Chemical master equation for Michaelis-Menten kinetics

1 Chemical master equation for Michaelis-Menten kinetics 

Enzyme-catalyzed reactions involve multi-molecular enzyme-substrate association. Therefore, even when the overall reaction is unimolecular, the enzyme mechanism is generally non-linear. If a system has more than one copy of the enzyme and a small number of the reactant molecules, then one needs the CME framework to represent the stochastic behavior of the system. Note that in cellular regulatory networks, the substrates themselves may be proteins that are present in small numbers of copies. Recall from Section 5.1, for example, that the mitogen-activated protein (MAP) is the substrate of MAP kinase, and the MAPK is the substrate of MAPKK. 

Stochastic biochemical systems and the chemical master equation 

Here to illustrate the procedure for non-linear systems, we work out the chemical master equation for the Michaelis-Menten reaction of Equation (11.19). We let p(m, n, t) be the probability that m, S, and n ES molecules occur in the enzyme reaction system at time t. The probability p(m, n, t) satisfies the CME 

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