Book of Abstracts: Albany 2007
June 19-23 2007
Stochastic Description of Enzymatic Reaction
The application of the deterministic approach for description of the enzymatic reaction kinetics is appropriate if a large number of molecules of substrate and enzyme are engaged in the enzymatic reaction. In that case, one can consider that the number of molecules in the substrate-enzyme complex is equivalent to its average value and almost no fluctuations of the enzymatic reaction rate occur. However, in a real cell system it is rather possible to deal with a small number of enzyme and substrate molecules and since the substrate-enzyme complex formation and dissolving processes depend upon random collision of the substrate molecules to the molecules of enzyme, then it is obvious that the number of the substrate-enzyme complex molecules fluctuates. This results in fluctuations of the enzymatic reaction rate. In such case, in order to present a rather realistic view of the process, the kinetics of the enzymatic reaction should be described using the theory of arbitrary processes. It is considered that there are no diffusion limitations. For the stochastic description of the enzymatic reaction, a probabilistic function is defined, which presents the defined number of substrate-enzyme complexes in the system at a given moment of time taking into account that the system possessed another amount of substrate-enzyme complexes at the previous moment of time.
The transitional probabilities and time dependences of the average rate of enzymatic reaction and its dispersion are easily determined in the case of the simplest one-substrate enzymatic reaction. In case of low substrate concentrations the dispersion of the enzymatic reaction rate linearly depends upon substrate concentration, while it decreases and tends to zero in the case of higher substrate concentrations. The Michaelis constant can be determined by maximum coordinate of the dispersion. Such an approach allows also determining correlation function and spectral density of the enzymatic reaction rate. It is shown that the correlation function of the enzymatic reaction rate depends upon time exponentially, while its spectral density has Lorentz form. The analysis of the correlation function and spectral density of enzymatic reaction rate shows that their comparison to the experimental correlation function and spectral density allows obtaining additional information on the bounding parameters of the substrate and enzyme, namely the values of the rate constants of substrate-enzyme complex formation as well as rate constants of its dissolving to substrate and free enzyme and to product and free enzyme.
Valeri B. Arakelyan1
1Yerevan Physics Institute