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A Model with Darwinian Dynamics on a Rugged Landscape

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Abstract

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.

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Notes

  1. If one allows for many mutations to exist, while still having a single dominant population at almost all times, a somewhat different regime is obtained [27]. Then Eq. (4) no longer holds due to the population ‘cloud’ of deleterious mutations. If these mutants do not reproduce (\(\lambda =0\)), Eq. (4) may be mended by considering an effective \(N_{eff}=N-N_{cloud}\), but for more general deleterious mutations a simple prescription is hard to give. However, this correction is small when mutation rates are low \(\mu \ll 1\) (but not necessarily very low \(\mu N\ll 1\)). More precisely, Eq. (4) holds when the fraction of deleterious mutations is small, \(\frac{\mu \lambda }{\lambda -\lambda _{del}}\ll 1\), where \(\lambda \) is the fitness of the dominant population and \(\lambda _{del}\) is a typical fitness of deleterious mutations.

  2. This entails that the width of the fitness distribution in the population at a given time is \(\sigma _{\lambda }^{2}\sim \frac{N}{L^{2}\tau _0}\sim \frac{1}{N^{2}}\) (following arguments as in [33, 34] ). The time-scale for a given spin flip is on average \(\tau _{point}=L\tau _0\). which corresponds also to the time-scale for an individual to shuffle its entire configuration.

  3. This phase diagram seem very similar to the one obtained by Neher and Shraiman, where recombination and epistasis play the roles of mutation and selection. See: Neher et al. [39].

References

  1. Kirkpatrick, S., Vecchi, M.P.: Optimization by simmulated annealing. Science 220(4598), 671–680 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Kirkpatrick, S.: Optimization by simulated annealing: quantitative studies. J. Stat. Phys. 34(5–6), 975–986 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  3. Mitchell, M.: An Introduction to Genetic Algorithms. MIT Press, Cambridge (1998)

    MATH  Google Scholar 

  4. Pedersen, J.B., Sibani, P.: The long time behavior of the rate of recombination. J. Chem. Phys. 75(11), 5368–5372 (1981)

    Article  ADS  Google Scholar 

  5. Sibani, P., Pedersen, A.: Evolution dynamics in terraced NK landscapes. EPL 48(3), 346 (1999)

    Article  ADS  Google Scholar 

  6. Seetharaman, S., Jain, K.: Evolutionary dynamics on strongly correlated fitness landscapes. Phys. Rev. E 82(3), 031109 (2010)

    Article  ADS  Google Scholar 

  7. Saakian, D., Hu, C.-K.: Eigen model as a quantum spin chain: exact dynamics. Phys. Rev. E 69(2), 021913 (2004)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  9. Saakian, D.B., Hu, C.-K.: Solvable biological evolution model with a parallel mutation-selection scheme. Phys. Rev. E 69(4), 046121 (2004)

    Article  ADS  Google Scholar 

  10. Franz, S., Peliti, L., Sellitto, M.: An evolutionary version of the random energy model. J. Phys. A 26(23), L1195 (1993)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  12. Giardina, C., et al.: Simulating rare events in dynamical processes. J. Stat. Phys. 145(4), 787–811 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Eigen, M., McCaskill, J., Schuster, P.: The molecular quasi-species. Adv. Chem. Phys 75, 149–263 (1989)

    Google Scholar 

  14. Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. Harper & Row, New York (1970)

    MATH  Google Scholar 

  15. Peliti, L.: Introduction to the statistical theory of Darwinian evolution. (1997). arXiv: cond-mat/9712027

  16. Sella, G., Hirsh, A.E.: The application of statistical physics to evolutionary biology. Proc. Natl. Acad. Sci. 102(27), 9541–9546 (2005)

    Article  ADS  Google Scholar 

  17. Mustonen, V., Lassig, M.: Fitness flux and ubiquity of adaptive evolution. Proc. Natl. Acad. Sci. 107(9), 4248–4253 (2010)

    Article  ADS  Google Scholar 

  18. Berg, J., Lassig, M.: Stochastic evolution of transcription factor binding sites. Biophysics 48(1), 36–44 (2003)

    Google Scholar 

  19. Berg, J., Willmann, S., Lassig, M.: Adaptive evolution of transcription factor binding sites. BMC Evol. Biol. 4(1), 42 (2004)

    Article  Google Scholar 

  20. Barton, N.H., Coe, J.B.: On the application of statistical physics to evolutionary biology. J. Theor. Biol. 259(2), 317–324 (2009)

    Article  MathSciNet  Google Scholar 

  21. Moran, P.A.P.: The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford (1962)

    MATH  Google Scholar 

  22. Kauffman, S.A.: Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22(3), 437–467 (1969)

    Article  MathSciNet  Google Scholar 

  23. Kingman, J.F.C.: A simple model for the balance between selection and mutation. J. Appl. Probab. 15, 1–12 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  24. Neher, R.A., Vucelja, M., Mezard, M., Shraiman, B.I.: Emergence of clones in sexual populations. J. Stat. Mech 2013(01), P01008 (2013)

    Article  MathSciNet  Google Scholar 

  25. Derrida, B.: Random-energy model: limit of a family of disordered models. Phys. Rev. Lett. 45(2), 79–82 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  26. Schuster, P., Sigmund, K.: Replicator dynamics. J. Theor. Biol. 100(3), 533–538 (1983)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  28. Rouzine, I.M., Wakeley, J., Coffin, J.M.: The solitary wave of asexual evolution. Proc. Natl. Acad. Sci. 100(2), 587–592 (2003)

    Article  ADS  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  31. Mustonen, V., Lassig, M.: Molecular evolution under fitness fluctuations. Phys. Rev. Lett. 100(10), 108101 (2008)

    Article  ADS  Google Scholar 

  32. Cammarota, C., Marinari, E.: Spontaneous energy-barrier formation in an entropy-driven glassy dynamics (2014). arXiv: 1410.2116

  33. Kessler, D.A., Levine, H., Ridgway, D., Tsimring, L.: Evolution on a smooth landscape. J. Stat. Phys. 87(3–4), 519–544 (1997)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  35. Iwasa, Y., Michor, F., Nowak, M.A.: Stochastic tunnels in evolutionary dynamics. Genetics 166(3), 1571–1579 (2004)

    Article  Google Scholar 

  36. Weissman, D.B., et al.: The rate at which asexual populations cross fitness valleys. Theor. Popul. Biol. 75(4), 286–300 (2009)

    Article  MATH  Google Scholar 

  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)

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  40. Brotto, T., Bunin, G., Kurchan, J.: arXiv:1507.07453 (unpublished)

  41. Evans, D.J., Searles, D.J.: The fluctuation theorem. Adv. Phys. 51(7), 1529–1585 (2002)

    Article  ADS  Google Scholar 

  42. Berthier, L., Kurchan, J.: Non-equilibrium glass transitions in driven and active matter. Nat. Phys. 9(5), 310–314 (2013)

    Article  Google Scholar 

  43. Kussell, E., Leibler, S.: Phenotypic diversity, population growth, and information in fluctuating environments. Science 309(5743), 2075–2078 (2005)

    Article  ADS  Google Scholar 

  44. Struik, L.C.E.: On the rejuvenation of physically aged polymers by mechanical deformation. Polymer 38(16), 4053–4057 (1997)

    Article  Google Scholar 

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Acknowledgments

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

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Authors and Affiliations

  1. Laboratoire de Physique Statistique, École Normale Supérieure, PSL Research University, Université Paris Diderot Sorbonne Paris-Cité, Sorbonne Universités UPMC Univ Paris 06, CNRS, 24 rue Lhomond, 75005, Paris, France

    Tommaso Brotto & Jorge Kurchan

  2. Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16, 20133, Milano, Italy

    Tommaso Brotto

  3. INFN, Sezione di Milano, Via Celoria 16, 20133, Milano, Italy

    Tommaso Brotto

  4. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Guy Bunin

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  1. Tommaso Brotto
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Correspondence to Jorge Kurchan.

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Brotto, T., Bunin, G. & Kurchan, J. A Model with Darwinian Dynamics on a Rugged Landscape. J Stat Phys 166, 1065–1077 (2017). https://doi.org/10.1007/s10955-016-1637-2

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