The expected number of viable autocatalytic sets in chemical reaction systems

TL;DR

This paper derives exact formulas for the expected number of self-sustaining autocatalytic networks in chemical systems, analyzing how catalysis and inhibition influence their formation, with implications for understanding the origins of life.

Contribution

It provides the first exact expressions for the expected number of autocatalytic networks in general chemical systems, including uninhibited ones, and explores catalysis-inhibition trade-offs.

Findings

Exact formulas for expected autocatalytic networks.

Patterns of catalysis and inhibition affecting network formation.

Trade-offs between catalysis and inhibition in network emergence.

Abstract

The emergence of self-sustaining autocatalytic networks in chemical reaction systems has been studied as a possible mechanism for modelling how living systems first arose. It has been known for several decades that such networks will form within systems of polymers (under cleavage and ligation reactions) under a simple process of random catalysis, and this process has since been mathematically analysed. In this paper, we provide an exact expression for the expected number of self-sustaining autocatalytic networks that will form in a general chemical reaction system, and the expected number of these networks that will also be uninhibited (by some molecule produced by the system). Using these equations, we are able to describe the patterns of catalysis and inhibition that maximise or minimise the expected number of such networks. We apply our results to derive a general theorem concerning the trade-off between catalysis and inhibition, and to provide some insight into the extent to which the expected number of self-sustaining autocatalytic networks coincides with the probability that at least one such system is present.