Hopf Bifurcations of Reaction Networks with Zero-One Stoichiometric Coefficients

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.

Absolute concentration robustness and multistationarity in reaction networks: Conditions for coexistence

“…Our works fit into a growing body of literature that explores the minimal conditions needed for various dynamical behaviours, including the two properties that are the focus of the current work: multistationarity [20,22,28] and ACR [23,24]. There are additional such studies on multistability [29] and Hopf bifurcations [5,6,30,31,34] (which generate periodic orbits). For instance, in analogy to Theorem 1.2 above, the presence of Hopf bifurcations requires an at-most-bimolecular network to have at least three species, four reactions, and dimension 3 [5,34].…”

“…We call this the stoichiometric subspace and denote it by S. The dimension of a network is the dimension of its stoichiometric subspace. (This dimension is sometimes called the "rank" [4,29].) In particular, if dim(S) = n (that is, S = R n ), we say that G is full-dimensional.…”