Stability conditions of chemical networks in a linear framework

TL;DR

This paper derives a linear method to determine the decay threshold in autocatalytic chemical networks with Metzler Jacobians, providing insights into network stability and design principles relevant to the Origin of Life.

Contribution

It introduces a linear approach to compute decay thresholds in chemical networks with Metzler matrices, simplifying stability analysis compared to eigenvalue methods.

Findings

Decay threshold depends on network size, directionality, and connectivity.

Linear problem formulation simplifies stability analysis.

Design principles for stable chemical networks are proposed.

Abstract

Autocatalytic chemical reaction networks can collectively replicate or maintain their constituents despite degradation reactions only above a certain threshold, which we refer to as the decay threshold. When the chemical network has a Jacobian matrix with the Metzler property, we leverage analytical methods developed for Markov processes to show that the decay threshold can be calculated by solving a linear problem, instead of the standard eigenvalue problem. We explore how this decay threshold depends on the network parameters, such as its size, the directionality of the reactions (reversible or irreversible), and its connectivity, then we deduce design principles from this that might be relevant to research on the Origin of Life.