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Equilibrium behavior of population genetic models with non-random mating. Part II: Pedigrees, Homozygosity and Stochastic Models

Published online by Cambridge University Press:  14 July 2016

Samuel Karlin
Samuel Karlin*
Affiliation:
Stanford University, California
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Wright (1921) computed various correlations of relatives by a rather cumbersome procedure called the “method of path coefficients”. Wright's method is basically a disguised form of the use of Bayes' rule and the law of total probabilities. Malecot (1948) reorganized Wright's calculations by introducing the fundamental concept of identity by descent and exploiting its properties. The method of identity by descent has been perfected and developed by Malécot and his students, especially Gillois, Jauquard and Bouffette. Kempthorne (1957) has applied the concept of identity by descent to the study of quantitative inheritance. Kimura (1963) elegantly employed the ideas of identity by descent in determining rates of approach to homozygosity in certain mating situations with finite population size. Later in this chapter we will extend and refine the results of Kimura (1963) to give a more complete study of rates of approach to homozygosity. Ellison (1966) established several important limit theorems corresponding to polyploid, multi-locus random mating infinite populations by judicious enlargement of the concepts of identity by descent. Kesten (unpublished)) has recently refined the technique of Ellison.

Type
Review Paper
Information
Journal of Applied Probability , Volume 5 , Issue 3 , December 1968 , pp. 487 - 566
Copyright
Copyright © Applied Probability Trust 1968 

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References

Bailey, N. T. J. (1961) Introduction to the Mathematical Theory of Genetic Linkage. Oxford University Press.Google Scholar
Bartlett, M. S. and Haldane, J. B. S. (1935) The theory of inbreeding with forced heterozygosis. J. Genet. 31, 327–340.CrossRefGoogle Scholar
Bennett, J. H. (1953) Junctions in inbreeding. Genetica 26, 392–406.Google Scholar
Bennett, J. H. (1954) The distribution of heterogeneity upon inbreeding. J. R. Statist. Soc. B 16, No. 1, 88–99.Google Scholar
Bennett, J. H. (1956) Lethal genes in inbred lines. Heredity 10, 263–270.Google Scholar
Bennett, J. H. (1958) The existence and stability of selective balanced polymorphism at a sex-linked locus. Aust. J. Biol. Sci. 11, 598–602.Google Scholar
Bennett, J. H. and Binet, F. E. (1956) Association between Mendelian factors with mixed selfing and random mating. Heredity 10, 51–55.Google Scholar
Bodmer, W. F. (1960a) The genetics of homostyly in populations of Primula vulgaris. Phil. Trans. Roy. Soc. B 242, 517–549.Google Scholar
Bodmer, W. F. (1960b) Discrete stochastic processes in population genetics. J. R. Statist. Soc. B 22, 218–236.Google Scholar
Bodmer, W. F., Feldman, M. W. and Karlin, S. (1970) Theoretical Population Genetics. Manuscript in preparation.Google Scholar
Crow, J. F. (1967) Genetic Notes. 6th Edition, Burgess Publishing Company.Google Scholar
Crow, J. F. and Kimura, M. (1969) Population Genetics. Harper and Row (in press).Google Scholar
Crow, J. F. and Morton, N. E. (1955) Measurement of gene frequency drift in small populations. Evolution 9, 202–214.CrossRefGoogle Scholar
Darwin, C. (1859) The Origin of Species.Google Scholar
East, E. M. and Mangelsdorf, A. J. (1925) A new interpretation of the hereditary behavior of self sterile plants. Proc. Nat. Acad. Sci USA 11, 166–171.CrossRefGoogle ScholarPubMed
Ellison, B. E. (1966) Limit theorems for random mating in infinite populations. J. Appl. Prob. 3, 129–141.CrossRefGoogle Scholar
Ewens, W. J. (1963) Numerical results and diffusion approximations in a genetic process. Biometrika 50, 241–249.Google Scholar
Ewens, W. J. (1964) The pseudo-transient distribution and its uses in genetics. J. Appl. Prob. 1, 141–156.Google Scholar
Feldman, M. W. (1966) On the offspring number distribution in a genetic population. J. Appl. Prob. 3, 129–141.Google Scholar
Finney, D. J. (1952) The equilibrium of a self incompatible polymorphic species. Genetica 26, 33–64.Google Scholar
Fisher, R. A. (1918) The correlation between relatives on the supposition of Mendelian inheritance. Trans. Roy. Soc. 52, II, No. 15, 399–433.Google Scholar
Fisher, R. A. (1949) The Theory of Inbreeding. Oliver and Boyd, Edinburgh.Google Scholar
Fisher, R. A. (1958) The Genetical Theory of Natural Selection. Second revised edition, Dover, New York.Google Scholar
Fisher, R. A. and Mather, K. (1943) The inheritance of style length in Lythrum salicaria. Ann. Eugenics 12, 1–23.Google Scholar
Gale, J. S. (1964) Some applications of the theory of junctions. Biometrics 20, 85–117.Google Scholar
Garber, M. J. (1951) Approach to genotypic equilibrium with varying percentage of self-fertilization. J. Heredity 42, 299–300.Google Scholar
Ghat, G. L. (1967) Loss of heterozygosity in populations under mixed random mating and selfing. J. Indian Soc. Agr. Statist. 18, 73–81.Google Scholar
Gillois, M. (1964) La Relation d'Identité en Génétique. Unpublished thesis, Faculty of Sciences, University of Paris.Google Scholar
Haldane, J. B. S. (1956) The conflict between inbreeding and selection. J. Genet. 54, 56–63.Google Scholar
Haldane, J. B. S. and Jayakar, S. D. (1962) An enumeration of some human relations. J. Genet. 58, 81–107.Google Scholar
Hayman, B. I. (1953) Mixed selfing and random mating when homozygotes are at a disadvantage. Heredity 7, 185–192.Google Scholar
Hayman, B. I. and Mather, K. (1953) Progress of inbreeding when homozygotes are at a disadvantage. Heredity 7, 165–183.CrossRefGoogle Scholar
Hayman, B. I. and Mather, K. (1956) Inbreeding when homozygotes are at a disadvantage: A reply (Comment). Heredity 10, Part, 2, 271–274.Google Scholar
Hill, W. G. and Robertson, A. (1966) The effect of linkage on limits to artificial selection. Genet. Res. 8, 269–294.Google Scholar
Jennings, H. S. (1916) The numerical results of diverse systems of breeding, Genetics 1, 53–89.Google Scholar
Kalmus, H. and Maynard-Smith, Sheila (1966) Some evolutionary consequences of pegmatypic mating systems (imprinting). Amer. Naturalist 100, 619–635.CrossRefGoogle Scholar
Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York and London.Google Scholar
Karlin, S. (1968) Rates of approach to homozygosity for finite stochastic models with variable population size. Amer. Naturalist 102, 443–455.Google Scholar
Karlin, S. and Feldman, M. W. (1968a) Analysis of models with homozygote x heterozygote matings. Genetics 59, 105–116.Google Scholar
Karlin, S. and Feldman, M. W. (1968b) Further analysis of negative assortative mating. Genetics 59, 117–136.CrossRefGoogle ScholarPubMed
Karlin, S. and Mcgregor, J. (1964a) On some stochastic models in gentics. Stochastic Models in Medicine and Biology. University of Wisconsin Press, Madison, 245–279.Google Scholar
Karlin, S. and Mcgregor, J. (1964b) Direct product branching processes and related Markoff chains. Proc. Nat. Acad. USA 51, 598–602.Google Scholar
Karlin, S. and Mcgregor, J. (1965) Direct product branching prosesses and related induced Markoff chains I. Calculations of rates of approach to homozygosity. Bernoulli, Bayes, Laplace Anniversary Volume, Springer-Verlag, 11–145.Google Scholar
Karlin, S. and Mcgregor, J. (1967) The number of mutant forms maintained in a population. Proc. 5th Berkeley Symp. Math. Stat. Prob. 1965, IV, 415–438.Google Scholar
Karlin, S. and Mcgregor, J. (1968a) The role of the Poisson progeny distribution in population genetics models. Math. Biosciences 2, 11–17.Google Scholar
Karlin, S. and Mcgregor, J. (1968b) Rates and probabilities of fixation for two locus random mating finite populations without selection. Genetics. 58, 141–159.Google Scholar
Karlin, S., Mcgregor, J. and Bodmer, W. F. (1967) The rate of production of recombinants between linked genes in finite populations. Proc. 5th Berkeley Symp. Math. Stat. Prob. 1965, IV 403–414.Google Scholar
Karlin, S. and Scudo, F. (1968) Assortative mating based on phenotype II: Two autosomal alleles without dominance. To appear.Google Scholar
Kempthorne, O. (1955) The correlations between relatives in random mating populations. Symp. Quant. Biol., Cold Spring Harbor, 22, 60–78.Google Scholar
Kempthorne, O. (1957) An Introduction to Statistical Genetics. Wiley, New York.Google Scholar
Kempthorne, O. (1967) The concept of identity of genes by descent. Proc. 5th Berkeley Symp. Math. Stat. Prob. 1965, IV, 333–348.Google Scholar
Kimura, M. (1955) Solution of a process of random genetic drift with a continuous model. Proc. Nat. Acad. Sci. USA 41, 144–150.Google Scholar
Kimura, M. (1956) A model of genetic system which leads to closer linkage by natural selection. Evolution 10, 278–287.Google Scholar
Kimura, M. (1957) Overdominant genes in a partially self-fertilizing population. Annual Report Nat. Inst. Genet. (Japan) 105–106.Google Scholar
Kimura, M. (1963) A probability method for treating inbreeding schemes especially with linked genes. Biometrics 19, 1–17.Google Scholar
Kimura, M. (1964) Diffusion models in population genetics. J. Appl. Prob. 1, 177–232.CrossRefGoogle Scholar
Kimura, M. (1965) Attainment of quasi linkage equilibrium when gene frequencies are changing by natural selection. Genetics 52, 875–890.Google Scholar
Kimura, M. and Crow, J. F. (1963a) On the maximum avoidance of inbreeding. Genet. Res. 4, 399–415.CrossRefGoogle Scholar
Kimura, M. and Crow, J. F. (1963b) The measurement of effective population number. Evolution 17, 279–288.Google Scholar
Kojima, K. and Kelleher, T. M. (1962) Survival of mutant genes. Amer. Naturalist 96, No. 891, 329–346.Google Scholar
Lerner, I. M. (1950) Population Genetics and Animal Improvement. Cambridge University Press.Google Scholar
Li, C. C. (1955) Population Genetics. The University Press, Chicago.Google Scholar
Lillestol, Jostein (1968) Another approach to some Markov chain models in population genetics. J. Appl. Prob. 5, 9–20.Google Scholar
Mainardi, D. (1964a) Relations between early experience and sexual preferences in female mice. Atti Ass. Genet. Ital. 9, 141.Google Scholar
Mainardi, D. (1964b) Early experience and sexual selection in Drosophila melanogaster. Ann. Zool. 4, 119.Google Scholar
Mainardi, D. (1968) La Scelta Sessuale Nell' Evoluzione Della Specie. Boringhieri, Torino.Google Scholar
Malécot, G. (1948) Les Mathématiques de l'Hérédité. Masson, Paris.Google Scholar
Malécot, G. (1951) Un traitement stochastique des problèmes linéaires (mutation, linkage, migration) en génétique de population. Ann. Univ. Lyon. Sciences, Section A 14, 79–100.Google Scholar
Malécot, G. (1959a) Les modèles stochastiques en génétique de population. Publ. Inst. Statist. U. Paris 8, fasc. 3.Google Scholar
Malécot, G. (1959b) La génétique de population. II. Modèles stochastiques. Publ. Inst. Statist. U. Paris 8, fasc. 3, 173–210.Google Scholar
Malécot, G. (1967) Identical loci and relationship. Proc. 5th Berkeley Symp. Math. Stat. Prob. 1965, IV, 317–332.Google Scholar
Mather, K. (1935) Reductional and equational separation of the chromosomes in bivalents and multivalents. J. Genet. 30, 53–78.Google Scholar
Mather, K. (1949) Biometrical Genetics. London: Methuen and Co.Google Scholar
Mather, K. and Hayman, B. I. (1952) The progress of inbreeding when heterozygotes are at an advantage. Biometrics 8, 176.Google Scholar
Moran, P. A. P. (1962) The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.Google Scholar
Moran, P. A. P. and Watterson, G. A. (1959) The genetic effects of family structure in natural populations. Aust. J. Biol. Sci. 12, 1–15.Google Scholar
O'Donald, P. (1959) Possibility of assortive mating in the Arctic Skua. Nature 182, 1210–1211.CrossRefGoogle Scholar
O'Donald, P. (1960a) Inbreeding as a result of imprinting. Heredity 15, 79.Google Scholar
O'Donald, P. (1960b) Assortive mating in a population in which two alleles are segregating. Heredity 15, 389–396.CrossRefGoogle Scholar
Page, A. R. and Hayman, B. I. (1960) Mixed sib and random mating when homozygotes are at a disadvantage. Heredity 14, Parts 1 and 2, 187–196.Google Scholar
Parsons, P. A. (1957) Selfing under conditions favouring heterozygosity. Heredity 11, Part 3, 411–421.Google Scholar
Parsons, P. A. (1962) The initial increase of a new gene under positive assortative mating. Heredity 17, 267–276.Google Scholar
Parsons, P. A. (1963) Complex polymorphisms where the coupling and repulsion double heterozygote viabilities differ. Heredity 18, 369–374.Google Scholar
Parsons, P. A. (1963) Polymorphisms and the balanced polygenic combination. Evolution 17, 564–574.Google Scholar
Parsons, P. A. (1965) Assortative mating for a metrical characteristic in drosophila. Heredity 20, 161–167.Google Scholar
Reeve, E. C. R. (1955) Inbreeding with homozygotes at a disadvantage. Ann. Hum. Genet. 19, 332–346.Google Scholar
Reiersöl, O. (1962) Genetic algebras studied recursively and by means of differential operators. Math. Scand. 10, 25–44.CrossRefGoogle Scholar
Richardson, W. H. (1964) Frequencies of genotypes of relatives as determined by stochastic matrices. Genetica 35, 323–354.Google Scholar
Schnell, F. W. (1961) Some general formulations of linkage effects in inbreeding. Genetics 46, 947–957.CrossRefGoogle ScholarPubMed
Scudo, F. M. (1964) Sex population genetics. Ri. Sci. 34, II–B, 93–146.Google Scholar
Scudo, F. M. (1967) L'accoppiamento assortativo basato sul fenotipo di parenti, alcune conseguenze in popolazioni. Genetica , Instituto Lombardo (Rend. Sc.) B 101, 435–455.Google Scholar
Scudo, F. M. and Karlin, S. (1968) Assortative mating based on phenotype; I. Two alleles with dominance. To appear.Google Scholar
Scudo, F. M., Mainardi, D. and Barbieri, D. (1965) Assortative mating based on early learning: population genetics. Acta Bio-medica 36, Fasc. 5, 583–605.Google Scholar
Shikata, M. (1965a) Study of genetic populations by generalized inbreeding coefficient, I. Homozygosity by descent in selfed population. Rep. Stat. Appl. Res., JUSE, 12, No. 3, 13–16.Google Scholar
Shikata, M. (1965b) A generalization of the inbreeding coefficient. Biometrics 21, 665–681.Google Scholar
Shikata, M. (1966) A note on group ring expression of self-fertilized population. Bull. Math. Biophys. 28, No. 2, 217–218.Google Scholar
Shikata, M. (1966) Cross-over effect in homozygosity by descent. J. Theoret. Biol. 10, 196–208.Google Scholar
Stern, Curt (1960) Principles of Human Genetics. 2nd edition. Freeman, San Francisco.Google Scholar
Watterson, G. A. (1959a) Non-random mating, and its effect on the rate of approach to homozygosity. Ann. Hum. Genet. 23, 204.Google Scholar
Watterson, G. A. (1959b) A new genetic population model and its rate of approach to homozygosity. Ann. Hum. Genet. 23, 221–232.Google Scholar
Watterson, G. A. (1960) Unpublished Ph.D. thesis, Australian National University.Google Scholar
Watterson, G. A. (1962) Some theoretical aspects of diffusion theory in population genetics. Ann. Math. Stat. 33, 939–957.Google Scholar
Watterson, G. A. (1964) The application of diffusion theory to two population genetic models of Moran. J. Appl. Prob. 1, 233–246.Google Scholar
Wentworth, E. H. and Remick, B. L. (1916) Some breeding properties of generalized Mendelian populations. Genetics 1, 608–616.Google Scholar
Workman, P. L. (1964) The maintenance of heterozygosity by partial negative assortative mating. Genetics 50, 1369–1382.Google Scholar
Wright, S. (1921) Systems of mating. Genetics 6, 111–178.Google Scholar
Wright, S. (1922) Coefficients of inbreeding and relationship. Amer. Nat. 56, 330–339.Google Scholar
Wright, S. (1931) Evolution in Mendelian populations. Genetics 16, 97–159.Google Scholar
Wright, S. (1939) The distribution of self-sterility alleles in populations. Genetics 24, 538–552.Google Scholar
Wright, S. (1951) The genetical structure of populations. Ann. Eugen. 15, 323–354.Google Scholar