Autocatalytic Networks: An Intimate Relation between Network Topology and Dynamics

Badal Joshi; Gheorghe Craciun

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arXiv · 2006.01384

Autocatalytic Networks: An Intimate Relation between Network Topology and Dynamics

Badal Joshi·Gheorghe Craciun

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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.

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Badal Joshi, Gheorghe Craciun. 2020-06-02. Autocatalytic Networks: An Intimate Relation between Network Topology and Dynamics. https://arxiv.org/abs/2006.01384

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