Electronic Journal of Probability

The fixation time of a strongly beneficial allele in a structured population

Andreas Greven, Peter Pfaffelhuber, Cornelia Pokalyuk, and Anton Wakolbinger

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Abstract

For a beneficial allele which enters a large unstructured population and eventually goes to fixation, it is known that the time to fixation is approximately $2\log (\alpha )/\alpha $ for a large selection coefficient $\alpha $. For a population that is distributed over finitely many colonies, with migration between these colonies, we detect various regimes of the migration rate $\mu $ for which the fixation times have different asymptotics as $\alpha \to \infty $.

If $\mu $ is of order $\alpha $, the allele fixes (as in the spatially unstructured case) in time $\sim 2\log (\alpha )/\alpha $. If $\mu $ is of order $\alpha ^\gamma , 0\leq \gamma \leq 1$, the fixation time is $\sim (2 + (1-\gamma )\Delta ) \log (\alpha )/\alpha $, where $\Delta $ is the number of migration steps that are needed to reach all other colonies starting from the colony where the beneficial allele appeared. If $\mu = 1/\log (\alpha )$, the fixation time is $\sim (2+S)\log (\alpha )/\alpha $, where $S$ is a random time in a simple epidemic model.

The main idea for our analysis is to combine a new moment dual for the process conditioned to fixation with the time reversal in equilibrium of a spatial version of Neuhauser and Krone’s ancestral selection graph.

Article information

Source
Electron. J. Probab. Volume 21, Number (2016), paper no. 61, 42 pp.

Dates
Received: 2 March 2014
Accepted: 19 September 2016
First available in Project Euclid: 4 October 2016

Permanent link to this document
http://projecteuclid.org/euclid.ejp/1475586182

Digital Object Identifier
doi:10.1214/16-EJP3355

Subjects
Primary: 92D15: Problems related to evolution
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx] 92D10: Genetics {For genetic algebras, see 17D92}

Keywords
interacting Wright–Fisher diffusions ancestral selection graph branching process approximation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Greven, Andreas; Pfaffelhuber, Peter; Pokalyuk, Cornelia; Wakolbinger, Anton. The fixation time of a strongly beneficial allele in a structured population. Electron. J. Probab. 21 (2016), paper no. 61, 42 pp. doi:10.1214/16-EJP3355. http://projecteuclid.org/euclid.ejp/1475586182.


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References

  • [1] Aldous, D. (1985). Exchangeability and related topics. In P. Hennequin (Ed.), École d’Été de Probabilités de Saint-Flour XIII–1983, Volume 1117 of Lecture Notes in Mathematics, Berlin, pp. 1–198. Springer.
  • [2] Athreya, K. and P. Ney (1972). Branching Processes. Springer.
  • [3] Athreya, S. and J. Swart (2005). Branching-coalescing particle systems. Prob. Theory Relat. Fields 131, 376–414.
  • [4] Dawson, D. (1993). Measure-valued Markov processes. In P. Hennequin (Ed.), École d’Été de Probabilités de Saint-Flour XXI–1991, Volume 1541 of Lecture Notes in Mathematics, Berlin, pp. 1–260. Springer.
  • [5] A. Depperschmidt, A. Greven, and P. Pfaffelhuber. Tree-valued Fleming–Viot dynamics with mutation and selection. Ann. Appl. Probab., 22(6):2560–2615, 2012.
  • [6] Etheridge, A., P. Pfaffelhuber, and A. Wakolbinger (2006). An approximate sampling formula under genetic hitchhiking. Ann. Appl. Probab. 16, 685–729.
  • [7] Ethier, S. and T. Kurtz (1986). Markov Processes. Characterization and Convergence. John Wiley, New York.
  • [8] Fearnhead, P. (2002). The common ancestor at a non-neutral locus. J. Appl. Probab. 39, 38–54.
  • [9] Feinberg, M. (1979). Lectures on chemical reaction networks. Notes of lectures given at the Mathematics Research Centre. University of Wisconsin.
  • [10] Harris, T. (1963). The Theory of Branching Processes. Springer.
  • [11] Hartfield, M. (2012). A framework for estimation the fixation time of an advantageous allele in stepping-stone models. J. Evol. Biol. 25, 1751–1764.
  • [12] Kaplan, N. L., R. R. Hudson, and C. H. Langley (1989). The ‘Hitchhiking effect’ revisited. Genetics 123, 887–899.
  • [13] Kim, Y. and T. Maruki (2011). Hitchhiking effect of a beneficial mutation spreading in a subdivided population. Genetics 189, 213–226.
  • [14] Krone, S. and C. Neuhauser (1997). Ancestral processes with selection. Theo. Pop. Biol. 51, 210–237.
  • [15] Mano, S. (2009). Duality, ancestral and diffusion processes in models with selection. Theo. Pop. Biol. 75, 164–175.
  • [16] Maynard Smith, J. and J. Haigh (1974). The hitch-hiking effect of a favorable gene. Genetic Research 23, 23–35.
  • [17] Nagylaki, T. (1982). Geographical invariance in population genetics. J. Theo. Biol. 99(1), 159–172.
  • [18] Neuhauser, C. and S. Krone (1997). The genealogy of samples in models with selection. Genetics 154, 519–534.
  • [19] Nielsen, R. (2005). Molecular Signatures of Natural Selection. Annu. Rev. Genet. 39, 197–218.
  • [20] Norris, J. R. (1998). Markov Chains. Cambridge University Press.
  • [21] Pfaffelhuber, P. and C. Pokalyuk (2013). The ancestral selection graph under strong directional selection. Theo. Pop. Biol. 87, 25–33.
  • [22] Sabeti, P., S. Schaffner, B. Fry, J. Lohmueller, P. Varilly, O. Shamovsky, A. Palma, T. Mikkelsen, D. Altshuler, and E. Lander (2006). Positive natural selection in the human lineage. Science 312, 1614–1620.
  • [23] Schweinsberg, J. and R. Durrett (2005). Random partitions approximating the coalescence of lineages during a selective sweep. Ann. Appl. Probab. 15, 1591–1651.
  • [24] Shiga, T. and K. Uchiyama (1986). Stationary states and their stability of the stepping stone model involving mutation and selection. Prob. Theo. Rel. Fields 73, 87–116.
  • [25] Slatkin, M. (1981). Fixation probabilities and fixation times in a subdivided population. Evolution 35, 477–488.
  • [26] Slatkin, M. (1976). The rate of spread of an advantageous allele in a subdivided population. Population Genetics and Ecology, 767–780.
  • [27] Stephan, W., T. H. E. Wiehe, and M. W. Lenz (1992). The effect of strongly selected substitutions on neutral polymorphism: Analytical results based on diffusion theory. Theo. Pop. Biol. 41, 237–254.
  • [28] Thornton, K., J. Jensen, C. Becquet, and P. Andolfatto (2007). Progress and prospects in mapping recent selection in the genome. Heredity 98, 340–348.
  • [29] Wakeley, J. and O. Sargsyan (2009). The conditional ancestral selection graph with strong balancing selection. Theo. Pop. Biol. 75, 355–364.
  • [30] Whitlock, M. C. (2003). Fixation probability and time in subdivided populations. Genetics 164(2), 767–779.