Rué Pau, Villà-Freixa Jordi, Burrage Kevin
Computational Biochemistry and Biophysics Group, Research Unit on Biomedical Informatics, IMIM/Universitat Pompeu Fabra, 08003 Barcelona, Catalonia, Spain.
BMC Syst Biol. 2010 Aug 11;4:110. doi: 10.1186/1752-0509-4-110.
With increasing computer power, simulating the dynamics of complex systems in chemistry and biology is becoming increasingly routine. The modelling of individual reactions in (bio)chemical systems involves a large number of random events that can be simulated by the stochastic simulation algorithm (SSA). The key quantity is the step size, or waiting time, tau, whose value inversely depends on the size of the propensities of the different channel reactions and which needs to be re-evaluated after every firing event. Such a discrete event simulation may be extremely expensive, in particular for stiff systems where tau can be very short due to the fast kinetics of some of the channel reactions. Several alternative methods have been put forward to increase the integration step size. The so-called tau-leap approach takes a larger step size by allowing all the reactions to fire, from a Poisson or Binomial distribution, within that step. Although the expected value for the different species in the reactive system is maintained with respect to more precise methods, the variance at steady state can suffer from large errors as tau grows.
In this paper we extend Poisson tau-leap methods to a general class of Runge-Kutta (RK) tau-leap methods. We show that with the proper selection of the coefficients, the variance of the extended tau-leap can be well-behaved, leading to significantly larger step sizes.
The benefit of adapting the extended method to the use of RK frameworks is clear in terms of speed of calculation, as the number of evaluations of the Poisson distribution is still one set per time step, as in the original tau-leap method. The approach paves the way to explore new multiscale methods to simulate (bio)chemical systems.
随着计算机性能的提升，模拟化学和生物学中复杂系统的动力学变得越来越常规。（生物）化学系统中单个反应的建模涉及大量随机事件，这些事件可以通过随机模拟算法（SSA）进行模拟。关键量是步长，即等待时间τ，其值反比于不同通道反应的倾向大小，并且在每次触发事件后都需要重新评估。这种离散事件模拟可能极其昂贵，特别是对于刚性系统，由于某些通道反应的快速动力学，τ可能非常短。已经提出了几种替代方法来增加积分步长。所谓的τ跳跃方法通过允许所有反应在该步长内从泊松或二项分布触发来采用更大的步长。尽管相对于更精确的方法，反应系统中不同物种的期望值得以维持，但随着τ的增加，稳态下的方差可能会出现较大误差。
在本文中，我们将泊松τ跳跃方法扩展到一类通用的龙格 - 库塔（RK）τ跳跃方法。我们表明，通过适当选择系数，扩展后的τ跳跃的方差可以表现良好，从而导致步长显著增大。
就计算速度而言，将扩展方法应用于RK框架的好处是显而易见的，因为与原始τ跳跃方法一样，每次时间步长内泊松分布的评估次数仍然是一组。该方法为探索模拟（生物）化学系统的新多尺度方法铺平了道路。
