Logical Reduction of Biological Networks to Their Most Determinative Components

Abstract

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.

Keywords: Biological information theory; Boolean networks; Linear operators; Mutual information; Network reduction; Numerical simulations; Sensitivity.

Fig. 1

(Color Figure online) Main numerical characteristics of the nodes of the fibroblast network as specified in each subplot

Fig. 2

(Color Figure online) Comparison of DP and $\mathit{\sigma }$ by nodes of the fibroblast network. The nodes are sorted by names in the top panels, while in the bottom panels they are ordered according to increasing DP, and $\mathit{\sigma }$, respectively, as indicated in the graphs. Note that the strength values seem to be slightly larger than the DP values

Fig. 3

(Color Figure online) Values of the upper bound in (25) for sub-networks chosen based on the top l values of DP and $\mathit{\sigma }$, respectively. The bottom panel shows the differences in the entropy that favor mostly $\mathit{\sigma }$ for $l>20$ with approximation

Fig. 4

(Color Figure online) Graphs of $I_A(f)$ and $\mathrm{MI}(f(X);X_A)$ computed with formulas (29) and (30) versus K for a network with $n=8$ nodes and $|A|=1,2,\dots ,n$ as specified in the subplots. The MI curve is very close to zero for most values of |A|, while the influence has much larger values in all cases. Recall also that MI is always a number in [0, 1]. The actual values increase with the increase in |A|. Notice also the expected symmetry as K crosses from values less than to values greater than $2^n/2$