We investigate oscillations in coupled systems. The methodology is based on the Hopf bifurcation theorem and a condition extended from the Routh–Hurwitz criterion. Such a condition leads to locating the bifurcation values of the parameters. With such an approach, we analyze a single-cell system modeling the minimal genetic negative feedback loop and the coupled-cell system composed by these single-cell systems. We study the oscillatory properties for these systems and compare these properties between the model with Hill-type repression and the one with protein-sequestration-based repression. As the parameters move from the Hopf bifurcation value for single cells to the one for coupled cells, we compute the eigenvalues of the linearized systems to obtain the magnitude of the collective frequency when the periodic solution of the coupled-cell system is generated. Extending from this information on the parameter values, we further compute and compare the collective frequency for the coupled-cell system and the average frequency of the decoupled individual cells. To compare these scenarios with other biological oscillators, we perform parallel analysis and computations on a segmentation clock model.

我们研究耦合系统中的振荡。该方法基于 Hopf 分岔定理和从 Routh-Hurwitz 准则扩展的条件。这种条件导致定位参数的分叉值。通过这种方法，我们分析了模拟最小遗传负反馈回路的单细胞系统和由这些单细胞系统组成的耦合细胞系统。我们研究了这些系统的振荡特性，并比较了具有希尔型抑制的模型和具有基于蛋白质螯合的抑制的模型之间的这些特性。随着参数从单个细胞的 Hopf 分叉值移动到耦合细胞的 Hopf 分岔值，当耦合单元系统的周期解产生时，我们计算线性化系统的特征值以获得集体频率的大小。根据参数值的这些信息，我们进一步计算和比较耦合单元系统的集体频率和解耦单个单元的平均频率。为了将这些场景与其他生物振荡器进行比较，我们对分段时钟模型进行了并行分析和计算。