General

In 1908, the mathematician Godfrey Hardy from England and the physician Wilhelm Weinberg from Germany proclaimed, independent from one another, the same law which was named after them. The Hardy-Weinberg equilibrium (HWE) law states that the alleles of a locus are independent and therefore in an infinite population the expected genotype frequencies remain constant and can be predicted by the allele frequencies according to the binomial distribution.

For a simple biallelic locus where the probability for the A allele is k, the expected frequencies of the three genotypes AA, Aa, and aa are k2, 2k(1 — k), and (1 — k)2, respectively (Table 1).

Several conditions are assumed for a population to comply with HWE (Table 2).

Given that these conditions can hardly ever be met in their totality, HWE is at best an approximation. Deviations may be due to any one or combinations of violations of the HWE assumptions. For example, nonran-dom mating may occur with loci related to some special characteristics (phenotypes) as deafness or specific

Table 1 Possible combination between maternal and paternal alleles for a biallelic locus

Paternal a

A

Maternal

a

aa

aA

A

aA

AA

Table 2

Assumptions of Hardy-Weinberg equilibrium

Random mating No natural selection No migration

No genetic drift (i.e., infinite population size for practical purposes) No mutations

Largely nonoverlapping generations Sexual reproduction Diploid organisms genotype; assortative mating describes the situation where individuals with the same phenotype or genotype have a higher or lower than random chance of mating (positive and negative assortative mating, respectively). In some societies and species, relatives may have a higher than random chance of mating with each other (inbreeding). In polygamous societies, males with disproportionate large number of matings will have their genotypes over-represented in the population pool. Moreover, no population is infinite and mating may be dictated to a good extent by geographical proximity. For small populations (e.g., isolated villages and other isolated populations), mating is not random. Moreover, with relatively small populations genetic drift can be very influential. Genetic drift means that even though the genotypes of the offspring are expected to reflect those of the parents, with small numbers there will be a deviation from the expected frequencies due to chance; the deviation may then grow larger with subsequent generations as a stochastic process. At the extreme, drift may eventually lead to the complete disappearance of an allele (allele loss) and the total predominance of its complementary allele (fixation). Genetic drift may also affect the distribution of genotypes even in populations that are currently very large (practically infinite), but had been very small at some point in the past. The action of genetic drift on the small sample size results in a bottleneck effect that may be carried forth even when the population becomes much larger again. Conversely, genetic migration (also known as gene flow) reflects the introduction of genetic variants from outside populations who have different genotype distributions than a local population of interest. Such genetic migration may also invalidate the HWE of a local population and a similar effect may be introduced (albeit typically at a very slow pace) by mutations. Finally, selection forces (e.g., survival or reproductive advantages of specific genotypes) will lead to preferential increase in the relative frequency of some genotypes in the population (and the relative decrease in others) and new mutations may generate additional new genetic diversity.

Deviation from HWE for a specific locus may indicate failure in one or more of these assumptions, or, indicate an association between the locus and a specific condition, for example, a disease. In addition, even if genuine population-based explanations are possible for deviations from

HWE, a common alternative explanation, especially in the epidemiological literature, is simple genotyping error. In fact, Hardy-Weinberg testing has been used as a quality control measure for epidemiological studies and deviations in general populations in such studies have often been considered a (nonspecific and questionably sensitive) signal of genotyping error.

The conditions under which the law is violated are often described with the summary terms of 'outbreeding' and 'inbreeding'; referring to an excess of heterozygotes and homozygotes respectively compared to the frequencies expected under HWE. The use of the term inbreeding in this sense should not be confused with the more narrow notion of inbreeding between relatives or small closed populations that is only one of many reasons of HWE deviation (as described above).

The genotype frequencies under HWE for a biallelic locus with allele probabilities rK1 and w2 ('K1 + = 1) can derived from the terms of the binomial expansion

+ ^2)2. With this consideration, HWE law can be expressed for the case of a multiallelic locus with probabilities ..., rK„ by considering the binomial expansion + +•••+ ^n)2. Similarily, for the case of polyploidy of dimension r (organisms that have up to r copies of each chromosome), the expected genotype frequencies under HWE can be derived by the multinomial expansion + ^2)r. In the general case of a multiallelic polyploid organism, it is + +••• + ^n)r.

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