Influence of time delay on dynamics of cell cycle

In this work, based on the Hill dynamics and Michaelis-Menten equation, a theoretical model is built to study the influence of time delay on the oscillation dynamics of a cyclin-dependent kinase 1 (CDK1)/ anaphase-promoting complex (APC) system. The theoretical model considers the time delay in the CDK1- polo-like protein kinase (P1K1)-APC pathway. We find that under different time delay conditions, the CDK1, Plk1 and APC exhibit periodic oscillation characteristics over time, indicating cell cycle progression. With the increase of time delay, the oscillation periods and amplitudes of CDK1, Plk1 and APC increase, which indicates that the time delay will change the dynamic characteristics of the system. It implies that during the cell cycle, the status of the CDK1/APC oscillation system will show a long-term correlation with the biochemical reaction time of each component, such as CDK1, Plk1 and APC. This correlation is influenced by its past, and there is a time-delay effect. The additional correction will be made due to time delay. By investigating the time-delay effect in Gaussian white noise environment, we find that in the Gaussian white noise environment, the noise disturbance obviously changes the dynamic characteristics of CDK1 evolution with time. In a low-noise environment, the CDK1/APC system changes the oscillation amplitude or period through self-adjusting time delay, so that the system can restore the stable periodic oscillation, while in a high noise environment, CDK1 exhibits a damped oscillation, indicating that the periodic oscillation dynamics of the CDK1/APC system will be significantly changed by strong noise. In the CDK1/APC system oscillation process, the system adjusts the physiological response through a feedback mechanism. There is a time delay between the perception of the noise effect and the establishment of an appropriate physiological response. By different time delays, the system can adjust complex non-periodic chaotic rhythms with different time delays, and recover to produce a stable periodic physiological process. Owing to the time delay, the CDK1/APC oscillation system changes from the original stable oscillation to a damped oscillation, but the original oscillation mode is difficult to recover. The theoretical results further reveal the time-delay effect in cell cycle processes such as Xenopus embryos, and provide a theoretical basis for designing pathway treatment plans that regulate cell cycle and block tumor transformation.