Mathematical modeling of circadian rhythms

Among biological rhythms, those with a circadian (close to 24h) period are conspicuous by their ubiquity and by the key role they play in allowing organisms to adapt to their periodically changing environment. Thanks to genetic and biochemical advances on the molecular bases of circadian rhythms in a variety of organisms, mathematical models closely related to experimental observations can be considered for the regulatory mechanisms of circadian clocks. In the best-studied organism Drosophila, circadian rhythms originate from the autoregulatory negative feedback exerted by a complex between the PER and TIM proteins on the expression of their genes. A model of this genetic regulatory network predicts the occurrence of sustained circadian oscillations in continuous darkness. Numerical integration of the differential equations governing the time evolution of the model show that the oscillations correspond to the evolution toward a limit cycle. When incorporating the effect of light, the model accounts for phase shifting of the rhythm by light pulses and for entrainment by light-dark cycles. The model can also explain the long-term suppression of circadian rhythms by a single pulse of light, and predicts the occurrence of autonomous chaotic behavior. Stochastic simulations show how circadian oscillations are affected by molecular noise. An extension of the model to circadian rhythms in mammals permits an investigation of the dynamical bases of certain physiological disorders of the sleep-wake cycle in humans.