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.
Similar content being viewed by others
Notes
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.
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.
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
Kirkpatrick, S., Vecchi, M.P.: Optimization by simmulated annealing. Science 220(4598), 671–680 (1983)
Kirkpatrick, S.: Optimization by simulated annealing: quantitative studies. J. Stat. Phys. 34(5–6), 975–986 (1984)
Mitchell, M.: An Introduction to Genetic Algorithms. MIT Press, Cambridge (1998)
Pedersen, J.B., Sibani, P.: The long time behavior of the rate of recombination. J. Chem. Phys. 75(11), 5368–5372 (1981)
Sibani, P., Pedersen, A.: Evolution dynamics in terraced NK landscapes. EPL 48(3), 346 (1999)
Seetharaman, S., Jain, K.: Evolutionary dynamics on strongly correlated fitness landscapes. Phys. Rev. E 82(3), 031109 (2010)
Saakian, D., Hu, C.-K.: Eigen model as a quantum spin chain: exact dynamics. Phys. Rev. E 69(2), 021913 (2004)
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)
Saakian, D.B., Hu, C.-K.: Solvable biological evolution model with a parallel mutation-selection scheme. Phys. Rev. E 69(4), 046121 (2004)
Franz, S., Peliti, L., Sellitto, M.: An evolutionary version of the random energy model. J. Phys. A 26(23), L1195 (1993)
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)
Giardina, C., et al.: Simulating rare events in dynamical processes. J. Stat. Phys. 145(4), 787–811 (2011)
Eigen, M., McCaskill, J., Schuster, P.: The molecular quasi-species. Adv. Chem. Phys 75, 149–263 (1989)
Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. Harper & Row, New York (1970)
Peliti, L.: Introduction to the statistical theory of Darwinian evolution. (1997). arXiv: cond-mat/9712027
Sella, G., Hirsh, A.E.: The application of statistical physics to evolutionary biology. Proc. Natl. Acad. Sci. 102(27), 9541–9546 (2005)
Mustonen, V., Lassig, M.: Fitness flux and ubiquity of adaptive evolution. Proc. Natl. Acad. Sci. 107(9), 4248–4253 (2010)
Berg, J., Lassig, M.: Stochastic evolution of transcription factor binding sites. Biophysics 48(1), 36–44 (2003)
Berg, J., Willmann, S., Lassig, M.: Adaptive evolution of transcription factor binding sites. BMC Evol. Biol. 4(1), 42 (2004)
Barton, N.H., Coe, J.B.: On the application of statistical physics to evolutionary biology. J. Theor. Biol. 259(2), 317–324 (2009)
Moran, P.A.P.: The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford (1962)
Kauffman, S.A.: Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22(3), 437–467 (1969)
Kingman, J.F.C.: A simple model for the balance between selection and mutation. J. Appl. Probab. 15, 1–12 (1978)
Neher, R.A., Vucelja, M., Mezard, M., Shraiman, B.I.: Emergence of clones in sexual populations. J. Stat. Mech 2013(01), P01008 (2013)
Derrida, B.: Random-energy model: limit of a family of disordered models. Phys. Rev. Lett. 45(2), 79–82 (1980)
Schuster, P., Sigmund, K.: Replicator dynamics. J. Theor. Biol. 100(3), 533–538 (1983)
Desai, M.M., Fisher, D.S.: Beneficial mutation-selection balance and the effect of linkage on positive selection. Genetics 176(3), 1759–1798 (2007)
Rouzine, I.M., Wakeley, J., Coffin, J.M.: The solitary wave of asexual evolution. Proc. Natl. Acad. Sci. 100(2), 587–592 (2003)
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)
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)
Mustonen, V., Lassig, M.: Molecular evolution under fitness fluctuations. Phys. Rev. Lett. 100(10), 108101 (2008)
Cammarota, C., Marinari, E.: Spontaneous energy-barrier formation in an entropy-driven glassy dynamics (2014). arXiv: 1410.2116
Kessler, D.A., Levine, H., Ridgway, D., Tsimring, L.: Evolution on a smooth landscape. J. Stat. Phys. 87(3–4), 519–544 (1997)
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)
Iwasa, Y., Michor, F., Nowak, M.A.: Stochastic tunnels in evolutionary dynamics. Genetics 166(3), 1571–1579 (2004)
Weissman, D.B., et al.: The rate at which asexual populations cross fitness valleys. Theor. Popul. Biol. 75(4), 286–300 (2009)
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)
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)
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)
Brotto, T., Bunin, G., Kurchan, J.: arXiv:1507.07453 (unpublished)
Evans, D.J., Searles, D.J.: The fluctuation theorem. Adv. Phys. 51(7), 1529–1585 (2002)
Berthier, L., Kurchan, J.: Non-equilibrium glass transitions in driven and active matter. Nat. Phys. 9(5), 310–314 (2013)
Kussell, E., Leibler, S.: Phenotypic diversity, population growth, and information in fluctuating environments. Science 309(5743), 2075–2078 (2005)
Struik, L.C.E.: On the rejuvenation of physically aged polymers by mechanical deformation. Polymer 38(16), 4053–4057 (1997)
Acknowledgments
We would like to thank JP Bouchaud, and D.A. Kessler for helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-016-1637-2