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Very preliminary everything

Abstract

biological replicates more than 1000 curves Product inhibition, 75°C activity, many different things found out. kinetics modeling

Introduction

Enzyme kinetics is a widely studied field in biology as from the kinetic parameters one could make many different predictions about the preferences of a certain enzyme. Often knowledge about the structure of the enzymes reveals in models about the models for the kinetics of the enzyme. Our approach is the other way around. Often for extracting the enzyme kinetical parameters, the reaction rate is measured over a certain time and substrate concentration is varied. This would have been too much work for such a large screening and thus we decided to record the process curves for each lysozyme. From the ###link assays we have obtained over 1000 degradation curves from diferent kinds of lysozymes and want to make predictions about the thermal stability of the different enzymes. For nearly all of the ###10### different lysozymes not only two biological replicates from different inductions but also at least 4 technical replicates for each temperature the data was measured. Therefore a widespread approach has been made to extract the relevant information, starting with easy fitting of the data and ending with testing of different enzyme kinetics models, evaluation by likelihood ratio tests and in the end identifying the relevent parameters.

Preceding measurements

In the beginning of the assays we needed to make certain calibrating measurements so that in the end we could model the enzyme kinetics, minimizing the influence of the measurement process.

OD to concentration calibration

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. ###Plot needed?### 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.

Pipeting

One crucial part for the modeling is the time between when lysozyme was added to the substrate and the first measurement in the platereader. This was measured and assumed, that it nearly took the same time for each measurement with normally distributed errors. As the platereader took about 1s for measuring one well, this delay was also taken into account.

Control

We measured degradation curves for two different controls many times. One was E.coli's lysate without any lysozyme expressed and the other was only the reaction buffer added to the substrate. We observed some slight decay in the curves, but the curves were clearly distinguishable from samples, where lysozyme was in there.

First try, easy fitting

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. [3] This way curves for the temperature decay can be measured for each kind of lysozyme and finally compared to each other. 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.

Fitting Michaelis-Menten kinetics to the concentration data

1. 1. 1. Actually don't know, where to put that part, but want to keep it ###

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 [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.

The basic differential equation for Michaelis-Menten kinetics [5] is:

   \[\frac{d\left[S\right]}{dt} = \frac{- V _{max} \left[S\right]}{K_m + \left[S\right]} \]

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:

   \[ \frac{K_m + \left[S\right]}{\left[S\right]}  \frac{d\left[S\right]}{dt} = -V_{max}\]

Which we can now solve by separation of variables and integration.

   \[ \int_{\left[S\right]_0}^{\left[S\right]_t} \left(\frac{K_m }{\left[S\right]} + 1\right) d\left[S\right] = \int_{0}^{t}-V_{max} dt'\]

This leads to,

   \[K_m \ln\left( \frac{\left[S\right]_t}{\left[S\right]_0} \right) + \left[S\right]_t - \left[S\right]_0  = -V_{max} t  \]

what we reform to a closed functional behaviour of time.

   \[t = - \frac{K_m \ln\left( \frac{\left[S\right]_t}{\left[S\right]_0} \right) + \left[S\right]_t - \left[S\right]_0 } {V_{max}}  \]

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. 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. So we can refine the functional behaviour as:

   \[t = - \frac{K_m \ln\left( \frac{ m ({\mathit{OD}}_t - a)}{ m (\mathit{OD}_0 - a)} \right) +  m \mathit{OD}_t -  m \mathit{OD}_0 } {V_{max}} \]

$OD_0$ is just the first measured OD we get, this parameter is not fitted. 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, [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. [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 \[ \left[S\right]_t = K_m \omega \left[ \left( \left[A\right]_0 / K_m \right) \exp \left( \left[A\right]_0 / K_m -V_{max} t / K_m \right) \right] \] 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: \[\frac{d\left[S\right]}{dt} = - V _{max} \left[S\right] \] which is solved by a simple exponential equation \[ \left[S\right]_{t} = \left[S\right]_0 e^{\left( - V_{max} t \right)} \] As we're measuring OD function fitted to the data results in: \[ \mathit{OD}_{t} = \left(\mathit{OD}_0 - a\right) e^{\left( - V_{max} t \right)} + a \] with a a parameter for the offset in OD due to the plate and the proteinmix. 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 d2d arFramework developed by Andreas Raue [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. In the next part we will explain how the model was refined to the lysozyme and how the results were obtained.

Modeling product inhibition

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. 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.

Lysozyme acting on M. lysodeicticus

Produkthemmung, Aufspalten, verschiedenste Effekte, ....

New model

Different modles tested

Final model

Results

References

[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). [4] Hommes, F. A. "The integrated Michaelis-Menten equation." Archives of biochemistry and biophysics 96.1 (1962): 28-31. [5] Purich, Daniel L. Contemporary Enzyme Kinetics and Mechanism: Reliable Lab Solutions. Academic Press, 2009. [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. [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. [8] Raue, A. et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE 8, (2013).