Accurate Computation of Likelihoods in the Coalescent with Recombination Via Parsimony

  • Rune B. Lyngsø
  • Yun S. Song
  • Jotun Hein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4955)

Abstract

Understanding the variation of recombination rates across a given genome is crucial for disease gene mapping and for detecting signatures of selection, to name just a couple of applications. A widely-used method of estimating recombination rates is the maximum likelihood approach, and the problem of accurately computing likelihoods in the coalescent with recombination has received much attention in the past. A variety of sampling and approximation methods have been proposed, but no single method seems to perform consistently better than the rest, and there still is great value in developing better statistical methods for accurately computing likelihoods. So far, with the exception of some two-locus models, it has remained unknown how the true likelihood exactly behaves as a function of model parameters, or how close estimated likelihoods are to the true likelihood. In this paper, we develop a deterministic, parsimony-based method of accurately computing the likelihood for multi-locus input data of moderate size. We first find the set of all ancestral configurations (ACs) that occur in evolutionary histories with at most k crossover recombinations. Then, we compute the likelihood by summing over all evolutionary histories that can be constructed only using the ACs in that set. We allow for an arbitrary number of crossing over, coalescent and mutation events in a history, as long as the transitions stay within that restricted set of ACs. For given parameter values, by gradually increasing the bound k until the likelihood stabilizes, we can obtain an accurate estimate of the likelihood. At least for moderate crossover rates, the algorithm-based method described here opens up a new window of opportunities for testing and fine-tuning statistical methods for computing likelihoods.

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References

  1. 1.
    Bafna, V., Bansal, V.: The number of recombination events in a sample history: conflict graph and lower bounds. IEEE/ACM Transactions on Computational Biology and Bioinformatics 1, 78–90 (2004)CrossRefGoogle Scholar
  2. 2.
    Bafna, V., Bansal, V.: Improved Recombination Lower Bounds for Haplotype Data. In: Miyano, S., Mesirov, J., Kasif, S., Istrail, S., Pevzner, P.A., Waterman, M. (eds.) RECOMB 2005. LNCS (LNBI), vol. 3500, pp. 569–584. Springer, Heidelberg (2005)Google Scholar
  3. 3.
    Beaumont, M.: Detecting population expansion and decline using microsatellites. Genetics 153, 2013–2029 (1999)Google Scholar
  4. 4.
    Bordewich, M., Semple, C.: Computing the minimum number of hybridization events for a consistent evolutionary history. Discrete Applied Mathematics 155, 914–928 (2007)MathSciNetMATHGoogle Scholar
  5. 5.
    De Iorio, M., Griffiths, R.C.: Importance sampling on coalescent histories. I. Adv. Appl. Prob. 36, 417–433 (2004)CrossRefMATHGoogle Scholar
  6. 6.
    De Iorio, M., Griffiths, R.C.: Importance sampling on coalescent histories. II: Subdivided population models. Adv. Appl. Prob. 36, 434–454 (2004)CrossRefMATHGoogle Scholar
  7. 7.
    Ethier, S.N., Griffiths, R.C.: The infinitely-many-sites model as a measure valued diffusion. Ann. Probab. 15, 515–545 (1987)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Ethier, S.N., Griffiths, R.C.: On the two-locus sampling distribution. J. Math. Biol. 29, 131–159 (1990)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Fearnhead, P., Donnelly, P.: Estimating recombination rates from population genetic data. Genetics 159, 1299–1318 (2001)Google Scholar
  10. 10.
    Fearnhead, P., Donnelly, P.: Approximate likelihood methods for estimating local recombination rates. J. R. Statist. Soc. B 64, 657–680 (2002)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Fearnhead, P., Smith, N.G.C.: A novel method with improved power to detect recombination hotspots from polymorphism data reveals multiple hotspots in human genes. Am. J. Hum. Genet. 77, 781–794 (2005)CrossRefGoogle Scholar
  12. 12.
    Griffiths, R.C., Marjoram, P.: Ancestral inference from samples of DNA sequences with recombination. J. Comput. Biol. 3, 479–502 (1996)CrossRefGoogle Scholar
  13. 13.
    Griffiths, R.C., Tavaré, S.: Ancestral inference in population genetics. Stat. Sci. 9, 307–319 (1994)CrossRefMATHGoogle Scholar
  14. 14.
    Griffiths, R.C., Tavaré, S.: Sampling theory for neutral alleles in a varying environment. Proc. R. Soc. London B. 344, 403–410 (1994)Google Scholar
  15. 15.
    Griffiths, R.C., Tavaré, S.: Simulating probability distributions in the coalescent. Theor. Popul. Biol. 46, 131–159 (1994)CrossRefMATHGoogle Scholar
  16. 16.
    Gusfield, D.: Optimal, efficient reconstruction of Root-Unknown phylogenetic networks with constrained recombination. J. Comput. Sys. Sci. 70, 381–398 (2005)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Gusfield, D., Eddhu, S., Langley, C.: The fine structure of galls in phylogenetic networks. INFORMS J. on Computing, special issue on Computational Biology 16, 459–469 (2004)MathSciNetGoogle Scholar
  18. 18.
    Gusfield, D., Eddhu, S., Langley, C.: Optimal, efficient reconstruction of phylogenetic networks with constrained recombination. J. Bioinf. Comput. Biol. 2, 173–213 (2004)CrossRefGoogle Scholar
  19. 19.
    Hein, J.: Reconstructing evolution of sequences subject to recombination using parsimony. Math. Biosci. 98, 185–200 (1990)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Hein, J.: A heuristic method to reconstruct the history of sequences subject to recombination. J. Mol. Evol. 36, 396–405 (1993)CrossRefGoogle Scholar
  21. 21.
    Hudson, R.R.: Generating Samples under the Wright-Fisher neutral model of genetic variation. Bioinformatics 18, 337–338 (2002)CrossRefGoogle Scholar
  22. 22.
    Hudson, R., Kaplan, N.: Statistical properties of the number of recombination events in the history of a sample of DNA sequences. Genetics 111, 147–164 (1985)Google Scholar
  23. 23.
    Hudson, R.R.: Two-locus sampling distributions and their application. Genetics 159, 1805–1817 (2001)Google Scholar
  24. 24.
    International HapMap Consortium. A haplotype map of the human genome 437, 1299–1320 (2005)Google Scholar
  25. 25.
    Kuhner, M.K., Yamato, J., Felsenstein, J.: Estimating effective population size and mutation rate from sequence data using metropolis-hastings sampling. Genetics 140, 1421–1430 (1995)Google Scholar
  26. 26.
    Kuhner, M.K., Yamato, J., Felsenstein, J.: Maximum likelihood estimation of recombination rates from population data. Genetics 156, 1393–1401 (2000)Google Scholar
  27. 27.
    Larribe, F., Lessard, S., Schork, N.J.: Gene Mapping via the Ancestral Recombination Graph. Theor. Popul. Biol. 62, 2150–2229 (2002)CrossRefGoogle Scholar
  28. 28.
    Li, N., Stephens, M.: Modeling linkage disequilibrium and identifying recombination hotspots using single-nucleotide polymorphism data. Genetics 165, 2213–2233 (2003)Google Scholar
  29. 29.
    Lyngsø, R.B., Song, Y.S., Hein, J.: Minimum recombination histories by branch and bound. In: Casadio, R., Myers, G. (eds.) WABI 2005. LNCS (LNBI), vol. 3692, pp. 239–250. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  30. 30.
    McVean, G., Awadalla, P., Fearnhead, P.: A coalescent-based method for detecting and estimating recombination from gene sequences. Genetics 160, 1231–1241 (2002)Google Scholar
  31. 31.
    McVean, G., Cardin, N.: Approximating the coalescent with recombination. Philos. Trans. R. Soc. Lond. B Biol. Sci. 360, 1387–1393 (2005)CrossRefGoogle Scholar
  32. 32.
    McVean, G.A.T., Myers, S., Hunt, S., Deloukas, P., Bentley, D.R., Donnelly, P.: The fine-scale structure of recombination rate variation in the human genome. Science 304, 581–584 (2004)CrossRefGoogle Scholar
  33. 33.
    Myers, S., Bottolo, L., Freeman, C., McVean, G., Donnelly, P.: A fine-scale map of recombination rates and hotspots across the human genome. Science 310, 321–324 (2005)CrossRefGoogle Scholar
  34. 34.
    Myers, S.R., Griffiths, R.C.: Bounds on the minimum number of recombination events in a sample history. Genetics 163, 375–394 (2003)Google Scholar
  35. 35.
    Simonsen, K.L., Churchill, G.A.: A Markov chain model of coalescence with recombination. Theor. Popul. Biol. 52, 43–59 (1997)CrossRefMATHGoogle Scholar
  36. 36.
    Song, Y.S., Hein, J.: Parsimonious reconstruction of sequence evolution and haplotype blocks: Finding the minimum number of recombination events. In: Proc. of Workshop on Algorithms in Bioinformatics 2003, Berlin, Germany. LNCS, pp. 287–302. Springer, Berlin (2003)Google Scholar
  37. 37.
    Song, Y.S., Hein, J.: On the minimum number of recombination events in the evolutionary history of DNA sequences. J. Math. Biol. 48, 160–186 (2004)CrossRefMathSciNetMATHGoogle Scholar
  38. 38.
    Song, Y.S., Hein, J.: Constructing minimal ancestral recombination graphs. J. Comput. Biol. 12, 147–169 (2005)CrossRefGoogle Scholar
  39. 39.
    Song, Y.S., Lyngsø, R.B., Hein, J.: Counting all possible ancestral configurations of sample sequences in population genetics. IEEE Transactions on Computational Biology and Bioinformatics 3(3), 239–251 (2006)CrossRefGoogle Scholar
  40. 40.
    Song, Y.S., Wu, Y., Gusfield, D.: Efficient computation of close lower and upper bounds on the minimum number of needed recombinations in the evolution of biological sequences. In: Proc. of ISMB 2005, Bioinformatics, vol. 21, pp. 413–422 (2005)Google Scholar
  41. 41.
    Stephens, M., Donnelly, P.: Inference in molecular population genetics. J.R. Stat. Soc. Ser. B 62, 605–655 (2000)CrossRefMathSciNetMATHGoogle Scholar
  42. 42.
    Wall, J.D.: A comparison of estimators of the population recombination rate. Mol. Biol. Evol. 17, 156–163 (2000)Google Scholar
  43. 43.
    Wang, L., Zhang, K., Zhang, L.: Perfect phylogenetic networks with recombination. J. Comput. Biol. 8, 69–78 (2001)CrossRefGoogle Scholar
  44. 44.
    Wilson, I.J., Balding, D.J.: Genealogical inference from microsatellite data. Genetics 150, 499–510 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Rune B. Lyngsø
    • 1
  • Yun S. Song
    • 2
  • Jotun Hein
    • 1
  1. 1.Department of StatisticsOxford UniversityOxfordUK
  2. 2.Computer Science Division and Department of StatisticsUniversity of CaliforniaBerkeleyUSA

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