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.
Highlights
Stochastic modelling is essential for studying key biological processes, such as signaling chemical pathways in a cell, when some molecular species are in low numbers
Since the Milstein method is of strong order of accuracy one, numerical solutions employing it on variable time-step meshes converge to the exact solution of the chemical Langevin equation, as the step size converges to zero [15]
In order to validate the accuracy of our variable step size strategy for the numerical solution of the chemical Langevin equation (CLE), we compare the histogram generated with our scheme for the CLE with that computed with the exact algorithm for the chemical master equation due to Gillespie [5, 6]
Summary
Stochastic modelling is essential for studying key biological processes, such as signaling chemical pathways in a cell, when some molecular species are in low numbers. Adaptive timestepping strategies reduce the step size in the regions where stiffness is present such that the error is maintained below the required tolerance, but relax the time-step as soon as the integration exists these regions This significantly reduces the computational cost of the simulation, while preserving the desired accuracy of the numerical solution. For the strong numerical solution of stochastic differential equations with multidimensional Wiener processes, most of the existing adaptive time-stepping strategies were designed for systems with commutative noise (see [17–19]). This paper proposes a variable time-stepping strategy for the strong numerical solution of the chemical Langevin equation, based on proportional integral- (PI-) control. The technique uses estimates of the local error, based on the work by Sotiropoulos and Kaznessis [20] This variable time-stepping method may be applied to any biochemical system which can be modelled with the chemical Langevin equation, having an arbitrary magnitude of the random fluctuations.
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