Stoechiometric and dynamical autocatalysis for diluted chemical reaction networks

TL;DR

This paper establishes a topological criterion for autocatalysis in diluted chemical reaction networks, linking stoichiometric conditions to dynamical exponential growth, and provides a mathematical framework for analyzing autocatalytic behavior.

Contribution

It introduces a necessary and sufficient topological condition for autocatalysis and connects stoichiometric autocatalysis with dynamical exponential amplification in diluted regimes.

Findings

Topological condition (Top) characterizes autocatalysis in diluted networks.

Autocatalytic networks exhibit exponential growth when degradation rates are low.

Auxiliary Markov chains are used to analyze autocatalytic dynamics.

Abstract

Autocatalysis underlies the ability of chemical and biochemical systems to replicate. Recently, Blokhuis et al. gave a stoechiometric definition of autocatalysis for reaction networks, stating the existence of a combination of reactions such that the balance for all autocatalytic species is strictly positive, and investigated minimal autocatalytic networks, called {\em autocatalytic cores}. By contrast, spontaneous autocatalysis -- namely, exponential amplification of all species internal to a reaction network, starting from a diluted regime, i.e. low concentrations -- is a dynamical property. We introduce here a topological condition (Top) for autocatalysis, namely: restricting the reaction network description to highly diluted species, we assume existence of a strongly connected component possessing at least one reaction with multiple products (including multiple copies of a single species). We find this condition to be necessary and sufficient for stoechiometric autocatalysis. When degradation reactions have small enough rates, the topological condition further ensures dynamical autocatalysis, characterized by a strictly positive Lyapunov exponent giving the instantaneous exponential growth rate of the system. The proof is generally based on the study of auxiliary Markov chains. We provide as examples general autocatalytic cores of Type I and Type III in the typology of Blokhuis et al. In a companion article, Lyapunov exponents and the behavior in the growth regime are studied quantitatively beyond the present diluted regime .