An Unstructured Mesh Reaction-Drift-Diffusion Master Equation with Reversible Reactions

TL;DR

This paper introduces a convergent reaction-drift-diffusion master equation (CRDDME) for modeling reversible reactions with spatial transport influenced by drift in complex geometries, ensuring detailed balance and equilibrium preservation.

Contribution

The paper develops a novel unstructured mesh reaction-drift-diffusion master equation that accurately models reversible reactions with drift, preserving detailed balance and equilibrium states.

Findings

CRDDME accurately models reaction-drift-diffusion processes.

The method preserves detailed balance at equilibrium.

Numerical examples demonstrate convergence and accuracy.

Abstract

We develop a convergent reaction-drift-diffusion master equation (CRDDME) to facilitate the study of reaction processes in which spatial transport is influenced by drift due to one-body potential fields within general domain geometries. The generalized CRDDME is obtained through two steps. We first derive an unstructured grid jump process approximation for reversible diffusions, enabling the simulation of drift-diffusion processes where the drift arises due to a conservative field that biases particle motion. Leveraging the Edge-Averaged Finite Element method, our approach preserves detailed balance of drift-diffusion fluxes at equilibrium, and preserves an equilibrium Gibbs-Boltzmann distribution for particles undergoing drift-diffusion on the unstructured mesh. We next formulate a spatially-continuous volume reactivity particle-based reaction-drift-diffusion model for reversible reactions of the form $\textrm{A} + \textrm{B} \leftrightarrow \textrm{C}$. A finite volume discretization is used to generate jump process approximations to reaction terms in this model. The discretization is developed to ensure the combined reaction-drift-diffusion jump process approximation is consistent with detailed balance of reaction fluxes holding at equilibrium, along with supporting a discrete version of the continuous equilibrium state. The new CRDDME model represents a continuous-time discrete-space jump process approximation to the underlying volume reactivity model. We demonstrate the convergence and accuracy of the new CRDDME through a number of numerical examples, and illustrate its use on an idealized model for membrane protein receptor dynamics in T cell signaling.