Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, China.
Department of mathematics, Yunnan Normal University, Kunming, 650092, China.
Sci Rep. 2017 Sep 13;7(1):11495. doi: 10.1038/s41598-017-11149-5.
Recent experimental evidence reports that oscillations of p38 MAPK (p38) activity would efficiently induce pro-inflammatory gene expression, which might be deleterious to immune systems and may even cause cellular damage and apoptosis. It is widely accepted now that transcriptional and translational delays are ubiquitous in gene expression, which can typically result in oscillatory responses of gene regulations. Consequently, delay-driven sustained oscillations in p38 activity (p38*) could in principle be commonplace. Nevertheless, so far the studies of the impact of such delays on p38* have been lacking both experimentally and theoretically. Here, we use experimental data to develop a delayed mathematical model, with the aim of understanding how such delays affect oscillatory behaviour on p38*. We analyze the stability and oscillation of the model with and without explicit time delays. We show that a sufficiently input stimulation strength is prerequisite for generating p38* oscillations, and that an optimal rate of model parameters is also essential to these oscillations. Moreover, we find that the time delays required for transcription and translation in mitogen-activated protein kinase phosphatase-1 (MKP-1) gene expression can drive p38* to be oscillatory even when the concentration of p38* level is at a stable state. Furthermore, the length of these delays can determine the amplitude and period of the oscillations and can enormously extend the oscillatory ranges of model parameters. These results indicate that time delays in MKP-1 synthesis are required, albeit not sufficient, for p38* oscillations, which may lead to new insights related to p38 oscillations.
最近的实验证据表明，p38 MAPK（p38）活性的振荡将有效地诱导促炎基因表达，这可能对免疫系统有害，甚至可能导致细胞损伤和凋亡。现在人们普遍认为，转录和翻译延迟在基因表达中普遍存在，这通常会导致基因调控的振荡反应。因此，p38 活性（p38*）的延迟驱动持续振荡在原则上可能是常见的。然而，到目前为止，关于这些延迟对 p38的影响的研究在实验和理论上都缺乏。在这里，我们使用实验数据来开发一个延迟的数学模型，目的是了解这些延迟如何影响 p38的振荡行为。我们分析了有无显式时间延迟的模型的稳定性和振荡。我们表明，足够的输入刺激强度是产生 p38振荡的前提条件，模型参数的最佳速率对于这些振荡也是必不可少的。此外，我们发现，丝裂原活化蛋白激酶磷酸酶-1（MKP-1）基因表达中转录和翻译所需的时间延迟可以驱动 p38即使在 p38水平处于稳定状态时也会发生振荡。此外，这些延迟的长度可以确定振荡的幅度和周期，并可以极大地扩展模型参数的振荡范围。这些结果表明，MKP-1 合成中的时间延迟是 p38振荡所必需的，尽管不是充分的，但可能会导致与 p38 振荡相关的新见解。
