Skip to content


  • Research Article
  • Open Access

Algorithms for Finding Small Attractors in Boolean Networks

  • 1Email author,
  • 2,
  • 2,
  • 1 and
  • 3
EURASIP Journal on Bioinformatics and Systems Biology20072007:20180

  • Received: 29 June 2006
  • Accepted: 13 February 2007
  • Published:


A Boolean network is a model used to study the interactions between different genes in genetic regulatory networks. In this paper, we present several algorithms using gene ordering and feedback vertex sets to identify singleton attractors and small attractors in Boolean networks. We analyze the average case time complexities of some of the proposed algorithms. For instance, it is shown that the outdegree-based ordering algorithm for finding singleton attractors works in time for , which is much faster than the naive time algorithm, where is the number of genes and is the maximum indegree. We performed extensive computational experiments on these algorithms, which resulted in good agreement with theoretical results. In contrast, we give a simple and complete proof for showing that finding an attractor with the shortest period is NP-hard.


  • System Biology
  • Boolean Network
  • Small Attractor


Authors’ Affiliations

Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Bioinformatics Center, Institute for Chemical Research, Kyoto University, Uji Kyoto, 611-0011, Japan
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong


  1. Celis JE, Kruhøffer M, Gromova I, et al.: Gene expression profiling: monitoring transcription and translation products using DNA microarrays and proteomics. FEBS Letters 2000, 480(1):2-16. 10.1016/S0014-5793(00)01771-3View ArticleGoogle Scholar
  2. Hughes TR, Mao M, Jones AR, et al.: Expression profiling using microarrays fabricated by an ink-jet oligonucleotide synthesizer. Nature Biotechnology 2001, 19(4):342-347. 10.1038/86730View ArticleGoogle Scholar
  3. Lipshutz RJ, Fodor SPA, Gingeras TR, Lockhart DJ: High density synthetic oligonucleotide arrays. Nature Genetics 1999, 21(supplement 1):20-24.View ArticleGoogle Scholar
  4. Lockhart DJ, Winzeler EA: Genomics, gene expression and DNA arrays. Nature 2000, 405(6788):827-836. 10.1038/35015701View ArticleGoogle Scholar
  5. Jong HD: Modeling and simulation of genetic regulatory systems: a literature review. Journal of Computational Biology 2002, 9(1):67-103. 10.1089/10665270252833208View ArticleGoogle Scholar
  6. Glass K, Kauffman SA: The logical analysis of continuous, nonlinear biochemical control networks. Journal of Theoretical Biology 1973, 39(1):103-129. 10.1016/0022-5193(73)90208-7View ArticleGoogle Scholar
  7. Kauffman SA: Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology 1969, 22(3):437-467. 10.1016/0022-5193(69)90015-0View ArticleGoogle Scholar
  8. Kauffman SA: Homeostasis and differentiation in random genetic control networks. Nature 1969, 224(215):177-178. 10.1038/224177a0View ArticleGoogle Scholar
  9. Kauffman SA: The large scale structure and dynamics of genetic control circuits: an ensemble approach. Journal of Theoretical Biology 1974, 44(1):167-190. 10.1016/S0022-5193(74)80037-8View ArticleMathSciNetGoogle Scholar
  10. Huang S: Gene expression profiling, genetic networks, and cellular states: an integrating concept for tumorigenesis and drug discovery. Journal of Molecular Medicine 1999, 77(6):469-480. 10.1007/s001099900023View ArticleGoogle Scholar
  11. Kauffman SA: The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, New York, NY, USA; 1993.Google Scholar
  12. Somogyi R, Sniegoski C: Modeling the complexity of genetic networks: understanding multigenic and pleiotropic regulation. Complexity 1996, 1(6):45-63.View ArticleMathSciNetGoogle Scholar
  13. Shmulevich I, Zhang W: Binary analysis and optimization-based normalization of gene expression data. Bioinformatics 2002, 18(4):555-565. 10.1093/bioinformatics/18.4.555View ArticleGoogle Scholar
  14. Thieffry D, Huerta AM, Pérez-Rueda E, Collado-Vides J: From specific gene regulation to genomic networks: a global analysis of transcriptional regulation in Escherichia coli . BioEssays 1998, 20(5):433-440. 10.1002/(SICI)1521-1878(199805)20:5<433::AID-BIES10>3.0.CO;2-2View ArticleGoogle Scholar
  15. Huang S: Cell state dynamics and tumorigenesis in Boolean regulatory networks. InterJournal Genetics MS: 416 []
  16. Drossel B: Number of attractors in random Boolean networks. Physical Review E 2005., 72(1): 5 pagesGoogle Scholar
  17. Drossel B, Mihaljev T, Greil F: Number and length of attractors in a critical Kauffman model with connectivity one. Physical Review Letters 2005., 94(8): 4 pagesGoogle Scholar
  18. Samuelsson B, Troein C: Superpolynomial growth in the number of attractors in Kauffman networks. Physical Review Letters 2003., 90(9): 4 pagesGoogle Scholar
  19. Socolar JES, Kauffman SA: Scaling in ordered and critical random Boolean networks. Physical Review Letters 2003., 90(6): 4 pagesGoogle Scholar
  20. Devloo V, Hansen P, Labbé M: Identification of all steady states in large networks by logical analysis. Bulletin of Mathematical Biology 2003, 65(6):1025-1051. 10.1016/S0092-8240(03)00061-2View ArticleGoogle Scholar
  21. Mochizuki A: An analytical study of the number of steady states in gene regulatory networks. Journal of Theoretical Biology 2005, 236(3):291-310. 10.1016/j.jtbi.2005.03.015View ArticleMathSciNetGoogle Scholar
  22. Pal R, Ivanov I, Datta A, Bittner ML, Dougherty ER: Generating Boolean networks with a prescribed attractor structure. Bioinformatics 2005, 21(21):4021-4025. 10.1093/bioinformatics/bti664View ArticleGoogle Scholar
  23. Zhou X, Wang X, Pal R, Ivanov I, Bittner M, Dougherty ER: A Bayesian connectivity-based approach to constructing probabilistic gene regulatory networks. Bioinformatics 2004, 20(17):2918-2927. 10.1093/bioinformatics/bth318View ArticleGoogle Scholar
  24. Akutsu T, Kuhara S, Maruyama O, Miyano S: A system for identifying genetic networks from gene expression patterns produced by gene disruptions and overexpressions. Genome Informatics 1998, 9: 151-160.Google Scholar
  25. Milano M, Roli A: Solving the satisfiability problem through Boolean networks. Proceedings of the 6th Congress of the Italian Association for Artificial Intelligence on Advances in Artificial Intelligence, Lecture Notes in Artificial Intelligence, Springer, Bologna, Italy, September 1999 1792: 72-83.MathSciNetGoogle Scholar
  26. Garey MR, Johnson DS: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York, NY, USA; 1979.MATHGoogle Scholar
  27. Even G, Naor J, Schieber B, Sudan M: Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 1998, 20(2):151-174. 10.1007/PL00009191View ArticleMathSciNetMATHGoogle Scholar
  28. Barabási A-L, Albert R: Emergence of scaling in random networks. Science 1999, 286(5439):509-512. 10.1126/science.286.5439.509View ArticleMathSciNetGoogle Scholar
  29. Guelzim N, Bottani S, Bourgine P, Képès F: Topological and causal structure of the yeast transcriptional regulatory network. Nature Genetics 2002, 31(1):60-63. 10.1038/ng873View ArticleGoogle Scholar
  30. Akutsu T, Miyano S, Kuhara S: Identification of genetic networks from a small number of gene expression patterns under the Boolean network model. Proceedings of the 4th Pacific Symposium on Biocomputing (PSB '99), Big Island of Hawaii, Hawaii, USA, January 1999 4: 17-28.Google Scholar
  31. Akutsu T, Kuhara S, Maruyama O, Miyano S: Identification of genetic networks by strategic gene disruptions and gene overexpressions under a Boolean model. Theoretical Computer Science 2003, 298(1):235-251. 10.1016/S0304-3975(02)00425-5View ArticleMathSciNetMATHGoogle Scholar
  32. Akutsu T, Hayashida M, Ching W-K, Ng MK: Control of Boolean networks: hardness results and algorithms for tree structured networks. Journal of Theoretical Biology 2007, 244(4):670-679. 10.1016/j.jtbi.2006.09.023View ArticleMathSciNetGoogle Scholar


© Shu-Qin Zhang et al. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.