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The Influence of Canalization on the Robustness of Boolean Networks

Claus Kadelka, Jack Kuipers, Reinhard Laubenbacher

TL;DR

These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.

Abstract

Time- and state-discrete dynamical systems are frequently used to model molecular networks. This paper provides a collection of mathematical and computational tools for the study of robustness in Boolean network models. The focus is on networks governed by $k$-canalizing functions, a recently introduced class of Boolean functions that contains the well-studied class of nested canalizing functions. The activities and sensitivity of a function quantify the impact of input changes on the function output. This paper generalizes the latter concept to $c$-sensitivity and provides formulas for the activities and $c$-sensitivity of general $k$-canalizing functions as well as canalizing functions with more precisely defined structure. A popular measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the $c$-sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.

The Influence of Canalization on the Robustness of Boolean Networks

TL;DR

These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.

Abstract

Time- and state-discrete dynamical systems are frequently used to model molecular networks. This paper provides a collection of mathematical and computational tools for the study of robustness in Boolean network models. The focus is on networks governed by -canalizing functions, a recently introduced class of Boolean functions that contains the well-studied class of nested canalizing functions. The activities and sensitivity of a function quantify the impact of input changes on the function output. This paper generalizes the latter concept to -sensitivity and provides formulas for the activities and -sensitivity of general -canalizing functions as well as canalizing functions with more precisely defined structure. A popular measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the -sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.

Paper Structure

This paper contains 6 sections, 6 theorems, 45 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Concept of Canalization
  3. Activities and Normalized Average Sensitivities
  4. Derrida Values of Networks Governed by $k$-canalizing functions
  5. Derrida Values of Stochastic Discrete Dynamical Systems
  6. Discussion

Key Result

Theorem 3.2

Let $f_k$ be a $k$-canalizing function of $n$ variables. By relabeling the variables if necessary, assume that $f_k$ is $k$-canalizing in the variable order $x_1, x_2, \ldots, x_k$. The expected activity of $x_j$ in $f_k$ is

Figures (2)

  • Figure 1: Derrida plot for networks of $N=100$ genes governed by NCFs with $n=5$ inputs and varying layer structure. A black dashed line shows the line $y=x$. The table shows the Derrida values for perturbations of size up to $3$ (first 8 rows), as well as average values for cases when only the layer number but not the exact layer structure is known (rows 9-12) and average values when both are unknown (last row).
  • Figure 2: For all different functions of $n$ variables with canalizing depth $k$, the impact of a single perturbation ($D(F,1)$) is plotted against the absolute bias of the function. In the left plot, the number of non-canalizing variables is constant ($n-k = 5$), while in the right plot, the total number of variables is constant ($n=7$). For visualisation purposes, the scatter points are connected by a line.

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Remark 2.5
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 13 more