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Boolean dynamics revisited through feedback interconnections

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Abstract

Boolean models of physical or biological systems describe the global dynamics of the system and their attractors typically represent asymptotic behaviors. In the case of large networks composed of several modules, it may be difficult to identify all the attractors. To explore Boolean dynamics from a novel viewpoint, we will analyse the dynamics emerging from the composition of two known Boolean modules. The state transition graphs and attractors for each of the modules can be combined to construct a new asymptotic graph which will (1) provide a reliable method for attractor computation with partial information; (2) illustrate the differences in dynamical behavior induced by the updating strategy (asynchronous, synchronous, or mixed); and (3) show the inherited organization/structure of the original network’s state transition graph.

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References

  • Abou-Jaoudé W, Traynard P, Monteiro P, Saez-Rodriguez J, Helikar T, Thieffry D, Chaouiya C (2016) Logical modeling and dynamical analysis of cellular networks. Front Genet 7:94

    Google Scholar 

  • Akutsu T, Melkman A, Tamura T, Yamamoto M (2011) Determining a singleton attractor of a boolean network with nested canalyzing functions. J Comput Biol 18(10):1275–1290

    MathSciNet  Google Scholar 

  • Baccala R, Kono D, Theofilopoulos A (2005) Interferons as pathogenic effectors in autoimmunity. Immunol Rev 204:9–26

    Google Scholar 

  • Baldazzi V, Ropers D, Markowicz Y, Kahn D, Geiselmann J, de Jong H (2010) The carbon assimilation network in Escherichia coli is densely connected and largely sign-determined by directions of metabolic fluxes. PLoS Comput Biol 6(6):e1000812

    MathSciNet  Google Scholar 

  • Chaves M, Albert R, Sontag E (2005) Robustness and fragility of boolean models for genetic regulatory networks. J Theor Biol 235(3):431–449

    MathSciNet  Google Scholar 

  • Chaves M, Carta A (2015) Attractor computation using interconnected boolean networks: testing growth rate models in E. coli. Theor Comput Sci 599:47–63

    MathSciNet  MATH  Google Scholar 

  • Chaves M, Preto M (2013) Hierarchy of models: from qualitative to quantitative analysis of circadian rhythms in cyanobacteria. Chaos 23(2):025113

    MathSciNet  MATH  Google Scholar 

  • Chaves M, Tournier L (2011) Predicting the asymptotic dynamics of large biological networks by interconnections of Boolean modules. In: Proceedings of 50th conference decision and control and European control conference, Orlando, Florida, USA

  • Chaves M, Tournier L (2018) Analysis tools for interconnected boolean networks with biological applications. Front Physiol 9:586

    Google Scholar 

  • Comet JP, Bernot G, Das A, Diener F, Massot C, Cessieux A (2012) Simplified models for the mammalian circadian clock. Procedia Comput Sci 11:127–138

    Google Scholar 

  • Crama Y, Hammer P (2011) Boolean functions: theory, algorithms, and applications. Cambridge University Press, New York

    MATH  Google Scholar 

  • Demongeot J, Elena A, Sené S (2008) Robustness in regulatory networks: a multi-disciplinary approach. Acta Biotheor 56(1):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(8):1–18

    Google Scholar 

  • Devloo V, Hansen P, Labbé M (2003) Identification of all steady states in large networks by logical analysis. Bull Math Biol 65:1025–1051

    MATH  Google Scholar 

  • Diestel R (2005) Graph theory, 3rd edn. Springer, Heidelberg

    MATH  Google Scholar 

  • Dong G, Yang Q, Wang Q, Kim YI, Wood T, Osteryoung K, van Oudenaarden A, Golden S (2010) Elevated atpase activity of kaic applies a circadian checkpoint on cell division in synechococcus elongatus. Cell 140:529–539

    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(14):124–131

    Google Scholar 

  • Feillet C, Krusche P, Tamanini F, Janssens R, Downey M, Martin P, Teboul M, Saito S, Lévi F, Bretschneider T, van der Horst G, Delaunay F, Rand D (2014) Phase locking and multiple oscillating attractors for the coupled mammalian clock and cell cycle. PNAS 111(27):9828–9833

    Google Scholar 

  • Feillet C, van der Horst G, Lévi F, Rand D, Delaunay F (2015) Coupling between the circadian clock and cell cycle oscillators: implication for healthy cells and malignant growth. Front Neurol 6:96

    Google Scholar 

  • García-Gomez M, Azpeitia E, Alvarez-Buylla E (2017) A dynamic genetic-hormonal regulatory network model explains multiple cellular behaviors of the root apical meristem of Arabidopsis thaliana. PLoS Comput Biol 13(4):e1005488

    Google Scholar 

  • Garg A, Di Cara A, Xenarios I, Mendoza L, De Micheli G (2008) Synchronous versus asynchronous modeling of gene regulatory networks. Bioinformatics 24(17):1917–1925

    Google Scholar 

  • Gerard C, Goldbeter A (2009) Temporal self-organization of the cyclin/Cdk network driving the mammalian cell cycle. PNAS 106(51):21643–21648

    Google Scholar 

  • Hong C, Hwang J, Cho KH, Shin I (2015) An efficient steady-state analysis method for large boolean networks with high maximum node connectivity. PLoS ONE 10(12):e0145734

    Google Scholar 

  • Irons D (2006) Improving the efficiency of attractor cycle identification in Boolean networks. Phys D 217:7–21

    MathSciNet  MATH  Google Scholar 

  • Jarrah A, Raposa B, Laubenbacher R (2007) Nested canalyzing, unate cascade, and polynomial functions. Phys D 233(2):167–174

    MathSciNet  MATH  Google Scholar 

  • Kauffman S, Peterson C, Samuelsson B, Troein C (2004) Genetic networks with canalyzing boolean rules are always stable. PNAS 101(49):17102–17107

    Google Scholar 

  • Klamt S, Saez-Rodriguez J, Lindquist J, Simeoni L, Gilles E (2006) A methodology for the structural and functional analysis of signaling and regulatory networks. BMC Bioinform 7(1):56

    Google Scholar 

  • Klarner H, Bockmayr A, Siebert H (2014) Computing symbolic steady states of boolean networks. In: Was J, Sirakoulis G, Bandini S (eds) Cellular automata, LNCS, vol 8751. Springer, Heidelberg, pp 561–570

    Google Scholar 

  • Klarner H, Siebert H (2015) Approximating attractors of boolean networks by iterative CTL model checking. Front Bioeng Biotechnol 3:130

    Google Scholar 

  • Mendoza L, Xenarios I (2006) A method for the generation of standardized qualitative dynamical systems of regulatory networks. Theor Biol Med Model 3(1):13

    Google Scholar 

  • Mori T, Flöttmann M, Krantz M, Akutsu T, Klipp E (2015) Stochastic simulation of boolean rxncon models: towards quantitative analysis of large signaling networks. BMC Syst Biol 9(45):1–9

    Google Scholar 

  • Ortiz-Gutiérrez E, García-Cruz K, Azpeitia E, Castillo A, Sánchez M, Alvarez-Buylla E (2015) A dynamic gene regulatory network model that recovers the cyclic behavior of Arabidopsis thaliana cell cycle. PLoS Comput Biol 11(9):e1004486

    Google Scholar 

  • Plikus MV, Vollmers C, de la Cruz D, Chaix A, Ramos R, Panda S, Chuong CM (2013) Local circadian clock gates cell cycle progression of transient amplifying cells during regenerative hair cycling. PNAS 110(23):E2106–E2115

    Google Scholar 

  • Purnick P, Weiss R (2009) The second wave of synthetic biology: from modules to systems. Nat Rev Mol Cell Biol 10(6):410

    Google Scholar 

  • Rust M, Markson J, Lane W, Fisher D, O’Shea E (2007) Ordered phosphorylation governs oscillation of a three-protein circadian clock. Science 318:809

    Google Scholar 

  • Saez-Rodriguez J, Simeoni L, Lindquist JA, Hemenway R, Bommhardt U, Arndt B, Haus UU, Weismantel R, Gilles ED, Klamt S, Schraven B (2007) A logical model provides insights into T cell receptor signaling. PLoS Comput Biol 3(8):e163

    MathSciNet  Google Scholar 

  • Shmulevich I, Dougherty E, Kim S, Zhang W (2002) Probabilistic boolean networks: a rule-based uncertainty model for gene regulatory networks. Bioinformatics 18(2):261–274

    Google Scholar 

  • Sontag E (1998) Mathematical control theory, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Stoll G, Caron B, Viara E, Dugourd A, Zinovyev A, Naldi A, Kroemer G, Barillot E, Calzone L (2017) Maboss 2.0: an environment for stochastic boolean modeling. Bioinformatics 33(4):2226–2228

    Google Scholar 

  • Tournier L, Chaves M (2013) Interconnection of asynchronous Boolean networks, asymptotic and transient dynamics. Automatica 49(4):884–893

    MathSciNet  MATH  Google Scholar 

  • Vecchio DD, Ninfa A, Sontag E (2008) Modular cell biology: retroactivity and insulation. Mol Syst Biol 4:161

    Google Scholar 

  • Veliz-Cuba A, Aguilar B, Hinkelmann F, Laubenbacher R (2014) Steady state analysis of boolean molecular network models via model reduction and computational algebra. BMC Bioinform 15:221

    Google Scholar 

  • Wang RS, Saadatpour A, Albert R (2012) Boolean modeling in systems biology: an overview of methodology and applications. Phys Biol 9:055001

    Google Scholar 

  • Zañudo J, Albert R (2013) An effective network reduction approach to find the dynamical repertoire of discrete dynamic networks. Chaos 23(2):025111

    MathSciNet  MATH  Google Scholar 

  • Zhang R, Shah M, Yang J, Nyland S, Liu X, Yun J, Albert R, Loughran TP Jr (2008) Network model of survival signaling in large granular lymphocyte leukemia. PNAS 105(42):16308–16313

    Google Scholar 

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Acknowledgements

M. Chaves was partially supported by the French agency for research through Project ICycle ANR-16-CE33-0016-01. D. Figueiredo and M. A. Martins are partially supported by the ERDF European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 and by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia within Projects POCI-01-0145-FEDER-016692, POCI-01-0145-FEDER-030947 and project UID/MAT/ 04106/2013 at CIDMA. D. Figueiredo acknowledges the support of FCT via the Ph.D scholarship PD/BD/114186/2016. This work was partially supported by a France-Portugal partnership PHC PESSOA 2018 between M. Chaves (Campus France #40823SD) and M. A. Martins.

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Authors and Affiliations

  1. Université Côte d’Azur, Inria, INRA, CNRS, Sorbonne Université, Biocore Team, Sophia Antipolis, France

    Madalena Chaves

  2. Department of Mathematics, CIDMA - Center for R&D Mathematics and Applications, University of Aveiro, Aveiro, Portugal

    Daniel Figueiredo & Manuel A. Martins

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  1. Madalena Chaves
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  2. Daniel Figueiredo
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  3. Manuel A. Martins
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Correspondence to Madalena Chaves.

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Chaves, M., Figueiredo, D. & Martins, M.A. Boolean dynamics revisited through feedback interconnections. Nat Comput 19, 29–49 (2020). https://doi.org/10.1007/s11047-018-9716-8

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