Robustness and relative stability of multiple attractors in\\ a stochastic Boolean network
Robustness and relative stability of multiple attractors in\\ a stochastic Boolean network
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DOI：
10.1360/n012017-00132
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发表时间：
2017-11
期刊：
影响因子：
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通讯作者：
Guo Yong-yi;Y. Zhiyi;Ge Hao
中科院分区：
文献类型：
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作者：
Guo Yong-yi;Y. Zhiyi;Ge Hao
生物网络中总是存在多个表型状态和动态路径，但它们对波动的稳健性和相对稳定性还没有得到充分的认识或仔细的分析。在这里，我们试图用一个广泛使用的随机布尔网络模型来解决这些问题。在这样一个随机布尔网络中，表型状态和动态路径的稳定性对应于吸引子的稳健性和相对稳定性，这可以用指数扰动马尔可夫链理论来分析。在指数扰动马氏链中，期望脱离吸引子的退出时间的对数与最小激活能障碍成正比，我们首先证明了在指数扰动的马尔可夫链中，吸引子之间具有最小激活能障碍的所有路径在零噪声极限下具有相同的概率权重，因此相对稳定性仅取决于最优路径的数目。这一理论的另一个重要含义是，一旦不可忽略的涨落相当小，就会出现相变现象：在一个参数区域中，共存的表型状态和路径的概率权重相互比较；而在其他一些参数区域中，某些表型状态或路径的概率权重甚至可以支配并变得全局吸引。最后，我们将该理论应用于一个人工模型和P53蛋白的Siah-1/β-catenin/p14/19 ARF环的动力学。我们的理论还可以确定表型态和平行路径之间的转换时间和最优转换路径的数量是如何依赖于所有参数的，并有助于识别生物网络中可能更关键的节点和它们之间的相互作用。
Multiple phenotypic states and dynamic pathways always exist in biological networks, but their robustness against fluctuations and relative stability have not been fully recognized or carefully analyzed yet. Here we try to address these issues with a widely used stochastic Boolean network model. The stability of phenotypic states and dynamical pathways corresponds to the robustness and relative stability of attractors in such a stochastic Boolean network, which can be analyzed using the theory of exponentially perturbed Markov chains. It is already known that the logarithm of the expected exit time escaping from an attractor is proportional to the ``minimum activation energy barrier" in the exponentially perturbed Markov chains. We first prove that within an exponentially perturbed Markov chain, all pathways between attractors with the minimum possible ``activation energy barrier" have the same probability weights in the zero-noise limit, and therefore the relative stability just depends on the number of optimal paths. Another important implication of this theory is that, once the non-neglectable fluctuations are rather low, a phase transition phenomenon emerges: In one parameter region the probability weights of the coexisted phenotypic states and pathways are comparable with each other; whereas in some other parameter regions, the probability weight of certain phenotypic state or pathway can even dominate and become globally attractive. Finally, we apply the theory to an artificial model and the Siah-1/beta-catenin/p14/19 ARF loop of protein p53 dynamics. Our theory can also determine how the transition time and the number of optimal transition paths between the phenotypic states and parallel pathways depend on all the parameters, and help to identify those possibly more crucial nodes and the interactions between them in a biological network.