Munsky Brian, Khammash Mustafa
Mechanical and Environmental Engineering, University of California-Santa Barbara, Santa Barbara, California 93106, USA.
J Chem Phys. 2006 Jan 28;124(4):044104. doi: 10.1063/1.2145882.
This article introduces the finite state projection (FSP) method for use in the stochastic analysis of chemically reacting systems. One can describe the chemical populations of such systems with probability density vectors that evolve according to a set of linear ordinary differential equations known as the chemical master equation (CME). Unlike Monte Carlo methods such as the stochastic simulation algorithm (SSA) or tau leaping, the FSP directly solves or approximates the solution of the CME. If the CME describes a system that has a finite number of distinct population vectors, the FSP method provides an exact analytical solution. When an infinite or extremely large number of population variations is possible, the state space can be truncated, and the FSP method provides a certificate of accuracy for how closely the truncated space approximation matches the true solution. The proposed FSP algorithm systematically increases the projection space in order to meet prespecified tolerance in the total probability density error. For any system in which a sufficiently accurate FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP is utilized to solve two examples taken from the field of systems biology, and comparisons are made between the FSP, the SSA, and tau leaping algorithms. In both examples, the FSP outperforms the SSA in terms of accuracy as well as computational efficiency. Furthermore, due to very small molecular counts in these particular examples, the FSP also performs far more effectively than tau leaping methods.
本文介绍了用于化学反应系统随机分析的有限状态投影（FSP）方法。人们可以用概率密度向量来描述此类系统的化学种群，这些向量根据一组称为化学主方程（CME）的线性常微分方程演化。与诸如随机模拟算法（SSA）或τ跳跃等蒙特卡罗方法不同，FSP直接求解或近似CME的解。如果CME描述的系统具有有限数量的不同种群向量，则FSP方法可提供精确的解析解。当可能存在无限或极大量的种群变化时，可以截断状态空间，并且FSP方法可提供关于截断空间近似与真实解的匹配程度的精度证明。所提出的FSP算法系统地增加投影空间，以便在总概率密度误差中满足预先指定的容差。对于存在足够精确的FSP的任何系统，FSP算法显示在有限步数内收敛。利用FSP求解了取自系统生物学领域的两个示例，并对FSP、SSA和τ跳跃算法进行了比较。在这两个示例中，FSP在准确性和计算效率方面均优于SSA。此外，由于这些特定示例中的分子数非常少，FSP的性能也比τ跳跃方法有效得多。
