Combinatorics on update digraphs in Boolean networks

  • Authors:

  • J. Aracena;E. Fanchon;M. Montalva;M. Noual

  • Affiliations:

  • CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Av. Esteban Iturra s/n, Casilla 160-C, Concepción, Chile;TIMC-IMAG, Université Joseph Fourier CNRS-UMR 5525, Domaine de la Merci, 38710 La Tronche, France;CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Av. Esteban Iturra s/n, Casilla 160-C, Concepción, Chile;Université de Lyon, íNS-Lyon, LIP, CNRS UMR5668, and IXXI, Institut rhône-alpin des systèmes complexes, 69007 Lyon, France

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2011

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Abstract

Boolean networks have been used as models of gene regulation and other biological networks. One key element in these models is the update schedule, which indicates the order in which states have to be updated. In Aracena et al. (2009) [1], the authors define equivalence classes that relate deterministic update schedules that yield the same update digraph and thus the same dynamical behavior of the network. In this paper we study algorithmical and combinatorial aspects of update digraphs. We show a polynomial characterization of these digraphs, which enables us to characterize the corresponding equivalence classes. We prove that the update digraphs are exactly the projections, on the respective subgraphs, of a complete update digraph with the same number of vertices. Finally, the exact number of complete update digraphs is determined, which provides upper and lower bounds on the number of equivalence classes.