The Annals of Applied Probability

An approximate sampling formula under genetic hitchhiking

Alison Etheridge, Peter Pfaffelhuber, and Anton Wakolbinger

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Abstract

For a genetic locus carrying a strongly beneficial allele which has just fixed in a large population, we study the ancestry at a linked neutral locus. During this “selective sweep” the linkage between the two loci is broken up by recombination and the ancestry at the neutral locus is modeled by a structured coalescent in a random background. For large selection coefficients α and under an appropriate scaling of the recombination rate, we derive a sampling formula with an order of accuracy of $\mathcal{O}((\log \alpha)^{-2})$ in probability. In particular we see that, with this order of accuracy, in a sample of fixed size there are at most two nonsingleton families of individuals which are identical by descent at the neutral locus from the beginning of the sweep. This refines a formula going back to the work of Maynard Smith and Haigh, and complements recent work of Schweinsberg and Durrett on selective sweeps in the Moran model.

Article information

Source
Ann. Appl. Probab. Volume 16, Number 2 (2006), 685-729.

Dates
First available in Project Euclid: 29 June 2006

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1151592248

Digital Object Identifier
doi:10.1214/105051606000000114

Mathematical Reviews number (MathSciNet)
MR2244430

Zentralblatt MATH identifier
1115.92044

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] 60K37: Processes in random environments 92D10: Genetics {For genetic algebras, see 17D92}

Keywords
Selective sweeps genetic hitchhiking approximate sampling formula random ancestral partition diffusion approximation structured coalescent Yule processes random background

Citation

Etheridge, Alison; Pfaffelhuber, Peter; Wakolbinger, Anton. An approximate sampling formula under genetic hitchhiking. The Annals of Applied Probability 16 (2006), no. 2, 685--729. doi:10.1214/105051606000000114. http://projecteuclid.org/euclid.aoap/1151592248.


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References

  • Barton, N. (1998). The effect of hitch-hiking on neutral genealogies. Gen. Res. 72 123--133.
  • Barton, N. H., Etheridge, A. M. and Sturm, A. (2004). Coalescence in a random background. Ann. Appl. Probab. 14 754--785.
  • Birkner, M., Blath, J., Capaldo, A., Etheridge, A., Möhle, M., Schweinsberg, J. and Wakolbinger, A. (2005). Alpha-stable branching and beta coalescents. Electron. J. Probab. 10 303--325.
  • Durrett, R. and Schweinsberg, J. (2004). Approximating selective sweeps. Theor. Popul. Biol. 66 129--138.
  • Durrett, R. and Schweinsberg, J. (2005). A coalescent model for the effect of advantageous mutations on the genealogy of a population. Stochastic Process. Appl. 115 1628--1657.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Evans, S. N. and O'Connell, N. (1994). Weighted occupation time for branching particle systems and a representation for the supercritical superprocess. Canad. Math. Bull. 37 187--196.
  • Ewens, W. J. (2004). Mathematical Population Genetics. I. Theoretical Introduction, 2nd ed. Springer, New York.
  • [Fis30] Fisher, R. A. (1930). The Genetical Theory of Natural Selection, 2nd ed. Clarendon Press.
  • Fristedt, B. and Gray, L. (1997). A Modern Approach to Probability Theory. Birkhäuser, Boston.
  • [Gri03] Griffiths, R. C. (2003). The frequency spectrum of a mutation and its age, in a general diffusion model. Theor. Popul. Biol. 64 241--251.
  • Griffiths, R. C. and Tavaré, S. (1998). The age of a mutation in a general coalescent tree. Stochastic Models 14 273--295.
  • Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. Wiley, New York.
  • Kaj, I. and Krone, S. M. (2003). The coalescent process in a population with stochastically varying size. J. Appl. Probab. 40 33--48.
  • [KHL89] Kaplan, N. L., Hudson, R. R. and Langley, C. H. (1989). The `Hitchhiking effect' revisited. Genetics 123 887--899.
  • Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.
  • Kurtz, T. G. (1971). Limit theorems for sequences of jupmp Markov processes approximating ordinary differential equations. J. Appl. Probab. 8 344--356.
  • [MSH74] Maynard Smith, J. and Haigh, J. (1974). The hitch-hiking effect of a favorable gene. Gen. Res. 23 23--35.
  • O'Connell, N. (1993). Yule process approximation for the skeleton of a branching process. J. Appl. Probab. 30 725--729.
  • Riordan, J. (1968). Combinatorial Identities. Wiley, New York.
  • Schweinsberg, J. and Durrett, R. (2005). Random partitions approximating the coalescence of lineages during a selective sweep. Ann. Appl. Probab. 15 1591--1651.
  • [SWL92] Stephan, W., Wiehe, T. H. E. and Lenz, M. W. (1992). The effect of strongly selected substitutions on neutral polymorphism: Analytical results based on diffusion theory. Theor. Popul. Biol. 41 237--254.