The influence of canalization on the robustness of Boolean networks

Highlights: • The Boolean algebra concept of sensitivity is generalized to $c$-sensitivity. • Activities and $c$-sensitivities of various Boolean canalizing functions are computed. • Derrida values of canalizing Boolean networks are weighted sums of $c$-sensitivities. 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 variable 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.