Abstract
Biochemical processes typically involve many chemical species, some in abundance and some in low molecule numbers. We first identify the rate constant limits under which the concentrations of a given set of species will tend to infinity (the abundant species) while the concentrations of all other species remains constant (the non-abundant species). Subsequently, we prove that, in this limit, the fluctuations in the molecule numbers of non-abundant species are accurately described by a hybrid stochastic description consisting of a chemical master equation coupled to deterministic rate equations. This is a reduced description when compared to the conventional chemical master equation which describes the fluctuations in both abundant and non-abundant species. We show that the reduced master equation can be solved exactly for a number of biochemical networks involving gene expression and enzyme catalysis, whose conventional chemical master equation description is analytically impenetrable. We use the linear noise approximation to obtain approximate expressions for the difference between the variance of fluctuations in the non-abundant species as predicted by the hybrid approach and by the conventional chemical master equation. Furthermore, we show that surprisingly, irrespective of any separation in the mean molecule numbers of various species, the conventional and hybrid master equations exactly agree for a class of chemical systems.
Highlights
The chemical master equation (CME) is the accepted stochastic description of chemical reaction systems.1 Since intrinsic noise roughly scales as the inverse square root of the mean number of molecules,1 it follows that the CME provides a more accurate description than deterministic rate equations (REs), when species are in low concentrations
We investigate the use of the Linear Noise Approximation (LNA) to obtain an estimate of the error made by the use of the reduced CME
The reduced CME can in this case be exactly solved in steady-state conditions and one obtains a binomial distribution with parameters ET and 1 − a describing the fluctuations in enzyme molecule numbers; for the case ET = 1, the binomial distribution reduces to the Bernoulli distribution found earlier for the open Michaelis Menten system with one enzyme molecule (see Eq (59))
Summary
The chemical master equation (CME) is the accepted stochastic description of chemical reaction systems. Since intrinsic noise roughly scales as the inverse square root of the mean number of molecules, it follows that the CME provides a more accurate description than deterministic rate equations (REs), when species are in low concentrations. The ratio of median protein to median mRNA lifetime is about 5.24 Clearly in these cases, abundance separation is significant while time scale separation is weak, and a method which takes advantage of the former appears to be ideal as a means to infer information about the stochastic dynamics of mRNA and of other proteins present in low copy numbers. In this limit, the marginal distributions of the non-abundant species given by the hybrid model converge to the same distributions given by the CME of the full system This fact is useful when the hybrid model can be solved analytically, which is the case in several examples that we study.
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