随机化学动力学中矩封闭近似的有效性条件
Validity conditions for moment closure approximations in stochastic chemical kinetics
摘要：
基于矩闭合（MA）的近似通常用于获得化学系统的平均分子数和分子数波动方差的估计。这种方法的优点是，它可以远远低于化学主方程的精确随机模拟计算昂贵。在这里，我们数值研究的条件下，MA方程产生的结果，反映了真正的随机动态系统。我们表明，对于具有确定性初始条件的振荡和振荡化学系统，MA方程的解可以被解释为CME的真实时刻的有效近似，只有当从化学主方程获得的稳态平均分子数落在一定的有限范围内。对于单稳态系统，同样的有效性准则意味着从化学主方程获得的稳态平均分子数必须高于一定的阈值。对于平均分子数的有效性范围之外，MA方程导致定性错误的振荡动力学或非物理的预测，如负方差的分子数或多个稳态矩的固定分布的初始条件是不同的。我们的研究结果澄清了MA方法的有效性范围，并表明，在解释结果的陷阱只能通过系统的比较，一定的顺序与高阶的MA方程的解决方案，可以克服。
英文摘要：
Approximations based on moment-closure (MA) are commonly used to obtain estimates of the mean molecule numbers and of the variance of fluctuations in the number of molecules of chemical systems. The advantage of this approach is that it can be far less computationally expensive than exact stochastic simulations of the chemical master equation. Here we numerically study the conditions under which the MA equations yield results reflecting the true stochastic dynamics of the system. We show that for bistable and oscillatory chemical systems with deterministic initial conditions, the solution of the MA equations can be interpreted as a valid approximation to the true moments of the CME, only when the steady-state mean molecule numbers obtained from the chemical master equation fall within a certain finite range. The same validity criterion for monostable systems implies that the steady-state mean molecule numbers obtained from the chemical master equation must be above a certain threshold. For mean molecule numbers outside of this range of validity, the MA equations lead to either qualitatively wrong oscillatory dynamics or to unphysical predictions such as negative variances in the molecule numbers or multiple steady-state moments of the stationary distribution as the initial conditions are varied. Our results clarify the range of validity of the MA approach and show that pitfalls in the interpretation of the results can only be overcome through the systematic comparison of the solutions of the MA equations of a certain order with those of higher orders.
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