The Influence of Canalization on the Robustness of Boolean Networks

TL;DR

This paper introduces mathematical tools to analyze the robustness of Boolean networks, especially those governed by k-canalizing functions, by deriving formulas for activities, sensitivities, and Derrida values, enabling efficient robustness assessment without simulation.

Contribution

It generalizes the concept of sensitivity to c-sensitivity for k-canalizing functions and provides formulas for calculating Derrida values efficiently.

Findings

Formulas for activities and c-sensitivity of k-canalizing functions.

Derrida value expressed as a weighted sum of c-sensitivities.

Efficient computation of network robustness without simulation.

Abstract

Time- and state-discrete dynamical systems are frequently used to model molecular networks. This paper provides a collection of mathematical and computational tools for the study of robustness in Boolean network models. The focus is on networks governed by $k$-canalizing functions, a recently introduced class of Boolean functions that contains the well-studied class of nested canalizing functions. The activities and sensitivity of a function quantify the impact of input changes on the function output. This paper generalizes the latter concept to $c$-sensitivity and provides formulas for the activities and $c$-sensitivity of general $k$-canalizing functions as well as canalizing functions with more precisely defined structure. A popular measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the $c$-sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.