Reaction rates for mesoscopic reaction-diffusion kinetics

TL;DR

This paper derives mesoscopic reaction rates for the RDME by matching it to microscopic Brownian dynamics, establishing limits for mesh resolution accuracy, and demonstrating convergence of mesoscopic to microscopic dynamics.

Contribution

It introduces a method to derive mesoscopic reaction rates by matching RDME statistics to microscopic BD models and analyzes the limits of mesh resolution for accuracy.

Findings

Derived mesoscopic reaction rates via statistical matching.

Identified fundamental mesh resolution limits for accuracy.

Showed convergence of mesoscopic to microscopic dynamics near limits.

Abstract

The mesoscopic reaction-diffusion master equation (RDME) is a popular modeling framework, frequently applied to stochastic reaction-diffusion kinetics in systems biology. The RDME is derived from assumptions about the underlying physical properties of the system, and it may produce unphysical results for models where those assumptions fail. In that case, other more comprehensive models are better suited, such as hard-sphere Brownian dynamics (BD). Although the RDME is a model in its own right, and not inferred from any specific microscale model, it proves useful to attempt to approximate a microscale model by a specific choice of mesoscopic reaction rates. In this paper we derive mesoscopic reaction rates by matching certain statistics of the RDME solution to statistics of the solution of a widely used microscopic BD model: the Smoluchowski model with a mixed boundary condition at the reaction radius of two molecules. We also establish fundamental limits for the range of mesh resolutions for which this approach yields accurate results, and show both theoretically and in numerical examples that as we approach the lower fundamental limit, the mesoscopic dynamics approach the microscopic dynamics.