Event Driven Approach for Simulating Gene Regulation Networks

  • Marco Berardi
  • Nicoletta Del Buono
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8584)


Gene regulatory networks can be described by continuous models in which genes are acting directly on each other. Genes are activated or inhibited by transcription factors which are direct gene products. The action of a transcription factor on a gene is modeled as a binary on-off response function around a certain threshold concentration. Different thresholds can regulate the behaviors of genes, so that the combined effect on a gene is generally assumed to obey Boolean-like composition rules. Analyzing the behavior of such network model is a challenging task in mathematical simulation, particularly when at least one variable is close to one of its thresholds, called switching domains. In this paper, we briefly review a particular class model for gene regulation networks, namely, the piece-wise linear model and we present an event-driven method to analyze the motion in switching domains.


Gene regulatory networks piecewise-linear differential equation event-driven method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and electronics. Lecture Notes in Applied and Computations Mechanics, vol. 35. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  2. 2.
    Aihara, K., Suzuki, H.: Theory of hybrid dynamical systems and its applications to biological and medical systems. Phil. Trans. R. Soc. A 368(1930), 4893–4914 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Berardi, M.: Rosenbrock-type methods applied to discontinuous differential systems. Mathematics and Computers in Simulation 95, 229–243 (2014)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Berardi, M., Lopez, L.: On the continuous extension of Adams-Bashforth methods and the event location in discontinuous ODEs. Applied Methematics Letters 25, 995–999 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    D’Abbicco, M., Calamita, G., Del Buono, N., Berardi, M., Gena, P., Lopez, L.: A model for the Hepatic Glucose Metabolism based on Hill and Step Functions (Preprint 2014)Google Scholar
  6. 6.
    de Jong, H., Geiselmann, J., Hernandez, C., Page, M.: Genetic Network Analyzer: qualitative simulation of genetic regulatory networks. Bioinformatics 19(3), 336–344 (2003)CrossRefGoogle Scholar
  7. 7.
    de Jong, H.: Modeling and Simulation of Genetic Regulatory Systems: A Literature Review. J. of Comp. Biol. 9(1), 67–103 (2002)CrossRefGoogle Scholar
  8. 8.
    Del Buono, N., Elia, C., Lopez, L.: On the equivalence between the sigmoidal approach and Utkin’s approach for piecewise-linear models of gene regulatory networks. To appear on SIAM J. Appl. Dynam. Syst. (2014)Google Scholar
  9. 9.
    Dieci, L., Elia, C., Lopez, L.: Sharp sufficient attractivity conditions for sliding on a codimension 2 discontinuity surface. To appear on: Mathematics and Computers in simulations (2014), doi:, ISSN: 1872-7166
  10. 10.
    Dieci, L., Elia, C., Lopez, L.: A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis. J. of Differential Equations 254, 1800–1832 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dieci, L., Lopez, L.: Numerical Solution of Discontinuous Differential Systems: Approaching the Discontinuity Surface from One-Side. Applied Numerical Mathematics 67, 98–110 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Dieci, L., Lopez, L.: A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side. Journal of Computational and Applied Mathematics 236(16), 3967–3991 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Dieci, L., Lopez, L.: Fundamental matrix solutions of piecewise smooth differential systems. Mathematics and Computers in Simulation 81(5), 932–953 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Dieci, L., Lopez, L.: Sliding motion on discontinuity surfaces of high co-dimension. A construction for selection a Filippov vector field. Numerische Mathematik 117, 779–811 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Dieci, L., Lopez, L.: On Filippov and Utkin Sliding Solution of Discontinuous Systems. In: DeBernardis, E., Spigler, R., Valente, V. (eds.) 9th Conference of the SIMAI, Rome, September 15-19, 2008. Applied and Industrial Mathematics in Italy III, Series on Advances in Mathematics for Applied Sciences, vol. 82, pp. 323–330 (2010)Google Scholar
  16. 16.
    Dieci, L., Lopez, L.: Sliding Motion in Filippov Differential Systems: Theoretical Results and a Computational Approach. SIAM J. Numer. Anal. 47(3), 2023–2051 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Filippov, A.F.: Differential Equations With Discontinuous Right hand Sides: Control Systems. Kluwer Academic Publisher (1988)Google Scholar
  18. 18.
    Ironi, L., Panzeri, L., Plahte, E., Simoncini, V.: Dynamics of actively regulated gene networks. Physica D 240, 779–794 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Leine, R.I.: Bifurcations in Discontinuous Mechanical Systems of Filippov’s type. PhD thesis, Techn. Univ. Eindhoven, The Netherlands (2000)Google Scholar
  20. 20.
    Libre, J., da Silva, P.R., Teixeira, M.A.: Regularization of discontinuous vector fields via singular perturbation. J. Dynam. Diff. Equat. 19, 309–331 (2009)CrossRefGoogle Scholar
  21. 21.
    Piiroinen, P.T., Kuznetsov, Y.A.: An event-driven method to simulate Filippov systems with accurate computing of sliding motions. ACM Trans. Math. Soft. 34(13), 1–24 (2008)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Utkin, V.: Sliding Modes in Control Optimization. Springer, New York (1992)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Marco Berardi
    • 1
    • 2
  • Nicoletta Del Buono
    • 2
  1. 1.IRSA-CNRBariItaly
  2. 2.Dipartimento di MatematicaUniversità degli Studi di Bari Aldo MoroBariItaly

Personalised recommendations