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.

翻译：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，在细胞信号传导中普遍存在，并展现出丰富的动力学行为，例如无限制的多稳态性。已知双位点磷酸化网络在磷酸化和去磷酸化以混合机制发生时，会以周期轨迹的形式表现出振荡：两个位点的磷酸化需要与激酶一次相遇，而两个位点的去磷酸化则需要与磷酸酶两次相遇。一个尚未解决的问题是，一个要求磷酸化和去磷酸化都需要两次相遇的机制是否也允许振荡。在这项工作中，我们通过排除任何包含其四个中间蛋白质复合物中三个的约简网络中的霍普夫分岔，为此网络不存在振荡提供了证据。我们的论证依赖于一种新颖的、能保持霍普夫分岔缺失性质的网络约简步骤，以及对从系统雅可比矩阵特征多项式的赫尔维茨行列式获得的、排除霍普夫分岔的半代数条件的详细分析。我们推测，移除米氏型机制中出现的某些逆反应不会对霍普夫分岔的存在与否产生影响。我们在某些有利场景下证明了该猜想的一个蕴含关系，并通过基于额外例子的证据支持该猜想。