Margarita Kostré | Searcharxiv

arXiv2025

Open reaction-diffusion systems: bridging probabilistic theory and simulations across scales

Reaction-diffusion processes are the foundational model for a diverse range of complex systems, ranging from biochemical reactions to social agent-based phenomena. The underlying dynamics of these systems occur at the individual particle/agent level, and in realistic applications, they often display interaction with their environment through energy or material exchange with a reservoir. This requires intricate mathematical considerations, especially in the case of material exchange since the varying number of particles/agents results in ``on-the-fly'' modification of the system dimension. In this work, we first overview the probabilistic description of reaction-diffusion processes at the particle level, which readily handles varying number of particles. We then extend this model to consistently incorporate interactions with macroscopic material reservoirs. Based on the resulting expressions, we bridge the probabilistic description with macroscopic concentration-based descriptions for linear and nonlinear reaction-diffusion systems, as well as for an archetypal open reaction-diffusion system. Using these mathematical bridges across scales, we finally develop numerical schemes for open reaction-diffusion systems, which we implement in two illustrative examples. This work establishes a methodological workflow to bridge particle-based probabilistic descriptions with macroscopic concentration-based descriptions of reaction-diffusion in open settings, laying the foundations for a multiscale theoretical framework upon which to construct theory and simulation schemes that are consistent across scales.

cond-mat.stat-mech↗

arXiv2020

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

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.

q-bio.QM↗