Quasi-Entropy Closure: a fast and reliable approach to close the moment equations of the Chemical Master Equation.

Abstract

The Chemical Master Equation is a stochastic approach to describe the evolution of a (bio)chemical reaction system. Its solution is a time-dependent probability distribution on all possible configurations of the system. As this number is typically large, the Master Equation is often practically unsolvable. The Method of Moments reduces the system to the evolution of a few moments, which are described by ordinary differential equations. Those equations are not closed, since lower order moments generally depend on higher order moments. Various closure schemes have been suggested to solve this problem. Two major problems with these approaches are first that they are open loop systems, which can diverge from the true solution, and second, some of them are computationally expensive. Here we introduce Quasi-Entropy Closure, a moment-closure scheme for the Method of Moments. It estimates higher order moments by reconstructing the distribution that minimizes the distance to a uniform distribution subject to lower order moment constraints. Quasi-Entropy Closure can be regarded as an advancement of Zero-Information Closure, which similarly maximizes the information entropy. Results show that both approaches outperform truncation schemes. Quasi-Entropy Closure is computationally much faster than Zero-Information Closure, although both methods consider solutions on the space of configurations and hence do not completely overcome the curse of dimensionality. In addition, our scheme includes a plausibility check for the existence of a distribution satisfying a given set of moments on the feasible set of configurations. All results are evaluated on different benchmark problems. Supplementary data are available at Bioinformatics online.