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On-line version ISSN 1678-4685
Genet. Mol. Biol. vol.28 no.3 São Paulo July/Sept. 2005
Alan E. Stark
University of New South Wales, School of Community Medicine, Kensington, Australia
Hardy-Weinberg genotypic proportions can be maintained in a population under non-random mating. A compact formula gives the proportions of mating pair types. These are illustrated by some simple examples.
Key words: Hardy-Weinberg equilibrium, non-random mating.
Consider a population with respect to a single locus having alleles a and A with respective frequencies q and p. Denote frequencies of genotypes aa, aA and AA by f0, f1 and f2. Denote the 3´3 matrix of mating-pair frequencies by [fij], i = 0, 1, 2; j = 0, 1, 2. Suppose the population has attained Hardy-Weinberg (H-W) proportions f0 = q2, f1 = 2pq, f2 = p2.
Then H-W proportions are maintained if the elements of [fij] are given by
where e0 = -p/q, e1 = 1 and e2 = -q/p.
Random mating is defined by matrix (1) with n = 0, i.e. fij = fi fj.
Under the stated conditions, the sum of elements in the first (zero) row of [fij] is f0 = q2. This sum is also the frequency of type aa in the offspring, as can be seen by summing the appropriately weighted terms of [fij], noting that f11 = 4f02.
The applicable interval of n, as a function of q, is governed by the need for the fij to be non-negative. Without loss of generality, consider q in the range 0 < q < 1/2. Then the interval containing permissible values of n is (-q2/p2, q/p). At the lower limit, reading from left to right and from top to bottom row, the elements of [fij] are 0, 2q3, q2(p - q), 2q3, 4q2(p - q), 2q(p3 + q3), q2(p - q), 2q(p3 + q3), (p - q)(p2 + q2). For the upper limit, the values are q3, 0, pq2, 0, 4pq2, 2pq(p - q), pq2, 2pq(p - q), p(p3 + q3). Thus it can be seen that the mating scheme defined by (1) gives a wide spectrum of non-random mating which maintains H-W proportions, i.e. the "Hardy-Weinberg Principle" is more general than is usually envisaged.
Stark (1980) gives a more general mating scheme which subsumes (1). Li (1988) showed that random mating is a sufficient condition, not a necessary one, for the attainment of the Hardy Weinberg proportions, but here we provide for the first time a truly generalized mathematical argumentation to prove the fact. Stark (1977 a,b) give a more detailed description of the underlying mating model and several diagrams which illustrate the ranges of applicability of formula (1).
Li CC (1988) Pseudo-random mating populations. In celebration of the 80th anniversary of the Hardy-Weinberg law. Genetics 119:731-737. [ Links ]
Stark AE (1977a) Models of correlation between mates and relatives and some applications. Unpublished Ph.D. Thesis, School of Community Medicine, The University of New South Wales, Kensington, Australia. [ Links ]
Stark AE (1977b) Dwa uogólnienia prawa Hardy'ego-Weinberga w genetyce populacji (The Hardy-Weinberg Law of Population Genetics and two Generalizations). Matematyka Stosowana IX:123-137. [ Links ]
Stark AE (1980) Inbreeding systems: Classification by a canonical form. J Math Biol 10:305. [ Links ]
Alan E. Stark
3/20 Seaview Street
2093 Balgowlah, NSW, Australia
Received: March 11, 2005; Accepted: May 5, 2005.
Editor: Fábio de Melo Sene