Autocatalytic Networks: An Intimate Relation between Network Topology and Dynamics

TL;DR

This paper explores how the structure of autocatalytic networks, called hyperchains, influences their dynamical behavior, with implications for understanding the origin of life and evolutionary processes.

Contribution

It establishes a direct link between network topology and the dynamical properties of hyperchains, generalizing previous models like hypercycles.

Findings

Network topology determines existence and stability of equilibria.

Certain graph properties predict system permanence.

Hyperchains relate to models in evolution and ecology.

Abstract

We study a family of networks of autocatalytic reactions, which we call hyperchains, that are a generalization of hypercycles. Hyperchains, and the associated dynamical system called replicator equations, are a possible mechanism for macromolecular evolution and proposed to play a role in abiogenesis, the origin of life from prebiotic chemistry. The same dynamical system also occurs in evolutionary game dynamics, genetic selection, and as Lotka-Volterra equations of ecology. An arrow in a hyperchain encapsulates the enzymatic influence of one species on the autocatalytic replication of another. We show that the network topology of a hyperchain, which captures all such enzymatic influences, is intimately related to the dynamical properties of the mass action system it generates. Dynamical properties such as existence, uniqueness and stability of a positive equilibrium as well as permanence, are determined by graph-theoretic properties such as existence of a spanning linear subgraph, being unrooted, being cyclic, and Hamiltonicity.