Coupling particle-based reaction-diffusion simulations with reservoirs mediated by reaction-diffusion PDEs

TL;DR

This paper introduces a novel numerical scheme that couples particle-based reaction-diffusion models with reaction-diffusion PDEs to simulate open biochemical systems interacting with dynamic, inhomogeneous reservoirs.

Contribution

It develops theoretical mean-field results for open particle systems and proposes a consistent coupling method with PDEs for realistic biochemical modeling.

Findings

Derived mean-field equations for open diffusing particles

Established mean-field for second-order reactions

Created a numerical scheme for coupled particle-PDE systems

Abstract

Open biochemical systems of interacting molecules are ubiquitous in life-related processes. However, established computational methodologies, like molecular dynamics, are still mostly constrained to closed systems and timescales too small to be relevant for life processes. Alternatively, particle-based reaction-diffusion models are currently the most accurate and computationally feasible approach at these scales. Their efficiency lies in modeling entire molecules as particles that can diffuse and interact with each other. In this work, we develop modeling and numerical schemes for particle-based reaction-diffusion in an open setting, where the reservoirs are mediated by reaction-diffusion PDEs. We derive two important theoretical results. The first one is the mean-field for open systems of diffusing particles; the second one is the mean-field for a particle-based reaction-diffusion system with second-order reactions. We employ these two results to develop a numerical scheme that consistently couples particle-based reaction-diffusion processes with reaction-diffusion PDEs. This allows modeling open biochemical systems in contact with reservoirs that are time-dependent and spatially inhomogeneous, as in many relevant real-world applications.