The number and probability of canalizing functions

Abstract

Canalizing functions have important applications in physics and biology. For example, they represent a mechanism capable of stabilizing chaotic behavior in Boolean network models of discrete dynamical systems. When comparing the class of canalizing functions to other classes of functions with respect to their evolutionary plausibility as emergent control rules in genetic regulatory systems, it is informative to know the number of canalizing functions with a given number of input variables. This is also important in the context of using the class of canalizing functions as a constraint during the inference of genetic networks from gene expression data. To this end, we derive an exact formula for the number of canalizing Boolean functions of n variables. We also derive a formula for the probability that a random Boolean function is canalizing for any given bias p of taking the value 1. In addition, we consider the number and probability of Boolean functions that are canalizing for exactly k variables. Finally, we provide an algorithm for randomly generating canalizing functions with a given bias p and any number of variables, which is needed for Monte Carlo simulations of Boolean networks.

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• Boolean nested canalizing functions: a comprehensive analysis
• Canalization and Symmetry in Boolean Models for Genetic Regulatory Networks
• Entropy of complex relevant components of Boolean networks
• The Number of Multistate Nested Canalyzing Functions
• The phase diagram of random Boolean networks with nested canalizing functions
• Boolean networks with veto functions
• Stratification and enumeration of Boolean functions by canalizing depth
• Dimension Reduction of Large AND-NOT Network Models
• The Role of Boolean Irreducibility in \textit{NK}-Kauffman Networks
• Multistate nested canalizing functions
• Nested Canalyzing, unate cascade, and polynomial functions