Theoretical investigation on models of circadian rhythms based on dimerization and proteolysis of PER and TIM

Circadian rhythms of physiology and behavior are widespread\break mechanisms in many organisms. The internal biological rhythms are driven by molecular clocks, which oscillate with a period nearly but not exactly 24 hours. Many classic models of circadian rhythms are based on a time-delayed negative feedback, suggested by the protein products inhibiting transcription of their own genes. In 1999, based on stabilization of PER upon dimerization, Tyson et al. [J. J. Tyson, C. I. Hong, C. D. Thron, B. Novak, Biophys. J. 77 (1999) 2411--2417] proposed a crucial positive feedback to the circadian oscillator. This idea was mathematically expressed in a three-dimensional model. By imposing assumptions that the dimerization reactions were fast and dimeric proteins were in rapid equilibrium, they reduced the model to a pair of nonlinear ordinary differential equations of mRNA and total protein concentrations. Then they used phase plane analysis tools to investigate circadian rhythms. In this paper, the original three-dimensional model is studied. We explore the existence of oscillations and their periods. Much attention is paid to investigate how the periods depend on model parameters. The numerical simulations are in good agreement with their reduced work.