Advertisement

Complexity of Maximum Fixed Point Problem in Boolean Networks

  • Florian BridouxEmail author
  • Nicolas Durbec
  • Kevin Perrot
  • Adrien Richard
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11558)

Abstract

A Boolean network (BN) with n components is a discrete dynamical system described by the successive iterations of a function \(f:\{{ \texttt {0}},{ \texttt {1}}\}^n \rightarrow \{{ \texttt {0}},{ \texttt {1}}\}^n\). This model finds applications in biology, where fixed points play a central role. For example in genetic regulation they correspond to cell phenotypes. In this context, experiments reveal the existence of positive or negative influences among components: component i has a positive (resp. negative) influence on component j, meaning that j tends to mimic (resp. negate) i. The digraph of influences is called signed interaction digraph (SID), and one SID may correspond to multiple BNs. The present work opens a new perspective on the well-established study of fixed points in BNs. Biologists discover the SID of a BN they do not know, and may ask: given that SID, can it correspond to a BN having at least k fixed points? Depending on the input, this problem is in \( \textsf {P}\) or complete for \(\textsf {NP}\), \(\textsf {NP}^\textsf {\#P}\) or \(\textsf {NEXPTIME}\).

Notes

Acknowledgments

The authors would like to thank for their support the Young Researcher project ANR-18-CE40-0002-01 “FANs”, project ECOS-CONICYT C16E01, and project STIC AmSud CoDANet 19-STIC-03 (Campus France 43478PD).

References

  1. 1.
    The Online Encyclopedia of Integer Sequences, founded in 1964 by N. J. A. Sloane. Sequence A006126. https://oeis.org/A006126
  2. 2.
    Akutsu, T., Miyano, S., Kuhara, S.: Identification of genetic networks from a small number of gene expression patterns under the boolean network model. In: Biocomputing’99, pp. 17–28. World Scientific (1999)Google Scholar
  3. 3.
    Albert, R.: Boolean modeling of genetic regulatory networks. In: Ben-Naim, E., Frauenfelder, H., Toroczkai, Z. (eds.) Complex Networks. LNP, vol. 650, pp. 459–481. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-44485-5_21zbMATHCrossRefGoogle Scholar
  4. 4.
    Aracena, J.: Maximum number of fixed points in regulatory Boolean networks. Bull. Math. Biol. 70(5), 1398–1409 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Aracena, J., Demongeot, J., Goles, E.: Fixed points and maximal independent sets in and-or networks. Discrete Appl. Math. 138(3), 277–288 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Aracena, J., Richard, A., Salinas, L.: Maximum number of fixed points in AND-OR-NOT networks. J. Comput. Syst. Sci. 80(7), 1175–1190 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Boros, E., Ibaraki, T., Makino, K.: Error-free and best-fit extensions of partially defined Boolean functions. Inf. Comput. 140(2), 254–283 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Crama, Y., Hammer, P.L.: Boolean Functions: Theory, Algorithms, and Applications. Cambridge University Press, Cambridge (2011)zbMATHCrossRefGoogle Scholar
  10. 10.
    Feder, T.: A new fixed point approach for stable networks and stable marriages. J. Comput. Syst. Sci. 45(2), 233–284 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Gadouleau, M., Richard, A., Riis, S.: Fixed points of Boolean networks, guessing graphs, and coding theory. SIAM J. Discrete Math. 29(4), 2312–2335 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Gadouleau, M., Riis, S.: Graph-theoretical constructions for graph entropy and network coding based communications. IEEE Trans. Inf. Theory 57(10), 6703–6717 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Goles, E., Salinas, L.: Sequential operator for filtering cycles in Boolean networks. Adv. Appl. Math. 45(3), 346–358 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kauffman, S.A.: Metabolic stability and epigenesis in randomly connected nets. J. Theor. Biol. 22, 437–467 (1969)CrossRefGoogle Scholar
  15. 15.
    Kauffman, S.A.: Origins of Order Self-Organization and Selection in Evolution. Oxford University Press, Oxford (1993)Google Scholar
  16. 16.
    Kaufman, M., Soulé, C., Thomas, R.: A new necessary condition on interaction graphs for multistationarity. J. Theoret. Biol. 248(4), 675–685 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kosub, S.: Dichotomy results for fixed-point existence problems for Boolean dynamical systems. Math. Comput. Sci. 1(3), 487–505 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Le Novère, N.: Quantitative and logic modelling of molecular and gene networks. Nat. Rev. Genet. 16, 146–158 (2015)CrossRefGoogle Scholar
  19. 19.
    Littman, M.L., Goldsmith, J., Mundhenk, M.: The computational complexity of probabilistic planning. J. Artif. Intell. Res. 9, 1–36 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    McCuaig, W.: Pólya’s permanent problem. Electron. J. Comb. 11(1), 79 (2004)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Montalva, M., Aracena, J., Gajardo, A.: On the complexity of feedback set problems in signed digraphs. Electron. Not. Discrete Math. 30, 249–254 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)Google Scholar
  23. 23.
    Paulevé, L., Richard, A.: Topological fixed points in Boolean networks. Comptes Rendus de l’Académie des Sci.-Ser. I-Math. 348(15–16), 825–828 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Richard, A.: Positive and negative cycles in Boolean networks. J. Theor. Biol. 463, 67–76 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Robert, F.: Discrete Iterations: A Metric Study. Springer Series in Computational Mathematics, vol. 6, p. 198. Springer, Heidelberg (1986).  https://doi.org/10.1007/978-3-642-61607-5CrossRefGoogle Scholar
  27. 27.
    Robertson, N., Seymour, P., Thomas, R.: Permanents, pfaffian orientations, and even directed circuits. Ann. Math. 150(3), 929–975 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Soulé, C.: Mathematical approaches to differentiation and gene regulation. C.R. Paris Biol. 329, 13–20 (2006)CrossRefGoogle Scholar
  29. 29.
    Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Thomas, R.: Boolean formalization of genetic control circuits. J. Theor. Biol. 42(3), 563–585 (1973).  https://doi.org/10.1016/0022-5193(73)90247-6CrossRefGoogle Scholar
  31. 31.
    Thomas, R., d’Ari, R.: Biological Feedback. CRC Press, Boca Raton (1990)zbMATHGoogle Scholar
  32. 32.
    Thomas, R., Kaufman, M.: Multistationarity, the basis of cell differentiation and memory. II. Logical analysis of regulatory networks in terms of feedback circuits. Chaos Interdisc. J. Nonlinear Sci. 11(1), 180–195 (2001)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Florian Bridoux
    • 1
    Email author
  • Nicolas Durbec
    • 1
  • Kevin Perrot
    • 1
  • Adrien Richard
    • 2
    • 3
  1. 1.Aix-Marseille Université, Université de Toulon, CNRS, LISMarseilleFrance
  2. 2.Laboratoire I3S, CNRSUniversité Côte d’AzurNiceFrance
  3. 3.CMM, UMI CNRS 2807, Universidad de ChileSantiagoChile

Personalised recommendations