# Flows, scaling, and the control of moment hierarchies for stochastic chemical reaction networks

## TL;DR

This paper develops a comprehensive mathematical framework to analyze the behavior of stochastic chemical reaction networks at all deficiency levels, revealing new insights into their moment hierarchies, scaling regimes, and steady-state properties.

## Contribution

It derives equations for all-order moments in CRNs with arbitrary deficiency, extending previous results limited to deficiency 0 or 1, and introduces a finite representation of the generator acting on factorial moments.

## Key findings

- Moment hierarchies interpolate between scaling regimes at steady state.
- Boundedness of high-order moments aids in solving for low-order moments in multi-stable systems.
- A $1/n$-expansion characterizes the contribution of certain network flows to moments.

## Abstract

Stochastic chemical reaction networks (CRNs) are complex systems which combine the features of concurrent transformation of multiple variables in each elementary reaction event, and nonlinear relations between states and their rates of change. Most general results concerning CRNs are limited to restricted cases where a topological characteristic known as deficiency takes value 0 or 1. Here we derive equations of motion for fluctuation moments at all orders for stochastic CRNs at general deficiency. We show, for the case of the mass-action rate law, that the generator of the stochastic process acts on the hierarchy of factorial moments with a finite representation. Whereas simulation of high-order moments for many-particle systems is costly, this representation reduces solution of moment hierarchies to a complexity comparable to solving a heat equation. At steady states, moment hierarchies for finite CRNs interpolate between low-order and high-order scaling regimes, which may be approximated separately by distributions similar to those for deficiency-0 networks, and connected through matched asymptotic expansions. In CRNs with multiple stable or metastable steady states, boundedness of high-order moments provides the starting condition for recursive solution downward to low-order moments, reversing the order usually used to solve moment hierarchies. A basis for a subset of network flows defined by having the same mean-regressing property as the flows in deficiency-0 networks gives the leading contribution to low-order moments in CRNs at general deficiency, in a $1/n$-expansion in large particle numbers. Our results give a physical picture of the different informational roles of mean-regressing and non-mean-regressing flows, and clarify the dynamical meaning of deficiency not only for first-moment conditions but for all orders in fluctuations.