# Complex Balancing Reconstructed to the Asymptotic Stability of Mass-action Chemical Reaction Networks with Conservation Laws

## TL;DR

This paper introduces a generalized Lyapunov function for analyzing the asymptotic stability of mass-action systems with conservation laws by reconstructing networks and examining their stability properties.

## Contribution

It develops a novel reconstruction method and a generalized pseudo-Helmholtz function to analyze stability in complex balanced and more general mass-action systems.

## Key findings

- Reverse reconstruction has a unique equilibrium in each positive class.
- Complex balanced reconstruction ensures asymptotic stability of the original network.
- The method provides a systematic way to compute reconstructions with conservation laws.

## Abstract

Motivated by the fact that the pseudo-Helmholtz function is a valid Lyapunov function for characterizing asymptotic stability of complex balanced mass action systems (MASs), this paper develops the generalized pseudo-Helmholtz function for stability analysis for more general MASs assisted with conservation laws. The key technique is to transform the original network into two different MASs, defined by reconstruction and reverse reconstruction, with an important aspect that the dynamics of the original network for free species is equivalent to that of the reverse reconstruction. Stability analysis of the original network is then conducted based on an analysis of how stability properties are retained from the original network to the reverse reconstruction. We prove that the reverse reconstruction possesses only an equilibrium in each positive stoichiometric compatibility class if the corresponding reconstruction is complex balanced. Under this complex balanced reconstruction strategy, the asymptotic stability of the reverse reconstruction, which also applies to the original network, is thus reached by taking the generalized pseudo-Helmholtz function as the Lyapunov function. To facilitate applications, we further provide a systematic method for computing complex balanced reconstructions assisted with conservation laws. Some representative examples are presented to exhibit the validity of the complex balanced reconstruction strategy.