done with a higher accuracy, because none of these linkers should be lost by error. Therefore here another function for calculating the distance from the protein to the connection is used, that is more time-consuming, but also more accurate, than the ones used normally. + 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 Å, we used this length as the minimal allowed distance. + Those allowed angles are stored for the coming linker generation. + + ==Generation of geometric paths== + 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. + {{:Team:Heidelberg/templates/image-quarter| + align=right| + caption=Figure 2) Linker going around torus| + 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. | + file=torus.png}} + + 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. + {{:Team:Heidelberg/templates/image-full| + align=right| + caption=Figure 3) The generation of the paths| + 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. | + file=figure3.png}} + + The linkers are built in a modular way, with blocks of well-defined size. From the [http://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 8Å, 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. + 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. + 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. + + ===Step 1=== + 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. + {{:Team:Heidelberg/templates/image-quarter| + align=right| + caption=Figure 4) Step 1| + descr= Only one single alpha helix connects the flexible ends of the protein. | + file=one_helix_flex_ends.png}} + 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. + 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 Å to the maximum length of the flexible part. + Then all possible straight segments between the points and the lastpoints are tested. If they are closer than 5 Å 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 Å, then the path is eventually saved. + + ===Step 2=== + 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. + + {{:Team:Heidelberg/templates/image-quarter| + align=right| + caption=Figure 5) Step 2| + descr= The linker forms an angle of 90°. | + file=figure6.png}} + 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. + + ===Step 3=== + 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). + + 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. + 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 Å, then the path is eventually saved. In the same way, if two points are directly closer than 0.75 Å, then the path is also saved. - ====Linker with one angle at flexible ends.==== - The other possibilty for taking flexible ends into account without blowing up the calculation time was to reduce the possibilities in the amount of angles in the helical pattern to one specified. - For the linkers with flexible ends and one angle are built as rectangular triangles. ###figure### The rectangular triangles are chosen, because a rectangle can be built well with the angles we have and the rectangle helps a lot in further calculation. - At first out of all possible linkerparts it is analyzed which triangles are possible to be built, by use of Pythagoras' Theorem. It is important that both legs can be built with our linker patterns and that the hypotenuse has the correct length to fit between the ends. - Afterwards all possible rectangular triangles constructed, that have the two edges of the hypothenuse on N- and C-terminus. For this purpose Thales' Theorem is used, as all the possible rectangles lie on a sphere of radius hypothenuses half. Then these rectangular triangles are analyzed, whether they can be built with our linkerparts, by comparing the angles with the amount of possible triangles from Pythagoras' Theorem. - These triangles are now all shiftet by each displacement of possible angles at the C-terminus, resulting in the rectanglepoints ###figure needed###. Now the connections to the possible points from the N-terminus are generated. At this step the triangles don't need to be rectangular anymore, but can have slightly different angles, but in the next steps the paths are analyzed, whether they still fit. At first they lengthes of the legs of the triangles are checked and then it is checked, that the paths don't disturb the protein. If they would disturbe the protein anyhow, they are just deleted. - ===Rigid paths=== - If the fist two possibilities for finding suitable paths didn't work the software provides also the possibility to find paths with up to three edges. Thus even one of the worst shapes for circularization, a torus with the two termini in the pits, could be circularized with our linker system, without needing infinitely long straights. On the other hand this were the maximum of possibilities which was still feasible to calculate. - Here now the flexible parts of the protein are estimated to point into the same direction  as the following helix. By this mean the amount of possibilities is kept fixed but of course this is quite some rough estimation. - Let's check the next paragraph together in detail - At first points from each end in the right distances are generated. ###figure needed, explains point names### Now for each point of the first points to all directions in 5 degrees angledifference next points are generated. Of these all points that don't fit are immediately sorted out. Now all connections from second points to the last points are checked whether they lie in the right distance. After this the normal sorting steps are made for these new connections. ==Sorting out of paths== ==Sorting out of paths== - The most calcualtion time is consumed while sorting out misleading paths. This is due to the fact, that every path needs to be checked. Making the brute-force ansatz the amount of possible paths is about 10^9, so this step consumes most time. + 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 [http://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 5Å away from any of the alpha helices, then the path is also rejected. - There are three main functions, that sort out paths that don't fit. The easiest just sorts paths out, that would require an angle, we can't produce with our angle patterns. But as we nearly can produce angles from ???20 -170??? degrees only very few paths are sorted out by this function. + - The next function just checks whether the endpoint is lying in the protein. If yes, the path is deleted. Otherwise the connection between the point coming from to this point is checked, whether it is passing too near at the protein. + ==Shifting paths to the patterns== ==Shifting paths to the patterns== - Because of rounding errors and other inaccuracies, it can always happen that the generated paths don't fit perfectly to the helical- and to the angle patterns. Therefore before translating them to the aminoacid sequence, the paths need to be refined. This is also done before the weighting, so that the paths don't change after weighting anymore. + 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. - Therefore the distances between the different points are calculated and then the points are shifted so far, that they fit into the patterns. The shifts never exceed a certain length so that no path then would pass through the protein after refinement, even though it didn't before. + ==Weighting of paths== ==Weighting of paths== - Before translating the paths to sequences and thus to expressible linkers, at first the value of each path needs to be checked. Therefore we have identified four contributions, that determine how good a path will work to enhance heatstability of a target protein by circularization. In the end all these contributions are summarized to get one final value for the goodness of the connection. For each of the single contributions, always lower weighting values refer to a better path. Therefore all contributions somehow need to be normed by some property of the protein, so that the contributions of each function on it's own are independent of the target protein and generality is achieved. + 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. - At first it is tested how long the linker is. We assumed that a shorter linker normally is better for restraining the ends, as a longer helix might give more flexibility to the ends. Because we want this value to be independent of the size of the protein, the length of the linker is normed by the distance of the two termini. It is clear that not only because the two termini are far from each other and clearly a longer linker is needed, this contribution exceeds all other contributions to the weighting function. + 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. - The next value is a measure for the goodness of the angles used in the linker. Each of our potential angle patterns is somehow distributed with a standard deviation from the mean. It is clear, that the narrower this distribution is, the likelier it is, that the pattern will in the end produce the assumed angle between the embracing helices. Therefore path-angles that fit perfectly with the angle patterns provided get a lower weighting value. + The second contribution relates to the angles used in the linker. We learned from the [http://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. - Then the distance from the protein is taken into account. Because the linkers should not disturb the protein in it's normal environment, linkers that pass near to the protein's surface are considered better linkers. Of course no linker is too near on the surface, so that it would interact with the protein too much. This value is normalized with the minimal distance a linker should have from the surface. + 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 Å and this distance was used for normalization of calculated distances. - After this the places a linker should ommit are calculated. Each protein binds some ligands or substrates. The user can specify, how big they are and where they attach. If a linker passes through a potential ligand, this value goes to infinity, so that a linker is discarded. Otherwise the farther a linker is from a potential ligand, the better this value gets. The user can also specify the importance of certain regions. In the end the total weighting is independent of the amount of binding sites. + 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. ===Calibrating the weighting function=== ===Calibrating the weighting function=== - ???keep?### + Every contribution has its own distribution. You can see an example in figure - Every contribution has it's own distribution. You can see an example in figure ###  [[figure histogram_length_lys.png]], but all of them have different shapes. The aim is to find the paths that minimize all of these distributions. Therefore in the weighting function the four mentioned contributions are combined in a linear manner: + {{:Team:Heidelberg/templates/image-quarter| - $W(p) = \alpha L(p) + \beta A(p) + \gamma D(p) + \delta u(p)$ + align=right| + caption=Figure 6) Distribution of length contribution of lysozyme| + 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.| + file=histogram_lengths_lys.png}} + , 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: + $W(p) = \alpha L(p) + \beta A(p) + \gamma D(p) + u(p)$ 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. 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. - The weighting constants were obtained from the [[linker-screening ###lin]] performed with lysozyme and the [[enzyme-modeling ###link]] + The weighting constants were obtained from the [http://2014.igem.org/Team:Heidelberg/Project/Linker_Screening linker screening] performed with lysozyme and the [http://2014.igem.org/Team:Heidelberg/Modeling/Enzyme_Modeling modeling of the enzyme activity]. Their calculation is presented in the results below. - Please see [[below ###link]] for the detailed explanation, how the values were obtained. + ==Translating paths to sequence== ==Translating paths to sequence== 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. 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. - A huge in silico screening for refining the preferences of the patterns was then set up using the [[iGEM@home|###Link]] system. For the complete description of search for suitable patterns, one can read the [[modeling|###Link to patternspart]] page. + A huge in silico screening for refining the preferences of the patterns was then set up using the [http://2014.igem.org/Team:Heidelberg/Software/igemathome distribution calculation] system. For the complete description of search for suitable patterns, one can read the [http://2014.igem.org/Team:Heidelberg/Modeling/Linker_Modeling modeling] page. - All the possible paths are now split up at the angles and compared with the possible patterns in the databases. ###Figure needed to explain, how the path is translated### 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. + 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. + ==Clustering of paths== ==Clustering of paths== 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. 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. Line 71: Line 110: =Results= =Results= ==DNMT1== ==DNMT1== - A major motivation of our effort to design rigid linkers with angles was the circularization of the DNA methyltranferase Dnmt1 (link). 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 imporved a lot, resulting in reduced calculation time to about 1 day for DNMT1. + A major motivation of our effort to design rigid linkers with angles was the [http://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. - + - ###DNMT1 could be made heatstable, still missing### + - + - + - From the [[Project/Linker_Screening | linker-screening]] and the [[Modeling/Enzyme_Modeling | enzyme_modeling]] we have obtained the activities after heat-shock of the different linkers. With these a calibration of the weighting function has been made, please see table 1. for the results. + + ==Feedback from wet lab== + The  results from the software for lysozyme were tested as described in the [http://2014.igem.org/Team:Heidelberg/Project/Linker_Screening linker-screening] part and evaluated as described in the [http://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. {| class="table table-hover" style="text-align: center;" {| class="table table-hover" style="text-align: center;" - |+ '''table 1''': Linker and their amino acid sequence. Green: attachment sequences to prevent the flexible regions from being perturbed; Blue: angle; Purple: extein. + |+ '''Table 1''': Linker and their amino acid sequence. Green: attachment sequences to prevent the flexible regions from being perturbed; Blue: angle; Purple: extein. ! Linker ! Linker ! Amino acid sequence ! Amino acid sequence Line 226: Line 262: |- |- |} |} - [[circ_lam_lys_nils.png]] comparison with [[nice_linker_lysozyme_flexible_ends.png]] + - ###About predictions of software### + 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. - In the end we obtained a ranking of the in vitro tested linkers from the [[linker-screening]] and chose the parameters $\alpha, \beta, \gamma, \delta$ of the weighting function in the way, that the ranking from the software represented the ranking from the assays. The final values were: ... + + {{:Team:Heidelberg/templates/image-half| + align=right| + file =circ_lam_lys_nils.png| + caption = Fig 7) Circular lambdalysozyme structure| + descr= A linker calculated by the software was modeled using modeller.}} + {{:Team:Heidelberg/templates/image-half| + align=left| + file =  nice_linker_lysozyme_flexible_ends.png| + caption = Fig 8) Path predicted by software| + 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.}} + + + + In the end we obtained a ranking of the in vitro tested linkers from the [http://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. =Discussion= =Discussion= - Will always be refined with more data from i@h... + 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. + 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. + 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. + =References= =References= - - [0] 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). [1] Thornton, J.M. & Sibanda, B.L. Amino and carboxy-terminal regions in globular proteins. Journal of molecular biology 167, 443-460 (1983). [1] Thornton, J.M. & Sibanda, B.L. Amino and carboxy-terminal regions in globular proteins. Journal of molecular biology 167, 443-460 (1983). + + [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).

# General procedure

Figure 0)

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.

# Background

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.

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 [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.

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.

We have set up a completely new strategy to design rigid linkers. As further detailed in the 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 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.

## PDB analysis

Figure 1) Accessible ends

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°

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 Å, we used this length as the minimal allowed distance. Those allowed angles are stored for the coming linker generation.

## Generation of geometric paths

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.

Figure 2) Linker going around torus

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.

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.

Figure 3) The generation of the paths

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.

The linkers are built in a modular way, with blocks of well-defined size. From the 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 8Å, 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. 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. 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.

### Step 1

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.

Figure 4) Step 1

Only one single alpha helix connects the flexible ends of the protein.

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. 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 Å to the maximum length of the flexible part. Then all possible straight segments between the points and the lastpoints are tested. If they are closer than 5 Å 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 Å, then the path is eventually saved.

### Step 2

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.

Figure 5) Step 2

The linker forms an angle of 90°.

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.

### Step 3

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).

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. 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 Å, then the path is eventually saved. In the same way, if two points are directly closer than 0.75 Å, then the path is also saved.

## Sorting out of paths

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 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 5Å away from any of the alpha helices, then the path is also rejected.

## Shifting paths to the patterns

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.

## Weighting of paths

### Calibrating the weighting function

Every contribution has its own distribution. You can see an example in figure

Figure 6) Distribution of length contribution of lysozyme

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.

, 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: $W(p) = \alpha L(p) + \beta A(p) + \gamma D(p) + u(p)$ 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. The weighting constants were obtained from the linker screening performed with lysozyme and the modeling of the enzyme activity. Their calculation is presented in the results below.

## Translating paths to sequence

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. A huge in silico screening for refining the preferences of the patterns was then set up using the distribution calculation system. For the complete description of search for suitable patterns, one can read the modeling page. 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.

## Clustering of paths

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.

# Results

## DNMT1

A major motivation of our effort to design rigid linkers with angles was the 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.

## Feedback from wet lab

The results from the software for lysozyme were tested as described in the linker-screening part and evaluated as described in the 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.

Table 1: Linker and their amino acid sequence. Green: attachment sequences to prevent the flexible regions from being perturbed; Blue: angle; Purple: extein.
Linker Amino acid sequence activity length- contribution angle- contribution binding site contribution distance from surface weightingvalue after calibration
sgt2 GGAEAAAKAAAHPEAAEAAAKRGTCWE 0.7477 1.9205 6.7789 0.002259 10.525 114912
rigid GGAEAAAKEAAAKAAPRGKCWE 0.9447
may1 GGAEAAAKEAAAKAAAAHPEAAEAAAK EAAAKAKTAAEAAAKEAAAKARGTCWE 0.7489 6.2225 13.19 0.00384 1095.2 196414
ord1 GGAEAAAKEAAAKATGDLAAEAAAKAARGTCWE 0.956 4.936 4.639 0.00055708 220.8 27985
ord3 GGAEAAAKEAAAKASLPAAAEAAAKEAAAKRGTCWE 1.390 4.949 7.116 0.000545 261.2 28557
sho1 GGRGTCWE 0.7087
sho2 GGAEAAAKRGTCWE 0.5743

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.

Fig 7) Circular lambdalysozyme structure

A linker calculated by the software was modeled using modeller.

Fig 8) Path predicted by software

A path the software predicted in step 1. The points resemble the turning points, the green cross shows the size of an alpha helix.

In the end we obtained a ranking of the in vitro tested linkers from the 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.

# Discussion

The software described here allowed us to design rigid linkers with well-defined angles. This represents a major advance compared to previous approaches like [2] as these linkers can circularize any protein of known structure with any complex geometry. 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. 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.

# References

[1] Thornton, J.M. & Sibanda, B.L. Amino and carboxy-terminal regions in globular proteins. Journal of molecular biology 167, 443-460 (1983).

[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).