The impossible challenge of estimating non-existent moments of the Chemical Master Equation

Abstract

Motivation: The Chemical Master Equation is a set of linear differential equations that describes the evolution of the probability distribution on all possible configurations of a (bio-)chemical reaction system. Since the number of configurations and therefore the dimension of the CME rapidly increases with the number of molecules, its applicability is restricted to small systems. A widely applied remedy for this challenge are moment-based approaches which consider the evolution of the first few moments of the distribution as summary statistics for the complete distribution. Here, we investigate the performance of two moment-estimation methods for reaction systems whose equilibrium distributions encounter heavy-tailedness and hence do not possess statistical moments. Results: We show that estimation via Stochastic Simulation Algorithm trajectories lose consistency over time and estimated moment values span a wide range of values even for large sample sizes. In comparison, the Method of Moments returns smooth moment estimates but is not able to indicate the non-existence of the allegedly predicted moments. We furthermore analyze the negative effect of a CME solution's heavy-tailedness on SSA run times and explain inherent difficulties. While moment estimation techniques are a commonly applied tool in the simulation of (bio-)chemical reaction networks, we conclude that they should be used with care, as neither the system definition nor the moment estimation techniques themselves reliably indicate the potential heavy-tailedness of the CME's solution.

A provably convergent control closure scheme for the Method of Moments of the Chemical Master Equation

“…The MoM and hence basically all moment closure approaches require the considered moments of the CME solution to exist for the time interval of interest, which we have assumed (and numerically tested) in this work. However, it should be noted that this is a delicate issue since reaction systems with solutions whose moments are not defined can easily be constructed (48). These systems can show wildly stochastic behavior, and in particular, SSA-based moment estimates do not converge for increasing sample sizes.…”

A Provably Convergent Control Closure Scheme for the Method of Moments of the Chemical Master Equation

“…However, it should be noted that this is a delicate issue since reaction systems with solutions whose moments are not defined can easily be constructed. 48 These systems can show wildly stochastic behavior, and in particular, SSA-based moment estimates do not converge for increasing sample sizes. Moment-based methods terribly fail when they are applied to such systems without caution.…”