Reaction rates for reaction-diffusion kinetics on unstructured meshes

TL;DR

This paper presents a new numerical method for accurately computing reaction rates in reaction-diffusion models on unstructured meshes, enabling better approximation of particle interactions in complex geometries.

Contribution

It introduces an efficient preprocessing scheme to estimate reaction rates on unstructured meshes, bridging the gap between coarse and fine mesh approximations.

Findings

Reaction rates converge to Smoluchowski model with mesh refinement

The method accurately predicts reaction dynamics in complex geometries

Numerical examples demonstrate improved accuracy over existing approaches

Abstract

The reaction-diffusion master equation is a stochastic model often utilized in the study of biochemical reaction networks in living cells. It is applied when the spatial distribution of molecules is important to the dynamics of the system. A viable approach to resolve the complex geometry of cells accurately is to discretize space with an unstructured mesh. Diffusion is modeled as discrete jumps between nodes on the mesh, and the diffusion jump rates can be obtained through a discretization of the diffusion equation on the mesh. Reactions can occur when molecules occupy the same voxel. In this paper, we develop a method for computing accurate reaction rates between molecules occupying the same voxel in an unstructured mesh. For large voxels, these rates are known to be well approximated by the reaction rates derived by Collins and Kimball, but as the mesh is refined, no analytical expression for the rates exists. We reduce the problem of computing accurate reaction rates to a pure preprocessing step, depending only on the mesh and not on the model parameters, and we devise an efficient numerical scheme to estimate them to high accuracy. We show in several numerical examples that as we refine the mesh, the results obtained with the reaction-diffusion master equation approach those of a more fine-grained Smoluchowski particle-tracking model.