A Combinatorial Exploration of Boolean Dynamics Generated by Isolated and Chorded Circuits

  • B. Mossé
  • É. RemyEmail author
Regular Article


Most studies of motifs of biological regulatory networks focus on the analysis of asymptotical behaviours (attractors, and even often only stable states), but transient properties are rarely addressed. In the line of our previous study devoted to isolated circuits (Remy et al. in Bioinformatics (Oxford, England) 19(Suppl. 2):172–178, 2003), we consider chorded circuits, that are motifs made of an elementary positive or negative circuit with a chord, possibly a self-loop. We provide detailed descriptions of the boolean dynamics of chorded circuits versus isolated circuits, under the synchronous and asynchronous updating schemes within the logical formalism. To this end, we address the description of the trajectories in the dynamics of isolated circuits with coding techniques and adapt them for chorded circuits. The use of the logical modeling gives access to mathematical tools (group actions, analysis of recurrent sequences, coding of trajectories, specific abacus...) allowing complete analytical analysis of basic yet important motifs. In particular, we show that whatever the chosen updating rule, the dynamics depends on a small number of parameters.


Boolean modeling Regulatory network Motifs Circuits Chorded circuits Discrete dynamical systems 



  1. Alon U (2007) Network motifs: theory and experimental approaches. Nat Rev Genetics 8(6):450–61CrossRefGoogle Scholar
  2. Baëza M, Viala S, Heim M, Dard A, Hudry B, Duffraisse M, Rogulja-Ortmann A, Brun C, Merabet S (2015) Inhibitory activities of short linear motifs underlie Hox interactome specificity in vivo. eLife 4:e06034CrossRefGoogle Scholar
  3. Bang-Jensen J, Gutin G (2008) Digraphs, theory, algorithms, applications. Springer, BerlinGoogle Scholar
  4. Bérenguier D, Chaouiya C, Monteiro P T, Naldi A, Remy E, Thieffry D, Tichit L (2013) Dynamical modeling and analysis of large cellular regulatory networks. Chaos (Woodbury, N.Y.) 23(2):025114CrossRefGoogle Scholar
  5. Comet J-P, Noual M, Richard A, Aracena J, Calzone L, Demongeot J, Kaufman M, Naldi A, Snoussi EH, Thieffry D (2013) On circuit functionality in Boolean networks. Bull Math Biol 75(6):906–19CrossRefGoogle Scholar
  6. Glass L (1975) Classification of biological networks by their qualitative dynamics. J Theor Biol 54(1):85–107CrossRefGoogle Scholar
  7. Hansen T, Mullen GL (1992) Primitive polynomials over finite fields. Math Comput 59(200):639–643CrossRefGoogle Scholar
  8. Lang S (1997) Algebra. Addison-we edition, BostonGoogle Scholar
  9. Lidl R, Niederreiter H (1986) Introduction to finite fields and their applicationsGoogle Scholar
  10. Remy É, Ruet P (2006) On differentiation and homeostatic behaviours of Boolean dynamical systems, vol 4230. LNBIGoogle Scholar
  11. Remy É, Mossé B, Chaouiya C, Thieffry D (2003) A description of dynamical graphs associated to elementary regulatory circuits. Bioinformatics (Oxford, England) 19(Suppl. 2):172–178Google Scholar
  12. Remy É, Mossé B, Thieffry D (2016) Boolean dynamics of compound regulatory circuits. In: Rogato A, Guarracino M, Zazzu V (eds) Dynamics of mathematical models in biology bringing mathematics to life. Springer, Berlin, pp 43–53CrossRefGoogle Scholar
  13. Robert F (1986) Discrete iterations. Springer, BerlinCrossRefGoogle Scholar
  14. Shih MH, Dong JL (2005) A combinatorial analogue of the Jacobian problem in automata networks. Adv Appl Math 34(1):30–46CrossRefGoogle Scholar
  15. Snoussi EH, Thomas R (1993) Logical identification of all steady states: the concept of feedback loop characteristic states. Bull Math Biol 55(5):973–991CrossRefGoogle Scholar
  16. Thomas R (1981) On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. Springer Series Synerget 9:180–193CrossRefGoogle Scholar
  17. Thomas R, D’Ari R (1990) Biological feedbackGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Aix Marseille Univ, CNRS, Centrale MarseilleMarseilleFrance

Personalised recommendations