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

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 nonexistence 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.