Department of Mathematics, Ryerson University, Toronto, Ontario M5B 2K3, Canada.
J Chem Phys. 2012 May 14;136(18):184101. doi: 10.1063/1.4711143.
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.
数学和计算建模是分析细胞和生物体中重要生物过程的关键工具。特别是，当不能再应用热力学极限的假设时，随机模型对于准确描述细胞动力学至关重要。然而，随机模型在计算上比传统的确定性模型更具挑战性。此外，应用中出现的许多生化系统具有多个时间尺度，这导致了数学上的刚性。在本文中，我们研究了一种化学 Langevin 方程的随机连续生化系统模型的数值解。化学 Langevin 方程是一个具有乘性、非交换噪声的随机微分方程。我们提出了一种基于局部误差估计的适用于 Langevin 区小噪声生化系统模型的自适应步长算法。所提出的数值方法是米尔斯坦方案。所提出的自适应方法在几个应用实例中进行了测试，与现有的固定步长方案相比，它显示出了提高的效率和准确性。
