Stability of the Kauffman model

Bilke, Sven; Sjunnesson, Fredrik

doi:10.1103/physreve.65.016129

Cited by 109 publications

(116 citation statements)

References 10 publications

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“…The N-K model has been studied extensively [16,17,18,19,20,21,22,23]. Certain features of real GRNs, including the ability of a single network to produce multiple cell types (which appear as multiple attractors for the network), are captured by the N-K model.…”

Section: Introductionmentioning

confidence: 99%

Canalization and symmetry in Boolean models for genetic regulatory networks

Reichhardt

Bassler

2007

J. Phys. A: Math. Theor.

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Abstract. Canalization of genetic regulatory networks has been argued to be favored by evolutionary processes due to the stability that it can confer to phenotype expression. We explore whether a significant amount of canalization and partial canalization can arise in purely random networks in the absence of evolutionary pressures. We use a mapping of the Boolean functions in the Kauffman N-K model for genetic regulatory networks onto a k−dimensional Ising hypercube (where k = K) to show that the functions can be divided into different classes strictly due to geometrical constraints. The classes can be counted and their properties determined using results from group theory and isomer chemistry. We demonstrate that partially canalizing functions completely dominate all possible Boolean functions, particularly for higher k. This indicates that partial canalization is extremely common, even in randomly chosen networks, and has implications for how much information can be obtained in experiments on native state genetic regulatory networks.

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Section: Introductionmentioning

confidence: 99%

Canalization and symmetry in Boolean models for genetic regulatory networks

Reichhardt

Bassler

2007

J. Phys. A: Math. Theor.

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“…Therefore, the existence of a critical connectivity K c that can maximize the fitness is also an evidence that networks with K c maintain maximum robustness and adaptability while performing complex computations. Although the critical connectivity of K c = 2 for the networks have been hypothesized and observed by many researchers [11,47,51,54,55,67], as far as we know, this is the first time that the critical connectivity has been established for networks in a concrete computational context, i.e., with specific tasks and not in a closed system (such as classical RBNs with no external inputs).…”

Section: Revised Mutation Schemementioning

confidence: 99%

On the Effect of Topology on Learning and Generalization in Random Automata Networks

Goudarzi

2000

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We extend the study of learning and generalization in feedforward Boolean networks [70,93] to random Boolean networks (RBNs). We explore the relationship between the learning capability and the network topology, the system size, the training sample size, and the complexity of the computational tasks. We show experimentally that there exists a critical connectivity K c that improves the generalization and adaptation in networks. In addition, we show that in finite size networks, the critical K is a power-law function of the system size N and the fraction of inputs used during the training. We explain why adaptation improves at this critical connectivity by showing that the network ensemble manifests maximal topological diversity near K c . Our work is partly motivated by self-assembled molecular and nanoscale electronics. Our findings allow to determine an automata network topology class for efficient and robust information processing.ii ACKNOWLEDGMENTS

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“…Its application to two relatively large signaling networks with more than 10 12 states in their state transition graphs demonstrated its ability to identify all attractors of the underlying systems and to make experimentally testable predictions about the long-term behaviors of the systems. Integration of our reduction method with the removal of leaf nodes (nodes with out-degree=0) as proposed in [2,11,14] can be very effective in simplifying biological regulatory networks.…”

Section: Clearly For Anymentioning

confidence: 99%

“…There have been several efforts to reduce the state space of Boolean models by simplifying the underlying networks. In [2,11,14] a network reduction method based on the removal of stable variables (i.e, variables that stabilize in an attracting state after a transient period, irrespective of updating strategy or initial conditions) and leaf nodes (i.e., nodes with out-degree = 0) was proposed. In another study, Naldi et al [12] proposed a reduction method for simplifying finite-state logical models by iteratively removing nodes without a self loop from the network.…”

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confidence: 99%

A Reduction Method for Boolean Network Models Proven to Conserve Attractors

Saadatpour¹,

Albert²,

Reluga³

2013

SIAM J. Appl. Dyn. Syst.

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Abstract. Boolean models, wherein each component is characterized with a binary (ON or OFF) variable, have been widely employed for dynamic modeling of biological regulatory networks. However, the exponential dependencse of the size of the state space of these models on the number of nodes in the network can be a daunting prospect for attractor analysis of large-scale systems. We have previously proposed a network reduction technique for Boolean models and demonstrated its applicability on two biological systems, namely, the abscisic acid signal transduction network as well as the T-LGL leukemia survival signaling network. In this paper, we provide a rigorous mathematical proof that this method not only conserves the fixed points of a Boolean network, but also conserves the complex attractors of general asynchronous Boolean models wherein at each time step a randomly selected node is updated. This method thus allows one to infer the long-term dynamic properties of a large-scale system from those of the corresponding reduced model. Key words. Boolean models, Network reduction, Asynchronous methods, Attractors, Biological regulatory networks AMS subject classifications. 92C42, 37G351. Introduction. The ever-accelerating pace of experimental data generation has laid the foundation for developing network models of biological systems wherein the components of a system are represented by nodes and the interactions among them by edges. Analyzing these network models and studying their dynamics can unravel unknown facets of the underlying biological systems. Among different dynamic modeling approaches, discrete models, in which each component is assumed to have a finite number of qualitative states, have been increasingly employed in modeling biological regulatory networks [10,18,19,20,23]. The simplest discrete dynamic models are the so-called Boolean models that assume only two states (ON or OFF) for each component [8,21].Since Boolean models are parameter free, they serve as a suitable starting point for modeling biological systems for which a detailed kinetic characterization of the interactions is not available. In particular, attractor analysis of these models is of immense biological importance as it can provide valuable insights into the long-term behaviors, i.e. observed phenotypes, of these systems in response to environmental stimuli and internal perturbations [1,5,6,9,15,16]. For example, it allows one to predict the long-term activity levels of components or to determine key components influencing different cellular traits. However, the exponential dependence of the size of the state space of Boolean models on the number of nodes in the network makes the identification of all attractors of even relatively small systems computationally intractable. In particular, it has been proven that determination of the existence of fixed points in Boolean networks is a strong NPcomplete problem [24]. There have been several efforts to reduce the state space of Boolean models by simplifying the underlying networks. In [2...

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Stability of the Kauffman model

Cited by 109 publications

References 10 publications

Canalization and symmetry in Boolean models for genetic regulatory networks

Canalization and symmetry in Boolean models for genetic regulatory networks

On the Effect of Topology on Learning and Generalization in Random Automata Networks

A Reduction Method for Boolean Network Models Proven to Conserve Attractors

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