The Hardy-Weinberg principle

EVOLUTIONARY GENETICS

SHORT COMMUNICATION

The Hardy-Weinberg principle

Alan E. Stark

University of New South Wales, School of Community Medicine, Kensington, Australia

^{Correspondence}

ABSTRACT

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 [f_ij], 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 [f_ij] 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. f_ij = f_i f_j.

Under the stated conditions, the sum of elements in the first (zero) row of [f_ij] 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 [f_ij], noting that f₁₁ = 4f₀₂.

The applicable interval of n, as a function of q, is governed by the need for the f_ij 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 [f_ij] 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).

Received: March 11, 2005; Accepted: May 5, 2005.

Editor: Fábio de Melo Sene

• Li CC (1988) Pseudo-random mating populations. In celebration of the 80^th anniversary of the Hardy-Weinberg law. Genetics 119:731-737.
• 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.
• 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.
• Stark AE (1980) Inbreeding systems: Classification by a canonical form. J Math Biol 10:305.

Publication Dates

• Publication in this collection
  05 Sept 2005
• Date of issue
  Sept 2005

History

• Accepted
  05 May 2005
• Received
  11 Mar 2005