Generic algebraic conditions for the occurrence of switch-like behavior of a chemical kinetic system of the hypoxia pathway

Abstract

Weakly reversible chemical reaction networks with zero deficiency associated with mass-action kinetics admit, within each positive stoichiometric compatibility class, one positive steady state which is locally asymptotically stable and this irrespective of the values of the kinetics constants. Networks which do not enjoy these structural properties potentially exhibit more diverse dynamical behaviors. In this article, we consider a chemical reaction network associated with mass-action kinetics which is not weakly reversible and has a deficiency larger than one. The chemical reactions are at most bimolecular, but inflow and outflow reactions are present. Our results are as follows. We establish the existence of positive steady-state solutions and obtain their analytic expressions in the concentration space and in convex coordinates. We show that the system fulfills necessary conditions for a saddle-node and for a bifurcation into a saddle and a node. We apply a constructive approach to obtain a set of numerical values for the state variables and kinetic parameters, not reported previously, such that the reduced Jacobian is characterized by a zero eigenvalue with all other eigenvalues having negative real parts. The bifurcation diagram confirms the presence of the switch-like behavior.