Boolean nested canalizing functions: a comprehensive analysis

TL;DR

This paper provides a comprehensive analysis of nested canalizing Boolean functions, introducing a new normal form, deriving formulas for their enumeration, and exploring their stability properties in molecular network models.

Contribution

It introduces a novel polynomial normal form, the concept of layers, and formulas for counting nested canalizing functions, advancing understanding of their structural and stability characteristics.

Findings

Average sensitivity of nested canalizing functions is between 0 and 2.

Layer number influences network stability.

Nested canalizing functions have properties making them suitable for biological modeling.

Abstract

Boolean network models of molecular regulatory networks have been used successfully in computational systems biology. The Boolean functions that appear in published models tend to have special properties, in particular the property of being nested canalizing, a concept inspired by the concept of canalization in evolutionary biology. It has been shown that networks comprised of nested canalizing functions have dynamic properties that make them suitable for modeling molecular regulatory networks, namely a small number of (large) attractors, as well as relatively short limit cycles. This paper contains a detailed analysis of this class of functions, based on a novel normal form as polynomial functions over the Boolean field. The concept of layer is introduced that stratifies variables into different classes depending on their level of dominance. Using this layer concept a closed form formula is derived for the number of nested canalizing functions with a given number of variables. Additional metrics considered include Hamming weight, the activity number of any variable, and the average sensitivity of the function. It is also shown that the average sensitivity of any nested canalizing function is between 0 and 2. This provides a rationale for why nested canalizing functions are stable, since a random Boolean function in n variables has average sensitivity n/2. The paper also contains experimental evidence that the layer number is an important factor in network stability.