Hopf Bifurcations of Reaction Networks with Zero-One Stoichiometric Coefficients

TL;DR

This paper establishes a lower bound on the network size for Hopf bifurcations in zero-one reaction networks and identifies specific subnetworks capable of oscillations, with applications to biological systems.

Contribution

It proves that zero-one networks with Hopf bifurcations must have at least four species and five reactions, providing a computational tool for identifying such networks.

Findings

Hopf bifurcations require at least four species and five reactions in zero-one networks.

Existence of rank-four subnetworks capable of oscillations in biological networks.

Development of a computational tool to identify zero-one networks with Hopf bifurcations.

Abstract

For the reaction networks with zero-one stoichiometric coefficients (or simply zero-one networks), we prove that if a network admits a Hopf bifurcation, then the rank of the stoichiometric matrix is at least four. As a corollary, we show that if a zero-one network admits a Hopf bifurcation, then it contains at least four species and five reactions. As applications, we show that there exist rank-four subnetworks, which have the capacity for Hopf bifurcations/oscillations, in two biologically significant networks: the MAPK cascades and the ERK network. We provide a computational tool for computing all four-species, five-reaction, zero-one networks that have the capacity for Hopf bifurcations.