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Population Genetics Theory - The Past and the Future

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Mathematical and Statistical Developments of Evolutionary Theory

Part of the book series: NATO ASI Series ((ASIC,volume 299))

Abstracts

Classical population genetics theory was largely directed towards processes relating to the future. Present theory, by contrast, focuses on the past, and in particular is motivated by the desire to make inferences about the evolutionary processes which have led to the presently observed patterns and nature of genetic variation. There are many connections between the classical prospective theory and the new retrospective theory. However, the retrospective theory introduces ideas not appearing in the classical theory, particularly those concerning the ancestry of the genes in a sample or in the entire population. It also introduces two important new distributions into the scientific literature, namely the Poisson-Dirichlet and the GEM: these are important not only in population genetics, but also in a very wide range in science and mathematics. Some of these are discussed. Population genetics theory has been greatly enriched by the introduction of many new concepts relating to the past evolution of biological populations.

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References

  • Aldous, D.J. (1985), Exchangeability and related topics, in: École d’été de probabilités de Saint-Flour XIII-1983 (P.L. Hennequin, éd.), Lecture Notes in Mathematics 1117, Springer-Verlag, Berlin, 2–198.

    Google Scholar 

  • Anderson, R. (1978), Some stochastic models in population genetics, Unpublished M.Sc. thesis, Monash University.

    Google Scholar 

  • Blackwell, D. and MacQueen, J.B. (1973). Ferguson distributions via Polya urn schemes, Ann. Statist. 1, 353–355.

    Article  Google Scholar 

  • Burville, P.J. (1974), Heaps: a concept in optimization, J. Inst. Math. Appl. 13, 263–278.

    Google Scholar 

  • Burville, P.J. and Kingman J.F.C. (1973), On a model for storage and search, J. Appl. Probab. 10, 697–701.

    Article  Google Scholar 

  • Cannings, C. (1974), The latent roots of certain Markov chains arising in genetics: a new approach. 1. Haploid models, Adv. in Appl. Probab. 6, 260–290.

    Google Scholar 

  • Caswell, H. (1976), Community structure: a neutral model analysis. Ecological Monographs 46, 327–353.

    Article  Google Scholar 

  • Constantini, D. (1987), Symmetry and the indistinguishability of classical particles, Phys. Lett. A 123, 433–436.

    Article  Google Scholar 

  • Crow, J.F. and Kimura, M. (1970), An Introduction to Population Genetics Theory, Harper and Row, New York.

    Google Scholar 

  • Donnelly, P. (1986a), Dual processes in population genetics, in: Stochastic Spatial Processes (P. Tautu, ed.), Lecture Notes in Mathematics 1212, Springer-Verlag, Berlin, 94–105.

    Chapter  Google Scholar 

  • Donnelly, P. (1986b), Partition structures, Polya urns, the Ewens sampling formula, and the ages of alleles. Theoret. Population Biol. 30, 271–288.

    Article  CAS  Google Scholar 

  • Donnelly, P. (1988), Heaps processes and size-biased permutations. Submitted.

    Google Scholar 

  • Donnelly, P. (1989), Weak convergence to a death process with an entrance boundary: ancestral processes in population genetics. To appear in Ann. Probab.

    Google Scholar 

  • Donnelly, P. and Joyce, P. (1989), Consistent ordered sampling distributions: characterization and convergence. Submitted.

    Google Scholar 

  • Donnelly, P. and Tavaré, S. (1986), The ages of alleles and a coalescent, Adv. in Appl. Probab. 18, 1–19.

    Article  Google Scholar 

  • Donnelly, P. and Tavaré, S. (1987), The population genealogy of the infinitely-many neutral alleles model, J. Math. Biol. 251, 381–391.

    Article  Google Scholar 

  • Engen, S. (1975), A note on the geometric series as a species frequency model, Biometrika 62, 694–699.

    Article  Google Scholar 

  • Ethier, S.N. (1989), The infinitely-many-neutral-alleles diffusion model with ages. Preprint.

    Google Scholar 

  • Ethier, S.N. and Kurtz, T.G. (1981), The infinitely-many-neutral- alleles diffusion model, Adv. in Appl. Probab. 13, 429–452.

    Article  Google Scholar 

  • Ethier, S.N. and Kurtz, T.G. (1986), Markov Processes: Characterization and Convergence, Wiley, New York.

    Google Scholar 

  • Ewens, W.J. (1972), The sampling theory of selectively neutral alleles, Theoret. Population Biol. 3, 87–112.

    Article  CAS  Google Scholar 

  • Ewens, W.J. (1979), Mathematical Population Genetics, Springer-Verlag, Berlin.

    Google Scholar 

  • Ewens, W.J. and Kirby, K (1975), The eigenvalues of the neutral alleles process, Theoret. Population Biol. 7, 212–220.

    Article  CAS  Google Scholar 

  • Ewens, W.J. and Padmadisastra, S. (1989), Asymptotic and numerical results for random functions. In preparation.

    Google Scholar 

  • Fisher, R.A. (1930), The Genetical Theory of Natural Selection, Clarendon Press, Oxford.

    Google Scholar 

  • Golomb, S.W. (1964), Random permutations. Bull. Amer. Math. Soc. 70, 747.

    Article  Google Scholar 

  • Goncharov, V. (1962), Du domaine d’analyse combinatoire, Amer. Math. Soc. Transl. (2) 19, 1–46.

    Google Scholar 

  • Griffiths, R.C. (1979a). A transition density expansion for a multi-allele diffusion model. Adv. in Appl. Probab. 11, 310–325.

    Article  Google Scholar 

  • Griffiths, R.C. (1979b), Exact sampling distributions from the infinite neutral alleles model, Adv. in Appl. Probab. 11, 326–354.

    Article  Google Scholar 

  • Griffiths, R.C. (1980), Lines of descent in the diffusion approximation of neutral Wright- Fisher models, Theoret. Population Biol. 17, 37–50.

    Article  CAS  Google Scholar 

  • Guess, H.A. and Ewens, W.J. (1972), Theoretical and simulation results relating to the neutral allele theory, Theoret. Population Biol. 3, 434–447.

    Article  CAS  Google Scholar 

  • Halmos, P.R. (1944), Random alms, Ann. Math. Statist. 15, 182–189.

    Article  Google Scholar 

  • Hendricks, W.J. (1972), The stationary distribution of an interesting Markov chain, J. Appl. Probab. 9, 231–233.

    Article  Google Scholar 

  • Hoppe, F.M. (1984), Polya-like urns and the Ewens sampling formula, J. Math. Biol. 20, 91–99.

    Article  Google Scholar 

  • Hoppe, F.M. (1986), Size-biased filtering of Poisson-Dirichlet samples with an application to partition structures in genetics, J. Appl. Probab. 23, 1008–1012.

    Article  Google Scholar 

  • Hoppe, F.M. (1987), The sampling theory of neutral alleles and an urn model in population genetics, J. Math. Biol. 25, 123–159.

    Article  PubMed  CAS  Google Scholar 

  • Hudson, R.R. and Kaplan, N.L. (1985), Statistical properties of the number of recombination events in the history of a sample of DNA sequences, Genetics 111, 147–164.

    PubMed  CAS  Google Scholar 

  • Hudson, R.R. and Kaplan, N.L. (1986), On the divergence of alleles in nested subsamples from finite populations, Genetics 113, 1057–1076.

    PubMed  CAS  Google Scholar 

  • Ignatov, T. (1982), On a constant arising in the theory of symmetric groups and on Poisson-Dirichlet measures, Theory Probab. Appl. 27, 136–147.

    Article  Google Scholar 

  • Joyce, P. (1988), Age-ordered distributions associated with some neutral population genetics models. Unpublished Ph.D. thesis. University of Utah.

    Google Scholar 

  • Joyce, P. and Tavaré, S. (1987), Cycles, permutations and the structures of the Yule process with immigration, Stochastic Process. Appl. 25, 309–314.

    Article  Google Scholar 

  • Kaplan, N.L. and Hudson, R.R. (1987), On the divergence of genes in multigene families, Theoret. Population Biol. 31, 178–194.

    Article  CAS  Google Scholar 

  • Kaplan, N.L. and Hudson, R.R. (1988), An evolutionary model for highly repeated interspersed DNA sequences, in: Mathematical Evolutionary Theory (M.W. Feldman, ed.), Princeton University Press, Princeton.

    Google Scholar 

  • Karlin, S. and McGregor, J.L. (1972), Addendum to a paper of W. Ewens, Theoret. Population Biol. 3, 113–116.

    Article  CAS  Google Scholar 

  • Keener, R., Rothman, E. and Starr, N. (1988), Distributions on partitions, Ann. Statist. 15, 1466–1481.

    Article  Google Scholar 

  • Kelly, F.P. (1979), Reversibility and Stochastic Networks, Wiley, New York.

    Google Scholar 

  • Kerov, S.V. and Vershik, A.M. (1986), Characters of infinite symmetric groups and probability properties of Robinson-Shenstead-Knuth’s algorithm, SIAM J. Algebraic Discrete Methods 7, 116–124.

    Article  Google Scholar 

  • Kimura, M. (1955), Solution of a process of random genetic drift with a continuous model, Proc. Nat. Acad. Sci. U.S.A. 41, 144–150.

    Article  CAS  Google Scholar 

  • Kimura, M. (1968), Evolutionary rate at the molecular level, Nature 217, 624–626.

    Article  Google Scholar 

  • Kimura, M. and Crow, J.F. (1964), The number of alleles that can be maintained in a finite population, Genetics 49, 725–738.

    PubMed  CAS  Google Scholar 

  • Kingman, J.F.C. (1975), Random discrete distributions, J. Roy. Statist. Soc. Ser. B. 37, 1–22.

    Google Scholar 

  • Kingman, J.F.C. (1977), The population structure associated with the Ewens sampling formula, Theoret. Population Biol. 11, 274–284.

    Article  CAS  Google Scholar 

  • Kingman, J.F.C. (1978a), Random partitions in population genetics, Proc. Roy. Soc. London Ser. A 361, 1–20.

    Article  Google Scholar 

  • Kingman, J.F.C. (1978b), The representaton of partition structures, J. Lond. Math. Soc. 18, 374–380.

    Article  Google Scholar 

  • Kingman, J.F.C. (1980), Mathematics of Genetic Diversity, SIAM, Philadelphia.

    Google Scholar 

  • Kingman, J.F.C. (1982a), On the genealogy of large populations, J. Appl. Probab. 19A, 27–43.

    Article  Google Scholar 

  • Kingman, J.F.C. (1982b), The coalescent, Stochastic Process. Appl. 13, 235–248.

    Article  Google Scholar 

  • Kingman, J.F.C. (1982c), Exchangeability and the evolution of large populations, in: Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.), North-Holland, Amsterdam, 97–112.

    Google Scholar 

  • Kirby, K. (1974), Unpublished Ph.D. thesis. Princeton University.

    Google Scholar 

  • Kolchin, V.F. (1976), A problem of the allocation of particles into cells and random mappings, Theory Probab. Appl. 21, 48–63.

    Article  Google Scholar 

  • Kruskal, M.D. (1954), The expected number of components under a random mapping function, Amer. Math. Monthly 61, 392–397.

    Article  Google Scholar 

  • Lambshead, P.J.D. (1986), Sub-catastrophic sewage and industrial waste contamination as revealed by marine neatode faunal analysis, Marine Ecology Progess Series 29, 247–260.

    Article  Google Scholar 

  • Lambshead, P.J.D. and Piatt, H.M. (1985), Structural patterns of marine benthic assemblages and their relationship with empirical statistical models, in: Nineteenth European Marine Biology Symposium, Cambridge University Press, Cambridge, 371–380.

    Google Scholar 

  • Malécot, G. (1984), Les mathématiques de l’hérédité, Masson, Paris.

    Google Scholar 

  • McCloskey, J.W. (1965), A model for the distribution of individuals by species in an environment. Unpublished Ph.D. thesis, Michigan State University.

    Google Scholar 

  • Moran, P.A.P. (1958), Random processes in genetics, Proc. Camb. Phil. Soc. 54, 69–71.

    Google Scholar 

  • Patil, G.P. and Taillie, C. (1977), Diversity as a concept and its implications for random communities, Bull. Inst. Internat. Statist. 47, 497–515.

    Google Scholar 

  • Pittel, B. (1983), On distributions relating to transitive closure of random finite mappings, Ann. Probab. 11, 428–441.

    Article  Google Scholar 

  • Platt, H.M. and Lambshead, P.J.D. (1985), Neutral model analysis of patterns of marine benthic species diversity. Marine Ecology Progress Series 24, 75–81.

    Article  Google Scholar 

  • Ross, S.M. (1981), A random graph. J. Appl. Probab. 18, 309–315.

    Article  Google Scholar 

  • Saunders, I.W., Tavaré, S. and Watterson, G.A. (1984), On the genealogy of nested subsamples from a haploid population, Adv. in Appl. Probab. 16, 471–491.

    Article  Google Scholar 

  • Sawyer, S. (1977), On the past history of an allele now known to have frequency p, J. Appl. Probab. 14, 439–450.

    Article  Google Scholar 

  • Shepp, L.A. and Lloyd, S.P. (1966), Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc. 121, 340–357.

    Article  Google Scholar 

  • Tavaré, S. (1984), Line-of-descent and genealogical processes and their applications in population genetics, Theoret. Population Biol. 26, 119–164.

    Article  Google Scholar 

  • Tavaré, S. (1987), The birth process with immigration, and the genealogical structure of large populations, J. Math. Biol. 25, 161–171.

    Article  PubMed  Google Scholar 

  • Tavaré, S., Ewens, W.J. and Joyce, P. (1988), Is knowing the age-order of alleles useful in testing neutrality? Preprint.

    Google Scholar 

  • Trajstman, A.C. (1974), On a conjecture of G.A. Watterson, Adv. in Appl. Probab. 6, 489–493.

    Google Scholar 

  • Vershik, A.M. (1986), The asymptotic distribution of factorizations of natural numbers into prime divisors, Soviet Math. Dokl. 34, 57–61.

    Google Scholar 

  • Vershik, A.M. and Shmidt, A.A. (1977), Limit measures arising in the theory of groups. I., Theory Probab. Appl. 22, 79–85.

    Article  Google Scholar 

  • Watterson, G.A. (1974), The sampling theory of selectively neutral alleles, Adv. in Appl. Probab. 6, 463–488.

    Article  Google Scholar 

  • Watterson, G.A. (1976a), Reversibility and the age of an allele. I. Moran’s infinitely- many neutral alleles model, Theoret. Population Biol. 10, 239–253.

    Article  CAS  Google Scholar 

  • Watterson, G.A. (1976b), The stationary distribution of the infinitely-many neutral alleles diffusion model, J. Appl. Probab. 13, 639–651.

    Article  Google Scholar 

  • Watterson, G.A. (1977), Heterosis or neutrality?, Genetics 85, 789–814.

    PubMed  CAS  Google Scholar 

  • Watterson, G.A. (1984), Lines of descent and the coalescent, Theoret. Population Biol. 10, 239–253.

    Article  Google Scholar 

  • Watterson, G.A. and Guess, H.A. (1977), Is the most frequent allele the oldest?, Theoret. Population Biol. 11, 141–160.

    Article  CAS  Google Scholar 

  • Wright, S. (1931), Evolution in Mendelian populations, Genetics 16, 97–159.

    PubMed  CAS  Google Scholar 

  • Wright, S. (1949), Genetics of populations, Encyclopaedia Britannica, 14th ed., vol. 10, 111.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Monash University, Clayton, Victoria, 3168, Australia

    W. J. Ewens

  2. Department of Biology, University of Pennsylvania, Philadelphia, PA, 19104, USA

    W. J. Ewens

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  1. W. J. Ewens
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Editors and Affiliations

  1. Départment de mathémataques et de statistique, Université de Montréal, C.P. 6128, Montréal, Qué H3C 3J7, Canada

    Sabin Lessard

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© 1990 Kluwer Academic Publishers

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Ewens, W.J. (1990). Population Genetics Theory - The Past and the Future. In: Lessard, S. (eds) Mathematical and Statistical Developments of Evolutionary Theory. NATO ASI Series, vol 299. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0513-9_4

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  • DOI: https://doi.org/10.1007/978-94-009-0513-9_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6717-1

  • Online ISBN: 978-94-009-0513-9

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