Revealing the canalizing structure of Boolean functions: Algorithms and applications

Dimitrova, Elena; Stigler, Brandilyn; Kadelka, Claus; Murrugarra, David

doi:10.1016/j.automatica.2022.110630

Cited by 9 publications

(10 citation statements)

References 25 publications

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“…While the canalizing depth provides a crude measure of the amount of canalization in a Boolean function, more detailed information is contained in the canalizing layer structure [17,18,30]. To investigate this, we compared the approximability of random networks, each governed entirely by 4-variable NCFs but with different layer structure.…”

Section: Resultsmentioning

confidence: 99%

“…The number of variables which become eventually canalizing is known as the canalizing depth [15]. Every non-zero Boolean function possesses a unique standard monomial form, from which the canalizing depth and the number of variables in each "layer" of canalization can be directly derived [17,18]. As the number of variables increases, canalizing and especially nested canalizing functions become increasingly rare [19][20][21].…”

Section: Introductionmentioning

confidence: 99%

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The nonlinearity of regulation in biological networks

Manicka

Johnson

Murrugarra

et al. 2021

Preprint

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Nonlinearity is a characteristic of complex biological regulatory networks that has implications ranging from therapy to control. To better understand its nature, we analyzed a suite of published Boolean network models, containing a variety of complex nonlinear interactions, with an approach involving a probabilistic generalization of Boolean logic that George Boole himself had proposed. Leveraging the continuous-nature of this formulation using Taylor-decomposition methods revealed the distinct layers of nonlinearity of the models. A comparison of the resulting series of model approximations with the corresponding sets of randomized ensembles furthermore revealed that the biological networks are relatively more linearly approximable. We hypothesize that this is a result of optimization by natural selection for the purpose of controllability.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The nonlinearity of regulation in biological networks

Manicka

Johnson

Murrugarra

et al. 2021

Preprint

Self Cite

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show abstract

“…2B). Four thousand eight hundred twenty-seven of the 5110 investigated update rules (94.4%) were even nested canalizing, meaning that all their variables become “eventually” canalizing ( 39 ). A comparison of the expected and observed proportion of canalizing and NCFs reveals the true significance of the overabundance of canalization in GRN models.…”

Section: Resultsmentioning

confidence: 99%

“…Each x i appears in exactly one of { M 1 , …, M r , p C }. The layer structure of f is the vector ( k 1 , k 2 , …, k r ) and describes the number of variables in each layer M i ( 36 , 39 ).…”

Section: Methodsmentioning

confidence: 99%

A meta-analysis of Boolean network models reveals design principles of gene regulatory networks

Kadelka,

Butrie,

Hilton

et al. 2024

Sci. Adv.

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Gene regulatory networks (GRNs) play a central role in cellular decision-making. Understanding their structure and how it impacts their dynamics constitutes thus a fundamental biological question. GRNs are frequently modeled as Boolean networks, which are intuitive, simple to describe, and can yield qualitative results even when data are sparse. We assembled the largest repository of expert-curated Boolean GRN models. A meta-analysis of this diverse set of models reveals several design principles. GRNs exhibit more canalization, redundancy, and stable dynamics than expected. Moreover, they are enriched for certain recurring network motifs. This raises the important question why evolution favors these design mechanisms.

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“…In Boolean models of biological networks a vast majority of the Boolean rules are nested canalyzing functions [28,[32][33][34][35]. Nested canalyzing functions are a set of Boolean functions in which every input variable is either canalyzing or conditionally canalyzing [36]. If a canalyzing input n i is fixed to a specific value x i = a i , then the function F (X Ii ) is fixed F = b i .…”

Section: Np-hard Problem We Follow Zañudo Et Al and Identifymentioning

confidence: 99%

Structure-based approach can identify driver nodes in ensembles of biologically-inspired Boolean networks

Newby¹,

Zañudo²,

Albert³

2023

Preprint

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Because the attractors of biological networks reflect stable behaviors (e.g., cell phenotypes), identifying control interventions that can drive a system towards its attractors (attractor control) is of particular relevance when controlling biological systems. Driving a network's feedback vertex set (FVS) by node-state override into a state consistent with a target attractor is proven to force every system trajectory to the target attractor, but in biological networks, the FVS is typically larger than can be realistically manipulated. External control of a subset of a biological network's FVS was proposed as a strategy to drive the network to its attractors utilizing fewer interventions; however, the effectiveness of this strategy was only demonstrated on a small set of Boolean models of biological networks. Here, we extend this analysis to ensembles of biologically-inspired Boolean networks. On these models, we use three structural metrics -PRINCE propagation, modified PRINCE propagation, and CheiRank -to rank FVS subsets by their predicted attractor control strength. We validate the accuracy of these rankings using three dynamical measures: To Control, Away Control, and logical domain of influence. We also calculate the propagation metrics on effective graphs, which incorporate each Boolean model's functional information into edge weights. While this additional information increases the predicting power of structural metrics, we find that the increase with respect to the unweighted network is limited. The propagation metrics in conjunction with the FVS can be used to identify realizable driver node sets by emulating the dynamics that are prevalent in biological networks. This approach only uses the network's structure, and the driver sets are shown to be robust to the specific dynamical model.

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Revealing the canalizing structure of Boolean functions: Algorithms and applications

Cited by 9 publications

References 25 publications

The nonlinearity of regulation in biological networks

The nonlinearity of regulation in biological networks

A meta-analysis of Boolean network models reveals design principles of gene regulatory networks

Structure-based approach can identify driver nodes in ensembles of biologically-inspired Boolean networks

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