Diffusion-dynamics laws in stochastic reaction networks

TL;DR

This paper investigates the influence of diffusion on stochastic reaction networks, revealing conditions under which diffusion effects can be ignored at steady state, especially in complex balanced and linear networks.

Contribution

It establishes a universal relation in the RDME framework showing diffusion independence in steady states of complex balanced networks and identifies conditions for diffusion effects to be negligible.

Findings

Steady-state distribution in complex balanced networks is diffusion-independent.

Networks with Poisson-like steady states can ignore diffusion at steady state.

RDME reduces to the chemical master equation for linear networks, unaffected by diffusion.

Abstract

Many biological activities are induced by cellular chemical reactions of diffusing reactants. The dynamics of such systems can be captured by stochastic reaction networks. A recent numerical study has shown that diffusion can significantly enhance the fluctuations in gene regulatory networks. However, the universal relation between diffusion and stochastic system dynamics remains veiled. Within the approximation of reaction-diffusion master equation (RDME), we find general relation that the steady-state distribution in complex balanced networks is diffusion-independent. Here, complex balance is the nonequilibrium generalization of detailed balance. We also find that for a diffusion-included network with a Poisson-like steady-state distribution, the diffusion can be ignored at steady state. We then derive a necessary and sufficient condition for networks holding such steady-state distributions. Moreover, we show that for linear reaction networks the RDME reduces to the chemical master equation, which implies that the stochastic dynamics of networks is unaffected by diffusion at any arbitrary time. Our findings shed light on the fundamental question of when diffusion can be neglected, or (if nonnegligible) its effects on the stochastic dynamics of the reaction network.