Minimal autocatalytic networks

TL;DR

This paper investigates the emergence of small autocatalytic networks using RAF theory, establishing computational complexity results, polynomial-time constructions, and bounds, and finds small networks are unlikely at the initial formation stage.

Contribution

It extends RAF theory analysis by providing complexity results, polynomial algorithms, and bounds for small autocatalytic networks, and compares with chemical organisation theory.

Findings

Finding that the problem of identifying the smallest RAF is NP-hard.

Polynomial-time methods to construct irreducible RAFs and identify minimal ones.

Small RAFs are unlikely to form at the initial catalysis threshold.

Abstract

Self-sustaining autocatalytic chemical networks represent a necessary, though not sufficient condition for the emergence of early living systems. These networks have been formalised and investigated within the framework of RAF theory, which has led to a number of insights and results concerning the likelihood of such networks forming. In this paper, we extend this analysis by focussing on how {\em small} autocatalytic networks are likely to be when they first emerge. First we show that simulations are unlikely to settle this question, by establishing that the problem of finding a smallest RAF within a catalytic reaction system is NP-hard. However, irreducible RAFs (irrRAFs) can be constructed in polynomial time, and we show it is possible to determine in polynomial time whether a bounded size set of these irrRAFs contain the smallest RAFs within a system. Moreover, we derive rigorous bounds on the sizes of small RAFs and use simulations to sample irrRAFs under the binary polymer model. We then apply mathematical arguments to prove a new result suggested by those simulations: at the transition catalysis level at which RAFs first form in this model, small RAFs are unlikely to be present. We also investigate further the relationship between RAFs and another formal approach to self-sustaining and closed chemical networks, namely chemical organisation theory (COT).