Numerical Studies On The Effect Of Time Delays On The Circadian Clock

Almost all living organisms display circadian rhythm in physiology and behavior which is called circadian clocks.In mammals,the main clock is located in the suprachiasmatic nucleus?SCN?,whose generation is based on the coordination expression of the core clock genes through the transcription-translation feedback loop.Disruption to circadian rhythms usually can lead to dysfunction even disease.Because the molecular mechanism of the circadian rhythm is not clearly understood,there is no effective treatment for these diseases.Therefore,it is a hot topic to study the regulation mechanism of circadian rhythm on the molecular level.With the development of biology,it is possible to study the mechanism of biological rhythm from the perspective of system biology.Many clock genes and proteins have been found.The clock gene regulation network?such as transcription-translation feedback loop?plays a key role.On the other hand,mathematical model has unique advantages in the integration of the experimental data and analysis of the dynamics of gene regulatory networks.A good mathematical model can not only help us understand the complex structure of the biological clock,but also predict new behaviors for further verification.This article mainly studied the influence of time delays on the mammals'clock.The time delays express the time of the modification of clock gene after translation.The two key factors used to describe the clock oscillator are system period and amplitude,so we mainly analyze the effects of these time delays on the period and amplitude.Our aim is to find the key time delays that affect the period and amplitude,which provids a theoretical basis for the regulation of the biological clock.In addition to the negative primary feedback loop in circadian oscillator,there are also some auxiliary loops which play essential roles in regulating the period and maintain its robustness.In this paper,the main research content is divided into two parts.On one hand,we studied the effect of time delays of the primary feedback loop in a detailed model on the circadian clock in mammals.One the other hand,we studied the effect of time delays in an auxiliary loop on the circadian clock in mammals.In the first part,with the help of numerical simulation,we studied the effect of time delays in a detailed circadian clock model.The results show that the period of the system is most sensitive to the time delay?₂and?₄,which express the time of transcription and translation modification of gene Per1 and Cry1 respectively.The amplitude of the system is most sensitive to the time delay?₂and?₆,which express the time of transcription and translation modification of gene Per1 and Rev-erb respectively.So?₂and?₄ are the key factors to influence the period,and?₂and?₆ are the key factors to influence the amplitude.The numerical results can provide theoretical guidance for the regulation of biological clock.In biological experiments,one can regulate the period of biological clock by changing the speed of the modification of gene Per1 and Cry1 after translation.Similarly,one can regulate the amplitude of biological clock by changing the speed of the modification of gene Per1 and Rev-erb after translation.In the second part,we studied the effect of time delays in the BMAL1/REV-ERB-?auxiliary loop.With delay differential equations,we gave a mathematical model of this loop and investigated its dynamical mechanisms.Specially,we theoretically provided the sufficient conditions for sustained oscillation of the loop with Hopf bifurcation theory.The total of delays determines the emergence of oscillators,which explains the crucial roles of delays in circadian clock revealed by biological experiments.Numerically,we studied the amplitude and period against the variations of delays and the degradation rates.The different sensitivities of amplitude and period on these factors provide ideas to adjust the amplitude or period of circadian oscillators.The research results of this paper will help us to further understand the production and the regulatory mechanism of biological clock.The analysis method can also be used to analyze other more complex biological clock model or analysis the role of time delays on other genetic regulatory networks which controls the basic life processes.