Simulating Stochasticities in Chemical Reactions with Deterministic Delay Differential Equations

Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics Tsinghua University, Beijing 100084, China

^⋆ Corresponding author. E-mail: jzlei@tsinghua.edu.cn

Abstract

The stochastic dynamics of chemical reactions can be accurately described by chemical master equations. An approximated time-evolution equation of the Langevin type has been proposed by Gillespie based on two explicit dynamical conditions. However, when numerically solve these chemical Langevin equations, we often have a small stopping time–a time point of having an unphysical solution–in the case of low molecular numbers. This paper proposes an approach to simulate stochasticities in chemical reactions with deterministic delay differential equations. We introduce a deterministic Brownian motion described by delay differential equations, and replace the Gaussian noise in the chemical Langevin equations by the solutions of these deterministic equations. This modification can largely increase the stopping time in simulations and regain the accuracy as in the chemical Langevin equations. The novel aspect of the present study is to apply the deterministic Brownian motion to chemical reactions. It suggests a possible direction of developing a hybrid method of simulating dynamic behaviours of complex gene regulation networks.