Construction and Bifurcation Analysis in a Mathematical Model for Circadian Oscillations of Clock Genes

在几乎所有生物体中，以24小时为周期的昼夜节律振荡都是明显的生物节律。在这项研究中，我们研究了在果蝇昼夜节律振荡器模型中产生的各种分岔现象。众所周知，在果蝇中，PER和TIM两种蛋白质水平的昼夜节律振荡是由PER-TIM复合物对编码这两种蛋白质的PER和TIM基因的表达施加的负反馈引起的。为了研究果蝇蛋白质的昼夜节律振荡，提出了一个数学模型。该模型不仅可以解释环境条件下的规律昼夜节律振荡，如持续的黑暗，还可以引起更复杂的振荡现象，包括混乱和出生节律。通过使用Kawakami的方法计算分岔，我们获得了与稳定和不稳定不变集相关的详细分岔图，并确定了模型产生复杂振荡和规则昼夜振荡的参数区域。此外，我们研究了在模型中观察到的分岔，并考虑了对光-暗（LD）周期的影响，并表明光诱导参数的周期变化波形对全局分岔结构或由LD周期引起的昼夜节律振荡器强迫项的动态行为类型有显著影响。

Circadian oscillations with a period of about 24 h are observed in nearly all living organisms as conspicuous biological rhythms. In this research, we investigate various kinds of bifurcation phenomena produced in a circadian oscillator model of Drosophila. In Drosophila, it is known that circadian oscillations in the levels of two proteins, PER and TIM, result from the negative feedback exerted by a PER-TIM complex on the expression of the per and tim genes that code for the two proteins. For studying circadian oscillations of proteins in Drosophila, a mathematical model has been proposed. The model cannot only account for regular circadian oscillations in environmental conditions such as constant darkness, but also give rise to more complex oscillatory phenomena including chaos and birhythmicity. By calculating bifurcations using Kawakami's method, we obtain detailed bifurcation diagrams related to stable and unstable invariant sets, and identify parameter regions in which the model generates complex oscillations as well as regular circadian oscillations. Moreover, we study bifurcations observed in the model incorporating the effect on a light-dark (LD) cycle and show that the waveform of the periodic variation in the light-induced parameter has a marked influence on the global bifurcation structure or the type of dynamic behavior resulting from the forcing term of the circadian oscillator by the LD cycles.