The number and probability of canalizing functions

TL;DR

This paper derives formulas for counting and probability of canalizing Boolean functions, which are crucial in modeling biological and physical systems, and provides an algorithm for generating such functions for simulations.

Contribution

It provides exact formulas for the number and probability of canalizing functions with any number of variables and bias, and introduces a method for random generation of these functions.

Findings

Exact formulas for counting canalizing functions

Probability formulas for random Boolean functions being canalizing

Algorithm for generating canalizing functions with specified bias

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.