On the stability of a class of Michaelis–Menten networks

We present a study of a class of closed Michaelis–Menten network models, which includes models of two categories of biochemical networks previously studied in the literature namely, processive and mixed mechanism phosphorylation futile cycle networks. The main focus of our study is on the uniqueness and stability of equilibrium points of this class of models. Firstly, we demonstrate that the total species concentration is a conserved quantity in models of this class. Next, we prove the existence of a unique positive equilibrium point in the set of points that correspond to a given total species concentration, using the intermediate value property of continuous functions. Finally, we demonstrate the asymptotic stability of this equilibrium point with respect to all initial conditions in the positive orthant that correspond to the same total species concentration as the equilibrium point, by constructing an appropriate Lyapunov function.