Kurosawa Gen, Mochizuki Atsushi, Iwasa Yoh
Department of Biology, Kyushu University, Fukuoka 812-8581, Japan.
J Theor Biol. 2002 May 21;216(2):193-208. doi: 10.1006/jtbi.2002.2546.
We study simple models for circadian rhythm, and examine the condition in which the equilibrium is unstable, generating a sustained oscillation. In the models, a clock gene(s) is transcribed to produce mRNAs, which are translated to produce proteins that suppress the transcription of the clock gene(s). First, using a Lyapunov function, we prove under very general conditions that two-variable models cannot generate a stable oscillation, implying that additional structures are needed for the model to generate a sustainable rhythm. By comparing several models of different complexities using the Routh-Hurwitz criteria of stability, we show that a sustained oscillation is more likely to occur if the cell is compartmentalized and the proteins need to be transported from the cytosol to the nucleus, if the proteins have to be modified before entering the nucleus, if the kinetics of transcription inhibition or the transport to the nucleus have cooperativity with a nonlinear dependence on the substrate concentration, or if the products of two clock genes form a heterodimer that suppresses both of their own genes. We discuss the implications of these results.
我们研究了昼夜节律的简单模型，并考察了平衡不稳定从而产生持续振荡的条件。在这些模型中，一个时钟基因转录产生信使核糖核酸（mRNA），mRNA被翻译产生蛋白质，这些蛋白质会抑制时钟基因的转录。首先，利用李雅普诺夫函数，我们在非常一般的条件下证明，双变量模型无法产生稳定振荡，这意味着模型需要额外的结构才能产生可持续的节律。通过使用稳定性的劳斯 - 赫尔维茨准则比较几个不同复杂度的模型，我们表明，如果细胞是分隔的且蛋白质需要从细胞质运输到细胞核，如果蛋白质在进入细胞核之前必须被修饰，如果转录抑制动力学或向细胞核的运输具有协同性且对底物浓度有非线性依赖，或者如果两个时钟基因的产物形成异源二聚体并抑制它们自身的基因，那么更有可能发生持续振荡。我们讨论了这些结果的意义。
