 | Theoretical Study for Circadian Rhythm in Drosophila: Condition for Generating A limit Cycle
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| | Gen Kurosawa1), Atsushi Mochizuki1) and Yoh Iwasa1) |
| | 1) Department of Biology, Faculty of Science, Kyushu University |
| | It has been shown that PER and TIM play a crucial role in the gene network to generate the circadian rhythm of Drosophila. They make heterodimers and inhibit the transcriptions of their own in nucleus. Leloup(1998) proposed several mathematical models for this phenomenon, and analysed them by numerical simulations. He claimed that the model in which PER-TIM complex regulates the transcription of both of their genes is easier to generate a stable oscillation than the model in which only PER does feedback control, as the parameter region for oscillation is larger in the former model than in the latter. In this paper, we present 3 simple models and compare the condition for generating a limit cycle. We introduce a parameter “cooperativity” indicating the degree of non-linearity in the gene transcription term. We found that a larger cooperativity generally tends to lose the local stability of equilibrium point and generate a limit cycle in all of the three models. Compared with “three variables model” (per mRNA, PER in cytosol, and PER in nucleus), “four variables model” (PER is modified before entering the nucleus) can generate a stable oscillation with a smaller cooperativity. Compared with these, our “PER-TIM model” can also generate oscillation with a smaller cooperativity. They show that the modification and heterodimerization of proteins are inportant to generate a limit cycle. |
| | Key Words: | circadian rhythm, limit cycle, PER, TIM |
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