Determining the long-term behavior of cell populations: A new procedure for detecting ergodicity in large stochastic reaction networks

摘要

A reaction network consists of a finite number of species, which interactthrough predefined reaction channels. Traditionally such networks were modeleddeterministically, but it is now well-established that when reactant copynumbers are small, the random timing of the reactions create internal noisethat can significantly affect the macroscopic properties of the system. Tounderstand the role of noise and quantify its effects, stochastic models arenecessary. In the stochastic setting, the population is described by aprobability distribution, which evolves according to a set of ordinarydifferential equations known as the Chemical Master Equation (CME). This set isinfinite in most cases making the CME practically unsolvable. In manyapplications, it is important to determine if the solution of a CME has aglobally attracting fixed point. This property is called ergodicity and itspresence leads to several important insights about the underlying dynamics. Thegoal of this paper is to present a simple procedure to verify ergodicity instochastic reaction networks. We provide a set of simple linear-algebraicconditions which are sufficient for the network to be ergodic. In particular,our main condition can be cast as a Linear Feasibility Problem (LFP) which isessentially the problem of determining the existence of a vector satisfyingcertain linear constraints. The inherent scalability of LFPs make our approachefficient, even for very large networks. We illustrate our procedure through anexample from systems biology.