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

TL;DR

This paper develops a theoretical framework to compute finite-volume corrections to enzyme-reaction network models, revealing that traditional rate equations underestimate substrate concentrations in small cellular compartments, with corrections depending on network topology and parameters.

Contribution

It introduces a mesoscopic approach to derive simple formulas for finite-volume corrections in enzyme networks, extending classical rate equations to small-volume regimes.

Findings

Rate equations underestimate substrate concentrations in small volumes.

Finite-volume corrections increase with decreasing compartment size and enzyme saturation.

Predictions match stochastic simulations for methylation and metabolism networks.

Abstract

Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtolitres. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concentrations. In this article we compute the finite-volume corrections (or equivalently the finite copy number corrections) to the solutions of the rate equations for chemical reaction networks composed of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small sub-cellular compartment. This is achieved by applying a mesoscopic version of the quasi-steady state assumption to the exact Fokker-Planck equation associated with the Poisson Representation of the chemical master equation. The procedure yields impressively simple and compact expressions for the finite-volume corrections. We prove that the predictions of the rate equations will always underestimate the actual steady-state substrate concentrations for an enzyme-reaction network confined in a small volume. In particular we show that the finite-volume corrections increase with decreasing sub-cellular volume, decreasing Michaelis-Menten constants and increasing enzyme saturation. The magnitude of the corrections depends sensitively on the topology of the network. The predictions of the theory are shown to be in excellent agreement with stochastic simulations for two types of networks typically associated with protein methylation and metabolism.