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.
As mentioned earlier, there has been much interest in the dynamics of phosphorylation systems @cite_28 . The mixed-mechanism network fits into the related literature as follows. The mixed network is a dual-site network situated between two extremes: the fully processive dual-site network -- in which the phosphorylation and dephosphorylation mechanisms are both processive -- and the fully distributive dual-site network. One might therefore expect the dynamics of the mixed-mechanism network to straddle those of the two networks. This is indeed the case. As summarized in Table , and reviewed in @cite_28 , fully processive networks are globally convergent to a unique steady state @cite_26 @cite_12 @cite_9 , while mixed-mechanism networks admit oscillations but not bistability @cite_3 , and fully distributive networks admit bistability @cite_16 (and the question of oscillations is open @cite_28 ).
@cite_26 @cite_28 @cite_9 @cite_3 @cite_16 @cite_12
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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. Multisite phosphorylation plays an important role in intracellular signaling. There has been much recent work aimed at understanding the dynamics of such systems when the phosphorylation dephosphorylation mechanism is distributive, that is, when the binding of a substrate and an enzyme molecule results in the addition or removal of a single phosphate group and repeated binding therefore is required for multisite phosphorylation. In particular, such systems admit bistability. Here, we analyze a different class of multisite systems, in which the binding of a substrate and an enzyme molecule results in the addition or removal of phosphate groups at all phosphorylation sites, that is, we consider systems in which the mechanism is processive, rather than distributive. We show that in contrast to distributive systems, processive systems modeled with mass-action kinetics do not admit bistability and, moreover, exhibit rigid dynamics: each invariant set contains a unique equilibrium, which is a global attractor. Additionally, we obtain a monomial parametrization of the steady states. Our proofs rely on a technique of Johnston for using “translated” networks to study systems with “toric steady states,” recently given sign conditions for the injectivity of polynomial maps, and a result from monotone systems theory due to Angeli and Sontag. Multi-site phosphorylation is ubiquitous in cell biology and has been widely studied experimentally and theoretically. The underlying chemical modification mechanisms are typically assumed to be distributive or processive. In this paper, we study the behaviour of mixed mechanisms that can arise either because phosphorylation and dephosphorylation involve different mechanisms or because phosphorylation and or dephosphorylation can occur through a combination of mechanisms. We examine a hierarchy of models to assess chemical information processing through different mixed mechanisms, using simulations, bifurcation analysis and analytical work. We demonstrate how mixed mechanisms can show important and unintuitive differences from pure distributive and processive mechanisms, in some cases resulting in monostable behaviour with simple dose–response behaviour, while in other cases generating new behaviour-like oscillations. Our results also suggest patterns of information processing that are relevant as the number of modification sites increases. Overall, our work creates a framework to examine information processing arising from complexities of multi-site modification mechanisms and their impact on signal transduction. 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.