Loss of bimolecular reactions in reaction-diffusion master equations is consistent with diffusion limited reaction kinetics in the mean field limit.

Abstract

We show that the resolution-dependent loss of bimolecular reactions in spatiotemporal Reaction-Diffusion Master Equations (RDMEs) is in agreement with the mean-field Collins-Kimball (C-K) theory of diffusion-limited reaction kinetics. The RDME is a spatial generalization of the chemical master equation, which enables studying stochastic reaction dynamics in spatially heterogeneous systems. It uses a regular Cartesian grid to partition space into locally well-mixed reaction compartments and treats diffusion as a jump reaction between neighboring grid cells. As the chance for reactants to be in the same grid cell decreases for smaller cell widths, the RDME loses bimolecular reactions in finer grids. We show that for a single homo-bimolecular reaction, the mesh spacing can be interpreted as the reaction radius of a well-mixed C-K rate. Then, the bimolecular reaction loss is consistent with diffusion-limited kinetics in the mean-field steady state. In this interpretation, the constant in a bimolecular reaction propensity is no longer the macroscopic reaction rate but the rate of the ballistic C-K step. For the same grid resolution, different diffusion models in RDME, such as those based on finite differences and Gaussian jumps, represent different reaction radii.