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Boolean nested canalizing functions: a comprehensive analysis

Yuan Li, John O. Adeyeye, David Murrugarra, Boris Aguilar, Reinhard Laubenbacher

Abstract

Boolean network models of molecular regulatory networks have been used successfully in computational systems biology. The Boolean functions that appear in published models tend to have special properties, in particular the property of being nested canalizing, a concept inspired by the concept of canalization in evolutionary biology. It has been shown that networks comprised of nested canalizing functions have dynamic properties that make them suitable for modeling molecular regulatory networks, namely a small number of (large) attractors, as well as relatively short limit cycles. This paper contains a detailed analysis of this class of functions, based on a novel normal form as polynomial functions over the Boolean field. The concept of layer is introduced that stratifies variables into different classes depending on their level of dominance. Using this layer concept a closed form formula is derived for the number of nested canalizing functions with a given number of variables. Additional metrics considered include Hamming weight, the activity number of any variable, and the average sensitivity of the function. It is also shown that the average sensitivity of any nested canalizing function is between 0 and 2. This provides a rationale for why nested canalizing functions are stable, since a random Boolean function in n variables has average sensitivity n/2. The paper also contains experimental evidence that the layer number is an important factor in network stability.

Boolean nested canalizing functions: a comprehensive analysis

Abstract

Boolean network models of molecular regulatory networks have been used successfully in computational systems biology. The Boolean functions that appear in published models tend to have special properties, in particular the property of being nested canalizing, a concept inspired by the concept of canalization in evolutionary biology. It has been shown that networks comprised of nested canalizing functions have dynamic properties that make them suitable for modeling molecular regulatory networks, namely a small number of (large) attractors, as well as relatively short limit cycles. This paper contains a detailed analysis of this class of functions, based on a novel normal form as polynomial functions over the Boolean field. The concept of layer is introduced that stratifies variables into different classes depending on their level of dominance. Using this layer concept a closed form formula is derived for the number of nested canalizing functions with a given number of variables. Additional metrics considered include Hamming weight, the activity number of any variable, and the average sensitivity of the function. It is also shown that the average sensitivity of any nested canalizing function is between 0 and 2. This provides a rationale for why nested canalizing functions are stable, since a random Boolean function in n variables has average sensitivity n/2. The paper also contains experimental evidence that the layer number is an important factor in network stability.

Paper Structure

This paper contains 7 sections, 12 theorems, 78 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Definitions and Notation
  4. A Detailed Categorization of Nested Canalizing Functions
  5. Hamming Weight, Activity, and Average Sensitivity
  6. Simulations of Network Dynamics
  7. Conclusion

Key Result

Proposition 3.4

The function $f$ is $\{\sigma:\alpha:\beta\}$ NCF if and only if $f$ is $\{\sigma:\alpha^{n}:\beta^{n}\}$ NCF.

Figures (1)

  • Figure 1: Derrida plots for different layer numbers. Here, $n$ is the number of nodes and $k$ is the connectivity. For each layer number, a Derrida plot was generated from 4096 random networks. The $x$-axis represents Hamming distance of pairs of states, and the $y$-axis represents Hamming distance of their images.

Theorems & Definitions (33)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • Example 3.5
  • Example 3.6
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Remark 4.3
  • ...and 23 more