Hardy–Weinberg principle

These influences include genetic drift, mate choice, assortative mating, natural selection, sexual selection, mutation, gene flow, meiotic drive, genetic hitchhiking, population bottleneck, founder effect, inbreeding and outbreeding depression.

In the absence of selection, mutation, genetic drift, or other forces, allele frequencies p and q are constant between generations, so equilibrium is reached.

Consider a population of monoecious diploids, where each organism produces male and female gametes at equal frequency, and has two alleles at each gene locus.

Organisms reproduce by random union of gametes (the "gene pool" population model).

Summing the elements of the Punnett square or the binomial expansion, we obtain the expected genotype proportions among the offspring after a single generation: These frequencies define the Hardy–Weinberg equilibrium.

This follows since the genotype frequencies of the next generation depend only on the allele frequencies of the current generation which, as calculated by equations (1) and (2), are preserved from the initial generation: For the more general case of dioecious diploids [organisms are either male or female] that reproduce by random mating of individuals, it is necessary to calculate the genotype frequencies from the nine possible matings between each parental genotype (AA, Aa, and aa) in either sex, weighted by the expected genotype contributions of each such mating.

[2] Equivalently, one considers the six unique diploid-diploid combinations: and constructs a Punnett square for each, so as to calculate its contribution to the next generation's genotypes.

These contributions are weighted according to the probability of each diploid-diploid combination, which follows a multinomial distribution with k = 3.

The diploid case is the binomial expansion of: and therefore the polyploid case is the binomial expansion of: where c is the ploidy, for example with tetraploid (c = 4): Whether the organism is a 'true' tetraploid or an amphidiploid will determine how long it will take for the population to reach Hardy–Weinberg equilibrium.

: Testing deviation from the HWP is generally performed using Pearson's chi-squared test, using the observed genotype frequencies obtained from the data and the expected genotype frequencies obtained using the HWP.

If this is the case, then the asymptotic assumption of the chi-squared distribution, will no longer hold, and it may be necessary to use a form of Fisher's exact test, which requires a computer to solve.

More recently a number of MCMC methods of testing for deviations from HWP have been proposed (Guo & Thompson, 1992; Wigginton et al. 2005) This data is from E. B. Ford (1971) on the scarlet tiger moth, for which the phenotypes of a sample of the population were recorded.

The 5% significance level for 1 degree of freedom is 3.84, and since the χ2 value is less than this, the null hypothesis that the population is in Hardy–Weinberg frequencies is not rejected.

In this way, the hypothesis of Hardy–Weinberg proportions is rejected if the number of heterozygotes is too large or too small.

Using this table, one must look up the significance level of the test based on the observed number of heterozygotes.

As is typical for Fisher's exact test for small samples, the gradation of significance levels is quite coarse.

denote the family of the genotype distributions under the assumption of Hardy Weinberg equilibrium.

can be rejected then the population is close to Hardy Weinberg equilibrium with a high probability.

The inbreeding coefficient is unstable as the expected value approaches zero, and thus not useful for rare and very common alleles.

Udny Yule (1902) argued against Mendelism because he thought that dominant alleles would increase in the population.

[10] The American William E. Castle (1903) showed that without selection, the genotype frequencies would remain stable.

[12] Reginald Punnett, unable to counter Yule's point, introduced the problem to G. H. Hardy, a British mathematician, with whom he played cricket.

Hardy was a pure mathematician and held applied mathematics in some contempt; his view of biologists' use of mathematics comes across in his 1908 paper where he describes this as "very simple":[13] The principle was thus known as Hardy's law in the English-speaking world until 1943, when Curt Stern pointed out that it had first been formulated independently in 1908 by the German physician Wilhelm Weinberg.

[14][15] William Castle in 1903 also derived the ratios for the special case of equal allele frequencies, and it is sometimes (but rarely) called the Hardy–Weinberg–Castle Law.

An example computation of the genotype distribution given by Hardy's original equations is instructive.

The reader may demonstrate that subsequent use of the second-generation values for a third generation will yield identical results.

babies are born with cystic fibrosis, this is about the frequency of homozygous individuals observed in Northern European populations.

It is possible to represent the distribution of genotype frequencies for a bi-allelic locus within a population graphically using a de Finetti diagram.

[16] The curved line in the diagram is the Hardy–Weinberg parabola and represents the state where alleles are in Hardy–Weinberg equilibrium.

[17] The de Finetti diagram was developed and used extensively by A. W. F. Edwards in his book Foundations of Mathematical Genetics.