The Number of Multistate Nested Canalyzing Functions

TL;DR

This paper derives a formula for counting multistate nested canalyzing functions, revealing that they form a very restrictive class compared to all possible functions, especially as the number of variables grows large.

Contribution

It provides the first explicit formula for the number of multistate nested canalyzing functions and analyzes their scarcity among all functions.

Findings

The ratio of nested canalyzing functions to all functions approaches zero as variables increase.

Nested canalyzing functions are highly restrictive, indicating special properties of molecular networks.

The study uses polynomial representations to characterize these functions.

Abstract

Identifying features of molecular regulatory networks is an important problem in systems biology. It has been shown that the combinatorial logic of such networks can be captured in many cases by special functions called nested canalyzing in the context of discrete dynamic network models. It was also shown that the dynamics of networks constructed from such functions has very special properties that are consistent with what is known about molecular networks, and that simplify analysis. It is important to know how restrictive this class of functions is, for instance for the purpose of network reverse-engineering. This paper contains a formula for the number of such functions and a comparison to the class of all functions. In particular, it is shown that, as the number of variables becomes large, the ratio of the number of nested canalyzing functions to the number of all functions converges to zero. This shows that the class of nested canalyzing functions is indeed very restrictive, indicating that molecular networks have very special properties. The principal tool used for this investigation is a description of these functions as polynomials and a parameterization of the class of all such polynomials in terms of relations on their coefficients.