Hardy-Weinberg principle

<math>(p_1 + ... + p_i)^2<math>

giving for all homozygotes:

<math>f(A_i A_i) = p_i^2<math>

and for all heterozygotes:

<math>f(A_i A_j) = 2p_ip_j<math>

Generalization for polyploidy

The Hardy–Weinberg principle may also be generalized to polyploid systems, that is to populations which have more than two copies of each chromosome. Consider again only two alleles. The diploid case is the binomial expansion of:

<math>(p + q)^2<math>

and therefore the polyploid case is the binomial expansion of:

<math>(p + q)^c<math>

where c is the ploidy, for example with tetraploid (c = 4):

Complete generalization

The completely generalized formula is the multinomial expansion of <math>(p_1 + \cdots + p_n)^n<math>:

<math>(p_1 + \cdots + p_n)^n = \sum_{k_1, \ldots, k_n, k_1 + \cdots +k_n=n} {n \choose k_1, \ldots, k_n}

 p_1^{k_1} \cdots p_n^{k_n} <math>

Applications

The Hardy–Weinberg principle may be applied in two ways, either a population is assumed to be in Hardy–Weinberg proportions, in which the genotype frequencies for can be calculated, or if the genotype frequencies of all three genotypes are known, the can be tested for deviations that are statistically significant.

Application to cases of complete dominance

Suppose that the phenotypes of AA and Aa are indistinguishable i.e. that there is complete dominance. Assuming that the Hardy–Weinberg principle applies to the population, then q can still be calculated from f(aa):

<math>q = \sqrt {f(aa)}<math>

and p can be calculated from q. And thus an estimate of f(AA) and f(Aa) derived from p^2 and 2pq respectively. Note however, such a population cannot be tested for equilibrium using the significance tests below because it is assumed a priori.

Significance tests for deviation

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. For systems where there are large numbers of alleles, this may result in data with many empty possible genotypes and low genotype counts, because there are often not enough individuals present in the sample to adequately represent all genotype classes. If this is the case, then the asymptotic assumption of the chi-square 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.

Example χ² test for deviation

These data are from E.B. Ford (1971) on the Scarlet tiger moth, for which the phenotypes of a sample of the population were recorded. Genotype-phenotype distinction is assumed to be negligibly small. The null hypothesis is that the population is in Hardy–Weinberg proportions, and the alternative hypothesis is that the population is not in Hardy–Weinberg proportions.

From which allele frequencies can be calculated:

<math>p<math>	<math>= {2 \times obs(AA) + obs(Aa) \over 2 \times (obs(AA) + obs (Aa) + obs(aa))}<math>
	<math>= {1469 \times 2 + 138 \over 2 \times (1469+138+5)}<math>
	<math>= { 3976 \over 3224} <math>
	<math>= 0.954<math>

and

<math>q<math>	<math>= 1 - p<math>
	<math>= 1 - 0.954<math>
	<math>= 0.046<math>

So the Hardy–Weinberg expectation is:

<math>Exp(AA) = p^2n = 0.954^2 \times 1612 = 1467.4<math>

<math>Exp(Aa) = 2pqn = 2 \times 0.954 \times 0.046 \times 1612 = 141.2<math>

<math>Exp(aa) = q^2n = 0.046^2 \times 1612< = 3.4<math>

Pearson's chi-square test states:

<math> \chi^2<math>	<math>= \sum {(O - E)^2 \over E}<math>
	<math> = {(1469 - 1467.4)^2 \over 1467.4} + {(138 - 141.2)^2 \over 141.2} + {(5 - 3.4)^2 \over 3.4}<math>
	<math> = 0.001 + 0.073 + 0.756 <math>
	<math> = 0.83 <math>

There is 1 degree of freedom. (degrees of freedom for χ² squared tests are normally n - 1, where n is the number of genotype classes. However, an extra degree of freedom is lost because the expected values were calculated from the observed values). 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 equilibrium is not rejected.

F-statistics

In F-statistics, the measure F is the observed frequency of heterozygotes over that expected from Hardy–Weinberg equilibrium:

<math> F = \frac{\operatorname{O}(f(\mathbf{Aa}))} {\operatorname{E}(f(\mathbf{Aa}))}, \!<math>

where the expected value from Hardy–Weinberg equilibrium is given by

<math> \operatorname{E}(f(\mathbf{Aa})) = 2\, p\, q, \!<math>

For example, for Ford's data above;

<math>F<math>	<math>= {138 \over 141.2}<math>
	<math>=0.977<math>

History

Mendelian genetics was rediscovered in 1900. However, it remained somewhat controversial for several years as it was not then known how it could cause continuous characters. Udny Yule (1902) argued against Mendelism because he thought that dominant alleles would increase in the population. The American William E. Castle (1903) showed that without selection, the genotype frequencies would remain stable. Karl Pearson (1903) found one equilibrium position with values of p = q = 0.5. 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".

To the Editor of Science: I am reluctant to intrude in a discussion concerning matters of which I have no expert knowledge, and I should have expected the very simple point which I wish to make to have been familiar to biologists. However, some remarks of Mr. Udny Yule, to which Mr. R. C. Punnett has called my attention, suggest that it may still be worth making...

Suppose that Aa is a pair of Mendelian characters, A being dominant, and that in any given generation the number of pure dominants (AA), heterozygotes (Aa), and pure recessives (aa) are as p:2q:r. Finally, suppose that the numbers are fairly large, so that mating may be regarded as random, that the sexes are evenly distributed among the three varieties, and that all are equally fertile. A little mathematics of the multiplication-table type is enough to show that in the next generation the numbers will be as (p+q)^2:2(p+q)(q+r):(q+r)^2, or as p1:2q1:r1, say.

The interesting question is — in what circumstances will this distribution be the same as that in the generation before? It is easy to see that the condition for this is q^2 = pr. And since q_1^2 = p_1r_1, whatever the values of p, q, and r may be, the distribution will in any case continue unchanged after the second generation

The principle was thus known as Hardy's law in the English-speaking world until Stern (1943) pointed out that it had first been formulated independently in 1908 by the German physician Wilhelm Weinberg (see Crow 1999).

References

• Castle, W. E. (1903). The laws of Galton and Mendel and some laws governing race improvement by selection. Proc. Amer. Acad. Arts Sci.. 35: 233–242.
• Crow, J.F. (1999). Hardy, Weinberg and language impediments. Genetics 152: 821-825. link (http://www.genetics.org/cgi/content/full/152/3/821?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&titleabstract=Punnett&searchid=1109675768213_849&stored_search=&FIRSTINDEX=0&journalcode=genetics)
• Ford, E.B. (1971). Ecological Genetics, London.
• Hardy, G. H. (1908). "Mendelian proportions in a mixed population". Science 28: 49–50. ESP copy (http://www.esp.org/foundations/genetics/classical/hardy.pdf)
• Pearson, K. (1904). Mathematical contributions to the theory of evolution. XI. On the influence of natural selection on the variability and correlation of organs. Philosophical Transactions of the Royal Society of London, Ser. A 200: 1–66.
• Stern, C. (1943). "The Hardy–Weinberg principle". Science 97: 137–138. JSTOR stable url (http://links.jstor.org/sici?sici=0036-8075%2819430205%293%3A97%3A2510%3C137%3ATHL%3E2.0.CO%3B2-8)
• Weinberg, W. (1908). "Über den Nachweis der Verebung beim Menschen". Jahreshefte des Vereins für vaterländische Naturkunde in Württemberg 64: 368–382.
• Yule, G. U. (1902). Mendel's laws and their probable relation to intra-racial heredity. New Phytol. 1: 193–207, 222–238.