López-Caamal Fernando, Middleton Richard H, Huber Heinrich J
Hamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland.
J Math Biol. 2014 Feb;68(3):609-45. doi: 10.1007/s00285-013-0644-z. Epub 2013 Jan 29.
Positive feedback loops are common regulatory elements in metabolic and protein signalling pathways. The length of such feedback loops determines stability and sensitivity to network perturbations. Here we provide a mathematical analysis of arbitrary length positive feedback loops with protein production and degradation. These loops serve as an abstraction of typical regulation patterns in protein signalling pathways. We first perform a steady state analysis and, independently of the chain length, identify exactly two steady states that represent either biological activity or inactivity. We thereby provide two formulas for the steady state protein concentrations as a function of feedback length, strength of feedback, as well as protein production and degradation rates. Using a control theory approach, analysing the frequency response of the linearisation of the system and exploiting the Small Gain Theorem, we provide conditions for local stability for both steady states. Our results demonstrate that, under some parameter relationships, once a biological meaningful on steady state arises, it is stable, while the off steady state, where all proteins are inactive, becomes unstable. We apply our results to a three-tier feedback of caspase activation in apoptosis and demonstrate how an intermediary protein in such a loop may be used as a signal amplifier within the cascade. Our results provide a rigorous mathematical analysis of positive feedback chains of arbitrary length, thereby relating pathway structure and stability.
正反馈回路是代谢和蛋白质信号通路中常见的调节元件。此类反馈回路的长度决定了对网络扰动的稳定性和敏感性。在此，我们对具有蛋白质产生和降解的任意长度正反馈回路进行数学分析。这些回路是蛋白质信号通路中典型调节模式的一种抽象。我们首先进行稳态分析，且与链长无关，精确识别出代表生物活性或无活性的两个稳态。由此，我们给出了作为反馈长度、反馈强度以及蛋白质产生和降解速率函数的稳态蛋白质浓度的两个公式。使用控制理论方法，分析系统线性化的频率响应并利用小增益定理，我们给出了两个稳态局部稳定性的条件。我们的结果表明，在某些参数关系下，一旦出现具有生物学意义的开稳态，它就是稳定的，而所有蛋白质均无活性的关稳态则变得不稳定。我们将结果应用于细胞凋亡中半胱天冬酶激活的三级反馈，并展示了此类回路中的中间蛋白如何在级联反应中用作信号放大器。我们的结果对任意长度的正反馈链进行了严格的数学分析，从而将通路结构与稳定性联系起来。
