Journal of Statistical Physics

, Volume 172, Issue 1, pp 175–207 | Cite as

Genealogical Properties of Subsamples in Highly Fecund Populations

  • Bjarki EldonEmail author
  • Fabian Freund


We consider some genealogical properties of nested samples. The complete sample is assumed to have been drawn from a natural population characterised by high fecundity and sweepstakes reproduction (abbreviated HFSR). The random gene genealogies of the samples are—due to our assumption of HFSR—modelled by coalescent processes which admit multiple mergers of ancestral lineages looking back in time. Among the genealogical properties we consider are the probability that the most recent common ancestor is shared between the complete sample and the subsample nested within the complete sample; we also compare the lengths of ‘internal’ branches of nested genealogies between different coalescent processes. The results indicate how ‘informative’ a subsample is about the properties of the larger complete sample, how much information is gained by increasing the sample size, and how the ‘informativeness’ of the subsample varies between different coalescent processes.


Coalescent High fecundity Nested samples Multiple mergers Time to most recent common ancestor 

Mathematics Subject Classification

92D15 60J28 



We thank Alison Etheridge for many and very valuable comments and suggestions, especially regarding Theorem 1. BE was funded by DFG grant STE 325/17-1 to Wolfgang Stephan through Priority Programme SPP1819: Rapid Evolutionary Adaptation. FF was funded by DFG grant FR 3633/2-1 through Priority Program 1590: Probabilistic Structures in Evolution.

Supplementary material


  1. 1.
    Agrios, G.: Plant Pathology. Academic Press, Amsterdam (2005)Google Scholar
  2. 2.
    Árnason, E., Halldórsdóttir, K.: Nucleotide variation and balancing selection at the Ckma gene in Atlantic cod: analysis with multiple merger coalescent models. PeerJ 3, e786 (2015). CrossRefGoogle Scholar
  3. 3.
    Arratia, R., Barbour, A.D., Tavaré, S.: Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society (EMS), Zürich (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Barney, B.T., Munkholm, C., Walt, D.R., Palumbi, S.R.: Highly localized divergence within supergenes in atlantic cod (gadus morhua) within the gulf of maine. BMC Genomics 18(1) (2017).
  5. 5.
    Barton, N.H., Etheridge, A.M., Véber, A.: Modelling evolution in a spatial continuum. J. Stat. Mech. 2013(01), P01,002 (2013).
  6. 6.
    Basu, A., Majumder, P.P.: A comparison of two popular statistical methods for estimating the time to most recent common ancestor (tmrca) from a sample of DNA sequences. J. Genet. 82(1–2), 7–12 (2003)CrossRefGoogle Scholar
  7. 7.
    Berestycki, J., Berestycki, N., Schweinsberg, J.: Beta-coalescents and continuous stable random trees. Ann. Probab. 35, 1835–1887 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Berestycki, J., Berestycki, N., Schweinsberg, J.: Small-time behavior of beta coalescents. Ann. Inst. H Poincaré Probab. Stat. 44, 214–238 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berestycki, N.: Recent progress in coalescent theory. Ensaios Mathématicos 16, 1–193 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bertoin, J.: Exchangeable coalescents. Cours d’école doctorale, pp. 20–24 (2010)Google Scholar
  11. 11.
    Bhaskar, A., Clark, A., Song, Y.: Distortion of genealogical properties when the sample size is very large. PNAS 111, 2385–2390 (2014)ADSCrossRefGoogle Scholar
  12. 12.
    Birkner, M., Blath, J.: Computing likelihoods for coalescents with multiple collisions in the infinitely many sites model. J. Math. Biol. 57, 435–465 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Birkner, M., Blath, J.: Coalescents and population genetic inference. Trends Stoch. Anal. 353, 329 (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Birkner, M., Blath, J., Capaldo, M., Etheridge, A.M., Möhle, M., Schweinsberg, J., Wakolbinger, A.: Alpha-stable branching and beta-coalescents. Electron. J. Probab. 10, 303–325 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Birkner, M., Blath, J., Eldon, B.: An ancestral recombination graph for diploid populations with skewed offspring distribution. Genetics 193, 255–290 (2013)CrossRefGoogle Scholar
  16. 16.
    Birkner, M., Blath, J., Eldon, B.: Statistical properties of the site-frequency spectrum associated with \(\varLambda \)-coalescents. Genetics 195, 1037–1053 (2013)CrossRefGoogle Scholar
  17. 17.
    Birkner, M., Blath, J., Möhle, M., Steinrücken, M., Tams, J.: A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks. ALEA Lat. Am. J. Probab. Math. Stat. 6, 25–61 (2009)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Birkner, M., Blath, J., Steinrücken, M.: Analysis of DNA sequence variation within marine species using Beta-coalescents. Theor. Popul. Biol. 87, 15–24 (2013)CrossRefzbMATHGoogle Scholar
  19. 19.
    Blath, J., Cronjäger, M.C., Eldon, B., Hammer, M.: The site-frequency spectrum associated with \(\varXi \)-coalescents. Theor. Popul. Biol. 110, 36–50 (2016). CrossRefzbMATHGoogle Scholar
  20. 20.
    Bolthausen, E., Sznitman, A.: On Ruelle’s probability cascades and an abstract cavity method. Commun. Math. Phys. 197, 247–276 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Capra, J.A., Stolzer, M., Durand, D., Pollard, K.S.: How old is my gene? Trends Genet. 29(11), 659–668 (2013)CrossRefGoogle Scholar
  22. 22.
    Desai, M.M., Walczak, A.M., Fisher, D.S.: Genetic diversity and the structure of genealogies in rapidly adapting populations. Genetics 193(2), 565–585 (2013)CrossRefGoogle Scholar
  23. 23.
    Dong, R., Gnedin, A., Pitman, J.: Exchangeable partitions derived from markovian coalescents. Ann. Appl. Probab. 17, 1172–1201 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Donnelly, P., Kurtz, T.G.: Particle representations for measure-valued population models. Ann. Probab. 27, 166–205 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Donnelly, P., Tavare, S.: Coalescents and genealogical structure under neutrality. Annu. Rev. Genet. 29(1), 401–421 (1995)CrossRefGoogle Scholar
  26. 26.
    Durrett, R.: Probability Models for DNA Sequence Evolution, 2nd edn. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  27. 27.
    Durrett, R., Schweinsberg, J.: Approximating selective sweeps. Theor. Popul. Biol. 66, 129–138 (2004)CrossRefzbMATHGoogle Scholar
  28. 28.
    Durrett, R., Schweinsberg, J.: A coalescent model for the effect of advantageous mutations on the genealogy of a population. Stoch. Proc. Appl. 115, 1628–1657 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Eldon, B.: Inference methods for multiple merger coalescents. In: Pontarotti, P. (ed.) Evolutionary Biology: Convergent Evolution, Evolution of Complex Traits, Concepts and Methods, pp. 347–371. Springer, New York (2016)CrossRefGoogle Scholar
  30. 30.
    Eldon, B., Birkner, M., Blath, J., Freund, F.: Can the site-frequency spectrum distinguish exponential population growth from multiple-merger coalescents. Genetics 199, 841–856 (2015)CrossRefGoogle Scholar
  31. 31.
    Eldon, B., Wakeley, J.: Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172, 2621–2633 (2006)CrossRefGoogle Scholar
  32. 32.
    Eldon, B., Wakeley, J.: Linkage disequilibrium under skewed offspring distribution among individuals in a population. Genetics 178, 1517–1532 (2008)CrossRefGoogle Scholar
  33. 33.
    Etheridge, A.: Some Mathematical Models from Population Genetics. Springer, Berlin (2011). CrossRefzbMATHGoogle Scholar
  34. 34.
    Etheridge, A., Griffiths, R.: A coalescent dual process in a Moran model with genic selection. Theor. Popul. Biol. 75, 320–330 (2009)CrossRefzbMATHGoogle Scholar
  35. 35.
    Etheridge, A.M., Griffiths, R.C., Taylor, J.E.: A coalescent dual process in a Moran model with genic selection, and the Lambda coalescent limit. Theor. Popul. Biol. 78, 77–92 (2010)CrossRefzbMATHGoogle Scholar
  36. 36.
    Ewens, W.J.: Mathematical Population Genetics 1: Theoretical Introduction, vol. 27. Springer, New York (2012)zbMATHGoogle Scholar
  37. 37.
    Freund, F., Möhle, M.: On the size of the block of 1 for \(\varXi \)-coalescents with dust. Modern Stoch. Theory Appl. 4(4), 407–425 (2017).
  38. 38.
    Freund, F., Siri-Jégousse, A.: Minimal clade size in the bolthausen-sznitman coalescent. J. Appl. Probab. 51(3), 657–668 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Goldschmidt, C., Martin, J.B.: Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Probab. 10(21), 718–745 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Griffiths, R.C., Tavare, S.: Monte carlo inference methods in population genetics. Math. Comput. Model. 23(8–9), 141–158 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Griffiths, R.C., Tavaré, S.: The age of a mutation in a general coalescent tree. Commun. Stat. Stoch. Model. 14, 273–295 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Griswold, C.K., Baker, A.J.: Time to the most recent common ancestor and divergence times of populations of common chaffinches (Fringilla coelebs) in Europe and North Africa: insights into Pleistocene refugia and current levels of migration. Evolution 56(1), 143–153 (2002)CrossRefGoogle Scholar
  43. 43.
    Halldórsdóttir, K., Árnason, E.: Whole-genome sequencing uncovers cryptic and hybrid species among Atlantic and Pacific cod-fish (2015).
  44. 44.
    Hintze, J.L., Nelson, R.D.: Violin plots: a box plot-density trace synergism. Am. Stat. 52(2), 181–184 (1998). Google Scholar
  45. 45.
    Hedgecock, D.: Does variance in reproductive success limit effective population sizes of marine organisms? In: Beaumont, A. (ed.) Genetics and Evolution of Aquatic Organisms, pp. 1222–1344. Chapman and Hall, London (1994)Google Scholar
  46. 46.
    Hedgecock, D., Pudovkin, A.I.: Sweepstakes reproductive success in highly fecund marine fish and shellfish: a review and commentary. Bull Mar. Sci. 87, 971–1002 (2011)CrossRefGoogle Scholar
  47. 47.
    Hedrick, P.: Large variance in reproductive success and the \({N}_e/{N}\) ratio. Evolution 59(7), 1596 (2005). CrossRefGoogle Scholar
  48. 48.
    Hénard, O.: The fixation line in the \({\varLambda }\)-coalescent. Ann. Appl. Probab. 25(5), 3007–3032 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Herriger, P., Möhle, M.: Conditions for exchangeable coalescents to come down from infinity. Alea 9(2), 637–665 (2012)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Hird, S., Kubatko, L., Carstens, B.: Rapid and accurate species tree estimation for phylogeographic investigations using replicated subsampling. Mol. Phylogenetics Evol. 57(2), 888–898 (2010)CrossRefGoogle Scholar
  51. 51.
    Hovmøller, M.S., Sørensen, C.K., Walter, S., Justesen, A.F.: Diversity of Puccinia striiformis on cereals and grasses. Annu. Rev. Phytopathol. 49, 197–217 (2011)CrossRefGoogle Scholar
  52. 52.
    Hudson, R.R.: Properties of a neutral allele model with intragenic recombination. Theor. Popul. Biol. 23, 183–201 (1983)CrossRefzbMATHGoogle Scholar
  53. 53.
    Huillet, T., Möhle, M.: On the extended Moran model and its relation to coalescents with multiple collisions. Theor. Popul. Biol. 87, 5–14 (2013)CrossRefzbMATHGoogle Scholar
  54. 54.
    Kaj, I., Krone, S.M.: The coalescent process in a population with stochastically varying size. J. Appl. Probab. 40(01), 33–48 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    King, L., Wakeley, J.: Empirical bayes estimation of coalescence times from nucleotide sequence data. Genetics 204(1), 249–257 (2016). CrossRefGoogle Scholar
  56. 56.
    Kingman, J.F.C.: The coalescent. Stoch. Proc. Appl. 13, 235–248 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Kingman, J.F.C.: Exchangeability and the evolution of large populations. In: Koch, G., Spizzichino, F. (eds.) Exchangeability in Probability and Statistics, pp. 97–112. North-Holland, Amsterdam (1982)Google Scholar
  58. 58.
    Kingman, J.F.C.: On the genealogy of large populations. J. Appl. Probab. 19A, 27–43 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Li, G., Hedgecock, D.: Genetic heterogeneity, detected by PCR-SSCP, among samples of larval Pacific oysters ( Crassostrea gigas ) supports the hypothesis of large variance in reproductive success. Can. J. Fish. Aquat. Sci. 55(4), 1025–1033 (1998). CrossRefGoogle Scholar
  60. 60.
    May, A.W.: Fecundity of Atlantic cod. J. Fish. Res. Board Can. 24, 1531–1551 (1967)CrossRefGoogle Scholar
  61. 61.
    Möhle, M.: Robustness results for the coalescent. J. Appl. Probab. 35(02), 438–447 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Möhle, M.: On sampling distributions for coalescent processes with simultaneous multiple collisions. Bernoulli 12(1), 35–53 (2006)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Möhle, M.: Coalescent processes derived from some compound Poisson population models. Electron. Commun. Probab. 16, 567–582 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Möhle, M., Sagitov, S.: A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29, 1547–1562 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Möhle, M., Sagitov, S.: Coalescent patterns in diploid exchangeable population models. J. Math. Biol. 47, 337–352 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Neher, R.A., Hallatschek, O.: Genealogies of rapidly adapting populations. Proc. Natl. Acad. Sci. 110(2), 437–442 (2013)ADSCrossRefGoogle Scholar
  67. 67.
    Niwa, H.S., Nashida, K., Yanagimoto, T.: Reproductive skew in japanese sardine inferred from DNA sequences. ICES J. Mar. Sci. 73(9), 2181–2189 (2016). CrossRefGoogle Scholar
  68. 68.
    Oosthuizen, E., Daan, N.: Egg fecundity and maturity of North Sea cod, Gadus morhua. Neth. J. Sea Res. 8(4), 378–397 (1974)CrossRefGoogle Scholar
  69. 69.
    Pettengill, J.B.: The time to most recent common ancestor does not (usually) approximate the date of divergence. PloS ONE 10(8), e0128,407 (2015)CrossRefGoogle Scholar
  70. 70.
    Pitman, J.: Coalescents with multiple collisions. Ann. Probab. 27, 1870–1902 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Sagitov, S.: The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36, 1116–1125 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Sagitov, S.: Convergence to the coalescent with simultaneous mergers. J. Appl. Probab. 40, 839–854 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Sargsyan, O., Wakeley, J.: A coalescent process with simultaneous multiple mergers for approximating the gene genealogies of many marine organisms. Theor. Popul. Biol. 74, 104–114 (2008)CrossRefzbMATHGoogle Scholar
  74. 74.
    Saunders, I.W., Tavaré, S., Watterson, G.A.: On the genealogy of nested subsamples from a haploid population. Adv. Appl. Probab. 16(3), 471 (1984). MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Schweinsberg, J.: Rigorous results for a population model with selection II: genealogy of the population. Electron. J. Probab. (2017)
  76. 76.
    Schweinsberg, J.: Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5, 1–50 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Schweinsberg, J.: A necessary and sufficient condition for the-coalescent to come down from the infinity. Electron. Commun. Probab. 5, 1–11 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    Schweinsberg, J.: Coalescent processes obtained from supercritical Galton-Watson processes. Stoch. Proc. Appl. 106, 107–139 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Simon, M., Cordo, C.: Inheritance of partial resistance to Septoria tritici in wheat (Triticum aestivum): limitation of pycnidia and spore production. Agronomie 17(6–7), 343–347 (1997)CrossRefGoogle Scholar
  80. 80.
    Slack, R.: A branching process with mean one and possibly infinite variance. Probab. Theory Relat. Fields 9(2), 139–145 (1968)MathSciNetzbMATHGoogle Scholar
  81. 81.
    Spouge, J.L.: Within a sample from a population, the distribution of the number of descendants of a subsample’s most recent common ancestor. Theor. Popul. Biol. 92, 51–54 (2014)CrossRefzbMATHGoogle Scholar
  82. 82.
    Tajima, F.: Evolutionary relationships of DNA sequences in finite populations. Genetics 105, 437–460 (1983)Google Scholar
  83. 83.
    Timm, A., Yin, J.: Kinetics of virus production from single cells. Virology 424(1), 11–17 (2012)CrossRefGoogle Scholar
  84. 84.
    Wakeley, J.: Coalescent Theory. Roberts & Co, Greenwood Village (2007)zbMATHGoogle Scholar
  85. 85.
    Wakeley, J., Takahashi, T.: Gene genealogies when the sample size exceeds the effective size of the population. Mol. Biol. Evol. 20, 208–2013 (2003)CrossRefGoogle Scholar
  86. 86.
    Waples, R.S.: Tiny estimates of the \({N_e}/{N}\) ratio in marine fishes: are they real? J. Fish Biol. 89(6), 2479–2504 (2016). CrossRefGoogle Scholar
  87. 87.
    Wiuf, C., Donnelly, P.: Conditional genealogies and the age of a neutral mutant. Theor. Popul. Biol. 56(2), 183–201 (1999).

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Authors and Affiliations

  1. 1.Museum für NaturkundeBerlinGermany
  2. 2.University of HohenheimStuttgartGermany

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