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## Genetics and Molecular Biology

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*Print version* ISSN 1415-4757*On-line version* ISSN 1678-4685

### Genet. Mol. Biol. vol.28 no.3 São Paulo July/Sept. 2005

#### http://dx.doi.org/10.1590/S1415-47572005000300027

**EVOLUTIONARY GENETICS SHORT COMMUNICATION**

**Alan E. Stark**

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

**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 *f _{0}*,

*f*and

_{1}*f*. Denote the 3´3 matrix of mating-pair frequencies by [

_{2}*f*],

_{ij}*i*= 0, 1, 2;

*j*= 0, 1, 2. Suppose the population has attained Hardy-Weinberg (H-W) proportions

*f*=

_{0}*q*,

^{2}*f*= 2

_{1}*pq*,

*f*=

_{2}*p*

^{2}.

Then H-W proportions are maintained if the elements of [*f _{ij}*] are given by

where *e _{0}* = -

*p/q*,

*e*= 1 and

_{1}*e*= -

_{2}*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

*f*=

_{0}*q*. This sum is also the frequency of type

^{2}*aa*in the offspring, as can be seen by summing the appropriately weighted terms of [

*f*], noting that

_{ij}*f*= 4

_{11}*f*.

_{02}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 (-

*q*,

^{2}/p^{2}*q/p*). At the lower limit, reading from left to right and from top to bottom row, the elements of [

*f*] are 0, 2

_{ij}*q*,

^{3}*q*(

^{2}*p*-

*q*), 2

*q*, 4

^{3}*q*(

^{2}*p*-

*q*), 2

*q*(

*p*+

^{3}*q*),

^{3}*q*(

^{2}*p*-

*q*), 2

*q*(

*p*+

^{3}*q*), (

^{3}*p*-

*q*)(

*p*+

^{2}*q*). For the upper limit, the values are

^{2}*q*, 0,

^{3}*pq*, 0, 4

^{2}*pq*, 2

^{2}*pq*(

*p*-

*q*),

*pq*, 2

^{2}*pq*(

*p*-

*q*),

*p*(

*p*+

^{3}*q*). 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,

^{3}*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).

**References**

Li CC (1988) Pseudo-random mating populations. In celebration of the 80^{th} 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 ]

**Correspondence to**

Alan E. Stark

3/20 Seaview Street

2093 Balgowlah, NSW, Australia

Email: ae_stark@ihug.com.au.

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

*Editor: Fábio de Melo Sene*