Multistability in Double Phosphorylation-Dephosphorylation Cycles

The aprioristic application of the Michaelis-Menten (or standard quasi-steady state) approximation to complex enzyme reaction networks, as prescribed by lots of Systems Biology literature, is the cause of serious incoherences between mathematical models and experimental data, bringing to what we could call the sequestration paradox, or negligible complex paradox. In this paper we study the double phosphorylation-dephosphorylation cycle, which is one of the most important reactions occurring inside a cell. The exact description of this phenomenon is not completely understood by a biochemical point of view and several models exist and need to be tested on experimental data. We therefore consider several models, using for each of them the full system of equations governing the dynamics and comparing our results with the ones that come from the acritical application of the quasi-steady state approximation that neglects the complexes. Moreover, we show that approaching the problem with the total quasi-steady approximation, on one side, and the full system, on the other one, solves the sequestration paradox. In particular we noticeably improve existing results on the system multistability, showing the correct range of the kinase concentrations giving bistability.