The spatiotemporal master equation: approximation of reaction-diffusion dynamics via Markov state modeling

Abstract

Accurate modeling and numerical simulation of reaction kinetics is a topic of steady interest. We present the spatiotemporal chemical master equation (ST-CME) as a model for stochastic reaction-diffusion systems that exhibit properties of metastability. By means of Markov state modelling, the space of motion is decomposed into metastable compartments and diffusive motion is approximated by jumps between these compartments. Treating these jumps as first-order reactions, simulation of the resulting stochastic system is possible by means of Gillespie’s stochastic simulation algorithm. We first heuristically derive the ST-CME starting from the most detailed model of particle-based reaction-diffusion dynamics, compare it to similar approaches to reaction-diffusion master equations (RDME), and finally give a rigorous justification of the ST-CME by means of the Galerkin approach to discretization. The resulting form of the ST-CME and its algorithmic realization is then numerically tested in application to a nuclear membrane transport problem in comparison to particle-based reaction-diffusion dynamics.