Mathematical and computational modeling are key tools in analyzing important biological processes in cells and living organisms. In particular, stochastic models are essential to accurately describe the cellular dynamics, when the assumption of the thermodynamic limit can no longer be applied. However, stochastic models are computationally much more challenging than the traditional deterministic models. Moreover, many biochemical systems arising in applications have multiple time-scales, which lead to mathematical stiffness. In this paper we investigate the numerical solution of a stochastic continuous model of well-stirred biochemical systems, the chemical Langevin equation. The chemical Langevin equation is a stochastic differential equation with multiplicative, non-commutative noise. We propose an adaptive stepsize algorithm for approximating the solution of models of biochemical systems in the Langevin regime, with small noise, based on estimates of the local error. The underlying numerical method is the Milstein scheme. The proposed adaptive method is tested on several examples arising in applications and it is shown to have improved efficiency and accuracy compared to the existing fixed stepsize schemes.
"This approach extends to a class of noncommutative Itô SDE, the chemical Langevin equation model, and the work on PI-controllers for ODE by Söderlind   and for commutative Stratonovich SDE by Burrage et al. . In addition, the PI-control of the step size outperforms the integral-(I-) control considered by the previous works for the chemical Langevin equation  . "
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] ABSTRACT: Stochastic modeling is essential for an accurate description of the biochemical network dynamics at the level of a single cell. Biochemically reacting systems often evolve on multiple time-scales, thus their stochastic mathematical models manifest stiffness. Stochastic models which, in addition, are stiff and computationally very challenging, therefore the need for developing effective and accurate numerical methods for approximating their solution. An important stochastic model of well-stirred biochemical systems is the chemical Langevin Equation. The chemical Langevin equation is a system of stochastic differential equation with multidimensional non-commutative noise. This model is valid in the regime of large molecular populations, far from the thermodynamic limit. In this paper, we propose a variable time-stepping strategy for the numerical solution of a general chemical Langevin equation, which applies for any level of randomness in the system. Our variable stepsize method allows arbitrary values of the time-step. Numerical results on several models arising in applications show significant improvement in accuracy and efficiency of the proposed adaptive scheme over the existing methods, the strategies based on halving/doubling of the stepsize and the fixed step-size ones.
The Journal of Chemical Physics 12/2012; 137(23):234110. DOI:10.1063/1.4771660 · 2.95 Impact Factor
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