Advertisement

Boolean Threshold Networks as Models of Genotype-Phenotype Maps

  • Chico Q. CamargoEmail author
  • Ard A. Louis
Conference paper
  • 76 Downloads
Part of the Springer Proceedings in Complexity book series (SPCOM)

Abstract

Boolean threshold networks (BTNs) are a class of mathematical models used to describe complex dynamics on networks. They have been used to study gene regulation, but also to model the brain, and are similar to artificial neural networks used in machine learning applications. In this paper we study BTNs from the perspective of genotype-phenotype maps, by treating the network’s set of nodes and connections as its genotype, and dynamic behaviour of the model as its phenotype. We show that these systems exhibit (1) Redundancy, that is many genotypes map to the same phenotypes; (2) Bias, the number of genotypes per phenotypes varies over many orders of magnitude; (3) Simplicity bias, simpler phenotypes are exponentially more likely to occur than complex ones; (4) Large robustness, many phenotypes are surprisingly robust to random perturbations in the parameters, and (5) this robustness correlates positively with the evolvability, the ability of the system to find other phenotypes by point mutations of the parameters. These properties should be relevant for the wide range of systems that can be modelled by BTNs.

Keywords

Boolean networks Gene regulatory networks Genotype-phenotype maps Input-output maps 

References

  1. 1.
    Ahnert, S.E.: Structural properties of genotype–phenotype maps. J. R. Soc. Interface 14(132), 20170275 (2017).  https://doi.org/10.1098/rsif.2017.0275CrossRefGoogle Scholar
  2. 2.
    Albert, R., Othmer, H.G.: The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster. J. Theor. Biol. 223(1), 1–18 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aral, N., Kabakçıoğlu, A.: Coherent regulation in yeast’s cell-cycle network. Phys. Biol. 12(3), 036002 (2015)ADSCrossRefGoogle Scholar
  4. 4.
    Aral, N., Kabakçıoğlu, A.: Coherent organization in gene regulation: a study on six networks. Phys. Biol. 13(2), 026006 (2016)ADSCrossRefGoogle Scholar
  5. 5.
    Azevedo, R.B., Lohaus, R., Srinivasan, S., Dang, K.K., Burch, C.L.: Sexual reproduction selects for robustness and negative epistasis in artificial gene networks. Nature 440(7080), 87 (2006)ADSCrossRefGoogle Scholar
  6. 6.
    Bergman, A., Siegal, M.L.: Evolutionary capacitance as a general feature of complex gene networks. Nature 424(6948), 549 (2003)ADSCrossRefGoogle Scholar
  7. 7.
    Boldhaus, G., Bertschinger, N., Rauh, J., Olbrich, E., Klemm, K.: Robustness of Boolean dynamics under knockouts. Phys. Rev. E 82(2), 021916 (2010)ADSCrossRefGoogle Scholar
  8. 8.
    Boldhaus, G., Klemm, K.: Regulatory networks and connected components of the neutral space. Eur. Phys. J. B-Condens. Matter Complex Syst. 77(2), 233–237 (2010)CrossRefGoogle Scholar
  9. 9.
    Borenstein, E., Krakauer, D.C.: An end to endless forms: epistasis, phenotype distribution bias, and nonuniform evolution. PLoS Comput. Biol. 4(10), e1000202 (2008)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, H., Wang, G., Simha, R., Du, C., Zeng, C.: Boolean models of biological processes explain cascade-like behavior. Sci. Rep. 7 (2016). Article number 20067. https://www.nature.com/articles/srep20067
  11. 11.
    Ciliberti, S., Martin, O.C., Wagner, A.: Innovation and robustness in complex regulatory gene networks. Proc. Natl. Acad. Sci. 104(34), 13591–13596 (2007) ADSCrossRefGoogle Scholar
  12. 12.
    Ciliberti, S., Martin, O.C., Wagner, A.: Robustness can evolve gradually in complex regulatory gene networks with varying topology. PLoS Comput. Biol. 3(2), e15 (2007)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Cowperthwaite, M.C., Economo, E.P., Harcombe, W.R., Miller, E.L., Meyers, L.A.: The ascent of the abundant: how mutational networks constrain evolution. PLoS Comput. Biol. 4(7), e1000110 (2008)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Davidich, M.I., Bornholdt, S.: Boolean network model predicts cell cycle sequence of fission yeast. PLoS One 3(2), e1672 (2008)ADSCrossRefGoogle Scholar
  15. 15.
    Davidich, M.I., Bornholdt, S.: Boolean network model predicts knockout mutant phenotypes of fission yeast. PLoS One 8(9), e71786 (2013)ADSCrossRefGoogle Scholar
  16. 16.
    Dingle, K., Camargo, C.Q., Louis, A.A.: Input-output maps are strongly biased towards simple outputs. Nat. Commun. 9(1), 761 (2018)ADSCrossRefGoogle Scholar
  17. 17.
    Dingle, K., Schaper, S., Louis, A.A.: The structure of the genotype-phenotype map strongly constrains the evolution of non-coding RNA. Interface Focus 5(6), 20150053 (2015)CrossRefGoogle Scholar
  18. 18.
    Espinosa-Soto, C., Padilla-Longoria, P., Alvarez-Buylla, E.R.: A gene regulatory network model for cell-fate determination during Arabidopsis thaliana flower development that is robust and recovers experimental gene expression profiles. Plant Cell 16(11), 2923–2939 (2004)CrossRefGoogle Scholar
  19. 19.
    Fauré, A., Naldi, A., Chaouiya, C., Thieffry, D.: Dynamical analysis of a generic Boolean model for the control of the mammalian cell cycle. Bioinformatics 22(14), e124–e131 (2006)CrossRefGoogle Scholar
  20. 20.
    Giacomantonio, C.E., Goodhill, G.J.: A Boolean model of the gene regulatory network underlying mammalian cortical area development. PLoS Comput. Biol. 6(9), e1000936 (2010)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Glass, L., Mackey, M.C.: From Clocks to Chaos: The Rhythms of Life. Princeton University Press, Princeton (1988)zbMATHGoogle Scholar
  22. 22.
    Greenbury, S.F., Ahnert, S.E.: The organization of biological sequences into constrained and unconstrained parts determines fundamental properties of genotype-phenotype maps. J. R. Soc. Interface 12(113), 20150724 (2015)CrossRefGoogle Scholar
  23. 23.
    Greenbury, S.F., Johnston, I.G., Louis, A.A., Ahnert, S.E.: A tractable genotype-phenotype map modelling the self-assembly of protein quaternary structure. J. R. Soc. Interface 11(95), 20140249 (2014)CrossRefGoogle Scholar
  24. 24.
    Greenbury, S.F., Schaper, S., Ahnert, S.E., Louis, A.A.: Genetic correlations greatly increase mutational robustness and can both reduce and enhance evolvability. PLoS Comput. Biol. 12(3), e1004773 (2016)ADSCrossRefGoogle Scholar
  25. 25.
    Helikar, T., Konvalina, J., Heidel, J., Rogers, J.A.: Emergent decision-making in biological signal transduction networks. Proc. Natl. Acad. Sci. 105(6), 1913–1918 (2008)ADSCrossRefGoogle Scholar
  26. 26.
    Hopfield, J.J., Tank, D.W.: Collective computation with continuous variables. In: Disordered Systems and Biological Organization, pp. 155–170. Springer (1986)Google Scholar
  27. 27.
    Hopfield, J.J., Tank, D.W.: Computing with neural circuits - a model. Science 233(4764), 625–633 (1986)ADSCrossRefGoogle Scholar
  28. 28.
    Kauffman, S.A.: Homeostasis and differentiation in random genetic control networks. Nature 224(5215), 177–178 (1969)ADSCrossRefGoogle Scholar
  29. 29.
    Kauffman, S.A.: The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, Oxford (1993)Google Scholar
  30. 30.
    Lempel, A., Ziv, J.: On the complexity of finite sequences. IEEE Trans. Inf. Theory 22(1), 75–81 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Li, F., Long, T., Lu, Y., Ouyang, Q., Tang, C.: The yeast cell-cycle network is robustly designed. Proc. Natl. Acad. Sci. U.S.A. 101(14), 4781–4786 (2004)ADSCrossRefGoogle Scholar
  32. 32.
    Li, H., Helling, R., Tang, C., Wingreen, N.: Emergence of preferred structures in a simple model of protein folding. Science 273(5275), 666 (1996)ADSCrossRefGoogle Scholar
  33. 33.
    Li, S., Assmann, S.M., Albert, R.: Predicting essential components of signal transduction networks: a dynamic model of guard cell abscisic acid signaling. PLoS Biol. 4(10), e312 (2006)CrossRefGoogle Scholar
  34. 34.
    May, R.M.: Models for two interacting populations. In: Theoretical Ecology: Principles and Applications, pp. 49–70 (1976)Google Scholar
  35. 35.
    McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys. 5(4), 115–133 (1943)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Nochomovitz, Y.D., Li, H.: Highly designable phenotypes and mutational buffers emerge from a systematic mapping between network topology and dynamic output. Proc. Natl. Acad. Sci. U.S.A. 103(11), 4180–4185 (2006) ADSCrossRefGoogle Scholar
  37. 37.
    Raman, K., Wagner, A.: The evolvability of programmable hardware. J. Roy. Soc. Interface 8(55), rsif20100212 (2010). https://royalsocietypublishing.org/doi/full/10.1098/rsif.2010.0212#ref-list-1 Google Scholar
  38. 38.
    Remy, E., Ruet, P., Mendoza, L., Thieffry, D., Chaouiya, C.: From logical regulatory graphs to standard petri nets: dynamical roles and functionality of feedback circuits. In: Transactions on Computational Systems Biology VII, pp. 56–72. Springer (2006)Google Scholar
  39. 39.
    Schaper, S., Louis, A.A.: The arrival of the frequent: how bias in genotype-phenotype maps can steer populations to local optima. PLoS One 9(2), e86635 (2014)ADSCrossRefGoogle Scholar
  40. 40.
    Schuster, P., Fontana, W., Stadler, P.F., Hofacker, I.L.: From sequences to shapes and back: a case study in RNA secondary structures. Proc. Roy. Soc. London B: Biol. Sci. 255(1344), 279–284 (1994)ADSCrossRefGoogle Scholar
  41. 41.
    Steiner, C.F.: Environmental noise, genetic diversity and the evolution of evolvability and robustness in model gene networks. PLoS One 7(12), e52204 (2012)ADSCrossRefGoogle Scholar
  42. 42.
    Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Valiant, L.: Probably Approximately Correct: Nature’s Algorithms for Learning and Prospering in a Complex World. Basic Books, New York (2013)Google Scholar
  44. 44.
    Valle-Pérez, G., Camargo, C.Q., Louis, A.A.: Deep learning generalizes because the parameter-function map is biased towards simple functions. In: International Conference on Learning Representations (ICLR) (2019)Google Scholar
  45. 45.
    Wagner, A.: Robustness and evolvability: a paradox resolved. Proc. Roy. Soc. London B: Biol. Sci. 275(1630), 91–100 (2008)CrossRefGoogle Scholar
  46. 46.
    Wagner, A.: Robustness and Evolvability in Living Systems. Princeton University Press, Princeton (2013)CrossRefGoogle Scholar
  47. 47.
    Watson, R.A., Szathmáry, E.: How can evolution learn? Trends Ecol. Evol. 31(2), 147–157 (2016)CrossRefGoogle Scholar
  48. 48.
    Zañudo, J.G., Aldana, M., Martínez-Mekler, G.: Boolean threshold networks: virtues and limitations for biological modeling. In: Information Processing and Biological Systems, pp. 113–151 (2011)Google Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Oxford Internet InstituteUniversity of OxfordOxfordUK
  2. 2.Rudolf Peierls Centre for Theoretical PhysicsUniversity of OxfordOxfordUK

Personalised recommendations