A First-Passage Kinetic Monte Carlo Method for Reaction-Drift-Diffusion Processes

TL;DR

This paper extends the First-Passage Kinetic Monte Carlo method to include drift from background potentials, enabling accurate simulation of reaction-drift-diffusion processes with convergence and application demonstrations.

Contribution

The authors develop a hybrid Dynamic Lattice FPKMC method that discretizes protective domains to handle drift in reaction-diffusion models, extending previous exact stochastic simulation techniques.

Findings

The method achieves numerical convergence and accuracy for 1D drift potentials.

It effectively models the impact of drift on reaction kinetics.

The approach is demonstrated with smooth and discontinuous potentials.

Abstract

Stochastic reaction-diffusion models are now a popular tool for studying physical systems in which both the explicit diffusion of molecules and noise in the chemical reaction process play important roles. The Smoluchowski diffusion-limited reaction model (SDLR) is one of several that have been used to study biological systems. Exact realizations of the underlying stochastic process described by the SDLR model can be generated by the recently proposed First-Passage Kinetic Monte Carlo (FPKMC) method. This exactness relies on sampling analytical solutions to one and two-body diffusion equations in simplified protective domains. In this work we extend the FPKMC to allow for drift arising from fixed, background potentials. As the corresponding Fokker-Planck equations that describe the motion of each molecule can no longer be solved analytically, we develop a hybrid method that discretizes the protective domains. The discretization is chosen so that the drift-diffusion of each molecule within its protective domain is approximated by a continuous-time random walk on a lattice. New lattices are defined dynamically as the protective domains are updated, hence we will refer to our method as Dynamic Lattice FPKMC or DL-FPKMC. We focus primarily on the one-dimensional case in this manuscript, and demonstrate the numerical convergence and accuracy of our method in this case for both smooth and discontinuous potentials. We also present applications of our method, which illustrate the impact of drift on reaction kinetics.