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A finite state projection algorithm for the stationary solution of the chemical master equation

机译：化学硕主方程固定解的有限状态投影算法

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The chemical master equation (CME) is frequently used in systems biology to quantify the effects of stochastic fluctuations that arise due to biomolecular species with low copy numbers. The CME is a system of ordinary differential equations that describes the evolution of probability density for each population vector in the state-space of the stochastic reaction dynamics. For many examples of interest, this state-space is infinite, making it difficult to obtain exact solutions of the CME. To deal with this problem, the Finite State Projection (FSP) algorithm was developed by Munsky and Khammash [J. Chem. Phys. 124(4), 044104 (2006)], to provide approximate solutions to the CME by truncating the state-space. The FSP works well for finite time-periods but it cannot be used for estimating the stationary solutions of CMEs, which are often of interest in systems biology. The aim of this paper is to develop a version of FSP which we refer to as the stationary FSP (sFSP) that allows one to obtain accurate approximations of the stationary solutions of a CME by solving a finite linear-algebraic system that yields the stationary distribution of a continuous-time Markov chain over the truncated state-space. We derive bounds for the approximation error incurred by sFSP and we establish that under certain stability conditions, these errors can be made arbitrarily small by appropriately expanding the truncated state-space. We provide several examples to illustrate our sFSP method and demonstrate its efficiency in estimating the stationary distributions. In particular, we show that using a quantized tensor-train implementation of our sFSP method, problems admitting more than 100 x 10(6) states can be efficiently solved. Published by AIP Publishing.
机译：化学母部方程（CME）经常用于系统生物学，以量化由于具有低拷贝数而导致的随机波动的影响。 CME是普通微分方程的系统，其描述了随机反应动态的状态空间中每个人群载体的概率密度的概率密度的演变。对于许多感兴趣的示例，这种状态空间是无限的，使得难以获得CME的精确解决方案。要解决这个问题，由Munsky和Khammash开发了有限状态投影（FSP）算法[J.化学。物理。 124（4），044104（2006）]，通过截断状态空间为CME提供近似解。 FSP适用于有限的时间周期，但不能用于估计CME的固定解，这通常是系统生物学的兴趣。本文的目的是开发一种版本的FSP，我们将其称为静止的FSP（SFSP），其允许通过求解有限的线性代数系统来获得CME的固定解的精确近似，该系统产生静止分布在截短的状态空间上连续时间马尔可夫链。我们从SFSP产生的近似误差产生的界限，并且我们在某些稳定性条件下建立，通过适当地扩展截断的状态空间，可以任意地进行这些错误。我们提供了几个例子来说明我们的SFSP方法，并展示其估计静止分布的效率。特别是，我们表明，使用我们的SFSP方法的量化张力列车实现，可以有效地解决录取超过100 x 10（6）个州的问题。通过AIP发布发布。

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