Exact lower and upper bounds on stationary moments in stochastic biochemical systems
Exact lower and upper bounds on stationary moments in stochastic biochemical systems
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DOI：
10.1088/1478-3975/aa75c6
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发表时间：
2017-08-01
期刊：
影响因子：
2
通讯作者：
Singh, Abhyudai
中科院分区：
文献类型：
--
作者：
Ghusinga, Khem Raj;Vargas-Garcia, Cesar A.;Singh, Abhyudai
在生化反应系统的随机描述中，物种种群计数统计矩的时间演化由线性动力系统描述。然而，除了一些理想情况（例如零阶和一阶反应动力学）之外，矩动力学是不确定的，因为低阶矩依赖于高阶矩。在这里，我们提出了一种新方法来找到给定的任意生化反应系统的静止力矩的精确下限和上限。该方法利用了以下事实：任何正值随机变量的统计矩必须满足一些通过矩矩阵的正半定性紧凑表示的约束。我们的分析表明，结合力矩矩阵的约束求解稳态下的力矩方程可以提供精确的力矩下限和上限。这些结果通过三个不同的例子来说明——常用的逻辑增长模型、具有自动调节的随机基因表达和激活子-阻遏子基因网络基序。有趣的是，在所有情况下，随着力矩方程扩展到包括高阶力矩，边界的准确性都会提高。我们的结果为开发近似方法提供了途径，这些方法为非线性随机系统的矩提供了明确的界限，否则这些系统在分析上是困难的。
In the stochastic description of biochemical reaction systems, the time evolution of statistical moments for species population counts is described by a linear dynamical system. However, except for some ideal cases (such as zero- and first-order reaction kinetics), the moment dynamics is underdetermined as lower-order moments depend upon higher-order moments. Here, we propose a novel method to find exact lower and upper bounds on stationary moments for a given arbitrary system of biochemical reactions. The method exploits the fact that statistical moments of any positive-valued random variable must satisfy some constraints that are compactly represented through the positive semidefiniteness of moment matrices. Our analysis shows that solving moment equations at steady state in conjunction with constraints on moment matrices provides exact lower and upper bounds on the moments. These results are illustrated by three different examples-the commonly used logistic growth model, stochastic gene expression with auto-regulation and an activator-repressor gene network motif. Interestingly, in all cases the accuracy of the bounds is shown to improve as moment equations are expanded to include higher-order moments. Our results provide avenues for development of approximation methods that provide explicit bounds on moments for nonlinear stochastic systems that are otherwise analytically intractable.