Multistationarity in triple-site mixed mechanism phosphorylation network

In this work, we investigate the existence of multistationarity for a triple-site mixed phosphorylation network, where the phosphorylation part contains distributive and processive components, while the dephosphorylation part is purely distributive. We obtain a simple inequality which defines a region in parameter space such that the parametric ordinary differential equations (ODE) system modeling the mixed network is multistationary, i.e., it has multiple positive steady states. We obtain a sufficient condition for uniqueness of the steady state in the form of parametric inequalities. Lastly, we show that the emergence of multistationarity is enabled by the catalytic constants regardless of the position of the processive part in the triple-site mixed mechanism phosphorylation network.