Nested canalyzing depth and network stability

TL;DR

This paper introduces the concept of nested canalyzing depth to measure how functions retain canalyzing structure, analyzing its impact on Boolean network stability and suggesting real biological networks have intermediate depth.

Contribution

It generalizes nested canalyzing functions by defining and characterizing nested canalyzing depth, linking it to network stability and biological relevance.

Findings

Higher canalyzing depth reduces sensitivity to input perturbations.

Networks with greater depth tend toward critical dynamics.

Nested canalyzing functions are not significantly more effective than sufficiently deep functions.

Abstract

We introduce the nested canalyzing depth of a function, which measures the extent to which it retains a nested canalyzing structure. We characterize the structure of functions with a given depth and compute the expected activities and sensitivities of the variables. This analysis quantifies how canalyzation leads to higher stability in Boolean networks. It generalizes the notion of nested canalyzing functions (NCFs), which are precisely the functions with maximum depth. NCFs have been proposed as gene regulatory network models, but their structure is frequently too restrictive and they are extremely sparse. We find that functions become decreasingly sensitive to input perturbations as the canalyzing depth increases, but exhibit rapidly diminishing returns in stability. Additionally, we show that as depth increases, the dynamics of networks using these functions quickly approach the critical regime, suggesting that real networks exhibit some degree of canalyzing depth, and that NCFs are not significantly better than functions of sufficient depth for many applications of the modeling and reverse engineering of biological networks.