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Oscillations play a major role in a number of biological systems, frompredator-prey models of ecology to circadian clocks. In this paper we focus onthe question of whether oscillations exist within dual-site phosphorylationsystems. Previously, Wang and Sontag showed, using monotone systems theory,that the Michaelis-Menten (MM) approximation of the distributive and sequentialdual-site phosphorylation system lacks oscillations. However, biologicalsystems are generally not purely distributive; there is generally someprocessive behavior as well. Accordingly, this paper focuses on the MMapproximation of a general sequential dual-site phosphorylation system thatcontains both processive and distributive components, termed the compositesystem. Expanding on the methods of Bozeman and Morales, we precludeoscillations in the MM approximation of the composite system. This implies thelack of oscillations in the MM approximations of the processive anddistributive systems, shown previously, as well as in the MM approximation ofthe partially processive and partially distributive mixed-mechanism system.

Precluding Oscillations in Michaelis-Menten Approximations of Dual-site Phosphorylation Systems