Adding species to chemical reaction networks: preserving rank preserves nondegenerate behaviours

TL;DR

This paper proves that adding species to a chemical reaction network without changing its rank preserves its ability to exhibit complex behaviors like multiple equilibria and oscillations, impacting biochemical modeling.

Contribution

It generalizes previous theorems by showing rank-preserving species additions maintain nondegenerate dynamics in CRNs.

Findings

Adding species with unchanged rank preserves multistationarity.

Enlargements can introduce new periodic orbits and bifurcations.

The theorem helps predict complex behaviors in extended CRNs.

Abstract

We show that adding new chemical species into the reactions of a chemical reaction network (CRN) in such a way that the rank of the network remains unchanged preserves its capacity for multiple nondegenerate equilibria and/or periodic orbits. One consequence is that any bounded nondegenerate behaviours which can occur in a CRN can occur in a CRN with bounded stoichiometric classes. The main result adds to a family of theorems which tell us which enlargements of a CRN preserve its capacity for nontrivial dynamical behaviours. It generalises some earlier claims, and complements similar claims involving the addition of reactions into CRNs. The result gives us information on how ignoring some chemical species, as is common in biochemical modelling, might affect the allowed dynamics in differential equation models of CRNs. We demonstrate the scope and limitations of the main theorem via several examples. These illustrate how we can use the main theorem to predict multistationarity and oscillation in CRNs enlarged with additional species; but also how the enlargements can introduce new behaviours such as additional periodic orbits and new bifurcations.