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Dynamical regimes in non-ergodic random Boolean networks

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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.

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Notes

  1. An attractor basin is the set of states whose evolution lead to the attractor, its size (or dimension) being the cardinality of the set.

  2. We remind that in disordered systems trajectories starting from nearby points typically lead to different attractors.

  3. The estimation is performed on many different initial conditions.

  4. In this case, the averages are around 1, as the ensemble considered is the first historical example of critical systems.

  5. An odd number of nodes avoids the cases with equal quantity of 0s and 1s.

  6. For the reasons discussed above.

  7. See (Serra et al. 2007b) for details on the calculation of these coefficients.

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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.

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Authors and Affiliations

  1. Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, Via Campi 213a, 41125, Modena, Italy

    Marco Villani, Davide Campioli & Roberto Serra

  2. European Centre for Living Technology, Ca’ Minich, S. Marco 2940, 30124, Venetia, Italy

    Marco Villani, Alessandro Filisetti & Roberto Serra

  3. Department of Informatics, Systems and Communication, University Milano-Bicocca, Viale Sarca 336, 20126, Milan, Italy

    Chiara Damiani

  4. Department of Computer Science and Engineering (DISI), Università di Bologna, Via Venezia 52, 47521, Cesena, Italy

    Andrea Roli

  5. Explora Srl., Via G Peroni 386, 00131, Rome, Italy

    Alessandro Filisetti

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  1. Marco Villani
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  2. Davide Campioli
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  3. Chiara Damiani
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  4. Andrea Roli
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  6. Roberto Serra
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Correspondence to Marco Villani.

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Villani, M., Campioli, D., Damiani, C. et al. Dynamical regimes in non-ergodic random Boolean networks. Nat Comput 16, 353–363 (2017). https://doi.org/10.1007/s11047-016-9552-7

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  • DOI: https://doi.org/10.1007/s11047-016-9552-7

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