Stability analysis of the Michaelis–Menten approximation of a mixed mechanism of a phosphorylation system

• Published in: Mathematical biosciences Vol. 301; pp. 159 - 166
• Main Author: Rao, Shodhan
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01.07.2018
• Subjects: Lyapunov methods; Mass action kinetics; Michaelis–Menten enzyme kinetics; Multisite phosphorylation; Poincaré Bendixson theorem; Steady state approach; Mass action kinetics; Steady state approach; Multisite phosphorylation; Poincaré Bendixson theorem; Michaelis–Menten enzyme kinetics; Lyapunov methods
• ISSN: 0025-5564; 1879-3134
• DOI: 10.1016/j.mbs.2018.05.001

•We carry out a Michaelis-Menten approximation of a mixed mechanism of a n-site phosphorylation system in which the mechanism of phosphorylation is distributive and that of dephosphorylation is processive.•We prove that the resulting chemical reaction network admits a unique equilibrium in every positive stoichiometric compatibility class.•If the Michaelis constants associated with the substrates in the phosphorylation reactions are equal, then the unique equilibrium in every positive stoichiometric compatibility class is shown to be asymptotically stable using Lyapunov method.•If there are only two sites of phosphorylation and dephosphorylation in the mixed mechanism, then the unique equilibrium in every positive stoichiometric compatibility class is shown to be asymptotically stable irrespective of the values of the parameters of the system. In this paper, we consider a mixed mechanism of a n-site phosphorylation system in which the mechanism of phosphorylation is distributive and that of dephosphorylation is processive. It is assumed that the concentrations of the substrates are much higher than those of the enzymes and their intermediate complexes. This assumption enables us to reduce the system using the steady-state approach to a Michaelis–Menten approximation of the system. It is proved that the resulting system of nonlinear ordinary differential equations admits a unique positive equilibrium in every positive stoichiometric compatibility class using the theory of quadratic equations. We then consider two special cases. In the first case, we assume that the Michaelis constants associated with the different substrates in the phosphorylation reactions are equal and construct a Lyapunov function to prove asymptotic stability of the system. In the second case, we assume that there are just two sites of phosphorylation and dephoshorylation and prove that the resulting system is asymptotically stable using Poincare´ Bendixson theorem.