A First-Passage Kinetic Monte Carlo method for reaction–drift–diffusion processes

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 processes 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 driftdiffusion 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.

On the Green's function of the partially diffusion-controlled reversible ABCD reaction for radiation chemistry codes

“…This property can be expressed mathematically by Chapman-Kolmogorov type equations, which hold for Markovian systems. As discussed in previous papers [14,17,18], these equations offer a way to validate the algorithms presented in this article. In this Appendix, six equations of this type are presented.…”

“…a Coulomb potential, it is necessary to use the Green's functions of the Debye-Smoluchowski equation [34]. In the reactions occurring during the radiolysis of water [14], most reactions of type ABCD involve charged particles on at least one side of the reaction. However, the Green's functions for the ABC and ABCD reactions cover a large number of chemical reactions occurring in biological systems.…”

“…More recently, this approach has been used in radiation chemistry codes to simulate the radiolysis of water and aqueous solutions [14,15], chemical dosimeters [16] and to study of interaction of ligand molecules with receptors [17]. In the radiolysis of water and aqueous solutions and in biological systems, many reactions are diffusion-controlled and partially diffusion-controlled.…”

Single molecule simulations in complex geometries with embedded dynamic one-dimensional structures

“…. , p 25 (13) X 1 and X 2 are random numbers sampled from a normal distribution with standard deviation 1. t split is chosen to be 0.01. Thus, the microtubule l i defined by the points (p i , −p i ) at time t, will be updated at time t + t by rotating p i and −p i around the x-and y-axis θ and φ degrees, with θ and φ given by Eq.…”

“…22 a method for simulating interactions with curved surfaces at the microscopic level is developed, and in Ref. 25 they develop an FPKMC method for simulating active transport due to chemical potentials on embedded lines of fixed length and with fixed configuration. A mesoscopic method that simulates interactions between molecules in free space and lines is considered in Ref.…”