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The Journal of Neuroscience, January 1, 1999, 19(1):40-47

A Mathematical Model for the Intracellular Circadian Rhythm Generator

Tjeerd olde Scheper¹, Don Klinkenberg², Cyriel Pennartz², and Jaap van Pelt²

¹ Oxford Brookes University, School for Computing and Math Science, Gipsy Lane Campus, OX3 0BP Headington Oxford, United Kingdom, and ² Graduate School Neurosciences Amsterdam, Netherlands Institute for Brain Research, 1105 AZ Amsterdam, The Netherlands

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.

Key words: SCN; circadian rhythm; molecular clock; entrainment; phase-response curves; models

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Copyright © 1999 Society for Neuroscience  0270-6474/99/19140-08$05.00/0

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