Electronic Journal of Probability

Fixation probability for competing selective sweeps

Charles Cuthbertson, Alison Etheridge, and Feng Yu

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We consider a biological population in which a beneficial mutation is undergoing a selective sweep when a second beneficial mutation arises at a linked locus. We investigate the probability that both mutations will eventually fix in the population.  Previous work has dealt with the case where the second mutation to arise confers a smaller benefit than the first. In that case population size plays almost no rôle. Here we consider the opposite case and observe that, by contrast, the probability of both mutations fixing can be heavily dependent on population size. Indeed the key parameter is $rN$, the product of the population size and the recombination rate between the two selected loci.  If $rN$ is small, the probability that both mutations fix can be reduced through interference to almost zero while for large $rN$ the mutations barely influence one another.  The main rigorous result is a method for calculating the fixation probability of a double mutant in the large population limit.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 31, 36 pp.

Accepted: 23 April 2012
First available in Project Euclid: 4 June 2016

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Zentralblatt MATH identifier

Primary: 92D15: Problems related to evolution

selective sweep fixation probability double mutant

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Cuthbertson, Charles; Etheridge, Alison; Yu, Feng. Fixation probability for competing selective sweeps. Electron. J. Probab. 17 (2012), paper no. 31, 36 pp. doi:10.1214/EJP.v17-1954. https://projecteuclid.org/euclid.ejp/1465062353

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  • Athreya, Krishna B.; Ney, Peter E. Branching processes. Die Grundlehren der mathematischen Wissenschaften, Band 196. Springer-Verlag, New York-Heidelberg, 1972. xi+287 pp.
  • N. H. Barton. Linkage and the limits to natural selection. Genetics, 140:821–841, 1995.
  • Burkholder, D. L. Distribution function inequalities for martingales. Ann. Probability 1 (1973), 19–42.
  • Etheridge, Alison; Pfaffelhuber, Peter; Wakolbinger, Anton. An approximate sampling formula under genetic hitchhiking. Ann. Appl. Probab. 16 (2006), no. 2, 685–729.
  • Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8
  • Ewens, Warren J. Mathematical population genetics. Biomathematics, 9. Springer-Verlag, Berlin-New York, 1979. xii+325 pp. ISBN: 3-540-09577-2.
  • Fisher, R. A. The genetical theory of natural selection. A complete variorum edition. Revised reprint of the 1930 original. Edited, with a foreword and notes, by J. H. Bennett. Oxford University Press, Oxford, 1999. xxii+332 pp. ISBN: 0-19-850440-3
  • J. H. Gillespie. Genetic drift in an infinite population: The pseudohitchiking model. Genetics, 155:909–919, 2000.
  • J. H. Gillespie. Is the population size of a species relevant to its evolution? Evolution, 55:2161–2169, 2001.
  • Grimmett, G. R.; Stirzaker, D. R. Probability and random processes. Second edition. The Clarendon Press, Oxford University Press, New York, 1992. xii+541 pp. ISBN: 0-19-853666-6; 0-19-853665-8
  • J. B. Haldane. The mathematical theory of natural and artificial selection. Proc. Camb. Philos. Soc., 23:838:844, 1927.
  • W. G. Hill and A. Robertson. The effect of linkage on limits to artificial selection. Genetics Research, 8:269–294, 1966.
  • Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3
  • Karlin, Samuel; Taylor, Howard M. A second course in stochastic processes. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. xviii+542 pp. ISBN: 0-12-398650-8.
  • Y. Kim. Allele frequency distribution under recurrent selective sweeps. Genetics, 172:1967–1978, 2006.
  • M. Kimura and T. Ohta. The average number of generations until fixation of a mutant gene in a finite population. Genetics, 61(3):763, 1969.
  • G. A. McVean and Charlesworth B. The effects of Hill-Robertson interference between weakly selected mutations on patterns of molecular evolution and recombination. Genetics, 155:929–44, 2000.
  • H. J. Muller. Some genetic aspects of sex. American Naturalist, 66:118–138, 1932.
  • S. P. Otto and N. H. Barton. The evolution of recombination: removing the limits to natural selection. Genetics, 147:879–906, 1997.
  • Feng Yu and A. M. Etheridge. The fixation probability of two competing beneficial mutations. Theor. Popul. Biol., 78:36–45, 2010.