Hopf Bifurcations of Reaction Networks with Zero-One Stoichiometric Coefficients

For 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 mitogen-activated-protein-kinase cascade and the extracellular signal-regulated kinase network. We provide a computational tool for computing all four-species, five-reaction, zero-one networks that have the capacity for Hopf bifurcations.