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.

21 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1111.7217

References in corpus (10)

• Random Boolean Network Models and the Yeast Transcriptional Network
• Genetic networks with canalyzing Boolean rules are always stable
• Canalizing Kauffman networks: non-ergodicity and its effect on their critical behavior
• The number and probability of canalizing functions
• Regulatory patterns in molecular interaction networks
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• Analysis of random Boolean networks using the average sensitivity
• Sensitivity versus block sensitivity of Boolean functions
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• Influences of monotone Boolean functions

Cited by in corpus (12)

• Identification of control targets in Boolean molecular network models via computational algebra
• A meta-analysis of Boolean network models reveals design principles of gene regulatory networks
• Molecular Network Control Through Boolean Canalization
• The Influence of Canalization on the Robustness of Boolean Networks
• Revealing the canalizing structure of Boolean functions: Algorithms and applications
• Bounds on the Average Sensitivity of Nested Canalizing Functions
• Compensatory interactions to stabilize multiple steady states or mitigate the effects of multiple deregulations in biological networks
• Stability of Linear Boolean Networks
• Nested canalizing functions minimize sensitivity and simultaneously promote criticality
• Stratification and enumeration of Boolean functions by canalizing depth
• Symmetry Properties of Nested Canalyzing Functions
• Certificate complexity and symmetry of nested canalizing functions