Harmonic Analysis of Boolean Networks: Determinative Power and Perturbations

TL;DR

This paper investigates how input nodes influence the states of nodes in large Boolean networks, using mutual information to quantify determinative power and analyzing the network's robustness and sensitivity, with applications to E. coli's regulatory network.

Contribution

It introduces a mutual information-based measure for determinative power in Boolean networks and explores its relation to sensitivity, especially for unate functions, with practical analysis of E. coli.

Findings

Mutual information quantifies input determinative power.

Large sensitivity does not always imply high determinative power.

E. coli network is robust to input perturbations.

Abstract

Consider a large Boolean network with a feed forward structure. Given a probability distribution on the inputs, can one find, possibly small, collections of input nodes that determine the states of most other nodes in the network? To answer this question, a notion that quantifies the determinative power of an input over the states of the nodes in the network is needed. We argue that the mutual information (MI) between a given subset of the inputs X = {X_1, ..., X_n} of some node i and its associated function f_i(X) quantifies the determinative power of this set of inputs over node i. We compare the determinative power of a set of inputs to the sensitivity to perturbations to these inputs, and find that, maybe surprisingly, an input that has large sensitivity to perturbations does not necessarily have large determinative power. However, for unate functions, which play an important role in genetic regulatory networks, we find a direct relation between MI and sensitivity to perturbations. As an application of our results, we analyze the large-scale regulatory network of Escherichia coli. We identify the most determinative nodes and show that a small subset of those reduces the overall uncertainty of the network state significantly. Furthermore, the network is found to be tolerant to perturbations of its inputs.