Skip to main content Accessibility help

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

  • Samuel Karlin (a1)


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 Kesten (unpublished)) has recently refined the technique of Ellison.



Hide All
Bailey, N. T. J. (1961) Introduction to the Mathematical Theory of Genetic Linkage. Oxford University Press.
Bartlett, M. S. and Haldane, J. B. S. (1935) The theory of inbreeding with forced heterozygosis. J. Genet. 31, 327340.
Bennett, J. H. (1953) Junctions in inbreeding. Genetica 26, 392406.
Bennett, J. H. (1954) The distribution of heterogeneity upon inbreeding. J. R. Statist. Soc. B 16, No. 1, 8899.
Bennett, J. H. (1956) Lethal genes in inbred lines. Heredity 10, 263270.
Bennett, J. H. (1958) The existence and stability of selective balanced polymorphism at a sex-linked locus. Aust. J. Biol. Sci. 11, 598602.
Bennett, J. H. and Binet, F. E. (1956) Association between Mendelian factors with mixed selfing and random mating. Heredity 10, 5155.
Bodmer, W. F. (1960a) The genetics of homostyly in populations of Primula vulgaris. Phil. Trans. Roy. Soc. B 242, 517549.
Bodmer, W. F. (1960b) Discrete stochastic processes in population genetics. J. R. Statist. Soc. B 22, 218236.
Bodmer, W. F., Feldman, M. W. and Karlin, S. (1970) Theoretical Population Genetics. Manuscript in preparation.
Crow, J. F. (1967) Genetic Notes. 6th Edition, Burgess Publishing Company.
Crow, J. F. and Kimura, M. (1969) Population Genetics. Harper and Row (in press).
Crow, J. F. and Morton, N. E. (1955) Measurement of gene frequency drift in small populations. Evolution 9, 202214.
Darwin, C. (1859) The Origin of Species.
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, 166171.
Ellison, B. E. (1966) Limit theorems for random mating in infinite populations. J. Appl. Prob. 3, 129141.
Ewens, W. J. (1963) Numerical results and diffusion approximations in a genetic process. Biometrika 50, 241249.
Ewens, W. J. (1964) The pseudo-transient distribution and its uses in genetics. J. Appl. Prob. 1, 141156.
Feldman, M. W. (1966) On the offspring number distribution in a genetic population. J. Appl. Prob. 3, 129141.
Finney, D. J. (1952) The equilibrium of a self incompatible polymorphic species. Genetica 26, 3364.
Fisher, R. A. (1918) The correlation between relatives on the supposition of Mendelian inheritance. Trans. Roy. Soc. 52, II, No. 15, 399433.
Fisher, R. A. (1949) The Theory of Inbreeding. Oliver and Boyd, Edinburgh.
Fisher, R. A. (1958) The Genetical Theory of Natural Selection. Second revised edition, Dover, New York.
Fisher, R. A. and Mather, K. (1943) The inheritance of style length in Lythrum salicaria. Ann. Eugenics 12, 123.
Gale, J. S. (1964) Some applications of the theory of junctions. Biometrics 20, 85117.
Garber, M. J. (1951) Approach to genotypic equilibrium with varying percentage of self-fertilization. J. Heredity 42, 299300.
Ghat, G. L. (1967) Loss of heterozygosity in populations under mixed random mating and selfing. J. Indian Soc. Agr. Statist. 18, 7381.
Gillois, M. (1964) La Relation d'Identité en Génétique. Unpublished thesis, Faculty of Sciences, University of Paris.
Haldane, J. B. S. (1956) The conflict between inbreeding and selection. J. Genet. 54, 5663.
Haldane, J. B. S. and Jayakar, S. D. (1962) An enumeration of some human relations. J. Genet. 58, 81107.
Hayman, B. I. (1953) Mixed selfing and random mating when homozygotes are at a disadvantage. Heredity 7, 185192.
Hayman, B. I. and Mather, K. (1953) Progress of inbreeding when homozygotes are at a disadvantage. Heredity 7, 165183.
Hayman, B. I. and Mather, K. (1956) Inbreeding when homozygotes are at a disadvantage: A reply (Comment). Heredity 10, Part, 2, 271274.
Hill, W. G. and Robertson, A. (1966) The effect of linkage on limits to artificial selection. Genet. Res. 8, 269294.
Jennings, H. S. (1916) The numerical results of diverse systems of breeding, Genetics 1, 5389.
Kalmus, H. and Maynard-Smith, Sheila (1966) Some evolutionary consequences of pegmatypic mating systems (imprinting). Amer. Naturalist 100, 619635.
Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York and London.
Karlin, S. (1968) Rates of approach to homozygosity for finite stochastic models with variable population size. Amer. Naturalist 102, 443455.
Karlin, S. and Feldman, M. W. (1968a) Analysis of models with homozygote x heterozygote matings. Genetics 59, 105116.
Karlin, S. and Feldman, M. W. (1968b) Further analysis of negative assortative mating. Genetics 59, 117136.
Karlin, S. and Mcgregor, J. (1964a) On some stochastic models in gentics. Stochastic Models in Medicine and Biology. University of Wisconsin Press, Madison, 245279.
Karlin, S. and Mcgregor, J. (1964b) Direct product branching processes and related Markoff chains. Proc. Nat. Acad. USA 51, 598602.
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, 11145.
Karlin, S. and Mcgregor, J. (1967) The number of mutant forms maintained in a population. Proc. 5th Berkeley Symp. Math. Stat. Prob. 1965, IV, 415438.
Karlin, S. and Mcgregor, J. (1968a) The role of the Poisson progeny distribution in population genetics models. Math. Biosciences 2, 1117.
Karlin, S. and Mcgregor, J. (1968b) Rates and probabilities of fixation for two locus random mating finite populations without selection. Genetics. 58, 141159.
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 403414.
Karlin, S. and Scudo, F. (1968) Assortative mating based on phenotype II: Two autosomal alleles without dominance. To appear.
Kempthorne, O. (1955) The correlations between relatives in random mating populations. Symp. Quant. Biol., Cold Spring Harbor, 22, 6078.
Kempthorne, O. (1957) An Introduction to Statistical Genetics. Wiley, New York.
Kempthorne, O. (1967) The concept of identity of genes by descent. Proc. 5th Berkeley Symp. Math. Stat. Prob. 1965, IV, 333348.
Kimura, M. (1955) Solution of a process of random genetic drift with a continuous model. Proc. Nat. Acad. Sci. USA 41, 144150.
Kimura, M. (1956) A model of genetic system which leads to closer linkage by natural selection. Evolution 10, 278287.
Kimura, M. (1957) Overdominant genes in a partially self-fertilizing population. Annual Report Nat. Inst. Genet. (Japan) 105106.
Kimura, M. (1963) A probability method for treating inbreeding schemes especially with linked genes. Biometrics 19, 117.
Kimura, M. (1964) Diffusion models in population genetics. J. Appl. Prob. 1, 177232.
Kimura, M. (1965) Attainment of quasi linkage equilibrium when gene frequencies are changing by natural selection. Genetics 52, 875890.
Kimura, M. and Crow, J. F. (1963a) On the maximum avoidance of inbreeding. Genet. Res. 4, 399415.
Kimura, M. and Crow, J. F. (1963b) The measurement of effective population number. Evolution 17, 279288.
Kojima, K. and Kelleher, T. M. (1962) Survival of mutant genes. Amer. Naturalist 96, No. 891, 329346.
Lerner, I. M. (1950) Population Genetics and Animal Improvement. Cambridge University Press.
Li, C. C. (1955) Population Genetics. The University Press, Chicago.
Lillestol, Jostein (1968) Another approach to some Markov chain models in population genetics. J. Appl. Prob. 5, 920.
Mainardi, D. (1964a) Relations between early experience and sexual preferences in female mice. Atti Ass. Genet. Ital. 9, 141.
Mainardi, D. (1964b) Early experience and sexual selection in Drosophila melanogaster. Ann. Zool. 4, 119.
Mainardi, D. (1968) La Scelta Sessuale Nell' Evoluzione Della Specie. Boringhieri, Torino.
Malécot, G. (1948) Les Mathématiques de l'Hérédité. Masson, Paris.
Malécot, G. (1951) Un traitement stochastique des problèmes linéaires (mutation, linkage, migration) en génétique de population. Ann. Univ. Lyon. Sciences, Section A 14, 79100.
Malécot, G. (1959a) Les modèles stochastiques en génétique de population. Publ. Inst. Statist. U. Paris 8, fasc. 3.
Malécot, G. (1959b) La génétique de population. II. Modèles stochastiques. Publ. Inst. Statist. U. Paris 8, fasc. 3, 173210.
Malécot, G. (1967) Identical loci and relationship. Proc. 5th Berkeley Symp. Math. Stat. Prob. 1965, IV, 317332.
Mather, K. (1935) Reductional and equational separation of the chromosomes in bivalents and multivalents. J. Genet. 30, 5378.
Mather, K. (1949) Biometrical Genetics. London: Methuen and Co.
Mather, K. and Hayman, B. I. (1952) The progress of inbreeding when heterozygotes are at an advantage. Biometrics 8, 176.
Moran, P. A. P. (1962) The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.
Moran, P. A. P. and Watterson, G. A. (1959) The genetic effects of family structure in natural populations. Aust. J. Biol. Sci. 12, 115.
O'Donald, P. (1959) Possibility of assortive mating in the Arctic Skua. Nature 182, 12101211.
O'Donald, P. (1960a) Inbreeding as a result of imprinting. Heredity 15, 79.
O'Donald, P. (1960b) Assortive mating in a population in which two alleles are segregating. Heredity 15, 389396.
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, 187196.
Parsons, P. A. (1957) Selfing under conditions favouring heterozygosity. Heredity 11, Part 3, 411421.
Parsons, P. A. (1962) The initial increase of a new gene under positive assortative mating. Heredity 17, 267276.
Parsons, P. A. (1963) Complex polymorphisms where the coupling and repulsion double heterozygote viabilities differ. Heredity 18, 369374.
Parsons, P. A. (1963) Polymorphisms and the balanced polygenic combination. Evolution 17, 564574.
Parsons, P. A. (1965) Assortative mating for a metrical characteristic in drosophila. Heredity 20, 161167.
Reeve, E. C. R. (1955) Inbreeding with homozygotes at a disadvantage. Ann. Hum. Genet. 19, 332346.
Reiersöl, O. (1962) Genetic algebras studied recursively and by means of differential operators. Math. Scand. 10, 2544.
Richardson, W. H. (1964) Frequencies of genotypes of relatives as determined by stochastic matrices. Genetica 35, 323354.
Schnell, F. W. (1961) Some general formulations of linkage effects in inbreeding. Genetics 46, 947957.
Scudo, F. M. (1964) Sex population genetics. Ri. Sci. 34, II–B, 93146.
Scudo, F. M. (1967) L'accoppiamento assortativo basato sul fenotipo di parenti, alcune conseguenze in popolazioni. Genetica , Instituto Lombardo (Rend. Sc.) B 101, 435455.
Scudo, F. M. and Karlin, S. (1968) Assortative mating based on phenotype; I. Two alleles with dominance. To appear.
Scudo, F. M., Mainardi, D. and Barbieri, D. (1965) Assortative mating based on early learning: population genetics. Acta Bio-medica 36, Fasc. 5, 583605.
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, 1316.
Shikata, M. (1965b) A generalization of the inbreeding coefficient. Biometrics 21, 665681.
Shikata, M. (1966) A note on group ring expression of self-fertilized population. Bull. Math. Biophys. 28, No. 2, 217218.
Shikata, M. (1966) Cross-over effect in homozygosity by descent. J. Theoret. Biol. 10, 196208.
Stern, Curt (1960) Principles of Human Genetics. 2nd edition. Freeman, San Francisco.
Watterson, G. A. (1959a) Non-random mating, and its effect on the rate of approach to homozygosity. Ann. Hum. Genet. 23, 204.
Watterson, G. A. (1959b) A new genetic population model and its rate of approach to homozygosity. Ann. Hum. Genet. 23, 221232.
Watterson, G. A. (1960) Unpublished Ph.D. thesis, Australian National University.
Watterson, G. A. (1962) Some theoretical aspects of diffusion theory in population genetics. Ann. Math. Stat. 33, 939957.
Watterson, G. A. (1964) The application of diffusion theory to two population genetic models of Moran. J. Appl. Prob. 1, 233246.
Wentworth, E. H. and Remick, B. L. (1916) Some breeding properties of generalized Mendelian populations. Genetics 1, 608616.
Workman, P. L. (1964) The maintenance of heterozygosity by partial negative assortative mating. Genetics 50, 13691382.
Wright, S. (1921) Systems of mating. Genetics 6, 111178.
Wright, S. (1922) Coefficients of inbreeding and relationship. Amer. Nat. 56, 330339.
Wright, S. (1931) Evolution in Mendelian populations. Genetics 16, 97159.
Wright, S. (1939) The distribution of self-sterility alleles in populations. Genetics 24, 538552.
Wright, S. (1951) The genetical structure of populations. Ann. Eugen. 15, 323354.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed