Combinatorics of polymer models of early metabolism

TL;DR

This paper provides exact and asymptotic counts of reactions in polymer models of early metabolism, including a novel rotational invariance assumption, enhancing understanding of prebiotic biochemical network formation.

Contribution

It offers the first exact enumeration of reactions in polymer models and introduces a new rotational invariance assumption with corresponding counts.

Findings

Exact reaction counts under common assumptions

Asymptotic counts closely match previous approximations

Rotational invariance model yields new enumeration results

Abstract

Polymer models are a widely used tool to study the prebiotic formation of metabolism at the origins of life. Counts of the number of reactions in these models are often crucial in probabilistic arguments concerning the emergence of autocatalytic networks. In the first part of this paper, we provide the first exact description of the number of reactions under widely applied model assumptions. Conclusions from earlier studies rely on either approximations or asymptotic counting, and we show that the exact counts lead to similar, though not always identical, asymptotic results. In the second part of the paper, we investigate a novel model assumption whereby polymers are invariant under spatial rotation. We outline the biochemical relevance of this condition and again give exact enumerative and asymptotic formulae for the number of reactions.