Natural Computing

, Volume 16, Issue 2, pp 353–363 | Cite as

Dynamical regimes in non-ergodic random Boolean networks

  • Marco Villani
  • Davide Campioli
  • Chiara Damiani
  • Andrea Roli
  • Alessandro Filisetti
  • Roberto Serra
Article

Abstract

Random boolean networks are a model of genetic regulatory networks that has proven able to describe experimental data in biology. Random boolean networks not only reproduce important phenomena in cell dynamics, but they are also extremely interesting from a theoretical viewpoint, since it is possible to tune their asymptotic behaviour from order to disorder. The usual approach characterizes network families as a whole, either by means of static or dynamic measures. We show here that a more detailed study, based on the properties of system’s attractors, can provide information that makes it possible to predict with higher precision important properties, such as system’s response to gene knock-out. A new set of principled measures is introduced, that explains some puzzling behaviours of these networks. These results are not limited to random Boolean network models, but they are general and hold for any discrete model exhibiting similar dynamical characteristics.

Keywords

Random Boolean networks Dynamical regimes Attractors Dynamical measures Gene knock-out 

Notes

Acknowledgments

We are deeply indebted to Stuart Kauffman for his inspiring ideas and for several discussions on various aspects of random Boolean networks. We also gratefully acknowledge useful discussions with David Lane and Alex Graudenzi.

Authors’ contributions

MV and RS conceived and designed the experiments and provided a first analysis of the results. DC developed the code and performed the experiments. CD performed further experiments. MV, RS, CD, AR and AF analysed and discussed the results. AR, MV, CD, and AF wrote the paper. All authors gave final approval for publication.

References

  1. Aldana M (2003) Boolean dynamics of networks with scale-free topology. Phys D 185(1):45–66MathSciNetCrossRefMATHGoogle Scholar
  2. Aldana M, Coppersmith S, Kadanoff LP (2003) Boolean dynamics with random couplings. In: Kaplan E, Marsden J, Sreenivasan KR (eds) Perspectives and problems in nonlinear science. Springer Applied Mathematical Sciences Series, Berlin, pp 23–90CrossRefGoogle Scholar
  3. Bagnoli F, Rechtman R, Ruffo S (1992) Damage spreading and lyapunov exponents in cellular automata. Phys Lett A 172(12):34–38CrossRefMATHGoogle Scholar
  4. Bastolla U, Parisi G (1998a) The modular structure of Kauffman networks. Phys D 115(3–4):219–233CrossRefMATHGoogle Scholar
  5. Bastolla U, Parisi G (1998b) Relevant elements, magnetization and dynamical properties in Kauffman networks: a numerical study. Phys D 115(3–4):203–218CrossRefMATHGoogle Scholar
  6. Benedettini S, Villani M, Roli A, Serra R, Manfroni M, Gagliardi A, Pinciroli C, Birattari M (2013) Dynamical regimes and learning properties of evolved boolean networks. Neurocomputing 99:111–123CrossRefGoogle Scholar
  7. Bornholdt S (2008) Boolean network models of cellular regulation: prospects and limitations. J R Soc Interface 5:S85–S94CrossRefGoogle Scholar
  8. Campioli D, Villani M, Poli I, Serra R (2011) Dynamical stability in random boolean networks. In: Apolloni B, Bassis S, Esposito A, Morabito FC (eds) Frontiers in Artificial Intelligence and Applications, WIRN, vol 234, 120th edn. IOS Press, Amsterdam Google Scholar
  9. Cheng X, Sun M, Socolar J (2012) Autonomous boolean modelling of developmental gene regulatory networks. J R Soc Interface 10:1–12CrossRefGoogle Scholar
  10. Damiani C, Serra R, Villani M, Kauffman S, Colacci A (2011) Cell-cell interaction and diversity of emergent behaviours. Syst Biol IET 5(2):137–144CrossRefGoogle Scholar
  11. Derrida B, Pomeau Y (1986) Random networks of automata: a simple annealed approximation. Europhys Lett 1 1(2):45–49CrossRefGoogle Scholar
  12. Derrida B, Weisbuch G (1986) Evolution of overlaps between configurations in random boolean networks. J Phys 47:1297–1303CrossRefGoogle Scholar
  13. Drossel B (2005) Number of attractors in random boolean networks. Phys Rev E 72(1):016110MathSciNetCrossRefGoogle Scholar
  14. Drossel B (2008) Random boolean networks. In: Schuster HG (ed) Reviews of nonlinear dynamics and complexity, vol 1. Wiley, New YorkGoogle Scholar
  15. Fretter C, Szejka A, Drossel B (2009) Perturbation propagation in random and evolved boolean networks. N J Phys 11(0905):0646Google Scholar
  16. Hughes T, Marton M, Jones A, Roberts C, Stoughton R, Armour C, Bennett H, Coffey E, Dai H, He Y, Kidd M, King A, Meyer M, Slade D, Lum P, Stepaniants S, Shoemaker D, Gachotte D, Chakraburtty K, Simon J, Bard M, Friend S (2000) Functional discovery via a compendium of expression profiles. Cell 102(1):109–126CrossRefGoogle Scholar
  17. Kauffman SA (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 22(3):437–467MathSciNetCrossRefGoogle Scholar
  18. Kauffman SA (1971) Gene regulation networks: a theory of their global structure and behaviour. Top Dev Biol 6:145–182Google Scholar
  19. Kauffman SA (1993) The origins of order. Oxford University Press, OxfordGoogle Scholar
  20. Kauffman SA (1995) At home in the universe. Oxford University Press, OxfordGoogle Scholar
  21. Luque B, Solé RV (2000) Lyapunov exponents in random boolean networks. Phys A 284:33–45CrossRefGoogle Scholar
  22. Mesot B, Teuscher C (2005) Deducing local rules for solving global tasks with random boolean networks. Phys D 211(12):88–106MathSciNetCrossRefMATHGoogle Scholar
  23. Moreira A, Amaral L (2005) Canalizing Kauffman networks: nonergodicity and its effect on their critical behavior. Phys Rev Lett 94(21):218702CrossRefGoogle Scholar
  24. Packard NH (1988) Adaptation toward the edge of chaos. In: Kelso JAS, Mandell AJ, Shlesinger MF (eds) Dynamic patterns in complex systems. World Scientific, Singapore, pp 293–301Google Scholar
  25. Ramo P, Kesseli J, Yli-Harja O (2006) Perturbation avalanches and criticality in gene regulatory networks. J Theor Biol 242(1):164–170MathSciNetCrossRefMATHGoogle Scholar
  26. Serra R, Villani M (2002) Perturbing the regular topology of cellular automata: Implications for the dynamics. In: Proceedings of the 5th international conference on cellular automata for research and industry, Springer, London, ACRI’01, pp 168–177Google Scholar
  27. Serra R, Villani M, Semeria A (2004) Genetic network models and statistical properties of gene expression data in knock-out experiments. J Theor Biol 227:149–157MathSciNetCrossRefGoogle Scholar
  28. Serra R, Villani M, Ingrami P, SAK, (2006) Coupled random boolean network forming an artificial tissue. In: LNCS 4173, pp 548–556Google Scholar
  29. Serra R, Villani M, Damiani C, Graudenzi A, Colacci A, Kauffman SA (2007a) Interacting random boolean networks. In: Jost J, Helbing D (eds) Proceedings of ECCS07: European Conference on Complex SystemsGoogle Scholar
  30. Serra R, Villani M, Graudenzi A, Kauffman SA (2007b) Why a simple model of genetic regulatory networks describes the distribution of avalanches in gene expression data. J Theor Biol 246(3):449–460MathSciNetCrossRefGoogle Scholar
  31. Serra R, Villani M, Damiani C, Graudenzi A, Colacci A (2008a) The diffusion of perturbations in a model of coupled random boolean networks. In: Umeo H, Morishiga S, Nishinari K, Komatsuzaki T, Banidini S (eds) Cellular Automata (proceedings of 8th International Conference on Cellular Auotomata ACRI 2008, Yokohama, September 2008). Springer Lecture Notes in Computer Science, Berlin, vol 5191, pp 315– 322. ISBN: 0302-9743Google Scholar
  32. Serra R, Villani M, Graudenzi A, Colacci A, Kauffman SA (2008b) The simulation of gene knock-out in scale-free random boolean models of genetic networks. Netw Heterog Media 2(3):333–343MathSciNetCrossRefMATHGoogle Scholar
  33. Serra R, Graudenzi A, Villani M (2009) Genetic regulatory networks and neural networks. In: New Directions in Neural Networks—18th Italian Workshop on Neural Networks: WIRN, pp 109–117Google Scholar
  34. Serra R, Villani M, Barbieri A, Kauffman S, Colacci A (2010) On the dynamics of random Boolean networks subject to noise: attractors, ergodic sets and cell types. J Theor Biol 265(2):185–193. doi:10.1016/j.jtbi.2010.04.012 MathSciNetCrossRefGoogle Scholar
  35. Shmulevich I, Kauffman S (2004) Activities and sensitivities in boolean network models. Phys Rev Lett 93(048701):1–4Google Scholar
  36. Shmulevich I, Dougherty E, Kim S, Zhang W (2002) Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks. Bioinformatics 18(2):261–274CrossRefGoogle Scholar
  37. Shmulevich I, Kauffman SA, Aldana M (2005) Eukaryotic cells are dynamically ordered or critical but not chaotic. PNAS 102(38):13439–13444CrossRefGoogle Scholar
  38. Socolar JES, Kauffman SA (2003) Scaling in ordered and critical random boolean networks. Phys Rev Lett 90(6):068702CrossRefGoogle Scholar
  39. Szejka A, Mihaljev T, Drossel B (2008) The phase diagram of random threshold networks. New J Phys 10(6):063009CrossRefGoogle Scholar
  40. Villani M, Serra R (2014) Attractors perturbations in biological modelling: avalanches and cellular differentiation. In: Cagnoni S, Mirolli M, Villani M (eds) Evolution, complexity and artificial life. Springer, Berlin, pp 59–76CrossRefGoogle Scholar
  41. Villani M, Barbieri A, Serra R (2011) A dynamical model of genetic networks for cell differentiation. PloS one 6(3):e17703CrossRefGoogle Scholar
  42. Villani M, Serra R, Barbieri A, Roli A, Kauffman S, Colacci A (2013) The influence of the introduction of a semi-permeable membrane in a stochastic model of catalytic reaction networks. In: ECCS 2013, European Conference on Complex Systems (poster presentation)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Marco Villani
    • 1
    • 2
  • Davide Campioli
    • 1
  • Chiara Damiani
    • 3
  • Andrea Roli
    • 4
  • Alessandro Filisetti
    • 2
    • 5
  • Roberto Serra
    • 1
    • 2
  1. 1.Department of Physics, Informatics and MathematicsUniversity of Modena and Reggio EmiliaModenaItaly
  2. 2.European Centre for Living Technology, Ca’ MinichVenetiaItaly
  3. 3.Department of Informatics, Systems and CommunicationUniversity Milano-BicoccaMilanItaly
  4. 4.Department of Computer Science and Engineering (DISI)Università di BolognaCesenaItaly
  5. 5.Explora Srl.RomeItaly

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