Reaction rates for a generalized reaction-diffusion master equation

TL;DR

This paper extends the reaction-diffusion master equation to include reactions between neighboring voxels, deriving optimal reaction rates to match detailed models and enabling accurate simulations beyond previous mesh size limitations.

Contribution

It introduces a generalized reaction-diffusion master equation allowing neighboring voxel reactions and derives reaction rates for improved accuracy in 2D and 3D.

Findings

The generalized algorithm accurately matches fine-grained models across various mesh sizes.

It reduces to the standard algorithm above a certain mesh size.

A fundamental lower bound for mesh size is established based on reaction radius.

Abstract

It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model, and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm, which, in both two dimensions and three dimensions, is on the order of the reaction radius of a reacting pair of molecules.