text: This work investigates the emergence of oscillations in one of the simplest cellular signaling networks exhibiting oscillations, namely, the dual-site phosphorylation and dephosphorylation network (futile cycle), in which the mechanism for phosphorylation is processive while the one for dephosphorylation is distributive (or vice-versa). The fact that this network yields oscillations was shown recently by Suwanmajo and Krishnan. Our results, which significantly extend their analyses, are as follows. First, in the three-dimensional space of total amounts, the border between systems with a stable versus unstable steady state is a surface defined by the vanishing of a single Hurwitz determinant. Second, this surface consists generically of simple Hopf bifurcations. Next, simulations suggest that when the steady state is unstable, oscillations are the norm. Finally, the emergence of oscillations via a Hopf bifurcation is enabled by the catalytic and association constants of the distributive part of the mechanism: if these rate constants satisfy two inequalities, then the system generically admits a Hopf bifurcation. Our proofs are enabled by the Routh-Hurwitz criterion, a Hopf-bifurcation criterion due to Yang, and a monomial parametrization of steady states.
  @cite_30: Abstract 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 Poincar e ´ Bendixson theorem.
  @cite_25: 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.
  @cite_24: Multisite phosphorylation is an important mechanism for fine-tuned regulation of protein function. Mathematical models developed over recent years have contributed to elucidation of the functional consequences of a variety of molecular mechanisms involved in processing of the phosphorylation sites. Here we review the results of such models, together with salient experimental findings on multisite protein phosphorylation. We discuss how molecular mechanisms that can be distinguished with respect to the order and processivity of phosphorylation, as well as other factors, regulate changes in the sensitivity and kinetics of the response, the synchronization of molecular events, signalling specificity, and other functional implications.
summary: Finally, we revisit Suwanmajo and Krishnan's claim mentioned earlier that the mixed-mechanism network is among the simplest enzymatic mechanisms with oscillations. In support of this claim, Tung proved that the simpler system obtained from the mixed-mechanism network by taking its (two-dimensional) Michaelis-Menten approximation, is not oscillatory @cite_25 . Moreover, Rao showed that this approximation is globally convergent to a unique steady state @cite_30 . The validity of the Michaelis-Menten approximation for phosphorylation systems has been called into question @cite_24 , and what we know about the mixed-mechanism system concurs: this system is oscillatory, but its Michaelis-Menten approximation is not.