Boolean network

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. ^[1]

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 Stuart A. Kauffman in 1969, as random models of genetic regulatory networks ^[5] but their mathematical understanding only started in the 2000s. ^{[6] [7]}

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 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	${\displaystyle \langle A\rangle \sim {\sqrt {N}}}$	${\displaystyle \langle \nu \rangle \sim {\sqrt {N}}}$
Bastolla/ Parisi [10]	1998	faster than a power law, ${\displaystyle \langle A\rangle >N^{x}\forall x}$	faster than a power law, ${\displaystyle \langle \nu \rangle >N^{x}\forall x}$	first numerical evidences
Bilke/ Sjunnesson [11]	2002		linear with system size, ${\displaystyle \langle \nu \rangle \sim N}$
Socolar/Kauffman [12]	2003		faster than linear, ${\displaystyle \langle \nu \rangle >N^{x}}$ with ${\displaystyle x>1}$
Samuelsson/Troein [13]	2003		superpolynomial growth, ${\displaystyle \langle \nu \rangle >N^{x}\forall x}$	mathematical proof
Mihaljev/Drossel [14]	2005	faster than a power law, ${\displaystyle \langle A\rangle >N^{x}\forall x}$	faster than a power law, ${\displaystyle \langle \nu \rangle >N^{x}\forall x}$
Fink/Sheldon [15]	2023		exponential growth, ${\displaystyle \langle \nu \rangle =(2/{\sqrt {e}})^{N}}$	mathematical proof

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 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 (${\displaystyle K_{c}}$), 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 (${\displaystyle N}$) in the network.

For N-K-model ^[16] the network is stable if ${\displaystyle K<K_{c}}$, critical if ${\displaystyle K=K_{c}}$, and unstable if ${\displaystyle K>K_{c}}$.

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

If ${\displaystyle p_{i}=p=const.}$ for every node, the transition between the stable and chaotic range depends on ${\displaystyle p}$. According to Bernard Derrida and Yves Pomeau ^[17] , the critical value of the average number of connections is ${\displaystyle K_{c}=1/[2p(1-p)]}$.

If ${\displaystyle K}$ is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by ${\displaystyle \langle K^{in}\rangle }$ ^{[18] [19] [20]} The network is stable if ${\displaystyle \langle K^{in}\rangle <K_{c}}$, critical if ${\displaystyle \langle K^{in}\rangle =K_{c}}$, and unstable if ${\displaystyle \langle K^{in}\rangle >K_{c}}$.

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: ${\displaystyle P(K)\propto K^{-\gamma }}$, and ${\displaystyle \langle K^{in}\rangle =\langle K^{out}\rangle }$, since every out-link from a node is an in-link to another. ^[21]

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks, ${\displaystyle q_{i}=2p_{i}(1-p_{i})}$. In the general case, stability of the network is governed by the largest eigenvalue ${\displaystyle \lambda _{Q}}$ of matrix ${\displaystyle Q}$, where ${\displaystyle Q_{ij}=q_{i}A_{ij}}$, and ${\displaystyle A}$ is the adjacency matrix of the network. ^[22] The network is stable if ${\displaystyle \lambda _{Q}<1}$, critical if ${\displaystyle \lambda _{Q}=1}$, unstable if ${\displaystyle \lambda _{Q}>1}$.

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. ^[23] One can distinguish the case where only in-degree distribution in power-law distributed, ^[24] 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, ^[25] different alternatives have been introduced. A common classification ^[26] 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. ^[27]
• 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) ^{[28] [29] [30] [31]} 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. ^{[32] [33]}

Application of Boolean Networks

Classification

• The Scalable Optimal Bayesian Classification ^[34] 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

• NK model

References

1. Naldi, A.; Monteiro, P. T.; Mussel, C.; Kestler, H. A.; Thieffry, D.; Xenarios, I.; Saez-Rodriguez, J.; Helikar, T.; Chaouiya, C. (25 January 2015). "Cooperative development of logical modelling standards and tools with CoLoMoTo". Bioinformatics. 31 (7): 1154–1159. doi: 10.1093/bioinformatics/btv013 . PMID   25619997.
2. 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. arXiv: q-bio/0311019 . Bibcode:2003JThBi.223....1A. CiteSeerX   10.1.1.13.3370 . doi:10.1016/S0022-5193(03)00035-3. PMC   6388622 . PMID   12782112.
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.
4. Drossel, Barbara (December 2009). "Random Boolean Networks". In Schuster, Heinz Georg (ed.). Chapter 3. Random Boolean Networks. Reviews of Nonlinear Dynamics and Complexity. Wiley. pp. 69–110. arXiv: 0706.3351 . doi:10.1002/9783527626359.ch3. ISBN   9783527626359. S2CID   119300231.
5. 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. PMID   5343519. S2CID   4179318.
6. Aldana, Maximo; Coppersmith, Susan; Kadanoff, Leo P. (2003). "Boolean Dynamics with Random Couplings". Perspectives and Problems in Nonlinear Science: A Celebratory Volume in Honor of Lawrence Sirovich. pp. 23–89. arXiv: nlin/0204062 . doi:10.1007/978-0-387-21789-5_2. ISBN   978-1-4684-9566-9. S2CID   15024306.
7. Gershenson, Carlos (2004). "Introduction to Random Boolean Networks". In Bedau, M., P. Husbands, T. Hutton, S. Kumar, and H. Suzuki (Eds.) Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX). Pp. 2004: 160–173. arXiv: nlin.AO/0408006 . Bibcode:2004nlin......8006G.
8. 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. Archived from the original on 4 February 2023. Retrieved 12 January 2016.
9. Greil, Florian (2012). "Boolean Networks as Modeling Framework". Frontiers in Plant Science. 3: 178. doi: 10.3389/fpls.2012.00178 . PMC   3419389 . PMID   22912642.
10. 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. S2CID   1585753.
11. Bilke, Sven; Sjunnesson, Fredrik (December 2001). "Stability of the Kauffman model". Physical Review E. 65 (1): 016129. arXiv: cond-mat/0107035 . Bibcode:2001PhRvE..65a6129B. doi:10.1103/PhysRevE.65.016129. PMID   11800758. S2CID   2470586.
12. Socolar, J.; Kauffman, S. (February 2003). "Scaling in Ordered and Critical Random Boolean Networks". Physical Review Letters. 90 (6): 068702. arXiv: cond-mat/0212306 . Bibcode:2003PhRvL..90f8702S. doi:10.1103/PhysRevLett.90.068702. PMID   12633339. S2CID   14392074.
13. Samuelsson, Björn; Troein, Carl (March 2003). "Superpolynomial Growth in the Number of Attractors in Kauffman Networks". Physical Review Letters. 90 (9): 098701. Bibcode:2003PhRvL..90i8701S. doi:10.1103/PhysRevLett.90.098701. PMID   12689263.
14. Mihaljev, Tamara; Drossel, Barbara (October 2006). "Scaling in a general class of critical random Boolean networks". Physical Review E. 74 (4): 046101. arXiv: cond-mat/0606612 . Bibcode:2006PhRvE..74d6101M. doi:10.1103/PhysRevE.74.046101. PMID   17155127. S2CID   17739744.
15. Fink, T; Sheldon, F (December 2023). "Number of attractors in the critical Kauffman model is exponential". Physical Review Letters. 131: 267402. doi:10.1103/PhysRevLett.131.267402.
16. Kauffman, S. A. (1969). "Metabolic stability and epigenesis in randomly constructed genetic nets". Journal of Theoretical Biology. 22 (3): 437–467. Bibcode:1969JThBi..22..437K. doi:10.1016/0022-5193(69)90015-0. PMID   5803332.
17. Derrida, B; Pomeau, Y (1986-01-15). "Random Networks of Automata: A Simple Annealed Approximation". Europhysics Letters. 1 (2): 45–49. Bibcode:1986EL......1...45D. doi:10.1209/0295-5075/1/2/001. S2CID   160018158. Archived from the original on 2020-05-17. Retrieved 2016-01-12.
18. Solé, Ricard V.; Luque, Bartolo (1995-01-02). "Phase transitions and antichaos in generalized Kauffman networks". Physics Letters A. 196 (5–6): 331–334. Bibcode:1995PhLA..196..331S. doi:10.1016/0375-9601(94)00876-Q.
19. 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. Bibcode:1997PhRvE..55..257L. doi:10.1103/PhysRevE.55.257.
20. 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. Bibcode:2001Chaos..11..809F. doi: 10.1063/1.1414882 . ISSN   1054-1500. PMID   12779520.
21. Aldana, Maximino; Cluzel, Philippe (2003-07-22). "A natural class of robust networks". Proceedings of the National Academy of Sciences. 100 (15): 8710–8714. Bibcode:2003PNAS..100.8710A. doi: 10.1073/pnas.1536783100 . ISSN   0027-8424. PMC   166377 . PMID   12853565.
22. 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. arXiv: 0901.4362 . Bibcode:2009PNAS..106.8209P. doi: 10.1073/pnas.0900142106 . ISSN   0027-8424. PMC   2688895 . PMID   19416903.
23. Aldana, Maximino (October 2003). "Boolean dynamics of networks with scale-free topology". Physica D: Nonlinear Phenomena. 185 (1): 45–66. arXiv: cond-mat/0209571 . Bibcode:2003PhyD..185...45A. doi:10.1016/s0167-2789(03)00174-x.
24. Drossel, Barbara; Greil, Florian (4 August 2009). "Critical Boolean networks with scale-free in-degree distribution". Physical Review E. 80 (2): 026102. arXiv: 0901.0387 . Bibcode:2009PhRvE..80b6102D. doi:10.1103/PhysRevE.80.026102. PMID   19792195. S2CID   2487442.
25. Harvey, Imman; Bossomaier, Terry (1997). "Time out of joint: Attractors in asynchronous random Boolean networks". In Husbands, Phil; Harvey, Imman (eds.). Proceedings of the Fourth European Conference on Artificial Life (ECAL97). MIT Press. pp. 67–75. ISBN   9780262581578. Archived from the original on 2023-02-04. Retrieved 2020-09-16.
26. Gershenson, Carlos (2002). "Classification of Random Boolean Networks". In Standish, Russell K; Bedau, Mark A (eds.). Proceedings of the Eighth International Conference on Artificial Life. Artificial Life. Vol. 8. Cambridge, Massachusetts, USA. pp. 1–8. arXiv: cs/0208001 . Bibcode:2002cs........8001G. ISBN   9780262692816. Archived from the original on 4 February 2023. Retrieved 12 January 2016.
27. Gershenson, Carlos; Broekaert, Jan; Aerts, Diederik (14 September 2003). "Contextual Random Boolean Networks". Advances in Artificial Life[7th European Conference, ECAL 2003]. Lecture Notes in Computer Science. Vol. 2801. Dortmund, Germany. pp. 615–624. arXiv: nlin/0303021 . doi:10.1007/978-3-540-39432-7_66. ISBN   978-3-540-39432-7. S2CID   4309400.
28. Imani, M.; Braga-Neto, U. M. (2017-01-01). "Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems". IEEE Transactions on Signal Processing. 65 (2): 359–371. arXiv: 1702.07269 . Bibcode:2017ITSP...65..359I. doi:10.1109/TSP.2016.2614798. ISSN   1053-587X. S2CID   178376.
29. Imani, M.; Braga-Neto, U. M. (2015). "Optimal state estimation for boolean dynamical systems using a boolean Kalman smoother". 2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP). pp. 972–976. doi:10.1109/GlobalSIP.2015.7418342. ISBN   978-1-4799-7591-4. S2CID   8672734.
30. Imani, M.; Braga-Neto, U. M. (2016). 2016 American Control Conference (ACC). pp. 227–232. doi:10.1109/ACC.2016.7524920. ISBN   978-1-4673-8682-1. S2CID   7210088.
31. Imani, M.; Braga-Neto, U. (2016-12-01). "Point-based value iteration for partially-observed Boolean dynamical systems with finite observation space". 2016 IEEE 55th Conference on Decision and Control (CDC). pp. 4208–4213. doi:10.1109/CDC.2016.7798908. ISBN   978-1-5090-1837-6. S2CID   11341805.
32. Zhang, Rui; Cavalcante, Hugo L. D. de S.; Gao, Zheng; Gauthier, Daniel J.; Socolar, Joshua E. S.; Adams, Matthew M.; Lathrop, Daniel P. (2009). "Boolean chaos". Physical Review E. 80 (4): 045202. arXiv: 0906.4124 . Bibcode:2009PhRvE..80d5202Z. doi:10.1103/PhysRevE.80.045202. ISSN   1539-3755. PMID   19905381. S2CID   43022955.
33. Cavalcante, Hugo L. D. de S.; Gauthier, Daniel J.; Socolar, Joshua E. S.; Zhang, Rui (2010). "On the origin of chaos in autonomous Boolean networks". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 368 (1911): 495–513. arXiv: 0909.2269 . Bibcode:2010RSPTA.368..495C. doi:10.1098/rsta.2009.0235. ISSN   1364-503X. PMID   20008414. S2CID   426841.
34. 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