Enzymatic Production and Linear Degradation

Let us analyze the stochastic dynamics of a system in which A molecules are produced, via an enzymatic reaction, out of the substrate X, and are directly degraded into molecules V. These processes can be summarized in the following set of chemical reactions (Lehninger et al. 2005 Houston 2001)  

In the above chemical reactions E stands for free enzyme Ex is the enzyme-substrate complex nx, , hex, nA, and respectively represent the X, E, Ex, A, and Y molecule counts and parameters kij denote reaction rates. 

Under the assumptions that nx and y are constant, and that + hex — nj, with nr constant, the system state is fully determined by the ( A . a) values. Let P nEx,nA, t) be the probability of having hex molecules Ex and ha molecules A at time t. To study the system stochastic dynamics one could write the master equation for P riEx,nA, t) and analyze it. However, the analysis can be simplified if we previously make a quasi-stationary approximation. Is it usually acknowledged that the reaction in (5.7) is much faster than those in (5.8) and (5.9). Since the reaction in (5.7) modifies the value of a but not of ha, we can directly apply the formalism developed in the Sect. 5.1. 

Regarding P ( xI a), according to Eq. (5.6) it is the stationary solution of the following master equation  

A comparison of Eqs. (5.14) and (3.17) reveals that they are equivalent, and so that the process modeled by Eq. (5.14) is that of r molecules flipping between states E and Ex, with constant transition rates for individual molecules. Thus, from the results in Chap. 3— Eqs. (3.13), (3.14), (3.18), and (3.19)—the stationary distribution P itiExlnA) happens to be a binomial distribution with parameters n = riT and p = + kEx)- Moreover, the average number of 

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