Abstract In practical gene regulatory networks, function perturbation often occurs due to gene mutation. This paper studies the function perturbation impact on the stability and set stability in distribution of probabilistic Boolean networks (PBNs) by using the semi-tensor product of matrices. Firstly, the stability and set stability in distribution of PBNs is recalled and the function perturbation problem is formulated. Secondly, when a given PBN is stable at an equilibrium (or a set) in distribution, based on the transition probability matrix and reachable set with positive probability, some necessary and sufficient conditions are presented to guarantee that the PBN is still stable at an equilibrium (or a set) in distribution after function perturbation. Finally, illustrative examples are worked out to support the obtained new results.

中文翻译：

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函数扰动对概率布尔网络分布稳定性的影响

摘要 在实际的基因调控网络中，经常会因基因突变而发生功能扰动。本文利用矩阵的半张量积研究了函数扰动对概率布尔网络(PBNs)分布稳定性和集合稳定性的影响。首先，回顾了PBNs分布的稳定性和集合稳定性，并制定了函数摄动问题。其次，当给定的PBN在分布的均衡（或集合）上稳定时，基于转移概率矩阵和概率为正的可达集，提出了一些充要条件，以保证PBN在均衡上仍然稳定（或一组）在函数扰动后分布。最后，制定了说明性示例以支持获得的新结果。