Graduate Program in Genetics, Bioinformatics and Computational Biology, Virginia Tech, Blacksburg, Virginia, United States of America.
Division of Systems Biology, Virginia Tech, Blacksburg, Virginia, United States of America.
PLoS Comput Biol. 2022 Mar 18;18(3):e1008340. doi: 10.1371/journal.pcbi.1008340. eCollection 2022 Mar.
Circadian rhythms in a wide range of organisms are mediated by molecular mechanisms based on transcription-translation feedback. In this paper, we use bifurcation theory to explore mathematical models of genetic oscillators, based on Kim & Forger's interpretation of the circadian clock in mammals. At the core of their models is a negative feedback loop whereby PER proteins (PER1 and PER2) bind to and inhibit their transcriptional activator, BMAL1. For oscillations to occur, the dissociation constant of the PER:BMAL1 complex, [Formula: see text], must be ≤ 0.04 nM, which is orders of magnitude smaller than a reasonable expectation of 1-10 nM for this protein complex. We relax this constraint by two modifications to Kim & Forger's 'single negative feedback' (SNF) model: first, by introducing a multistep reaction chain for posttranscriptional modifications of Per mRNA and posttranslational phosphorylations of PER, and second, by replacing the first-order rate law for degradation of PER in the nucleus by a Michaelis-Menten rate law. These modifications increase the maximum allowable [Formula: see text] to 2 nM. In a third modification, we consider an alternative rate law for gene transcription to resolve an unrealistically large rate of Per2 transcription at very low concentrations of BMAL1. Additionally, we studied extensions of the SNF model to include a second negative feedback loop (involving REV-ERB) and a supplementary positive feedback loop (involving ROR). Contrary to Kim & Forger's observations of these extended models, we find that, with our modifications, the supplementary positive feedback loop makes the oscillations more robust than observed in the models with one or two negative feedback loops. However, all three models are similarly robust when accounting for circadian rhythms (24 h period) with [Formula: see text] ≥ 1 nM. Our results provide testable predictions for future experimental studies.
生物体内广泛的生物钟节律是由基于转录-翻译反馈的分子机制介导的。在本文中，我们使用分岔理论来探索基于 Kim 和 Forger 对哺乳动物生物钟解释的遗传振荡器的数学模型。在他们的模型核心是一个负反馈环，其中 PER 蛋白（PER1 和 PER2）与转录激活因子 BMAL1 结合并抑制其活性。为了使振荡发生，PER:BMAL1 复合物的离解常数[Formula: see text]必须≤0.04 nM，这比该蛋白复合物 1-10 nM 的合理预期小几个数量级。我们通过对 Kim 和 Forger 的“单负反馈”（SNF）模型的两个修改来放宽这个约束：首先，通过引入多步反应链来进行 Per mRNA 的转录后修饰和 PER 的翻译后磷酸化，其次，通过用米氏-门坦速率定律代替核内 PER 降解的一级速率定律。这些修改将最大允许的[Formula: see text]增加到~2 nM。在第三个修改中，我们考虑了一种替代的基因转录速率定律，以解决在 BMAL1 浓度非常低时 Per2 转录率过高的问题。此外，我们还研究了 SNF 模型的扩展，包括第二个负反馈环（涉及 REV-ERB）和一个补充的正反馈环（涉及 ROR）。与 Kim 和 Forger 对这些扩展模型的观察结果相反，我们发现，通过我们的修改，补充的正反馈环使振荡比在具有一个或两个负反馈环的模型中观察到的更稳健。然而，当考虑到具有[Formula: see text]≥1 nM 的 circadian rhythms 时，所有三个模型都具有类似的稳健性。我们的结果为未来的实验研究提供了可测试的预测。
