A route to the hydrodynamic limit of a reaction-diffusion master equation using gradient structures

TL;DR

This paper presents a novel approach using gradient structures to rigorously derive the hydrodynamic limit of the reaction-diffusion master equation, connecting stochastic lattice models to continuous reaction-diffusion PDEs.

Contribution

It extends a gradient structure method from well-mixed systems to spatially extended reaction-diffusion models under detailed balance, providing a new analytical framework.

Findings

Established a gradient structure for the RDME.

Derived a gradient structure for the reaction-diffusion PDE limit.

Provided a rigorous link between stochastic and continuous models.

Abstract

The reaction-diffusion master equation (RDME) is a lattice-based stochastic model for spatially resolved cellular processes. It is often interpreted as an approximation to spatially continuous reaction-diffusion models, which, in the limit of an infinitely large population, may be described by means of reaction-diffusion partial differential equations (RDPDEs). Analyzing and understanding the relation between different mathematical models for reaction-diffusion dynamics is a research topic of steady interest. In this work, we explore a route to the hydrodynamic limit of the RDME which uses gradient structures. Specifically, we elaborate on a method introduced in [J. Maas, A. Mielke: Modeling of chemical reactions systems with detailed balance using gradient structures. J. Stat. Phys. (181), 2257-2303 (2020)] in the context of well-mixed reaction networks by showing that, once it is complemented with an appropriate limit procedure, it can be applied to spatially extended systems with diffusion. Under the assumption of detailed balance, we write down a gradient structure for the RDME and use the method to produce a gradient structure for its hydrodynamic limit, namely, for the corresponding RDPDE.