Saddle-node bifurcations in biochemical reaction networks with mass action kinetics and application to a double-phosphorylation mechanism

In Systems Biology quantitative models in form of parameter--dependent Ordinary Differential Equations are encountered frequently, when the dynamical behaviour of a biochemical reaction network is described. The question whether or not a given network topology is able to exhibit bistability or some other form of multistationarity is of particular interest. We show that for certain network structures it is possible to determine analytically 'critical states' and 'critical parameters' where the necessary conditions for a saddle--node bifurcation of codimension 1 are satisfied. Moreover, we derive sufficient conditions for these saddle--node bifurcations to occur. For the network related to the double-phosphorylation of an enzyme, we give a parametrization of all critical states and parameters and show that bistability is possible