The Reaction-Diffusion Master Equation as an Asymptotic Approximation of Diffusion to a Small Target

Abstract

The reaction-diffusion master equation (RDME) has recently been used as a model for biological systems in which both noise in the chemical reaction process and diffusion in space of the reacting molecules is important. In the RDME, space is partitioned by a mesh into a collection of voxels. There is an unanswered question as to how solutions depend on the mesh spacing. To have confidence in using the RDME to draw conclusions about biological systems, we would like to know that it approximates a reasonable physical model for appropriately chosen mesh spacings. This issue is investigated by studying the dependence on mesh spacing of solutions to the RDME in $\mathbb{R}^{3}$ for the bimolecular reaction $\mathrm{A}+\mathrm{B}\to\varnothing$, with one molecule of species $\mathrm{A}$ and one molecule of species $\mathrm{B}$ present initially. We prove that in the continuum limit the molecules never react and simply diffuse relative to each other. Nevertheless, we show that the RDME with nonzero lattice spacing yields an asymptotic approximation to a specific spatially continuous diffusion limited reaction (SCDLR) model. We demonstrate that for realistic biological parameters it is possible to find mesh spacings such that the relative error between asymptotic approximations to the solutions of the RDME and the SCDLR models is less than one percent.