Journal of Statistical Physics

, Volume 166, Issue 3–4, pp 1065–1077 | Cite as

A Model with Darwinian Dynamics on a Rugged Landscape

  • Tommaso Brotto
  • Guy Bunin
  • Jorge Kurchan


We discuss the population dynamics with selection and random diffusion, keeping the total population constant, in a fitness landscape associated with Constraint Satisfaction, a paradigm for difficult optimization problems. We obtain a phase diagram in terms of the size of the population and the diffusion rate, with a glass phase inside which the dynamics keeps searching for better configurations, and outside which deleterious ‘mutations’ spoil the performance. The phase diagram is analogous to that of dense active matter in terms of temperature and drive.


Population dynamics Glasses Genetic algorithm Glassy landscape 



We would like to thank JP Bouchaud, and D.A. Kessler for helpful discussions.


  1. 1.
    Kirkpatrick, S., Vecchi, M.P.: Optimization by simmulated annealing. Science 220(4598), 671–680 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kirkpatrick, S.: Optimization by simulated annealing: quantitative studies. J. Stat. Phys. 34(5–6), 975–986 (1984)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Mitchell, M.: An Introduction to Genetic Algorithms. MIT Press, Cambridge (1998)zbMATHGoogle Scholar
  4. 4.
    Pedersen, J.B., Sibani, P.: The long time behavior of the rate of recombination. J. Chem. Phys. 75(11), 5368–5372 (1981)ADSCrossRefGoogle Scholar
  5. 5.
    Sibani, P., Pedersen, A.: Evolution dynamics in terraced NK landscapes. EPL 48(3), 346 (1999)ADSCrossRefGoogle Scholar
  6. 6.
    Seetharaman, S., Jain, K.: Evolutionary dynamics on strongly correlated fitness landscapes. Phys. Rev. E 82(3), 031109 (2010)ADSCrossRefGoogle Scholar
  7. 7.
    Saakian, D., Hu, C.-K.: Eigen model as a quantum spin chain: exact dynamics. Phys. Rev. E 69(2), 021913 (2004)ADSCrossRefGoogle Scholar
  8. 8.
    Saakian, D.B., Fontanari, J.F.: Evolutionary dynamics on rugged fitness landscapes: exact dynamics and information theoretical aspects. Phys. Rev. E 80(4), 041903 (2009)ADSCrossRefGoogle Scholar
  9. 9.
    Saakian, D.B., Hu, C.-K.: Solvable biological evolution model with a parallel mutation-selection scheme. Phys. Rev. E 69(4), 046121 (2004)ADSCrossRefGoogle Scholar
  10. 10.
    Franz, S., Peliti, L., Sellitto, M.: An evolutionary version of the random energy model. J. Phys. A 26(23), L1195 (1993)ADSCrossRefGoogle Scholar
  11. 11.
    Anderson, J.B.: Quantum chemistry by random walk. H 2P, H+ 3D3h1A? 1, H23?+ u, H41?+ g, Be 1S. J. Chem. Phys. 65(10), 4121–4127 (1976)ADSCrossRefGoogle Scholar
  12. 12.
    Giardina, C., et al.: Simulating rare events in dynamical processes. J. Stat. Phys. 145(4), 787–811 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Eigen, M., McCaskill, J., Schuster, P.: The molecular quasi-species. Adv. Chem. Phys 75, 149–263 (1989)Google Scholar
  14. 14.
    Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. Harper & Row, New York (1970)zbMATHGoogle Scholar
  15. 15.
    Peliti, L.: Introduction to the statistical theory of Darwinian evolution. (1997). arXiv: cond-mat/9712027
  16. 16.
    Sella, G., Hirsh, A.E.: The application of statistical physics to evolutionary biology. Proc. Natl. Acad. Sci. 102(27), 9541–9546 (2005)ADSCrossRefGoogle Scholar
  17. 17.
    Mustonen, V., Lassig, M.: Fitness flux and ubiquity of adaptive evolution. Proc. Natl. Acad. Sci. 107(9), 4248–4253 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    Berg, J., Lassig, M.: Stochastic evolution of transcription factor binding sites. Biophysics 48(1), 36–44 (2003)Google Scholar
  19. 19.
    Berg, J., Willmann, S., Lassig, M.: Adaptive evolution of transcription factor binding sites. BMC Evol. Biol. 4(1), 42 (2004)CrossRefGoogle Scholar
  20. 20.
    Barton, N.H., Coe, J.B.: On the application of statistical physics to evolutionary biology. J. Theor. Biol. 259(2), 317–324 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Moran, P.A.P.: The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford (1962)zbMATHGoogle Scholar
  22. 22.
    Kauffman, S.A.: Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22(3), 437–467 (1969)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kingman, J.F.C.: A simple model for the balance between selection and mutation. J. Appl. Probab. 15, 1–12 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Neher, R.A., Vucelja, M., Mezard, M., Shraiman, B.I.: Emergence of clones in sexual populations. J. Stat. Mech 2013(01), P01008 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Derrida, B.: Random-energy model: limit of a family of disordered models. Phys. Rev. Lett. 45(2), 79–82 (1980)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Schuster, P., Sigmund, K.: Replicator dynamics. J. Theor. Biol. 100(3), 533–538 (1983)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Desai, M.M., Fisher, D.S.: Beneficial mutation-selection balance and the effect of linkage on positive selection. Genetics 176(3), 1759–1798 (2007)CrossRefGoogle Scholar
  28. 28.
    Rouzine, I.M., Wakeley, J., Coffin, J.M.: The solitary wave of asexual evolution. Proc. Natl. Acad. Sci. 100(2), 587–592 (2003)ADSCrossRefGoogle Scholar
  29. 29.
    Rouzine, I.M., Brunet, Ã., Wilke, C.O.: The traveling-wave approach to asexual evolution: Muller’s ratchet and speed of adaptation. Theor. Popul. Biol. 73(1), 24–46 (2008)CrossRefzbMATHGoogle Scholar
  30. 30.
    Good, B.H., Rouzine, I.M., Balick, D.J., Hallatschek, O., Desai, M.M.: Distribution of fixed beneficial mutations and the rate of adaptation in asexual populations. Proc. Natl. Acad. Sci. 109(13), 4950–4955 (2012)ADSCrossRefGoogle Scholar
  31. 31.
    Mustonen, V., Lassig, M.: Molecular evolution under fitness fluctuations. Phys. Rev. Lett. 100(10), 108101 (2008)ADSCrossRefGoogle Scholar
  32. 32.
    Cammarota, C., Marinari, E.: Spontaneous energy-barrier formation in an entropy-driven glassy dynamics (2014). arXiv: 1410.2116
  33. 33.
    Kessler, D.A., Levine, H., Ridgway, D., Tsimring, L.: Evolution on a smooth landscape. J. Stat. Phys. 87(3–4), 519–544 (1997)ADSCrossRefzbMATHGoogle Scholar
  34. 34.
    Ridgway, D., Levine, H., Kessler, D.A.: Evolution on a smooth landscape: the role of bias. J. Stat. Phys. 90(1–2), 191–210 (1998)ADSCrossRefzbMATHGoogle Scholar
  35. 35.
    Iwasa, Y., Michor, F., Nowak, M.A.: Stochastic tunnels in evolutionary dynamics. Genetics 166(3), 1571–1579 (2004)CrossRefGoogle Scholar
  36. 36.
    Weissman, D.B., et al.: The rate at which asexual populations cross fitness valleys. Theor. Popul. Biol. 75(4), 286–300 (2009)CrossRefzbMATHGoogle Scholar
  37. 37.
    Krzakala, F., Montanari, A., Ricci-Tersenghi, F., Semerjian, G., Zdeborova, L.: Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. Natl. Acad. Sci. 104(25), 10 318–10 323 (2007)Google Scholar
  38. 38.
    Kirkpatrick, T.R., Thirumalai, D., Wolynes, P.G.: Scaling concepts for the dynamics of viscous liquids near an ideal glassy state. Phys. Rev. A 40(2), 1045 (1989)ADSCrossRefGoogle Scholar
  39. 39.
    Neher, R.A., Shraiman, B.I.: Competition between recombination and epistasis can cause a transition from allele to genotype selection. Proc. Natl. Acad. Sci. 106(16), 6866–6871 (2009)ADSCrossRefGoogle Scholar
  40. 40.
    Brotto, T., Bunin, G., Kurchan, J.: arXiv:1507.07453 (unpublished)
  41. 41.
    Evans, D.J., Searles, D.J.: The fluctuation theorem. Adv. Phys. 51(7), 1529–1585 (2002)ADSCrossRefGoogle Scholar
  42. 42.
    Berthier, L., Kurchan, J.: Non-equilibrium glass transitions in driven and active matter. Nat. Phys. 9(5), 310–314 (2013)CrossRefGoogle Scholar
  43. 43.
    Kussell, E., Leibler, S.: Phenotypic diversity, population growth, and information in fluctuating environments. Science 309(5743), 2075–2078 (2005)ADSCrossRefGoogle Scholar
  44. 44.
    Struik, L.C.E.: On the rejuvenation of physically aged polymers by mechanical deformation. Polymer 38(16), 4053–4057 (1997)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Laboratoire de Physique Statistique, École Normale SupérieurePSL Research University, Université Paris Diderot Sorbonne Paris-Cité, Sorbonne Universités UPMC Univ Paris 06, CNRSParisFrance
  2. 2.Dipartimento di FisicaUniversità degli Studi di MilanoMilanoItaly
  3. 3.INFN, Sezione di MilanoMilanoItaly
  4. 4.Department of PhysicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations