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Modular Construction of Boolean Networks

Matthew Wheeler, Claus Kadelka, Alan Veliz-Cuba, David Murrugarra, Reinhard Laubenbacher

TL;DR

All Boolean networks can be obtained from a collection of simple Boolean networks as building blocks and a formula for the number of extensions of given simple networks and, in some cases, provides a parametrization of those extensions is provided.

Abstract

Boolean networks have been used in a variety of settings, as models for general complex systems as well as models of specific systems in diverse fields, such as biology, engineering, and computer science. Traditionally, their properties as dynamical systems have been studied through simulation studies, due to a lack of mathematical structure. This paper uses a common mathematical technique to identify a class of Boolean networks with a "simple" structure and describes an algorithm to construct arbitrary extensions of a collection of simple Boolean networks. In this way, all Boolean networks can be obtained from a collection of simple Boolean networks as building blocks. The paper furthermore provides a formula for the number of extensions of given simple networks and, in some cases, provides a parametrization of those extensions. This has potential applications to the construction of networks with particular properties, for instance in synthetic biology, and can also be applied to develop efficient control algorithms for Boolean network models.

Modular Construction of Boolean Networks

TL;DR

All Boolean networks can be obtained from a collection of simple Boolean networks as building blocks and a formula for the number of extensions of given simple networks and, in some cases, provides a parametrization of those extensions is provided.

Abstract

Boolean networks have been used in a variety of settings, as models for general complex systems as well as models of specific systems in diverse fields, such as biology, engineering, and computer science. Traditionally, their properties as dynamical systems have been studied through simulation studies, due to a lack of mathematical structure. This paper uses a common mathematical technique to identify a class of Boolean networks with a "simple" structure and describes an algorithm to construct arbitrary extensions of a collection of simple Boolean networks. In this way, all Boolean networks can be obtained from a collection of simple Boolean networks as building blocks. The paper furthermore provides a formula for the number of extensions of given simple networks and, in some cases, provides a parametrization of those extensions. This has potential applications to the construction of networks with particular properties, for instance in synthetic biology, and can also be applied to develop efficient control algorithms for Boolean network models.
Paper Structure (13 sections, 10 theorems, 41 equations, 2 figures)

This paper contains 13 sections, 10 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Boolean functions
  4. Boolean Networks
  5. Restrictions and extensions of Boolean functions and networks
  6. Restrictions and extensions of Boolean functions
  7. Restrictions and extensions of Boolean networks
  8. Simple Boolean networks
  9. Parametrizing Network Extensions
  10. Graphical Boolean networks
  11. Nested canalizing Boolean networks
  12. General Boolean networks
  13. Discussion

Key Result

Theorem 2.1

Every Boolean function $f(x_1,\ldots,x_n)\not\equiv 0$ can be uniquely written as where each $M_i = \prod_{j=1}^{k_i} (x_{i_j} + a_{i_j})$ is a nonconstant extended monomial, $p_C$ is the core polynomial of $f$, and $k = \sum_{i=1}^r k_i$ is the canalizing depth. Each $x_i$ appears in exactly one of $\{M_1,\ldots,M_r,p_C\}$, and the only restrictions are the following "exceptiona When $f$ is not

Figures (2)

  • Figure 1: Boolean network decomposition. (a) Wiring diagram of a non-strongly connected Boolean network $F$. (b-c) Wiring diagram of $F$ restricted to (b) $\{x_1,x_2\}$ and (c) $\{x_3,x_4\}$. These strongly connected components are the wiring diagrams of the simple networks of $F$.
  • Figure 2: Graphical representation of the iterative process to find all possible nested canalizing extensions of a two-variable NCF. Each diagram represents a layer structure, each block represents a variable, each column represents a layer with the leftmost column representing the outer layer. The central diagram displays the only possible layer structure of a two-variable NCF. The colored blocks highlight the newly added variable; the first (second) added variable is shown in blue (orange) and within a layer the newly added variable is always shown on top. For each diagram, a representative nested canalizing extension is shown. Edge labels show the number of extensions sharing the given layer structure. Note that for each extended layer structure, the arbitrary choice of canalizing input value for the newly added variables supplies an additional factor of 2, which the edge labels do not include.

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.1: he2016stratification
  • Definition 2.4
  • Example 2.1
  • Definition 2.5
  • Definition 2.6
  • Example 2.2
  • Definition 3.1
  • ...and 34 more