On the number of non-degenerate canalizing Boolean functions

TL;DR

This paper develops recursive formulas to count non-degenerate canalizing Boolean functions, enhancing understanding of their prevalence and role in biological networks by quantifying their combinatorial properties.

Contribution

It introduces a novel recursive enumeration method for non-degenerate canalizing Boolean functions, extending previous results and providing a foundation for analyzing their biological significance.

Findings

Derived recursive formulas for counting Boolean functions with specific canalizing properties.

Quantified the frequency of canalization in random Boolean functions.

Supported the biological relevance of canalization in gene regulatory networks.

Abstract

Canalization is a key organizing principle in complex systems, particularly in gene regulatory networks. It describes how certain input variables exert dominant control over a function's output, thereby imposing hierarchical structure and conferring robustness to perturbations. Degeneracy, in contrast, captures redundancy among input variables and reflects the complete dominance of some variables by others. Both properties influence the stability and dynamics of discrete dynamical systems, yet their combinatorial underpinnings remain incompletely understood. Here, we derive recursive formulas for counting Boolean functions with prescribed numbers of essential variables and given canalizing properties. In particular, we determine the number of non-degenerate canalizing Boolean functions -- that is, functions for which all variables are essential and at least one variable is canalizing. Our approach extends earlier enumeration results on canalizing and nested canalizing functions. It provides a rigorous foundation for quantifying how frequently canalization occurs among random Boolean functions and for assessing its pronounced over-representation in biological network models, where it contributes to both robustness and to the emergence of distinct regulatory roles.