Journal of Statistical Physics

, Volume 172, Issue 1, pp 156–174 | Cite as

Lines of Descent Under Selection

  • Ellen BaakeEmail author
  • Anton Wakolbinger


We review recent progress on ancestral processes related to mutation-selection models, both in the deterministic and the stochastic setting. We mainly rely on two concepts, namely, the killed ancestral selection graph and the pruned lookdown ancestral selection graph. The killed ancestral selection graph gives a representation of the type of a random individual from a stationary population, based upon the individual’s potential ancestry back until the mutations that define the individual’s type. The pruned lookdown ancestral selection graph allows one to trace the ancestry of individuals from a stationary distribution back into the distant past, thus leading to the stationary distribution of ancestral types. We illustrate the results by applying them to a prototype model for the error threshold phenomenon.


Mutation-selection model Killed ancestral selection graph Pruned lookdown ancestral selection graph Error threshold 

Mathematics Subject Classification

60J27 60J75 92D15 05C80 



It is our pleasure to thank Fernando Cordero, Sebastian Hummel, and Ute Lenz for fruitful discussions. This project received financial support from Deutsche Forschungsgemeinschaft (Priority Programme SPP 1590 Probabilistic Structures in Evolution, Grant Nos. BA 2469/5-1 and WA 967/4-1).


  1. 1.
    Athreya, K.B., Ney, P.E.: Branching Processes. Springer, New York (1970)zbMATHGoogle Scholar
  2. 2.
    Athreya, S.R., Swart, J.M.: Branching-coalescing particle systems. Prob. Theory Relat. Fields 131, 376–414 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baake, E., Baake, M., Wagner, H.: The Ising quantum chain is equivalent to a model of biological evolution, Phys. Rev. Lett. 78, 559–562 (1997), and Erratum Phys. Rev. Lett. 79, 1782 (1997)Google Scholar
  4. 4.
    Baake, E., Baake, M., Bovier, A., Klein, M.: An asymptotic maximum principle for essentially linear evolution models. J. Math. Biol. 50, 83–114 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baake, E., Cordero, F., Hummel, S.: A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent. arXiv:1710.04573 (submitted)
  6. 6.
    Baake, E., Gabriel, W.: Biological evolution through mutation, selection, and drift: an introductory review. In: Stauffer, D. (ed.) Annual Reviews of Computational Physics, vol. 7, pp. 203–264. World Scientific, Singapore (2000)Google Scholar
  7. 7.
    Baake, E., Georgii, H.-O.: Mutation, selection, and ancestry in branching models: a variational approach. J. Math. Biol. 54, 257–303 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Baake, E., Lenz, U., Wakolbinger, A.: The common ancestor type distribution of a \(\Lambda \)-Wright-Fisher process with selection and mutation. Electron. Commun. Probab. 21, 1–16 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Baake, E., Wakolbinger, A.: Feller’s contributions to mathematical biology. In: Schilling, R.L., Vondracek, Z., Woyczyński, W.A. (eds.) Selected Works of William Feller, vol. 2, pp. 25–43. Springer, Berlin (2015)Google Scholar
  10. 10.
    Bürger, R.: The Mathematical Theory of Selection, Recombination, and Mutation. Wiley, Chichester (2000)zbMATHGoogle Scholar
  11. 11.
    Cordero, F.: The deterministic limit of the Moran model: a uniform central limit theorem. Markov Processes Relat. Fields 23, 313–324 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cordero, F.: Common ancestor type distribution: a Moran model and its deterministic limit. Stoch. Proc. Appl. 127, 590–621 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Crow, J.F., Kimura, M.: Some genetic problems in natural populations. In: J. Neyman (ed.) Proceedings of the Third Berkeley Symposium on Probability and Mathematical Statistics, vol. 4. University of California Press, Berkeley, pp. 1–22 (1956)Google Scholar
  14. 14.
    Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. Harper & Row, New York (1970)zbMATHGoogle Scholar
  15. 15.
    Donnelly, P., Kurtz, T.G.: Genealogical processes for Fleming-Viot models with selection and recombination. Ann. Appl. Probab. 9, 1091–1148 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Durrett, R.: Probability Models for DNA Sequence Evolution, 2nd edn. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  17. 17.
    Eigen, M.: Selforganization of matter and the evolution of biological macromolecules. Naturwiss. 58, 465–523 (1971)ADSCrossRefGoogle Scholar
  18. 18.
    Eigen, M., McCaskill, J., Schuster, P.: The molecular quasi-species. Adv. Chem. Phys. 75, 149–263 (1989)Google Scholar
  19. 19.
    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986). reprint 2005CrossRefzbMATHGoogle Scholar
  20. 20.
    Ewens, W.J.: Mathematical Population Genetics I. Theoretical Introduction, 2nd edn. Springer, New York (2004)CrossRefzbMATHGoogle Scholar
  21. 21.
    Fearnhead, P.: The common ancestor at a nonneutral locus. J. Appl. Probab. 39, 38–54 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Feller, W.: Diffusion processes in genetics. In: J. Neyman (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability 1950, pp. 227–246. University of California Press, Berkeley (1951)Google Scholar
  23. 23.
    Fisher, R.A.: The Genetical Theory of Natural Selection. Clarendon Press, Oxford (1930)CrossRefzbMATHGoogle Scholar
  24. 24.
    Garske, T.: Error thresholds in a mutation-selection model with Hopfield-type fitness. Bull. Math. Biol. 68, 1715–1746 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Garske, T., Grimm, U.: Maximum principle and mutation thresholds for four-letter sequence evolution. Bull. Math. Biol. 66, 397–421 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Georgii, H.O., Baake, E.: Supercritical multitype branching processes: the ancestral types of typical individuals. Adv. Appl. Probab. 35, 1090–1110 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hermisson, J., Redner, O., Wagner, H., Baake, E.: Mutation-selection balance: ancestry, load, and maximum principle. Theor. Popul. Biol. 62, 9–46 (2002)CrossRefzbMATHGoogle Scholar
  28. 28.
    Hoppe, F.: Polya-like urns and the Ewens’ sampling formula. J. Math. Biol. 20, 91–94 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jagers, P., Nerman, O.: The stable doubly infinite pedigree process of supercritical branching populations. Z. für Wahrscheinlichkeitstheorie und verwandte Gebiete 65, 445–460 (1984)CrossRefzbMATHGoogle Scholar
  30. 30.
    Jagers, P.: General branching processes as Markov fields. Stoch. Proc. Appl. 32, 183–242 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Jagers, P.: Stabilities and instabilities in population dynamics. J. Appl. Probab. 29, 770–780 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Jansen, S., Kurt, N.: On the notion(s) of duality for Markov processes. Probab. Surv. 11, 59–120 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kimura, M.: On the probability of fixation of mutant genes in a population. Genetics 47, 713719 (1962)Google Scholar
  34. 34.
    Kingman, J.F.C.: The coalescent. Stoch. Proc. Appl. 13, 235–248 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kingman, J.F.C.: On the genealogy of large populations. J. Appl. Probab. 19A, 27–43 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Krone, S.M., Neuhauser, C.: Ancestral processes with selection. Theor. Popul. Biol. 51, 210–237 (1997)CrossRefzbMATHGoogle Scholar
  37. 37.
    Kurtz, T.G.: Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Probab. 8, 344–356 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Leibler, S., Kussell, E.: Individual histories and selection in heterogeneous populations. Proc. Natl. Acad. Sci. USA 107, 13183–13188 (2010)ADSCrossRefGoogle Scholar
  39. 39.
    Lenz, U., Kluth, S., Baake, E., Wakolbinger, A.: Looking down in the ancestral selection graph: a probabilistic approach to the common ancestor type distribution. Theor. Popul. Biol. 103, 27–37 (2015)CrossRefzbMATHGoogle Scholar
  40. 40.
    Leuthäusser, I.: An exact correspondence between Eigen’s evolution model and a two-dimensional Ising system. J. Chem. Phys. 84, 1884–1885 (1986)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Leuthäusser, I.: Statistical mechanics of Eigen’s evolution model. J. Stat. Phys. 48, 343–360 (1987)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Liggett, T.M.: Continuous Time Markov Processes: An Introduction. AMS, Providence (2010)CrossRefzbMATHGoogle Scholar
  43. 43.
    Malécot, G.: Les Mathématiques de l’Hérédité. Masson, Paris (1948)zbMATHGoogle Scholar
  44. 44.
    Moran, P.A.P.: Random processes in genetics. Proc. Camb. Philos. Soc. 54, 60–71 (1958)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Nagylaki, T.: Gustave Malécot and the transition from classical to modern population genetics. Genetics 122, 253–268 (1989)Google Scholar
  46. 46.
    Peliti, L.: Quasispecies evolution in general mean-field landscapes. Europhys. Lett. 57, 745–751 (2002)ADSCrossRefGoogle Scholar
  47. 47.
    Sughiyama, Y., Kobayashi, T.J.: Steady-state thermodynamics for population growth in fluctuating environments. Phys. Rev. E 95, 012131 (2017)ADSCrossRefGoogle Scholar
  48. 48.
    Tarazona, P.: Error threshold for molecular quasispecies as phase transition: from simple landscapes to spin glass models. Phys. Rev. A 45, 6038–6050 (1992)ADSCrossRefGoogle Scholar
  49. 49.
    Taylor, J.E.: The common ancestor process for a Wright-Fisher diffusion. Electron. J. Probab. 12, 808–847 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Wright, S.: Evolution in Mendelian populations. Genetics 16, 97–159 (1931)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Faculty of TechnologyBielefeld UniversityBielefeldGermany
  2. 2.Institute of MathematicsGoethe-Universität FrankfurtFrankfurt am MainGermany

Personalised recommendations