Application of the Reachability Analysis for the Iron Homeostasis Study

  • Alexandre Rocca
  • Thao Dang
  • Eric Fanchon
  • Jean-Marc Moulis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9957)


Our work is motivated by a model of the mammalian cellular Iron Homeostasis, which was analysed using simulations in [9]. The result of this analysis is a characterization of the parameters space such that the model satisfies a set of constraints, proposed by biologists or coming from experimental results. We now propose an approach to hypothesis validation which can be seen as a complement to the approach based on simulation. It uses reachability analysis (that is set-based simulation) to formally validate a hypothesis. For polynomials systems, reachability analysis using the Bernstein expansion is an appropriate technique. Moreover, the Bernstein technique allows us to tackle uncertain parameters at a small cost. In this work, we extend the reachability analysis method presented in [7] to handle polynomial fractions. Furthermore, to tackle the complexity of the Iron Homeostasis model, we use a piecewise approximation of the dynamics and propose a reachability method to deal with the resulting hybrid dynamics. These approximations and adaptations allowed us to validate a hypothesis stated in [9], with an exhaustive analysis over uncertain parameters and initial conditions.


Parametric ODE Reachability analysis Non-linear systems Biological systems 



This work is partially supported by the ANR CADMIDIA project (ANR-13-CESA-0008-03) and the ANR MALTHY project (ANR-12-INSE-003).


  1. 1.
    Batt, G., Yordanov, B., Weiss, R., Belta, C.: Robustness analysis and tuning of synthetic gene networks. Bioinformatics 23(18), 2415–2422 (2007)CrossRefGoogle Scholar
  2. 2.
    Berman, S., Halász, Á., Kumar, V.: MARCO: a reachability algorithm for multi-affine systems with applications to biological systems. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 76–89. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-71493-4_9 CrossRefGoogle Scholar
  3. 3.
    Brim, L., Fabriková, J., Drazan, S., Safranek, D.: Reachability in biochemical dynamical systems by quantitative discrete approximation (2011). arXiv preprint: arXiv:1107.5924
  4. 4.
    Chen, X., Ábrahám, E., Sankaranarayanan, S.: Flow*: an analyzer for non-linear hybrid systems. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 258–263. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    Dang, T., Maler, O., Testylier, R.: Accurate hybridization of nonlinear systems. In: Proceedings of the 13th ACM International Conference on Hybrid Systems: Computation and Control, pp. 11–20. ACM (2010)Google Scholar
  6. 6.
    Dang, T., Testylier, R.: Reachability analysis for polynomial dynamical systems using the bernstein expansion. Reliable Comput. 17(2), 128–152 (2012)MathSciNetGoogle Scholar
  7. 7.
    Dreossi, T., Dang, T., Piazza, C.: Parallelotope bundles for polynomial reachability. In: Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control, pp. 297–306. ACM (2016)Google Scholar
  8. 8.
    Fuchs, H., Kedem, Z.M., Naylor, B.F.: On visible surface generation by a priori tree structures. ACM Siggraph Comput. Graph. 14, 124–133 (1980). ACMCrossRefGoogle Scholar
  9. 9.
    Mobilia, N.: Méthodologie semi-formelle pour l’étude de systèmes biologiques: application à l’homéostasie du fer. Ph.D. thesis, Université Joseph Fourier, Grenoble (2015)Google Scholar
  10. 10.
    Narkawicz, A., Garloff, J., Smith, A.P., Munoz, C.A.: Bounding the range of a rational functiom over a box. Reliable Comput. 17, 34–39 (2012)MathSciNetGoogle Scholar
  11. 11.
    Platzer, A., Quesel, J.-D.: KeYmaera: a hybrid theorem prover for hybrid systems (system description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 171–178. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Sassi, M.A.B.: Analyse et contrôle des systèmes dynamiques polynomiaux. Ph.D. thesis, Université de Grenoble (2013)Google Scholar
  13. 13.
    Smith, A.P.: Fast construction of constant bound functions for sparse polynomials. J. Glob. Optim. 43(2–3), 445–458 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Alexandre Rocca
    • 1
    • 2
  • Thao Dang
    • 1
  • Eric Fanchon
    • 2
  • Jean-Marc Moulis
    • 3
  1. 1.VERIMAG/CNRSSaint Martin D’HèresFrance
  2. 2.Université Grenoble-Alpes - Grenoble 1/CNRS, TIMC-IMAG, UMR 5525GrenobleFrance
  3. 3.Université Grenoble-Alpes - Grenoble 1, Laboratoire de Bioénergétique Fondamentale et Appliquée (LBFA) - Inserm U1055GrenobleFrance

Personalised recommendations