Adaptive Time-Stepping Using Control Theory for the Chemical Langevin Equation

Abstract

Stochastic modeling of biochemical systems has been the subject of intense research in recent years due to the large number of important applications of these systems. A critical stochastic model of well-stirred biochemical systems in the regime of relatively large molecular numbers, far from the thermodynamic limit, is the chemical Langevin equation. This model is represented as a system of stochastic differential equations, with multiplicative and noncommutative noise. Often biochemical systems in applications evolve on multiple time-scales; examples include slow transcription and fast dimerization reactions. The existence of multiple time-scales leads to mathematical stiffness, which is a major challenge for the numerical simulation. Consequently, there is a demand for efficient and accurate numerical methods to approximate the solution of these models. In this paper, we design an adaptive time-stepping method, based on control theory, for the numerical solution of the chemical Langevin equation. The underlying approximation method is the Milstein scheme. The adaptive strategy is tested on several models of interest and is shown to have improved efficiency and accuracy compared with the existing variable and constant-step methods.