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Attractor landscapes in Boolean networks with firing memory: a theoretical study applied to genetic networks

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Abstract

In this paper we study the dynamical behavior of Boolean networks with firing memory, namely Boolean networks whose vertices are updated synchronously depending on their proper Boolean local transition functions so that each vertex remains at its firing state a finite number of steps. We prove in particular that these networks have the same computational power than the classical ones, i.e. any Boolean network with firing memory composed of m vertices can be simulated by a Boolean network by adding vertices. We also prove general results on specific classes of networks. For instance, we show that the existence of at least one delay greater than 1 in disjunctive networks makes such networks have only fixed points as attractors. Moreover, for arbitrary networks composed of two vertices, we characterize the delay phase space, i.e. the delay values such that networks admits limit cycles or fixed points. Finally, we analyze two classical biological models by introducing delays: the model of the immune control of the \(\lambda \)-phage and that of the genetic control of the floral morphogenesis of the plant Arabidopsis thaliana.

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References

  • Ahmad J, Roux OF, Bernot G, Comet J-P, Richard A (2008) Analysing formal models of genetic regulatory networks with delays. Int J Bioinf Res Appl 4:240–262

    Google Scholar 

  • Albert R, Othmer HG (2003) The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster. J Theor Biol 223:1–18

    MathSciNet  MATH  Google Scholar 

  • Aracena J, Goles E, Moreira A, Salinas L (2009) On the robustness of update schedules in Boolean networks. Biosystems 97:1–8

    Google Scholar 

  • Aracena J, Fanchon É, Montalva M, Noual M (2011) Combinatorics on update digraphs in Boolean networks. Discrete Appl Math 159:401–409

    MathSciNet  MATH  Google Scholar 

  • Bernot G, Comet J-P, Richard A, Guespin J (2004) Application of formal methods to biological regulatory networks: extending Thomas’ asynchronous logical approach with temporal logic. J Theor Biol 229:339–347

    MathSciNet  Google Scholar 

  • Brualdi RA, Ryser HJ (1991) Combinatorial matrix theory. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Choi M, Shi J, Jung SH, Chen X, Cho K-H (2012) Attractor landscape analysis reveals feedback loops in the p53 network that control the cellular response to DNA damage. Sci Signal 5:ra83

    Google Scholar 

  • Coen ES, Meyerowitz EM (1991) The war of the whorls: genetic interactions controlling flower development. Nature 353:31–37

    Google Scholar 

  • Davidich MI, Bornholdt S (2008) Boolean network model predicts cell cycle sequence of fission yeast. PLoS ONE 3:e1672

    Google Scholar 

  • Demongeot J, Elena A, Sené S (2008) Robustness in regulatory networks: a multi-disciplinary approach. Acta Biotheor 56:27–49

    Google Scholar 

  • Demongeot J, Goles E, Morvan M, Noual M, Sené S (2010) Attraction basins as gauges of robustness against boundary conditions in biological complex systems. PLoS ONE 5:e11793

    Google Scholar 

  • Dennunzio A, Formenti E, Manzoni L, Mauri G (2013) \(m\)-Asynchronous cellular automata: from fairness to quasi-fairness. Nat Comput 12:561–572

    MathSciNet  MATH  Google Scholar 

  • Fatès N (2014) A guided tour of asynchronous cellular automata. J Cell Autom 9:387–416

    MathSciNet  MATH  Google Scholar 

  • Fatès N, Morvan M (2005) An experimental study of robustness to asynchronism for elementary cellular automata. Complex Syst 16:1–27

    MathSciNet  MATH  Google Scholar 

  • Fauré A, Naldi A, Chaouiya C, Thieffry D (2006) Dynamical analysis of a generic Boolean model for the control of the mammalian cell cycle. Bioinformatics 22:e124–e131

    Google Scholar 

  • Fromentin J, Éveillard D, Roux O (2010) Hybrid modeling of biological networks: mixing temporal and qualitative biological properties. BMC Syst Biol 4:79

    Google Scholar 

  • Goles E (1980) Comportement oscillatoire d’une famille d’automates cellulaires non uniformes, Ph.D. thesis. Université Joseph Fourier et Institut national polytechnique de Grenoble

  • Goles E, Noual M (2010) Block-sequential update schedules and Boolean automata circuits. In: Proceedings of AUTOMATA’10, DMTCS, pp 41–50

  • Goles E, Salinas L (2008) Comparison between parallel and serial dynamics of Boolean networks. Theor Comput Sci 396:247–253

    MathSciNet  MATH  Google Scholar 

  • Goles E, Montalva M, Ruz GA (2013) Deconstruction and dynamical robustness of regulatory networks: application to the yeast cell cycle networks. Bull Math Biol 2:939–966

    MathSciNet  MATH  Google Scholar 

  • Graudenzi A, Serra R (2010) A new model of genetic networks: the gene protein Boolean network. In: Proceedings of WIVACE’08, pp 283–292

  • Graudenzi A, Serra R, Villani M, Colacci A, Kauffman SA (2011) Robustness analysis of a Boolean model of gene regulatory network with memory. J Comput Biol 18:559–577

    MathSciNet  Google Scholar 

  • Graudenzi A, Serra R, Villani M, Damiani C, Colacci A, Kauffman SA (2011) Dynamical properties of a Boolean model of gene regulatory network with memory. J Comput Biol 18:1291–1303

    MathSciNet  Google Scholar 

  • Kauffman SA (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 22:437–467

    MathSciNet  Google Scholar 

  • Leifeld T, Zhang Z, Zhang P (2018) Identification of Boolean network models from time series data incorporating prior knowledge. Front Physiol 9:695

    Google Scholar 

  • Li F, Long L, Lu Y, Ouyang Q, Tang C (2004) The yeast cell-cycle network is robustly designed. Proc Natl Acad Sci USA 101:4781–4786

    Google Scholar 

  • Mangan S, Alon U (2003) Structure and function of the feed-forward loop network motif. Proc Natl Acad Sci USA 100:11980–11985

    Google Scholar 

  • McCulloch WS, Pitts W (1943) A logical calculus of the ideas immanent in nervous activity. J Math Biophys 5:115–133

    MathSciNet  MATH  Google Scholar 

  • Mendoza L, Alvarez-Buylla ER (1998) Dynamics of the genetic regulatory network for Arabidopsis thaliana flower morphogenesis. J Theor Biol 193:307–319

    Google Scholar 

  • Méndez A, Mendoza L (2016) A network model to describe the terminal differentiation of B cells. PLoS Comput Biol 12:e1004696

    Google Scholar 

  • Mendoza L, Thieffry D, Alvarez-Buylla ER (1999) Genetic control of flower morphogenesis in Arabidopsis thaliana: a logical analysis. Bioinformatics 15:593–606

    Google Scholar 

  • Meyerowitz EM (1994) The genetics of flower development. Sci Am 271:56–65

    Google Scholar 

  • Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii S, Alon U (2002) Network motifs: simple building blocks of complex networks. Science 298:824–827

    Google Scholar 

  • Noual M, Sené S (2018) Synchronism vs asynchronism in monotonic Boolean automata networks. Nat Comput 17:393–402

    MathSciNet  Google Scholar 

  • Regnault D, Schabanel N, Thierry E (2009) Progresses in the analysis of stochastic 2D cellular automata: a study of asynchronous 2D minority. Theor Comput Sci 410:4844–4855

    MathSciNet  MATH  Google Scholar 

  • Remy Mossé É, Chaouiya C, Thieffry D (2003) A description of dynamical graphs associated to elementary regulatory circuits. Bioinformatics 19:172–178

    MATH  Google Scholar 

  • Ren F, Cao J (2008) Asymptotic and robust stability of genetic regulatory networks with time-varying delays. Neurocomputing 71:834–842

    Google Scholar 

  • Ribeiro T, Magnin M, Inoue K, Sakama C (2014) Learning delayed influences of biological systems. Front Bioeng Biotechnol 2:81

    Google Scholar 

  • Richard A, Comet J-P (2007) Necessary conditions for multistationarity in discrete dynamical systems. Discrete Appl Math 155:2403–2413

    MathSciNet  MATH  Google Scholar 

  • Robert F (1986) Discrete iterations: a metric study, volume 6 of Springer Series in Computational Mathematics. Springer, Berlin

    Google Scholar 

  • Ruz GA, Goles E (2010) Learning gene regulatory networks with predefined attractors for sequential updating schemes using simulated annealing. In: Proceedings of ICMLA’10, pp 889–894

  • Ruz GA, Goles E, Montalva M, Fogel GB (2014) Dynamical and topological robustness of the mammalian cell cycle network: a reverse engineering approach. Biosystems 115:23–32

    Google Scholar 

  • Ruz GA, Timmermann T, Goles E (2015) Reconstruction of a GRN model of salt stress response in Arabidopsis using genetic algorithms. In: Proceedings of CIBCB’15, pp 1–8

  • Thieffry D, Thomas R (1995) Dynamical behaviour of biological regulatory networks—II. Immunity control in bacteriophage Lambda. Bull Math Biol 57:277–297

    MATH  Google Scholar 

  • Thomas R (1973) Boolean formalization of genetic control circuits. J Theor Biol 42:563–585

    Google Scholar 

  • Thomas R (1991) Regulatory networks seen as asynchronous automata: a logical description. J Theor Biol 153:1–23

    Google Scholar 

  • Thomas R, Richelle J (1988) Positive feedback loops and multstationarity. Discrete Appl Math 19:381–396

    MathSciNet  MATH  Google Scholar 

  • Thomas R, Thieffry D, Kaufman M (1995) Dynamical behaviour of biological regulatory networks—I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state. Bull Math Biol 57:247–276

    MATH  Google Scholar 

  • Veliz-Cuba A, Stigler B (2011) Boolean models can explain bistability in the lac Operon. J Comput Biol 18:783–794

    MathSciNet  Google Scholar 

  • Yeger-Lotem E, Sattath S, Kashtan N, Itzkovitz S, Milo R, Pinter RY, Alon U, Margalit H (2004) Network motifs in integrated cellular networks of transcription–regulation and protein–protein interaction. Proc Natl Acad Sci USA 101:5934–5939

    Google Scholar 

  • Zhang R, Shah MV, Yang J, Nyland SB, Liu X, Yun JK, Albert R, Loughran TP (2008) Network model of survival signaling in large granular lymphocyte leukemia. Proc Natl Acad Sci USA 105:16308–16313

    Google Scholar 

  • Zúñiga A, Donoso RA, Ruiz D, Ruz GA, González B (2017) Quorum-Sensing systems in the plant growth-promoting bacterium paraburkholderia phytofirmans PsJN exhibit cross-regulation and are involved in biofilm formation. Mol Plant Microbe Interact 30:557–565

    Google Scholar 

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Acknowledgements

This work has been supported by ECOS-CONICYT C16E01 (EG, FL, GR, SS), FONDECYT 1140090 (EG, FL), Basal Project CMM (GR, EG), FANs program ANR-18-CE40-0002-01 (SS) and PACA Fri Project 2015_01134 (SS).

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Authors and Affiliations

  1. Facultad de Ingenería y Ciencias, Universidad Adolfo Ibañez, Santiago, Chile

    Eric Goles, Fabiola Lobos & Gonzalo A. Ruz

  2. Center of Applied Ecology and Sustainability (CAPES), Santiago, Chile

    Gonzalo A. Ruz

  3. LIS, CNRS, Université de Toulon, Université d’Aix-Marseille, Marseille, France

    Sylvain Sené

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  1. Eric Goles
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  2. Fabiola Lobos
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  3. Gonzalo A. Ruz
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  4. Sylvain Sené
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Correspondence to Eric Goles.

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Goles, E., Lobos, F., Ruz, G.A. et al. Attractor landscapes in Boolean networks with firing memory: a theoretical study applied to genetic networks. Nat Comput 19, 295–319 (2020). https://doi.org/10.1007/s11047-020-09789-0

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