Expected Number of Fixed Points in Boolean Networks with Arbitrary Topology

Mori, Fumito

doi:10.1103/physrevlett.119.028301

Cited by 16 publications

(21 citation statements)

References 47 publications

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“…Moreover, another fixed point emerges at s = −s * corresponding to a drive of −1 in the open loop system. Since the aforementioned condition is met with probability half we have an expected number of 2 × 1 2 = 1 fixed points in agreement with [38].…”

Section: Appendix B: Quasi Fixed Pointsmentioning

confidence: 75%

“…Appendix C: Fixed points are always there, regardless of topology Defined by equation (1), our system turns out to be a special case of the Theorem in Ref. [38] which states that the expectation EM of number of fixed points M in random Boolean networks is one, subject to a condition on the distribution of random Boolean functions that holds in our setting. Specifically, to comply with [38] an ensemble of random Boolean functions φ i (s) defining the network dynamics via s i (t+1) = φ i (s(t)), must have a set of neutral links, removal of which renders the network acyclic.…”

Section: Appendix B: Quasi Fixed Pointsmentioning

confidence: 99%

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Scale free topology as an effective feedback system

Rivkind

Schreier

Brenner

et al. 2019

Preprint

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Biological networks are often heterogeneous in their connectivity pattern, with degree distributions featuring a heavy tail of highly connected hubs. The implications of this heterogeneity on dynamical properties are a topic of much interest. Here we introduce a novel approach to analyze such networks the lumped hub approximation. Based on the observation that in finite networks a small number of hubs have a disproportionate effect on the entire system, we construct an approximation by lumping these nodes into a single effective hub, and replacing the rest by a homogeneous bulk. We use this approximation to study dynamics of networks with scale-free degree distributions, focusing on their probability of convergence to fixed points. We find that the approximation preserves convergence statistics over a wide range of settings. Our mapping provides a parametrization of scale free topology which is predictive at the ensemble level and also retains properties of individual realizations. Specifically for outgoing scale-free distributions, the role of the effective hub on the network can be elucidated by feedback analysis. We show that outgoing hubs have an organizing role that can drive the network to convergence, in analogy to suppression of chaos by an external drive. In contrast, incoming hubs have no such property, resulting in a marked difference between the behavior of networks with outgoing vs. incoming scale free degree distribution. Combining feedback analysis with mean field theory predicts a transition between convergent and divergent dynamics which is corroborated by numerical simulations. Our results show how interpreting topology as a feedback circuit can provide novel insights on dynamics. Furthermore, we highlight the effect of a handful of outlying hubs, rather than of the connectivity distribution law as a whole, on network dynamics.

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Section: Appendix B: Quasi Fixed Pointsmentioning

confidence: 75%

Section: Appendix B: Quasi Fixed Pointsmentioning

confidence: 99%

Scale free topology as an effective feedback system

Rivkind

Schreier

Brenner

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…This meaning is important in the sense that it establishes a connection between regulation graphs (or R-graphs) and state graphs. We take the meaning proposed by Naldi et al ( 2007 ) (and used also by Richard et al, 2012 and Mori and Mochizuki, 2017 ). According to this definition, a regulation is functional if it is sufficient to modify the activity of the regulated variable in a non-empty set of (molecular) contexts.…”

Section: Resultsmentioning

confidence: 99%

“…Because in Griffin the regulation graph plays a prominent role, the definition of a regulation is fundamental. Griffin uses the definition of Naldi et al ( 2007 ), Richard et al ( 2012 ), and Mori and Mochizuki ( 2017 ).…”

Section: Introductionmentioning

confidence: 99%

Griffin: A Tool for Symbolic Inference of Synchronous Boolean Molecular Networks

et al. 2018

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Boolean networks are important models of biochemical systems, located at the high end of the abstraction spectrum. A number of Boolean gene networks have been inferred following essentially the same method. Such a method first considers experimental data for a typically underdetermined “regulation” graph. Next, Boolean networks are inferred by using biological constraints to narrow the search space, such as a desired set of (fixed-point or cyclic) attractors. We describe Griffin, a computer tool enhancing this method. Griffin incorporates a number of well-established algorithms, such as Dubrova and Teslenko's algorithm for finding attractors in synchronous Boolean networks. In addition, a formal definition of regulation allows Griffin to employ “symbolic” techniques, able to represent both large sets of network states and Boolean constraints. We observe that when the set of attractors is required to be an exact set, prohibiting additional attractors, a naive Boolean coding of this constraint may be unfeasible. Such cases may be intractable even with symbolic methods, as the number of Boolean constraints may be astronomically large. To overcome this problem, we employ an Artificial Intelligence technique known as “clause learning” considerably increasing Griffin's scalability. Without clause learning only toy examples prohibiting additional attractors are solvable: only one out of seven queries reported here is answered. With clause learning, by contrast, all seven queries are answered. We illustrate Griffin with three case studies drawn from the Arabidopsis thaliana literature. Griffin is available at: http://turing.iimas.unam.mx/griffin.

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“…However finding all attractors (including cyclic and complex attractors) is challenging due to the complex dynamics of networks [12]. Thanks to the strong advances in understanding network structure [13][14][15][16], a promising way to tackle this problem is to try to find the attractors based on the network topology and the key features of the network's dynamics, instead of from its detailed dynamics [17]. For example, R. Thomas related the conditions of multi-stability and cyclic attractors to positive and negative feedback loops, respectively [18].…”

Section: Introductionmentioning

confidence: 99%

General method to find the attractors of discrete dynamic models of biological systems

Gan

Albert

2018

Phys. Rev. E

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Analyzing the long-term behaviors (attractors) of dynamic models of biological networks can provide valuable insight. We propose a general method that can find the attractors of multilevel discrete dynamical systems by extending a method that finds the attractors of a Boolean network model. The previous method is based on finding stable motifs, subgraphs whose nodes' states can stabilize on their own. We extend the framework from binary states to any finite discrete levels by creating a virtual node for each level of a multilevel node, and describing each virtual node with a quasi-Boolean function. We then create an expanded representation of the multilevel network, find multilevel stable motifs and oscillating motifs, and identify attractors by successive network reduction. In this way, we find both fixed point attractors and complex attractors. We implemented an algorithm, which we test and validate on representative synthetic networks and on published multilevel models of biological networks. Despite its primary motivation to analyze biological networks, our motif-based method is general and can be applied to any finite discrete dynamical system.

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Expected Number of Fixed Points in Boolean Networks with Arbitrary Topology

Cited by 16 publications

References 47 publications

Scale free topology as an effective feedback system

Scale free topology as an effective feedback system

Griffin: A Tool for Symbolic Inference of Synchronous Boolean Molecular Networks

General method to find the attractors of discrete dynamic models of biological systems

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