A Convergent Reaction-Diffusion Master Equation

TL;DR

This paper introduces a new convergent reaction-diffusion master equation (CRDME) that overcomes the limitations of the traditional RDME by ensuring convergence to continuous models, especially for bimolecular reactions, through finite volume discretization.

Contribution

The authors derive a convergent RDME (CRDME) using finite volume discretization, providing a more accurate approximation to spatially-continuous reaction-diffusion models with bimolecular reactions.

Findings

CRDME demonstrates numerical convergence of reaction time statistics.

For large lattice spacings or slow reactions, CRDME approximates RDME.

RDME can be viewed as an asymptotic approximation to CRDME.

Abstract

The reaction-diffusion master equation (RDME) is a lattice stochastic reaction-diffusion model that has been used to study spatially distributed cellular processes. The RDME is often interpreted as an approximation to spatially-continuous models in which molecules move by Brownian motion and react by one of several mechanisms when sufficiently close. In the limit that the lattice spacing approaches zero, in two or more dimensions, the RDME has been shown to lose bimolecular reactions. The RDME is therefore not a convergent approximation to any spatially-continuous model that incorporates bimolecular reactions. In this work we derive a new convergent RDME (CRDME) by finite volume discretization of a spatially-continuous stochastic reaction-diffusion model popularized by Doi. We demonstrate the numerical convergence of reaction time statistics associated with the CRDME. For sufficiently large lattice spacings or slow bimolecular reaction rates, we also show the reaction time statistics of the CRDME may be approximated by those from the RDME. The original RDME may therefore be interpreted as an approximation to the CRDME in several asymptotic limits.