A mathematical model for the intracellular circadian rhythm generator has been studied, based on a negative feedback of protein products on the transcription rate of their genes. The study is an attempt at examining minimal but biologically realistic requirements for a negative molecular feedback loop involving considerably faster reactions, to produce (slow) circadian oscillations. The model included mRNA and protein production and degradation, along with a negative feedback of the proteins upon mRNA production. The protein production process was described solely by its total duration and a nonlinear term, whereas also the feedback included nonlinear interactions among protein molecules. This system was found to produce robust oscillations in protein and mRNA levels over a wide range of parameter values. Oscillations were slow, with periods much longer than the time constants of any of the individual system parameters. Circadian oscillations were obtained for realistic values of the parameters. The system was readily entrainable to external periodic perturbations. Two distinct classes of phase response curves were found, viz. with or without a time domain within the circadian cycle in which external perturbations fail to induce a phase shift ("dead zone"). The delay and nonlinearity in the protein production and the cooperativity in the negative feedback (Hill coefficient) were for this model found to be necessary and sufficient to generate robust circadian oscillations. The similarities between model outcomes and empirical findings establish that circadian rhythmicity at the cellular level can plausibly emerge from interactions among molecular systems which are not in themselves rhythmic.

A fundamental question in the field of circadian rhythms concerns the biochemical and molecular nature of the oscillator. There is strong evidence that circadian oscillators are cell autonomous and rely on periodic gene expression. In Drosophila, Neurospora, Aplysia, and vertebrates, circadian oscillators are thought to be based on molecular autoregulatory loops composed of transcription, translation, and negative feedback by proteins on nuclear transcription. By studying a mathematical model of molecular clocks based on this general concept, the authors sought to determine which features such clocks must have to generate robust and stable oscillations and to allow entrainment by external stimuli such as light. The model produced circadian oscillations as an emergent property even though a time delay in protein synthesis and rate constants of the feedback loop were much shorter than 24 h. Along with the delay in protein production, strong nonlinear interactions in macromolecular synthesis and nuclear feedback appeared to be required for the model to show well-behaved oscillatory behavior. Realistic phase-shifting patterns induced by external stimuli could be achieved by multiple mechanisms-namely, up- and downward perturbations of protein or mRNA synthesis or degradation rates. The model makes testable predictions about interactions between clock elements and mechanisms of entrainment and may help to understand the functions of the intricate molecular interactions governing circadian rhythmogenesis.