Communications in Mathematical Sciences

Volume 14 (2016)

Number 6

A discrete stochastic formulation for reversible bimolecular reactions via diffusion encounter

Pages: 1741 – 1772

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n6.a13

Authors

Mauricio J. del Razo (Department of Applied Mathematics, University of Washington, Seattle, Wa., U.S.A.)

Hong Qian (University of Washington, Seattle, Washington, U.S.A.)

Abstract

The classical models for irreversible diffusion-influenced reactions can be derived by introducing absorbing boundary conditions to over-damped continuous Brownian motion (BM) theory. As there is a clear corresponding stochastic process, the mathematical description takes both Kolmogorov forward equation for the evolution of the probability distribution function and the stochastic sample trajectories. This dual description is a fundamental characteristic of stochastic processes and allows simple particle-based simulations to accurately match the expected statistical behavior. However, in the traditional theory using the back-reaction boundary condition to model reversible reactions with geminate recombinations, several subtleties arise: It is unclear what the underlying stochastic process is, which causes complications in producing accurate simulations; and it is non-trivial how to perform an appropriate discretization for numerical computations. In this work, we derive a discrete stochastic model that recovers the classical models and their boundary conditions in the continuous limit. In the case of reversible reactions, we recover the back-reaction boundary condition, unifying the back-reaction approach with those of current simulation packages. Furthermore, all the complications encountered in the continuous models become trivial in the discrete model. Our formulation brings to attention the question: With computations in mind, can we develop a discrete reaction kinetics model that is more fundamental than its continuous counterpart?

Keywords

stochastic reaction-diffusion, diffusion-influenced reactions, reversible reactions, chemical kinetics, Markov chain, Brownian motion, absorption boundary, back-reaction boundary

2010 Mathematics Subject Classification

60J10, 60J22, 60J50, 60J70, 65C35, 65C40, 92C40

Full Text (PDF format)

Published 12 August 2016