The decay of linkage disequilibrium under random union of gametes: How to calculate Bennett’s principal components. (English) Zbl 1011.92038

Summary: How rapidly does an arbitrary pattern of statistical association among a set of loci decay under meiosis and random union of gametes? This problem is non-trivial, even in the case of an infinitely large population where selection and other forces are absent. J.H. Bennett [Ann. Hum. Genet. 18, 311-317 (1954) found that, for an arbitrary number of loci with an arbitrary linkage map, it is possible to define measures of linkage disequilibrium that decay geometrically with time. He found a recursive method for deriving expressions for these variables in terms of “allelic moments” (the factorial moments about the origin of the “allelic indicators”), and expressions for the allelic moments in terms of his new variables. However, Bennett nowhere stated his recursive algorithm explicitly, nor did he give a general formula for his measures of linkage disequilibrium, for an arbitrary number of loci. Recursive definitions of Bennett’s variables were obtained by Yu.I. Lyubich [see “Mathematical structures in population genetics.” (1992; Zbl 0747.92019)]. However, the expressions generated by these recursions are not the same as those found by Bennett. (They do not express Bennett’s variables as functions of the allelic moments.) Lyubich’s derivations employ genetic algebras.
Here, I present a method for obtaining explicit expressions for Bennett’s variables in terms of the allelic moments. I show that the transformation from the allelic moments to Bennett’s variables and the inverse transformation always have the form that Bennett claimed. (This transformation and its inverse have essentially the same form.) I present general recursions for calculating the coefficients in the forward transformation and the coefficients in the inverse transformation. My derivations involve combinatorial arguments and ordinary algebra only. The special case of unlinked loci is briefly discussed.


92D10 Genetics and epigenetics
62P10 Applications of statistics to biology and medical sciences; meta analysis
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[1] Abraham, V.M., Linearizing quadratic transformations in genetic algebras, Proc. London math. soc. (3), 40, 346-363, (1980) · Zbl 0388.17007
[2] Abraham, V.M., The induced linear transformation in a genetic algebra, Proc. London math. soc. (3), 40, 364-384, (1980) · Zbl 0393.17008
[3] Baake, E. 2000, Mutation and recombination with tight linkage, submitted for publication.
[4] Barton, N.H., The effects of linkage and density-dependent regulation on gene flow, Heredity, 57, 415-426, (1986)
[5] Barton, N.H.; Turelli, M., Natural and sexual selection on many loci, Genetics, 127, 229-255, (1991)
[6] Bateson, W.; Saunders, E.R.; Punnett, R.C., Experimental studies in the psychology of heredity, Rep. evol. commun. R. soc., 2, 1-55, (1905)
[7] Bennett, J.H., On the theory of random mating, Ann. hum. genet., 18, 311-317, (1954)
[8] Berge, C., Principles of combinatorics, (1971), Academic Press New York · Zbl 0227.05002
[9] Bürger, R., Moments, cumulants and polygenic dynamics, J. math. biol., 30, 199-213, (1991) · Zbl 0760.92014
[10] Campos, T.M.M.; Machado, S.D.A.; Holgate, P., Sex dependent genetic recombination rates for several loci, Linear algebra appl., 136, 165-172, (1990) · Zbl 0711.92012
[11] Etherington, I.M.H., Genetic algebras, Proc. R. soc. Edinburgh, 59, 242-258, (1939) · Zbl 0027.29402
[12] Geiringer, H., On the probability theory of linkage in Mendelian heredity, Ann. math. statist., 15, 25-57, (1944) · Zbl 0063.01560
[13] Geiringer, H., Further remarks on linkage in Mendelian heredity, Ann. math. statist., 16, 390-393, (1945) · Zbl 0063.01561
[14] Geiringer, H., On the mathematics of random mating in case of different recombination values between males and females, Genetics, 33, 548-564, (1948)
[15] Haldane, J.B.S., Theoretical genetics of autopolyploids, J. genet., 22, 359-372, (1930)
[16] Hill, W.G., Disequilibrium among several linked genes in finite population I. Mean changes in disequilibrium, Theor. popul. biol., 5, 366-392, (1974) · Zbl 0291.92024
[17] Holgate, P., Sequences of powers in geneic algebras, J. London math. soc., 42, 489-496, (1967) · Zbl 0163.03103
[18] Holgate, P., The genetic algebra of k linked loci, Proc. London math. soc. (3), 18, 315-327, (1968) · Zbl 0157.26703
[19] Holgate, P., Canonical multiplication in the genetic algebra for linked loci, Linear algebra appl., 26, 281-286, (1979) · Zbl 0408.92004
[20] Holgate, P., Conditions for the linearization of the squaring operation in genetic algebras—A review, Cahiers math., 38, 23-33, (1989) · Zbl 0760.17025
[21] Holgate, P., The multi-locus hardy – weinberg law, J. math. biol., 31, 101-106, (1992) · Zbl 0757.92013
[22] Jennings, H.S., The numerical results of diverse systems of breeding with respect to two pairs of characters, linked or independent, with special relation to the effect of linkage, Genetics, 2, 97-154, (1917)
[23] Karlin, S.; Liberman, U., Theoretical recombination processes incorporating interference effects, Theor. popul. biol., 46, 198-231, (1994) · Zbl 0823.92014
[24] Kimura, M., Attainment of quasi-linkage equilibirum when gene frequencies are changing by natural selection, Genetics, 52, 875-890, (1965)
[25] Lyubich, Yu.I., Basic concepts and theorems of the evolutionary genetics of free populations, Russ. math. surv., 26, 51-123, (1971)
[26] Lyubich, Yu.I., Mathematical structures in population genetics, Biomathematics, 22, (1992), Springer-Verlag Berlin · Zbl 0593.92011
[27] McHale, D.; Ringwood, G.A., Haldane linearization of baric algebras, J. London math. soc. (2), 28, 17-26, (1983) · Zbl 0515.17010
[28] Morgan, T.H., The physical basis of heredity, (1919), Lippincott Philadelphia
[29] Nagylaki, T., Evolution of one- and two-locus systems, Genetics, 83, 583-600, (1976)
[30] Nagylaki, T., The evolution of multilocus systems under weak selection, Genetics, 134, 627-647, (1993)
[31] Nagylaki, T.; Hofbauer, J.; Brunovsky, P., Convergence of multilocus systems under weak epistasis or weak selection, J. math. biol., 38, 103-133, (1999) · Zbl 0981.92019
[32] Pearson, K., Mathematical contributions to the theory of evolution. XII. on a generalized theory of alternative inheritance, with special reference to Mendel’s laws, Philos. trans. R. soc. London ser. A, 203, 53-86, (1904) · JFM 35.0242.02
[33] Reiersol, O., Genetic algebras studied recursively and by means of differential operators, Math. scand., 10, 25-44, (1962) · Zbl 0286.17006
[34] Ringwood, G.A., Hypergeometric algebras and Mendelian genetics, Nieuw arch. wisk. (4), 3, 69-83, (1985) · Zbl 0565.17014
[35] Robbins, R.B., Some applications of mathematics to breeding problems. III, Genetics, 3, 375-389, (1918)
[36] Schnell, F.W., Some general formulations of linkage effects in inbreeding, Genetics, 46, 947-957, (1961)
[37] Slatkin, M., On treating the chromosome as the unit of selection, Genetics, 72, 157-168, (1972)
[38] Stuart, A.; Ord, J.K., Kendall’s advanced theory of statistics. vol. I. distribution theory, (1987), Griffin London
[39] Tietze, H., Uber das schicksal gemischter populationen nach den mendelschen vererbungsgesctzen beim menschen, Z. angew. math. mech., 3, 362-393, (1923) · JFM 49.0728.02
[40] Turelli, M.; Barton, N.H., Genetic and statistical analysis of strong selection on polygenic traits: what, me normal?, Genetics, 138, 913-941, (1994)
[41] Visconti, N.; Delbruck, M., The mechanism of genetic recombination in phage, Genetics, 38, 5-33, (1953)
[42] Weinberg, W., On the laws of inheritance in man. II. special part. III, appendix to general part, Z. indukt. abstamm. vererbungsl., 2, 276-330, (1909)
[43] Weinberg, W., Weitere beitrage zur theorie der verebung (further contributions to the theory of inheritance), Arch. rassen. ges. biol., 7, 35-49, (1910)
[44] Wörz-Buskeros, A., Algebras in genetics, Lecture notes in biomathematics, 36, (1980), Springer-Verlag Berlin
[45] Wright, S., “surfaces” of selective value, Proc. natl. acad. sci. USA, 58, 165-172, (1967)
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