# Solving Moment Hierarchies for Chemical Reaction Networks

## TL;DR

This paper develops a recursive method to solve the moment hierarchy for chemical reaction networks with non-zero deficiency, enabling accurate prediction of steady-state moments even when traditional factorization results do not apply.

## Contribution

It introduces a recursive approach to analyze the full moment hierarchy for non-zero-deficiency CRNs, extending understanding beyond existing zero-deficiency results.

## Key findings

- Accurately predicts moments for complex CRNs
- Recursively relates lower and higher moments
- Works for non-factorizable steady states

## Abstract

The study of Chemical Reaction Networks (CRN's) is a very active field. Earlier well-known results \cite{Feinberg:def_01:87, Anderson:product_dist:10} identify a topological quantity called deficiency, for any CRN, which, when exactly equal to zero, leads to a unique factorized steady-state for these networks. No results exist however for the steady states of non-zero-deficiency networks. In this paper, we show how to write the full moment-hierarchy for any non-zero-deficiency CRN obeying mass-action kinetics, in terms of equations for the factorial moments (FM). Using these, we can recursively predict values for lower moments from higher moments, reversing the procedure usually used to solve moment hierarchies. We show, for non-trivial examples, that in this manner we can predict to high accuracy, any moment of interest, for CRN's with non-zero deficiency and non-factorizable steady states.