  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 preclude
oscillations in the MM approximation of the composite system. This implies the
lack of 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.
