Boolean network

Theory

Network types

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Clique Component Cut Cycle Data structure Edge Loop Neighborhood Path Vertex Adjacency list / matrix Incidence list / matrix
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Bipartite Complete Directed Hyper Multi Random Weighted

Models

Topology
Random graph Erdős–Rényi Barabási–Albert Watts–Strogatz Exponential random (ERGM) Hyperbolic (HGN) Hierarchical Stochastic block model
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Boolean network agent based Epidemic/SIR

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State space of a Boolean Network with N=4 nodes and K=1 links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.

A Boolean network consists of a discrete set of Boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.

Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly capture the correct pattern of expressed and suppressed genes.^[1]^[2] The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.^[3]

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. Boolean networks are related to cellular automata.

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 Stuart A. Kauffman in 1969, as random models of genetic regulatory networks^[4] but their mathematical understanding only started in the 2000's.^[5]^[6]

Attractors

Since a Boolean network has only 2^N 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 cycle. In the literature in this field, each cycle is also 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^[7] and the dynamics of the network flow from these states towards attractors. The time it takes to reach an attractor is called transient time.^[3]

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.^[8]

Author	Year	Mean attractor length	Mean attractor number	comment
Kauffmann ^[4]	1969	$\langle A\rangle \sim {\sqrt {N}}$	$\langle \nu \rangle \sim {\sqrt {N}}$
Bastolla/ Parisi^[9]	1998	faster than a power law, $\langle A\rangle >N^{x}\forall x$	faster than a power law, $\langle \nu \rangle >N^{x}\forall x$	first numerical evidences
Bilke/ Sjunnesson^[10]	2002		linear with system size, $\langle \nu \rangle \sim N$
Socolar/Kauffman^[11]	2003		faster than linear, $\langle \nu \rangle >N^{x}$ with $x>1$
Samuelsson/Troein^[12]	2003		superpolynomial growth, $\langle \nu \rangle >N^{x}\forall x$	mathematical proof
Mihaljev/Drossel^[13]	2005	faster than a power law, $\langle A\rangle >N^{x}\forall x$	faster than a power law, $\langle \nu \rangle >N^{x}\forall x$

Stability

The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes ( $K_{c}$ ), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In this, with "initially close states" one means that the Hamming distance is small compared with the number of nodes ( $N$ ) in the network.

For N-K-model^[14] the network is stable if $K<K_{c}$ , critical if $K=K_{c}$ , and unstable if $K>K_{c}$ .

The state of a given node $n_{i}$ is updated according to its truth table, whose outputs are randomly populated. $p_{i}$ denotes the probability of assigning an off output to a given series of input signals.

If $p_{i}=p=const.$ for every node, the transition between the stable and chaotic range depends on $p$ . The critical value of the average number of connections is $K_{c}=1/[2p(1-p)]$ .^[15]

If $K$ is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by $\langle K^{in}\rangle$ ^[16]^[17]^[18] The network is stable if $\langle K^{in}\rangle <K_{c}$ , critical if $\langle K^{in}\rangle =K_{c}$ , and unstable if $\langle K^{in}\rangle >K_{c}$ .

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: $P(K)\propto K^{-\gamma }$ , and $\langle K^{in}\rangle =\langle K^{out}\rangle$ , since every out-link from a node is an in-link to another.^[19]

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks, $q_{i}=2p_{i}(1-p_{i})$ . In the general case, stability of the network is governed by the largest eigenvalue $\lambda _{Q}$ of matrix $Q$ , where $Q_{ij}=q_{i}A_{ij}$ , and $A$ is the adjacency matrix of the network.^[20] The network is stable if $\lambda _{Q}<1$ , critical if $\lambda _{Q}=1$ , unstable if $\lambda _{Q}>1$ .

Variations of the model

Other topologies

One theme is to study different underlying graph topologies.

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.^[21] One can distinguish the case where only in-degree distribution in power-law distributed,^[22] 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 usually not simultaneously changing their state,^[23] different alternatives have been introduced. A common classification^[24] 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 ≡ Q_i (mod P_i) where t is the time step.^[25]
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.

References

↑ Albert, Réka; Othmer, Hans G (July 2003). "The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster". Journal of Theoretical Biology. 223 (1): 1–18. doi:10.1016/S0022-5193(03)00035-3.
↑ Li, J.; Bench, A. J.; Vassiliou, G. S.; Fourouclas, N.; Ferguson-Smith, A. C.; Green, A. R. (30 April 2004). "Imprinting of the human L3MBTL gene, a polycomb family member located in a region of chromosome 20 deleted in human myeloid malignancies". Proceedings of the National Academy of Sciences. 101 (19): 7341–7346. Bibcode:2004PNAS..101.7341L. doi:10.1073/pnas.0308195101. PMC 409920. PMID 15123827.
1 2 Drossel, Barbara (December 2009). Schuster, Heinz Georg, ed. "Chapter 3. Random Boolean Networks". Reviews of Nonlinear Dynamics and Complexity. Reviews of Nonlinear Dynamics and Complexity. Wiley: 69–110. arXiv:0706.3351. doi:10.1002/9783527626359.ch3.
1 2 Kauffman, Stuart (11 October 1969). "Homeostasis and Differentiation in Random Genetic Control Networks". Nature. 224 (5215): 177–178. Bibcode:1969Natur.224..177K. doi:10.1038/224177a0.
↑ Aldana, Maximo; Coppersmith, Susan; Kadanoff, Leo P. (2003). "Boolean Dynamics with Random Couplings". Perspectives and Problems in Nonlinear Sciences: 23–89. doi:10.1007/978-0-387-21789-5_2. Retrieved 8 January 2016.
↑ Gershenson, C. (2004). Introduction to random Boolean networks. In Bedau, M., Husbands, P., Hutton, T., Kumar, S., and Suzuki, H., editors, Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX), pages 160–173, Boston, MA. http://arxiv.org/abs/nlin.AO/0408006
↑ Wuensche, Andrew (2011). Exploring discrete dynamics : [the DDLab manual : tools for researching cellular automata, random Boolean and multivalue neworks [sic] and beyond]. Frome, England: Luniver Press. p. 16. ISBN 9781905986316. Retrieved 12 January 2016.
↑ Greil, Florian (2012). "Boolean Networks as Modeling Framework". Frontiers in Plant Science. 3. doi:10.3389/fpls.2012.00178.
↑ Bastolla, U.; Parisi, G. (May 1998). "The modular structure of Kauffman networks". Physica D: Nonlinear Phenomena. 115 (3-4): 219–233. arXiv:cond-mat/9708214. Bibcode:1998PhyD..115..219B. doi:10.1016/S0167-2789(97)00242-X.
↑ Bilke, Sven; Sjunnesson, Fredrik (December 2001). "Stability of the Kauffman model". Physical Review E. 65 (1). arXiv:cond-mat/0107035. Bibcode:2002PhRvE..65a6129B. doi:10.1103/PhysRevE.65.016129.
↑ Socolar, J.; Kauffman, S. (February 2003). "Scaling in Ordered and Critical Random Boolean Networks". Physical Review Letters. 90 (6). arXiv:cond-mat/0212306. Bibcode:2003PhRvL..90f8702S. doi:10.1103/PhysRevLett.90.068702.
↑ Samuelsson, Björn; Troein, Carl (March 2003). "Superpolynomial Growth in the Number of Attractors in Kauffman Networks". Physical Review Letters. 90 (9). Bibcode:2003PhRvL..90i8701S. doi:10.1103/PhysRevLett.90.098701.
↑ Mihaljev, Tamara; Drossel, Barbara (October 2006). "Scaling in a general class of critical random Boolean networks". Physical Review E. 74 (4). arXiv:cond-mat/0606612. Bibcode:2006PhRvE..74d6101M. doi:10.1103/PhysRevE.74.046101.
↑ Kauffman, S. A. (1969). "Metabolic stability and epigenesis in randomly constructed genetic nets". Journal of Theoretical Biology. 22 (3): 437–467. doi:10.1016/0022-5193(69)90015-0. PMID 5803332.
↑ Derrida, B; Pomeau, Y (1986-01-15). "Random Networks of Automata: A Simple Annealed Approximation". Europhysics Letters (EPL). 1 (2): 45–49. doi:10.1209/0295-5075/1/2/001.
↑ Solé, Ricard V.; Luque, Bartolo (1995-01-02). "Phase transitions and antichaos in generalized Kauffman networks". Physics Letters A. 196 (5–6): 331–334. doi:10.1016/0375-9601(94)00876-Q.
↑ Luque, Bartolo; Solé, Ricard V. (1997-01-01). "Phase transitions in random networks: Simple analytic determination of critical points". Physical Review E. 55 (1): 257–260. doi:10.1103/PhysRevE.55.257.
↑ Fox, Jeffrey J.; Hill, Colin C. (2001-12-01). "From topology to dynamics in biochemical networks". Chaos: An Interdisciplinary Journal of Nonlinear Science. 11 (4): 809–815. doi:10.1063/1.1414882. ISSN 1054-1500.
↑ Aldana, Maximino; Cluzel, Philippe (2003-07-22). "A natural class of robust networks". Proceedings of the National Academy of Sciences. 100 (15): 8710–8714. doi:10.1073/pnas.1536783100. ISSN 0027-8424. PMC 166377. PMID 12853565.
↑ Pomerance, Andrew; Ott, Edward; Girvan, Michelle; Losert, Wolfgang (2009-05-19). "The effect of network topology on the stability of discrete state models of genetic control". Proceedings of the National Academy of Sciences. 106 (20): 8209–8214. doi:10.1073/pnas.0900142106. ISSN 0027-8424. PMC 2688895. PMID 19416903.
↑ Aldana, Maximino (October 2003). "Boolean dynamics of networks with scale-free topology". Physica D: Nonlinear Phenomena. Elsevier. 185 (1): 45–66. arXiv:cond-mat/0209571. doi:10.1016/s0167-2789(03)00174-x.
↑ Drossel, Barbara; Greil, Florian (4 August 2009). "Critical Boolean networks with scale-free in-degree distribution". Physical Review E. 80 (2). arXiv:0901.0387. doi:10.1103/PhysRevE.80.026102.
↑ Harvey, Imman; Bossomaier, Terry (1997). Husbands, Phil; Harvey, Imman, eds. "Time out of joint: Attractors in asynchronous random Boolean networks". Proceedings of the Fourth European Conference on Artificial Life (ECAL97). MIT Press: 67–75.
↑ Gershenson, Carlos (2002). Standish, Russell K; Bedau, Mark A, eds. "Classification of Random Boolean Networks". Proceedings of the eighth international conference on Artificial life. Artificial Life. Cambridge, Massachusetts, USA. 8: 1–8. Retrieved 12 January 2016.
↑ Gershenson, Carlos; Broekaert, Jan; Aerts, Diederik (14 September 2003). "Contextual Random Boolean Networks" [7th European Conference, ECAL 2003]. Advances in Artificial Life. Lecture Notes in Computer Science. Dortmund, Germany. 2801: 615–624. doi:10.1007/978-3-540-39432-7_66. ISBN 978-3-540-39432-7. Retrieved 12 January 2016.

Dubrova, E., Teslenko, M., Martinelli, A., (2005). *Kauffman Networks: Analysis and Applications, in "Proceedings of International Conference on Computer-Aided Design", pages 479-484.

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