Nested Canalyzing Depth and Network Stability

Layne, Lori; Dimitrova, Elena; Macauley, Matthew

doi:10.1007/s11538-011-9692-y

Cited by 35 publications

(49 citation statements)

References 19 publications

(31 reference statements)

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“…, x σ(n) ) is a Boolean function on n−k variables. When g is not a canalizing function itself, the integer k is the canalizing depth of f (as in [14]), and if g is in addition not constant, it is called the core function of f , denoted by f C . Remark 2.4.…”

Section: Definition 23mentioning

confidence: 99%

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The Influence of Canalization on the Robustness of Boolean Networks

Kadelka

Kuipers

Laubenbacher

2016

Preprint

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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.

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Section: Definition 23mentioning

confidence: 99%

“…. , x k , inputs a i and outputs b i , 1 ≤ i ≤ k. We will use a similar argument as in [14] to find the expected activities of f . By definition, the activity of x j in f is the probability that a change in x j changes the output of f .…”

Section: Appendixmentioning

confidence: 99%

The Influence of Canalization on the Robustness of Boolean Networks

Kadelka

Kuipers

Laubenbacher

2016

Preprint

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“…In reality, functions may be only partially canalizing, allowing for multiple, but not necessarily all inputs, to be canalizing in a cascading fashion as discussed, for example, in Layne et al. (2012) or Dimitrova et al. (2015).…”

Section: Final Commentsmentioning

confidence: 99%

Logical Reduction of Biological Networks to Their Most Determinative Components

Matache

2016

Bull Math Biol

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Boolean networks have been widely used as models for gene regulatory networks, signal transduction networks, or neural networks, among many others. One of the main difficulties in analyzing the dynamics of a Boolean network and its sensitivity to perturbations or mutations is the fact that it grows exponentially with the number of nodes. Therefore, various approaches for simplifying the computations and reducing the network to a subset of relevant nodes have been proposed in the past few years. We consider a recently introduced method for reducing a Boolean network to its most determinative nodes that yield the highest information gain. The determinative power of a node is obtained by a summation of all mutual information quantities over all nodes having the chosen node as a common input, thus representing a measure of information gain obtained by the knowledge of the node under consideration. The determinative power of nodes has been considered in the literature under the assumption that the inputs are independent in which case one can use the Bahadur orthonormal basis. In this article, we relax that assumption and use a standard orthonormal basis instead. We use techniques of Hilbert space operators and harmonic analysis to generate formulas for the sensitivity to perturbations of nodes, quantified by the notions of influence, average sensitivity, and strength. Since we work on finite-dimensional spaces, our formulas and estimates can be and are formulated in plain matrix algebra terminology. We analyze the determinative power of nodes for a Boolean model of a signal transduction network of a generic fibroblast cell. We also show the similarities and differences induced by the alternative complete orthonormal basis used. Among the similarities, we mention the fact that the knowledge of the states of the most determinative nodes reduces the entropy or uncertainty of the overall network significantly. In a special case, we obtain a stronger result than in previous works, showing that a large information gain from a set of input nodes generates increased sensitivity to perturbations of those inputs.

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“…These functions are quite new in the literature, and they are still being studied [38]. Notice that the NCFs are precisely the functions of canalizing depth n. Sometimes, canalizing functions of depth d, where 1 ≤ d ≤ n, are called partially nested canalizing.…”

Section: Canalizing Depthmentioning

confidence: 99%