A variational approach to moment-closure approximations for the kinetics of biomolecular reaction networks

TL;DR

This paper introduces a variational framework for moment-closure approximations in biomolecular reaction networks, improving their accuracy and understanding by addressing divergence issues and extending to multi-time distributions.

Contribution

It provides a new variational derivation of moment-closure equations, uses mixture distributions to prevent divergences, and extends entropic matching to a broader class of processes.

Findings

Variational derivation clarifies properties and failure modes of moment-closure.

Mixture of product-Poisson distributions prevents divergences at low system sizes.

Extended entropic matching as a special case of variational moment closure.

Abstract

Approximate solutions of the chemical master equation and the chemical Fokker-Planck equation are an important tool in the analysis of biomolecular reaction networks. Previous studies have highlighted a number of problems with the moment-closure approach used to obtain such approximations, calling it an ad-hoc method. In this article, we give a new variational derivation of moment-closure equations which provides us with an intuitive understanding of their properties and failure modes and allows us to correct some of these problems. We use mixtures of product-Poisson distributions to obtain a flexible parametric family which solves the commonly observed problem of divergences at low system sizes. We also extend the recently introduced entropic matching approach to arbitrary ansatz distributions and Markov processes, demonstrating that it is a special case of variational moment closure. This provides us with a particularly principled approximation method. Finally, we extend the above approaches to cover the approximation of multi-time joint distributions, resulting in a viable alternative to process-level approximations which are often intractable.