Dynamical regimes in non-ergodic random Boolean networks
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.
KeywordsRandom Boolean networks Dynamical regimes Attractors Dynamical measures Gene knock-out
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.
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.
- 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
- Drossel B (2008) Random boolean networks. In: Schuster HG (ed) Reviews of nonlinear dynamics and complexity, vol 1. Wiley, New YorkGoogle Scholar
- Fretter C, Szejka A, Drossel B (2009) Perturbation propagation in random and evolved boolean networks. N J Phys 11(0905):0646Google Scholar
- 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
- Kauffman SA (1971) Gene regulation networks: a theory of their global structure and behaviour. Top Dev Biol 6:145–182Google Scholar
- Kauffman SA (1993) The origins of order. Oxford University Press, OxfordGoogle Scholar
- Kauffman SA (1995) At home in the universe. Oxford University Press, OxfordGoogle Scholar
- 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
- 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
- Serra R, Villani M, Ingrami P, SAK, (2006) Coupled random boolean network forming an artificial tissue. In: LNCS 4173, pp 548–556Google Scholar
- 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
- 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
- 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
- Shmulevich I, Kauffman S (2004) Activities and sensitivities in boolean network models. Phys Rev Lett 93(048701):1–4Google Scholar
- 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