Computational Statistics

, Volume 28, Issue 1, pp 19–36 | Cite as

Attractors in Boolean networks: a tutorial

  • Martin Hopfensitz
  • Christoph Müssel
  • Markus Maucher
  • Hans A. Kestler
Original Paper


Boolean networks are a popular class of models for the description of gene-regulatory networks. They model genes as simple binary variables, being either expressed or not expressed. Simulations of Boolean networks can give insights into the dynamics of cellular systems. In particular, stable states and cycles in the networks are thought to correspond to phenotypes. This paper presents approaches to identify attractors in synchronous, asynchronous and probabilistic Boolean networks and gives examples of their usage in the BoolNet R package.


Systems biology Boolean networks Attractors 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Albert R, Othmer H (2003) The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster. J Theor Biol 223(1): 1–18MathSciNetCrossRefGoogle Scholar
  2. Aldana M (2003) Boolean dynamics of networks with scale-free topology. Physica D: Nonlinear Phenomena 185(1): 45–66MathSciNetzbMATHCrossRefGoogle Scholar
  3. Bornholdt S (2005) Systems biology. Less is more in modeling large genetic networks. Science 310(5747): 449–451CrossRefGoogle Scholar
  4. Bornholdt S (2008) Boolean network models of cellular regulation: prospects and limitations. J R Soc Interface 5(Suppl. 1): S85–S94CrossRefGoogle Scholar
  5. Calzone L, Tournier L, Fourquet S, Thieffry D, Zhivotovsky B, Barillot E, Zinovyev A (2010) Mathematical modelling of cell-fate decision in response to death receptor engagement. PLoS Comput Biol 6(3): e1000702MathSciNetCrossRefGoogle Scholar
  6. de Jong H (2002) Modeling and simulation of genetic regulatory systems: a literature review. J Comput Biol 9(1): 67–103CrossRefGoogle Scholar
  7. Dojer N, Gambin A, Mizera A, Wilczyński B, Tiuryn J (2006) Applying dynamic Bayesian networks to perturbed gene expression data. BMC Bioinform 7: 249CrossRefGoogle Scholar
  8. Fauré A, Naldi A, Chaouiya C, Thieffry D (2006) Dynamical analysis of a generic Boolean model for the control of the mammalian cell cycle. Bioinformatics 22(14): e124–e131CrossRefGoogle Scholar
  9. Garg A, Banerjee D, De Micheli G (2008) Implicit methods for probabilistic modeling of gene regulatory networks. In: Proceedings of the 30th annual international IEEE EMBS conference, pp 4621–4627Google Scholar
  10. Garg A, Di Cara A, Xenarios I, Mendoza L, De Micheli G (2008) Synchronous versus asynchronous modeling of gene regulatory networks. Bioinformatics 24(17): 1917–1925CrossRefGoogle Scholar
  11. Glass L, Kauffman SA (1973) The logical analysis of continuous, non-linear biochemical networks. J Theor Biol 39(1): 103–129CrossRefGoogle Scholar
  12. Goodwin B (1963) Temporal organization in cells: a dynamic theory of cellular control processes. Academic Press, LondonGoogle Scholar
  13. Grzegorczyk M, Husmeier D, Rahnenführer J (2010) Modelling non-stationary dynamic gene regulatory processes with the BGM model. Comput Stat 26(2): 199–218CrossRefGoogle Scholar
  14. Harvey I, Bossomaier T (1997) Time out of joint: attractors in asynchronous random Boolean networks. In: Proceedings of the forth European conference on artificial life, pp 67–75Google Scholar
  15. Huang S (1999) Gene expression profiling, genetic networks, and cellular states: an integrating concept for tumorigenesis and drug discovery. J Mol Med 77(6): 469–480CrossRefGoogle Scholar
  16. Kauffman SA (1969) Metabolic stability and epigensis in randomly constructed genetic nets. J Theor Biol 22(3): 437–467CrossRefGoogle Scholar
  17. Kauffman SA (1993) The origins of order: self-organization and selection in evolution. Oxford University Press, OxfordGoogle Scholar
  18. Kauffman S, Peterson C, Samuelsson B, Troein C (2004) Genetic networks with canalyzing Boolean rules are always stable. PNAS 101(49): 17,102–17,107CrossRefGoogle Scholar
  19. Lähdesmäki H, Shmulevich I, Yli-Harja O (2003) On learning gene regulatory networks under the Boolean network model. Mach Learn 52(1–2): 147–167zbMATHCrossRefGoogle Scholar
  20. Li F, Long T, Ouyang Q, Tang C (2004) The yeast cell-cycle network is robustly designed. PNAS 101: 4781–4786CrossRefGoogle Scholar
  21. Liang S, Fuhrman S, Somogyi R (1998) REVEAL, a general reverse engineering algorithm for inference of genetic network architectures. Pac Symp Biocomput 3: 18–29Google Scholar
  22. Lynch J (1995) On the threshold of chaos in random Boolean cellular automata. Random Struct Algorithms 6(2–3): 239–260MathSciNetzbMATHCrossRefGoogle Scholar
  23. Müssel C, Hopfensitz M, Kestler HA (2010) BoolNet—an R package for generation, reconstruction and analysis of Boolean networks. Bioinformatics 26(10): 1378–1380CrossRefGoogle Scholar
  24. Orlando DA, Lin CY, Bernard A, Wang JY, Socolar JES, Iversen ES, Hartemink AJ, Haase SB (2008) Global control of cell-cycle transcription by coupled CDK and network oscillators. Nature 453(7197): 944–947CrossRefGoogle Scholar
  25. Sahin O, Fröhlich H, Löbke C, Korf U, Burmester S, Majety M, Mattern J, Schupp I, Chaouiya C, Thieffry D, Poustka A, Wiemann S, Beissbarth T, Arlt D (2009) Modeling ERBB receptor-regulated G1/S transition to find novel targets for de novo trastuzumab resistance. BMC Syst Biol 3(1): 1CrossRefGoogle Scholar
  26. Samuelsson B, Troein C (2003) Superpolynomial growth in the number of attractors in Kauffman networks. Phys Rev Lett 90(9): 098701MathSciNetCrossRefGoogle Scholar
  27. Shmulevich I, Dougherty ER, Kim S, Zhang W (2002) Probabilistic Boolean networks: a rule-based uncertainty model for gene-regulatory networks. Bioinformatics 18(2): 261–274CrossRefGoogle Scholar
  28. Socolar JE, Kauffman SA (2003) Scaling in ordered and critical random Boolean networks. Phys Rev Lett 90(6): 068–702CrossRefGoogle Scholar
  29. Thomas R (1991) Regulatory networks seen as asynchronous automata: a logical description. J Theor Biol 153(1): 1–23CrossRefGoogle Scholar
  30. Wawra C, Kühl M, Kestler HA (2007) Extended analyses of the Wnt/β-catenin pathway: Robustness and oscillatory behaviour. FEBS Lett 581(21): 4043–4048CrossRefGoogle Scholar
  31. Xiao Y, Dougherty ER (2007) The impact of function perturbations in Boolean networks. Bioinformatics 23(10): 1265–1273CrossRefGoogle Scholar
  32. Zhou D, Müssel C, Lausser L, Hopfensitz M, Kühl M, Kestler HA (2009) Boolean networks for modeling and analysis of gene regulation. Ulmer Informatik-Bericht 2009–2010, Ulm UniversityGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Martin Hopfensitz
    • 1
  • Christoph Müssel
    • 1
  • Markus Maucher
    • 1
  • Hans A. Kestler
    • 1
  1. 1.Research Group Bioinformatics and Systems Biology, Institute of Neural Information ProcessingUniversity of UlmUlmGermany

Personalised recommendations