Bounding Transient Moments for a Class of Stochastic Reaction Networks Using Kolmogorov's Backward Equation

TL;DR

This paper introduces a method to compute guaranteed upper and lower bounds on transient moments in stochastic reaction networks using Kolmogorov's backward equation, avoiding moment closure issues.

Contribution

The authors develop a dual formulation based on Kolmogorov's backward equation that yields finite-dimensional bounds on transient moments for stochastic reaction networks.

Findings

Provides a systematic way to compute bounds for multiple initial conditions.

Avoids the moment closure problem by shifting to a dual formulation.

Enables efficient evaluation of moment bounds without recomputing the system.

Abstract

Stochastic chemical reaction networks (SRNs) in cellular systems are commonly modeled as continuous-time Markov chains (CTMCs) describing the dynamics of molecular copy numbers. The exact evaluation of transient copy number statistics is, however, often hindered by a non-closed hierarchy of moment equations. In this paper, we propose a method for computing theoretically guaranteed upper and lower bounds on transient moments based on the Kolmogorov's backward equation, which provides a dual representation of the CME, the governing equation for the probability distribution of the CTMC. This dual formulation avoids the moment closure problem by shifting the source of infinite dimensionality to the dependence on the initial state. We show that, this dual formulation, combined with the monotonicity of the CTMC generator, leads to a finite-dimensional linear time-invariant system that provides bounds on transient moments. The resulting system enables efficient evaluation of moment bounds across multiple initial conditions by simple inner-product operations without recomputing the bounding system. Further, for certain classes of SRNs, the bounding ODEs admit explicit construction from the reaction model, providing a systematic and constructive framework for computing provable bounds.