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For the reaction networks with zero-one stoichiometric coefficients (orsimply zero-one networks), we prove that if a network admits a Hopfbifurcation, then the rank of the stoichiometric matrix is at least four. As acorollary, we show that if a zero-one network admits a Hopf bifurcation, thenit contains at least four species and five reactions. As applications, we showthat there exist rank-four subnetworks, which have the capacity for Hopfbifurcations/oscillations, in two biologically significant networks: the MAPKcascades and the ERK network. We provide a computational tool for computing allfour-species, five-reaction, zero-one networks that have the capacity for Hopfbifurcations.

Hopf Bifurcations of Reaction Networks with Zero-One Stoichiometric Coefficients