Multistate Nested Canalizing Functions and Their Networks

TL;DR

This paper develops mathematical tools to analyze the robustness and stability of multistate gene regulatory networks modeled by nested canalizing functions, providing formulas for their enumeration and properties.

Contribution

It introduces a canonical polynomial form for multistate NCFs, derives formulas for their sensitivities, and analyzes their enumeration and equivalence classes.

Findings

Derived a formula for normalized average c-sensitivities of multistate NCFs

Provided a closed-form count and asymptotic estimate for NCFs

Calculated the number of equivalence classes under variable permutation

Abstract

This paper provides a collection of mathematical and computational tools for the study of robustness in nonlinear gene regulatory networks, represented by time- and state-discrete dynamical systems taking on multiple states. The focus is on networks governed by nested canalizing functions (NCFs), first introduced in the Boolean context by S. Kauffman. After giving a general definition of NCFs we analyze the class of such functions. We derive a formula for the normalized average $c$-sensitivities of multistate NCFs, which enables the calculation of the Derrida plot, a popular measure of network stability. We also provide a unique canonical parametrized polynomial form of NCFs. This form has several consequences. We can easily generate NCFs for varying parameter choices, and derive a closed form formula for the number of such functions in a given number of variables, as well as an asymptotic formula. Finally, we compute the number of equivalence classes of NCFs under permutation of variables. Together, the results of the paper represent a useful mathematical framework for the study of NCFs and their dynamic networks.