Boolean Networks: Beyond Generalized Asynchronicity

  • Thomas Chatain
  • Stefan Haar
  • Loïc PaulevéEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10875)


Boolean networks are commonly used in systems biology to model dynamics of biochemical networks by abstracting away many (and often unknown) parameters related to speed and species activity thresholds. It is then expected that Boolean networks produce an over-approximation of behaviours (reachable configurations), and that subsequent refinements would only prune some impossible transitions.

However, we show that even generalized asynchronous updating of Boolean networks, which subsumes the usual updating modes including synchronous and fully asynchronous, does not capture all transitions doable in a multi-valued or timed refinement.

We define a structural model transformation which takes a Boolean network as input and outputs a new Boolean network whose asynchronous updating simulates both synchronous and asynchronous updating of the original network, and exhibits even more behaviours than the generalized asynchronous updating. We argue that these new behaviours should not be ignored when analyzing Boolean networks, unless some knowledge about the characteristics of the system explicitly allows one to restrict its behaviour.


  1. 1.
    Aracena, J.: Maximum number of fixed points in regulatory Boolean networks. Bull. Math. Biol. 70(5), 1398–1409 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aracena, J., Demongeot, J., Goles, E.: Positive and negative circuits in discrete neural networks. IEEE Trans. Neural Netw. 15, 77–83 (2004)CrossRefGoogle Scholar
  3. 3.
    Aracena, J., Goles, E., Moreira, A., Salinas, L.: On the robustness of update schedules in Boolean networks. Biosystems 97(1), 1–8 (2009)CrossRefGoogle Scholar
  4. 4.
    Aracena, J., Richard, A., Salinas, L.: Number of fixed points and disjoint cycles in monotone Boolean networks. SIAM J. Discrete Math. 31(3), 1702–1725 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baetens, J., der Weeën, P.V., Baets, B.D.: Effect of asynchronous updating on the stability of cellular automata. Chaos, Solitons Fractals 45(4), 383–394 (2012)CrossRefGoogle Scholar
  6. 6.
    Baldan, P., Corradini, A., Montanari, U.: Contextual Petri nets, asymmetric event structures, and processes. Inf. Comput. 171(1), 1–49 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bernot, G., Cassez, F., Comet, J.P., Delaplace, F., Müller, C., Roux, O.: Semantics of biological regulatory networks. Electron. Notes Theoret. Comput. Sci. 180(3), 3–14 (2007)CrossRefGoogle Scholar
  8. 8.
    Busi, N., Pinna, G.M.: Non sequential semantics for contextual P/T nets. In: Billington, J., Reisig, W. (eds.) ICATPN 1996. LNCS, vol. 1091, pp. 113–132. Springer, Heidelberg (1996). Scholar
  9. 9.
    Chaouiya, C., Naldi, A., Remy, E., Thieffry, D.: Petri net representation of multi-valued logical regulatory graphs. Nat. Comput. 10(2), 727–750 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chatain, T., Haar, S., Jezequel, L., Paulevé, L., Schwoon, S.: Characterization of reachable attractors using petri net unfoldings. In: Mendes, P., Dada, J.O., Smallbone, K. (eds.) CMSB 2014. LNCS, vol. 8859, pp. 129–142. Springer, Cham (2014). Scholar
  11. 11.
    Chatain, T., Haar, S., Koutny, M., Schwoon, S.: Non-atomic transition firing in contextual nets. In: Devillers, R., Valmari, A. (eds.) PETRI NETS 2015. LNCS, vol. 9115, pp. 117–136. Springer, Cham (2015). Scholar
  12. 12.
    Garg, A., Di Cara, A., Xenarios, I., Mendoza, L., De Micheli, G.: Synchronous versus asynchronous modeling of gene regulatory networks. Bioinformatics 24(17), 1917–1925 (2008)CrossRefGoogle Scholar
  13. 13.
    Janicki, R., Koutny, M.: Semantics of inhibitor nets. Inf. Comput. 123(1), 1–16 (1995). Scholar
  14. 14.
    Janicki, R., Koutny, M.: Fundamentals of modelling concurrency using discrete relational structures. Acta Inf. 34, 367–388 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kauffman, S.A.: Metabolic stability and epigenesis in randomly connected nets. J. Theor. Biol. 22, 437–467 (1969)CrossRefGoogle Scholar
  16. 16.
    Mai, Z., Liu, H.: Boolean network-based analysis of the apoptosis network: irreversible apoptosis and stable surviving. J. Theor. Biol. 259(4), 760–769 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Martínez-Sosa, P., Mendoza, L.: The regulatory network that controls the differentiation of T lymphocytes. Biosystems 113(2), 96–103 (2013)CrossRefGoogle Scholar
  18. 18.
    Noual, M., Sené, S.: Synchronism versus asynchronism in monotonic Boolean automata networks. Natural Comput. (2017)Google Scholar
  19. 19.
    Palma, E., Salinas, L., Aracena, J.: Enumeration and extension of non-equivalent deterministic update schedules in Boolean networks. Bioinformatics 32(5), 722–729 (2016)CrossRefGoogle Scholar
  20. 20.
    Remy, E., Ruet, P., Thieffry, D.: Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework. Adv. Appl. Math. 41(3), 335–350 (2008)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Richard, A.: Negative circuits and sustained oscillations in asynchronous automata networks. Adv. Appl. Math. 44(4), 378–392 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schönfisch, B., de Roos, A.: Synchronous and asynchronous updating in cellular automata. Biosystems 51(3), 123–143 (1999)CrossRefGoogle Scholar
  23. 23.
    Steggles, L.J., Banks, R., Shaw, O., Wipat, A.: Qualitatively modelling and analysing genetic regulatory networks: a petri net approach. Bioinformatics 23(3), 336–343 (2007)CrossRefGoogle Scholar
  24. 24.
    Thieffry, D., Thomas, R.: Dynamical behaviour of biological regulatory networks - II. Immunity control in bacteriophage lambda. Bull. Math. Biol. 57, 277–297 (1995)zbMATHGoogle Scholar
  25. 25.
    Thomas, R.: Boolean formalization of genetic control circuits. J. Theor. Biol. 42(3), 563–585 (1973)CrossRefGoogle Scholar
  26. 26.
    Traynard, P., Fauré, A., Fages, F., Thieffry, D.: Logical model specification aided by model-checking techniques: application to the mammalian cell cycle regulation. Bioinformatics 32(17), i772–i780 (2016)CrossRefGoogle Scholar
  27. 27.
    Vogler, W.: Partial order semantics and read arcs. Theoret. Comput. Sci. 286(1), 33–63 (2002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Winkowski, J.: Processes of contextual nets and their characteristics. Fundamenta Informaticae 36(1), 71–101 (1998)MathSciNetzbMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2018

Authors and Affiliations

  1. 1.LSV, ENS Paris-Saclay, INRIA, CNRSCachanFrance
  2. 2.CNRS & LRI UMR 8623, Univ. Paris-Sud – CNRS, Université Paris-SaclayOrsayFrance

Personalised recommendations