Numerical methods for particle-based stochastic reaction-diffusion systems
Embargo Date
2025-02-01
OA Version
Citation
Abstract
Particle-based stochastic reaction-diffusion (PBSRD) systems are used in computational biology to model the dynamics of particles which react, diffuse, and interact in space. PBSRD systems range in complexity, from purely diffusive models of particle motion to systems with volume exclusion (hard-core repulsion), potential interactions (e.g., electrostatic forces and soft-core repulsion), or topological relationships between particles modulating both particle motion and reaction rates. As such, there are a variety of software packages implementing a range of numerical methods for solving PBSRD systems at these varying levels of detail. In this thesis, we develop and analyze numerical methods for PBSRD systems, first focusing on the convergent reaction-diffusion master equation (CRDME) framework and later presenting a macroscopic stochastic partial-integro differential equation (SPIDE) approximation to the particle system. In particular, we extend the CRDME to include particle-particle interactions by pairing a tensor product finite element method discretization with a specific quadrature rule for drift-diffusion PDEs. We show how our method can be used to develop detailed balance satisfying schemes for reversible reactions. We conduct empirical, and later analytical, convergence studies for our method and validate it against existing software for accuracy and efficiency. We also derive new detailed balance satisfying schemes for reversible reactions between pairs of particles, discussing practical implementation details including a new rejection framework. Finally, we compare statistics computed from PBSRD systems simulated using the CRDME to the numerical solution of a macroscopic SPIDE model for the stochastic fluctuations of the PBSRD systems around its mean field limit. As the number of particles increases, the variance of the PBSRD model can be shown to converge to the variance of the SPIDE solution in a rigorous central limit theorem. Numerical experiments show the efficiency of the SPIDE solver compared with the CRDME and its accuracy even for relatively modest particle numbers.