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Register a new userOn the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle
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Protein phosphorylation cycles are important mechanisms of the post translational modification of a protein and as such an integral part of intracellular signaling and control. We consider the sequential phosphorylation and dephosphorylation of a protein at two binding sites. While it is known that proteins where phosphorylation is processive and dephosphorylation is distributive admit oscillations (for some value of the rate constants and total concentrations) it is not known whether or not this is the case if both phosphorylation and dephosphorylation are distributive. We study four simplified mass action models of sequential and distributive phosphorylation and show that for each of those there do not exist rate constants and total concentrations where a Hopf bifurcation occurs. To arrive at this result we use convex parameters to parameterize the steady state and Hurwitz matrices.
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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.
The lactose operon in Escherichia coli was the first known gene regulatory network, and it is frequently used as a prototype for new modeling paradigms. Historically, many of these modeling frameworks use differential equations. More recently, Stigler and Veliz-Cuba proposed a Boolean network model that captures the bistability of the system and all of the biological steady states. In this paper, we model the well-known arabinose operon in E. coli with a Boolean network. This has several complex features not found in the lac operon, such as a protein that is both an activator and repressor, a DNA looping mechanism for gene repression, and the lack of inducer exclusion by glucose. For 11 out of 12 choices of initial conditions, we use computational algebra and Sage to verify that the state space contains a single fixed point that correctly matches the biology. The final initial condition, medium levels of arabinose and no glucose, successfully predicts the systems bistability. Finally, we compare the state space under synchronous and asynchronous update, and see that the former has several artificial cycles that go away under a general asynchronous update.
The living cell is an open nonequilibrium biochemical system, where ATP hydrolysis serves as the energy source for a wide range of intracellular processes including the assurance for decision-making. In the fission yeast cell cycle, the transition from G2 phase to M phase is triggered by the activation of Cdc13/Cdc2 and Cdc25, and the deactivation of Wee1. Each of these three events involves a phosphorylation-dephosphorylation (PdP) cycle, and together they form a regulatory circuit with feedback loops. Almost all quantitative models for cellular networks in the past have invalid thermodynamics due to the assumption of irreversible enzyme kinetics. We constructed a thermodynamically realistic kinetic model of the G2/M circuit, and show that the phosphorylation energy ($Delta G$), which is determined by the cellular ATP/ADP ratio, critically controls the dynamics and the bistable nature of Cdc2 activation. Using fission yeast nucleoplasmic extract (YNPE), we are able to experimentally verify our model prediction that increased , being synergistic to the accumulation of Cdc13, drives the activation of Cdc2. Furthermore, Cdc2 activation exhibits bistability and hysteresis in response to changes in phosphorylation energy. These findings suggest that adequate maintenance of phosphorylation energy ensures the bistability and robustness of the activation of Cdc2 in the G2/M transition. Free energy might play a widespread role in biological decision-making processes, connecting thermodynamics with information processing in biology.
We apply tools from real algebraic geometry to the problem of multistationarity of chemical reaction networks. A particular focus is on the case of reaction networks whose steady states admit a monomial parametrization. For such systems we show that in the space of total concentrations multistationarity is scale invariant: if there is multistationarity for some value of the total concentrations, then there is multistationarity on the entire ray containing this value (possibly for different rate constants) -- and vice versa. Moreover, for these networks it is possible to decide about multistationarity independent of the rate constants by formulating semi-algebraic conditions that involve only concentration variables. These conditions can easily be extended to include total concentrations. Hence quantifier elimination may give new insights into multistationarity regions in the space of total concentrations. To demonstrate this, we show that for the distributive phosphorylation of a protein at two binding sites multistationarity is only possible if the total concentration of the substrate is larger than either the total concentration of the kinase or the total concentration of the phosphatase. This result is enabled by the chamber decomposition of the space of total concentrations from polyhedral geometry. Together with the corresponding sufficiency result of Bihan et al. this yields a characterization of multistationarity up to lower dimensional regions.
Fission yeast G2/M transition is regulated by a biochemical reaction networks which contains four components: Cdc13, Cdc2, Wee1, and Cdc25. This circuit is characterized by the ultrasensitive responses of Wee1 or Cdc25 to Cdc13/Cdc2 activity, and the bistability of Cdc2 activation. Previous work has shown that this bistability is governed by phosphorylation energy. In this article, we developed the kinetic model of this circuit and conducted further thermodynamic analysis on the role of phosphorylation energy (&[Delta]G). We showed that level &[Delta]G shapes the response curves of Wee1 or Cdc25 to Cdc2 and governs the intrinsic noise level of Cdc2 activity. More importantly, the mutually antagonistic chemical fluxes around the PdP cycles in G2/M circuit were shown to act as a stabilizer of Cdc2 activity against &[Delta]G fluctuations. These results suggests the fundamental role of free energy and chemical fluxes on the sensitivity, bistability and robustness of G2/M transition.
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