Boolean network
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A Boolean network consists of a discrete set of
Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly convey the correct pattern of expressed and suppressed genes.[2][3] The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.[4]
Classical model
A Boolean network is a particular kind of sequential dynamical system, where time and states are discrete, i.e. both the set of variables and the set of states in the time series each have a bijection onto an integer series.
A random Boolean network (RBN) is one that is randomly selected from the set of all possible boolean networks of a particular size, N. One then can study statistically, how the expected properties of such networks depend on various statistical properties of the ensemble of all possible networks. For example, one may study how the RBN behavior changes as the average connectivity is changed.
The first Boolean networks were proposed by
Attractors
Since a Boolean network has only 2N possible states, a trajectory will sooner or later reach a previously visited state, and thus, since the dynamics are deterministic, the trajectory will fall into a steady state or cycle called an attractor (though in the broader field of dynamical systems a cycle is only an attractor if perturbations from it lead back to it). If the attractor has only a single state it is called a point attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that lead to an attractor is called the basin of the attractor. States which occur only at the beginning of trajectories (no trajectories lead to them), are called garden-of-Eden states[8] and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.[4]
With growing computer power and increasing understanding of the seemingly simple model, different authors gave different estimates for the mean number and length of the attractors, here a brief summary of key publications.[9]
Author | Year | Mean attractor length | Mean attractor number | comment |
---|---|---|---|---|
Kauffman [5] | 1969 | |||
Bastolla/ Parisi[10] | 1998 | faster than a power law, | faster than a power law, | first numerical evidences |
Bilke/ Sjunnesson[11] | 2002 | linear with system size, | ||
Socolar/Kauffman[12] | 2003 | faster than linear, with | ||
Samuelsson/Troein[13] | 2003 | superpolynomial growth, | mathematical proof | |
Mihaljev/Drossel[14] | 2005 | faster than a power law, | faster than a power law, |
Stability
In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean networks depends on the connections of their
For N-K-model[15] the network is stable if , critical if , and unstable if .
The state of a given node is updated according to its truth table, whose outputs are randomly populated. denotes the probability of assigning an off output to a given series of input signals.
If for every node, the transition between the stable and chaotic range depends on . According to Bernard Derrida and Yves Pomeau[16] , the critical value of the average number of connections is .
If is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by [17][18][19] The network is stable if , critical if , and unstable if .
The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution: , and , since every out-link from a node is an in-link to another.[20]
Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks, . In the general case, stability of the network is governed by the largest eigenvalue of matrix , where , and is the adjacency matrix of the network.[21] The network is stable if , critical if , unstable if .
Variations of the model
Other topologies
One theme is to study different underlying
- The homogeneous case simply refers to a grid which is simply the reduction to the famous Ising model.
- Scale-free topologies may be chosen for Boolean networks.[22] One can distinguish the case where only in-degree distribution in power-law distributed,[23] or only the out-degree-distribution or both.
Other updating schemes
Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously,[24] different alternatives have been introduced. A common classification[25] is the following:
- Deterministic asynchronous updated Boolean networks (DRBNs) are not synchronously updated but a deterministic solution still exists. A node i will be updated when t ≡ Qi (mod Pi) where t is the time step.[26]
- The most general case is full stochastic updating (GARBN, general asynchronous random boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
- The Partially-Observed Boolean Dynamical System (POBDS)[27][28][29][30] signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.
- Autonomous Boolean networks (ABNs) are updated in continuous time (t is a real number, not an integer), which leads to race conditions and complex dynamical behavior such as deterministic chaos.[31][32]
Application of Boolean Networks
Classification
- The Scalable Optimal Bayesian Classification[33] developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.
See also
References
- PMID 25619997.
- PMID 12782112.
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- ^ S2CID 119300231.
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- )
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- ISBN 9781905986316. Archivedfrom the original on 4 February 2023. Retrieved 12 January 2016.
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- from the original on 2020-05-17. Retrieved 2016-01-12.
- .
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- .
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- ISBN 9780262581578. Archived from the original on 2023-02-04. Retrieved 2020-09-16.)
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- ^ Hajiramezanali, E. & Imani, M. & Braga-Neto, U. & Qian, X. & Dougherty, E.. Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty. ACMBCB'18. https://dl.acm.org/citation.cfm?id=3233689 Archived 2021-03-22 at the Wayback Machine
- Dubrova, E., Teslenko, M., Martinelli, A., (2005). *Kauffman Networks: Analysis and Applications, in "Proceedings of International Conference on Computer-Aided Design", pages 479-484.
External links
- Analysis of Dynamic Algebraic Models (ADAM) v1.1
- bioasp/bonesis: Synthesis of Most Permissive Boolean Networks from network architecture and dynamical properties
- CoLoMoTo (Consortium for Logical Models and Tools)
- DDLab
- NetBuilder Boolean Networks Simulator
- Open Source Boolean Network Simulator
- JavaScript Kauffman Network
- Probabilistic Boolean Networks (PBN)
- RBNLab
- A SAT-based tool for computing attractors in Boolean Networks