An algebraic characterization of self-generating chemical reaction networks using semigroup models

TL;DR

This paper introduces an algebraic framework using semigroup models to analyze self-generating chemical reaction networks, providing new insights into their structure and dynamics relevant to origin-of-life studies.

Contribution

It develops a semigroup-based algebraic characterization of self-generating chemical networks, including a structure theorem and a tree-based representation method.

Findings

Semigroup models capture the catalytic functions of chemicals.

The lattice of self-generating sets has a proven structure theorem.

Self-generating sets imply the CRS is not nilpotent.

Abstract

The ability of a chemical reaction network to generate itself by catalyzed reactions from constantly present environmental food sources is considered a fundamental property in origin-of-life research. Based on Kaufmann's autocatalytic sets, Hordijk and Steel have constructed the versatile formalism of catalytic reaction systems (CRS) to model and to analyze such self-generating networks, which they named reflexively autocatalytic and food generated (RAF). Previously, it was established that the subsequent and simultaenous catalytic functions of the chemicals of a CRS give rise to an algebraic structure, termed a semigroup model. The semigroup model allows to naturally consider the function of any subset of chemicals on the whole CRS. This gives rise to a generative dynamics by iteratively applying the function of a subset to the externally supplied food set. The fixed point of this dynamics yields the maximal self-generating set of chemicals. Moreover, the lattice of all functionally closed self-generating sets of chemicals is discussed and a structure theorem for this lattice is proven. It is also shown that a CRS which contains self-generating sets of chemicals cannot be nilpotent and thus a useful link to the combinatorial theory of finite semigroups is established. The main technical tool introduced and utilized in this work is the representation of the semigroup elements as decorated rooted trees, allowing to translate the generation of chemicals from a given set of resources into the semigroup language.