Network reduction and absence of Hopf Bifurcations in dual phosphorylation networks with three Intermediates

Elisenda Feliu, Nidhi Kaihnsa
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Abstract

Phosphorylation networks, representing the mechanisms by which proteins are phosphorylated at one or multiple sites, are ubiquitous in cell signalling and display rich dynamics such as unlimited multistability. Dual-site phosphorylation networks are known to exhibit oscillations in the form of periodic trajectories, when phosphorylation and dephosphorylation occurs as a mixed mechanism: phosphorylation of the two sites requires one encounter of the kinase, while dephosphorylation of the two sites requires two encounters with the phosphatase. A still open question is whether a mechanism requiring two encounters for both phosphorylation and dephosphorylation also admits oscillations. In this work we provide evidence in favor of the absence of oscillations of this network by precluding Hopf bifurcations in any reduced network comprising three out of its four intermediate protein complexes. Our argument relies on a novel network reduction step that preserves the absence of Hopf bifurcations, and on a detailed analysis of the semi-algebraic conditions precluding Hopf bifurcations obtained from Hurwitz determinants of the characteristic polynomial of the Jacobian of the system. We conjecture that the removal of certain reverse reactions appearing in Michaelis-Menten-type mechanisms does not have an impact on the presence or absence of Hopf bifurcations. We prove an implication of the conjecture under certain favorable scenarios and support the conjecture with additional example-based evidence.
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具有三个中间体的双重磷酸化网络的网络缩小和霍普夫分岔的缺失
磷酸化网络代表了蛋白质在一个或多个位点被磷酸化的机制,在细胞信号中无处不在,并显示出丰富的动态性,如无限的多稳定性。当磷酸化和去磷酸化以混合机制发生时,已知双位点磷酸化网络会以周期性轨迹的形式表现出振荡:两个位点的磷酸化需要与激酶相遇一次,而两个位点的去磷酸化则需要与磷酸酶相遇两次。一个仍然悬而未决的问题是,磷酸化和去磷酸化都需要两次相遇的机制是否也会导致振荡。在这项工作中,我们通过排除由四个中间蛋白复合物中的三个组成的任何还原网络中的霍普夫分岔,提供了该网络不存在振荡的证据。我们的论证依赖于一个新颖的网络还原步骤,该步骤保留了霍普夫分岔的不存在,还依赖于对从系统的雅各布多项式的特征多项式的胡尔维茨行列式中获得的排除霍普夫分岔的半代数条件的详细分析。我们猜想,Michaelis-Menten 型机制中出现的某些反向反应的去除不会影响霍普夫分岔的存在与否。我们证明了该猜想在某些有利情况下的含义,并通过更多基于实例的证据来支持该猜想。
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