ZIP调节系统的全局Hopf分支
Global Hopf Bifurcation in the ZIP regulatory system
摘要：
拟南芥根系锌吸收的调控最近被建立在一个常微分方程的基础上，该方程包括锌的吸收、转运蛋白的表达以及激活剂和抑制剂之间的相互作用。对于某些参数的选择，该模型的稳态变得不稳定时，在外部锌浓度的变化。数值结果表明，周期轨道出现在两个临界值的外部锌浓度。在这里，我们证明了存在一个全局的Hopf分支与一个连续的家庭稳定的周期轨道之间的两个Hopf歧点。通过导出规范形分析了轨道在歧点邻域内的稳定性，通过计算Floquet乘子分析了轨道在全局延拓中的稳定性.从生物学的角度来看，稳定的周期轨道导致植物细胞中潜在的有毒锌峰。缓冲被认为是处理锌供应中的强瞬态变化的有效方式。我们通过缓冲反应扩展了该模型，并分析了该反应的性质依赖于稳态的稳定性。我们发现缓冲反应的平衡常数足够大可以稳定稳态并防止振荡的发展。因此，我们的研究结果表明，缓冲在植物细胞锌稳态的动态中具有关键作用。
英文摘要：
Regulation of zinc uptake in roots of Arabidopsis thaliana has recently been modeled by a system of ordinary differential equations based on the uptake of zinc, expression of a transporter protein and the interaction between an activator and inhibitor. For certain parameter choices the steady state of this model becomes unstable upon variation in the external zinc concentration. Numerical results show periodic orbits emerging between two critical values of the external zinc concentration. Here we show the existence of a global Hopf bifurcation with a continuous family of stable periodic orbits between two Hopf bifurcation points. The stability of the orbits in a neighborhood of the bifurcation points is analyzed by deriving the normal form, while the stability of the orbits in the global continuation is shown by calculation of the Floquet multipliers. From a biological point of view, stable periodic orbits lead to potentially toxic zinc peaks in plant cells. Buffering is believed to be an efficient way to deal with strong transient variations in zinc supply. We extend the model by a buffer reaction and analyze the stability of the steady state in dependence of the properties of this reaction. We find that a large enough equilibrium constant of the buffering reaction stabilizes the steady state and prevents the development of oscillations. Hence, our results suggest that buffering has a key role in the dynamics of zinc homeostasis in plant cells.
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