Extensive experimental evidence has been demonstrated that the dynamics of CDK1-APC feedback loop play crucial roles in regulating cell cycle processes, but the dynamical mechanisms underlying the regulation of this loop are still not completely understood. Here, the authors systematically investigated the stability and bifurcation criteria for a delayed CDK1-APC feedback loop. They showed that the maximum reaction rate of CDK1 inactivation by APC can drive sustained oscillations of CDK1 activity ( ) and APC activity ( ), and the amplitude of these oscillations is increasing with the increase of the reaction rate over a wide range; a certain range of the self-activation rate for CDK1 is also significant for generating these oscillations, for too high or too low rates the oscillations cannot be generated. Moreover, they derived the sufficient conditions to determine the stability and Hopf bifurcations, and found that the sum of time delays required for activating CDK1 and APC can induce and to be oscillatory, even when the and settle in a definite stable steady state. Furthermore, they presented an explicit algorithm for the properties of periodic oscillations. Finally, numerical simulations have been presented to justify the validity of theoretical analysis.

中文翻译：

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延迟 CDK1-APC 反馈环路的分岔和振荡动力学


大量实验证据表明，CDK1-APC反馈环路的动力学在调节细胞周期过程中发挥着至关重要的作用，但该环路调节背后的动力学机制仍不完全清楚。在这里，作者系统地研究了延迟 CDK1-APC 反馈环路的稳定性和分叉标准。他们表明，APC灭活CDK1的最大反应速率可以驱动CDK1活性（ ）和APC活性（ ）持续振荡，并且这些振荡的幅度在很宽的范围内随着反应速率的增加而增加； CDK1的自激活速率在一定范围内对于产生这些振荡也很重要，过高或过低的速率都无法产生振荡。此外，他们推导了确定稳定性和Hopf分岔的充分条件，并发现激活CDK1和APC所需的时间延迟之和可以诱导 和 振荡，即使当 和 稳定在确定的稳定状态时也是如此。此外，他们还提出了一种用于周期性振荡特性的显式算法。最后，通过数值模拟证明了理论分析的有效性。