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Chemical reactions inside cells occur in compartment volumes in the range ofatto- to femtolitres. Physiological concentrations realized in such smallvolumes imply low copy numbers of interacting molecules with the consequence ofconsiderable fluctuations in the concentrations. In contrast, rate equationmodels are based on the implicit assumption of infinitely large numbers ofinteracting molecules, or equivalently, that reactions occur in infinitevolumes at constant macroscopic concentrations. In this article we compute thefinite-volume corrections (or equivalently the finite copy number corrections)to the solutions of the rate equations for chemical reaction networks composedof arbitrarily large numbers of enzyme-catalyzed reactions which are confinedinside a small sub-cellular compartment. This is achieved by applying amesoscopic version of the quasi-steady state assumption to the exactFokker-Planck equation associated with the Poisson Representation of thechemical master equation. The procedure yields impressively simple and compactexpressions for the finite-volume corrections. We prove that the predictions ofthe rate equations will always underestimate the actual steady-state substrateconcentrations for an enzyme-reaction network confined in a small volume. Inparticular we show that the finite-volume corrections increase with decreasingsub-cellular volume, decreasing Michaelis-Menten constants and increasingenzyme saturation. The magnitude of the corrections depends sensitively on thetopology of the network. The predictions of the theory are shown to be inexcellent agreement with stochastic simulations for two types of networkstypically associated with protein methylation and metabolism.

Stochastic theory of large-scale enzyme-reaction networks: Finite copy number corrections to rate equation models