The objective of this study is to evaluate the role of cooperativity, captured by the Hill coefficient, in a minimal mathematical model describing the interactions between p53 and miR-34a. The model equations are analyzed for negative, none and normal cooperativity using a specific version of bifurcation theory and they are solved numerically. Special attention is paid to the sign of so-called first Lyapunov value. Interpretations of the results are given, both according to dynamic theory and in biological terms. In terms of cell signaling, we propose the hypothesis that when the outgoing signal of a system spends a physiologically significant amount of time outside of its equilibrium state, then the value of that signal can be sampled at any point along the trajectory towards that equilibrium and indeed, at multiple points. Coupled with non-linear behavior, such as that caused by cooperativity, this feature can account for a complex and varied response, which p53 is known for. From dynamical point of view, we found that when cooperativity is negative, the system has only one stable equilibrium point. In the absence of cooperativity, there is a single unstable equilibrium point with a critical boundary of stability. In the case with normal cooperativity, the system can have one, two, or three steady states with both, bi-stability and bi-instability occurring.

中文翻译：

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协同性在 p53-miR34 动力学数学模型中的作用。

本研究的目的是评估由 Hill 系数捕获的协同作用在描述 p53 和 miR-34a 之间相互作用的最小数学模型中的作用。使用特定版本的分岔理论分析模型方程的负协同性、无协同性和正协同性，并对其进行数值求解。特别注意所谓的第一李雅普诺夫值的符号。根据动态理论和生物学术语给出了对结果的解释。在细胞信号方面，我们提出了这样的假设，即当系统的输出信号在其平衡状态之外花费生理上显着的时间量时，该信号的值可以在沿着朝向该平衡的轨迹的任何点采样，并且事实上，在多个点。再加上非线性行为，例如由协同性引起的行为，这个特征可以解释复杂多变的响应，p53 就是众所周知的。从动力学的角度，我们发现当协同性为负时，系统只有一个稳定的平衡点。在缺乏协同性的情况下，存在一个具有临界稳定性边界的不稳定平衡点。在正常协同的情况下，系统可以有一个、两个或三个稳态，同时出现双稳态和双不稳定性。存在一个具有临界稳定性边界的不稳定平衡点。在正常协同的情况下，系统可以有一个、两个或三个稳态，同时出现双稳态和双不稳定性。存在一个具有临界稳定性边界的不稳定平衡点。在正常协同的情况下，系统可以有一个、两个或三个稳态，同时出现双稳态和双不稳定性。