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

Oscillations play a major role in a number of biological systems, from predator-prey models of ecology to circadian clocks. In this paper we focus on the question of whether oscillations exist within dual-site phosphorylation systems. Previously, Wang and Sontag showed, using monotone systems theory, that the Michaelis-Menten (MM) approximation of the distributive and sequential dual-site phosphorylation system lacks oscillations. However, biological systems are generally not purely distributive; there is generally some processive behavior as well. Accordingly, this paper focuses on the MM approximation of a general sequential dual-site phosphorylation system that contains both processive and distributive components, termed the composite system. Expanding on the methods of Bozeman and Morales, we find a necessary condition for the existence of oscillations in the MM approximation of the composite system. This condition precludes oscillations in the MM approximations of the processive and distributive systems, shown previously, as well as in the MM approximation of the partially processive and partially distributive mixed-mechanism system.