Matache Mihaela T, Matache Valentin
Department of Mathematics, University of Nebraska at Omaha, Omaha, NE, 68182-0243, USA.
Bull Math Biol. 2016 Jul;78(7):1520-45. doi: 10.1007/s11538-016-0193-x. Epub 2016 Jul 14.
Boolean networks have been widely used as models for gene regulatory networks, signal transduction networks, or neural networks, among many others. One of the main difficulties in analyzing the dynamics of a Boolean network and its sensitivity to perturbations or mutations is the fact that it grows exponentially with the number of nodes. Therefore, various approaches for simplifying the computations and reducing the network to a subset of relevant nodes have been proposed in the past few years. We consider a recently introduced method for reducing a Boolean network to its most determinative nodes that yield the highest information gain. The determinative power of a node is obtained by a summation of all mutual information quantities over all nodes having the chosen node as a common input, thus representing a measure of information gain obtained by the knowledge of the node under consideration. The determinative power of nodes has been considered in the literature under the assumption that the inputs are independent in which case one can use the Bahadur orthonormal basis. In this article, we relax that assumption and use a standard orthonormal basis instead. We use techniques of Hilbert space operators and harmonic analysis to generate formulas for the sensitivity to perturbations of nodes, quantified by the notions of influence, average sensitivity, and strength. Since we work on finite-dimensional spaces, our formulas and estimates can be and are formulated in plain matrix algebra terminology. We analyze the determinative power of nodes for a Boolean model of a signal transduction network of a generic fibroblast cell. We also show the similarities and differences induced by the alternative complete orthonormal basis used. Among the similarities, we mention the fact that the knowledge of the states of the most determinative nodes reduces the entropy or uncertainty of the overall network significantly. In a special case, we obtain a stronger result than in previous works, showing that a large information gain from a set of input nodes generates increased sensitivity to perturbations of those inputs.
布尔网络已被广泛用作基因调控网络、信号转导网络或神经网络等多种网络的模型。分析布尔网络的动态特性及其对扰动或突变的敏感性的主要困难之一在于，它会随着节点数量呈指数级增长。因此，在过去几年中已经提出了各种简化计算并将网络简化为相关节点子集的方法。我们考虑一种最近引入的方法，该方法可将布尔网络简化为产生最高信息增益的最具决定性的节点。节点的决定性能力是通过对所有以所选节点为公共输入的节点上的所有互信息量求和来获得的，因此代表了通过了解所考虑的节点而获得的信息增益的度量。在输入独立的假设下，文献中已经考虑了节点的决定性能力，在这种情况下，可以使用巴哈杜尔正交基。在本文中，我们放宽了该假设，转而使用标准正交基。我们使用希尔伯特空间算子和谐波分析技术来生成节点对扰动的敏感性公式，这些公式通过影响、平均敏感性和强度等概念来量化。由于我们在有限维空间中工作，我们的公式和估计可以并且是以普通矩阵代数术语来表述的。我们分析了通用成纤维细胞信号转导网络的布尔模型中节点的决定性能力。我们还展示了所使用的替代完全正交基所引起的异同。在相似之处中，我们提到最具决定性的节点的状态知识会显著降低整个网络的熵或不确定性这一事实。在一种特殊情况下，我们得到了比以前的工作更强的结果，表明从一组输入节点获得的大量信息增益会导致对这些输入扰动的敏感性增加。
