Network Reduction and Absence of Hopf Bifurcations in Dual Phosphorylation Networks With Three Intermediates

ABSTRACT We show that the removal of a type of reverse reactions preserves the absence of Hopf bifurcations in mass‐action reaction networks, or more precisely, the absence of single pairs of purely imaginary eigenvalues of the Jacobian of the system. The motivation is to answer the open question of whether the dual phosphorylation mechanism requiring two encounters of the enzyme and the substrate for both phosphorylation and dephosphorylation admits periodic trajectories. With the tools of model reduction, we provide evidence in favor of the absence of oscillations of this network by precluding Hopf bifurcations in any reduced network comprising three out of its four intermediate protein complexes. Our argument relies on a detailed analysis of the semi‐algebraic conditions precluding Hopf bifurcations obtained from Hurwitz determinants of the characteristic polynomial of the Jacobian of the reduced network.