Hardy-Weinberg principle

In population genetics, the Hardy–Weinberg principle (HWP) (also Hardy–Weinberg equilibrium (HWE), or Hardy–Weinberg law), named after G. H. Hardy and Wilhelm Weinberg, states that, under certain conditions, after one generation of random mating, the genotype frequencies at a single gene locus will become fixed at a particular equilibrium value. It also specifies that those equilibrium frequencies can be represented as a simple function of the allele frequencies at that locus.

In the simplest case of a single locus with two alleles A and a with allele frequencies of p and q, respectively, the HWP predicts that the genotypic frequencies for the AA homozygote to be p2, the Aa heterozygote to be 2pq and the other aa homozygote to be q2. The Hardy–Weinberg principle is an expression of the notion of a population in "genetic equilibrium" and is a basic principle of population genetics.

Assumptions
The original assumptions for Hardy–Weinberg equilibrium (HWE) were that the organism under consideration is:
 * Diploid, and the trait under consideration is not on a chromosome that has different copy numbers for different sexes, such as the X chromosome in humans
 * Sexually reproducing, either monoecious or dioecious
 * Discrete generations

In addition, the population under consideration is idealised, that is:


 * Random mating within a single population
 * Infinite population size (or sufficiently large so as to minimize the effect of genetic drift)

and experiences:


 * No selection
 * No mutation
 * No migration (gene flow)

The first group of assumptions are required for the mathematics involved. It is relatively easy to expand the definition of HWE to include modifications of these, such as for sex-linked traits. The other assumptions are inherent in the Hardy-Weinberg principle.

A Hardy-Weinberg population is used as a reference population when discussing various factors. It is not surprising that these populations are static.

Derivation
A better, but equivalent, probabilistic description for the HWP is that the alleles for the next generation for any given individual are chosen randomly and independent of each other. Consider two alleles, A and a, with frequencies p and q, respectively, in the population. The different ways to form new genotypes can be derived using a Punnett square, where the fraction in each cell is equal to the product of the row and column probabilities.

The final three possible genotypic frequencies in the offspring become:


 * $$f(\mathbf{AA}) = p^2\,$$
 * $$f(\mathbf{Aa}) = 2pq\,$$
 * $$f(\mathbf{aa}) = q^2\,$$

These frequencies are called Hardy-Weinberg frequencies (or Hardy-Weinberg proportions). This is achieved in one generation, and only requires the assumption of random mating with an infinite population size.

Sometimes, a population is created by bringing together males and females with different allele frequencies. In this case, the assumption of a single population is violated until after the first generation, so the first generation will not have Hardy-Weinberg equilibrium. Successive generations will have Hardy-Weinberg equilibrium.

Deviations from Hardy-Weinberg equilibrium
Violations of the Hardy–Weinberg assumptions can cause deviations from expectation. How this affects the population depends on the assumptions that are violated.


 * Random mating. The HWP states the population will have the given genotypic frequencies (called Hardy-Weinberg proportions) after a single generation of random mating within the population.  When violations of this provision occur, the population will not have Hardy-Weinberg proportions.  Three such violations are:
 * Inbreeding, which causes an increase in homozygosity for all genes.
 * Assortative mating, which causes an increase in homozygosity only for those genes involve with the trait that is assortatively mated (and genes in linkage disequilibrium with them).
 * Small population size, which causes a random change in genotypic frequencies, particularly if the population is very small. This is due to a sampling effect, and is called genetic drift.

The remaining assumptions affect the allele frequencies, but do not, in themselves, affect random mating. If a population violates one of these, the population will continue to have Hardy-Weinberg proportions each generation, but the allele frequencies will change with that force.


 * Selection, in general, causes allele frequencies to change, often quite rapidly. While directional selection eventually leads to the loss of all alleles except the favored one, some forms of selection, such as balancing selection, lead to equilibrium without loss of alleles.
 * Mutation will have a very subtle effect on allele frequencies. Mutation rates are of the order 10-4 to 10-8, and the change in allele frequency will be, at most, the same order.  Recurrent mutation will maintain alleles in the population, even if there is strong selection against them.
 * Migration genetically links two or more populations together. In general, allele frequencies will become more homogeneous among the populations.  Some models for migration inherently include nonrandom mating (Wahlund effect, for example).  For those models, the Hardy-Weinberg proportions will normally not be valid.

How these violations affect formal statistical tests for HWE is discussed later.

Unfortunately, violations of assumptions in the Hardy-Weinberg principle does not mean the population will violate HWE. For example, balancing selection leads to an equilibrium population with Hardy-Weinberg proportions. This property with selection vs. mutation is the basis for many estimates of mutation rate (call mutation-selection balance).

Sex linkage
Where the A gene is sex-linked, the heterogametic sex (e.g., human males) have only one copy of the gene (and are termed hemizygous), while the homogametic sex (e.g., human females) have two copies. The genotype frequencies at equilibrium are $$p$$ and $$q$$ for the heterogametic sex but $$p^2$$, $$2pq$$ and $$q^2$$ for the homogametic sex.

For example, in humans red-green colorblindness is an X-linked recessive trait. In western European males, the trait affects about 1 in 12, ($$q = 0.083$$) whereas it affects about 1 in 200 females ($$0.005$$, compared to $$ q^2 = 0.0070$$), very close to Hardy-Weinberg proportions.

If a population is brought together with males and females with different allele frequencies, the allele frequency of the male population follows that of the female population because each receives its X chromosome from its mother. The population converges on equilibrium very quickly.

Generalizations
The simple derivation above can be generalized for more than two alleles and polyploidy.

Generalization for more than two alleles
Consider an extra allele frequency, $$r$$. The two-allele case is the binomial expansion of $$(p+q)^2$$, and thus the three-allele case is the trinomial expansion of $$(p+q+r)^2$$.


 * $$(p+q+r)^2=p^2 + r^2 + q^2 + 2pq +2pr + 2qr$$

More generally, consider the alleles A1, ... Ai given by the allele frequencies $$p_1$$ to $$p_i$$;


 * $$(p_1 + \cdots + p_i)^2$$

giving for all homozygotes:


 * $$f(A_i A_i) = p_i^2$$

and for all heterozygotes:


 * $$f(A_i A_j) = 2p_ip_j$$

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


 * $$(p + q)^2$$

and therefore the polyploid case is the binomial expansion of:


 * $$(p + q)^c$$

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

Depending on 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.

Complete generalization
The completely generalized formula is the multinomial expansion of $$(p_1 + \cdots + p_n)^c$$:


 * $$(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} $$

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 can be calculated, or if the genotype frequencies of all three genotypes are known, they 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., there is complete dominance. Assuming that the Hardy–Weinberg principle applies to the population, then $$q$$ can still be calculated from f(aa):


 * $$q = \sqrt {f(aa)}$$

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:


 * $$p$$
 * $$= {2 \times \mathrm{obs}(AA) + \mathrm{obs}(Aa) \over 2 \times (\mathrm{obs}(AA) + \mathrm{obs}(Aa) + \mathrm{obs}(aa))}$$
 * $$= {1469 \times 2 + 138 \over 2 \times (1469+138+5)}$$
 * $$= { 3076 \over 3224} $$
 * $$= 0.954$$
 * }
 * $$= { 3076 \over 3224} $$
 * $$= 0.954$$
 * }
 * $$= 0.954$$
 * }
 * }

and




 * $$q$$
 * $$= 1 - p$$
 * $$= 1 - 0.954$$
 * $$= 0.046$$
 * }
 * $$= 0.046$$
 * }
 * $$= 0.046$$
 * }

So the Hardy–Weinberg expectation is:




 * $$\mathrm{Exp}(AA) = p^2n = 0.954^2 \times 1612 = 1467.4$$
 * }
 * }




 * $$\mathrm{Exp}(Aa) = 2pqn = 2 \times 0.954 \times 0.046 \times 1612 = 141.2$$
 * }
 * }




 * $$\mathrm{Exp}(aa) = q^2n = 0.046^2 \times 1612 \leq 3.4$$
 * }
 * }

Pearson's chi-square test states:




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

There is 1 degree of freedom. (degrees of freedom for χ2 squared tests are normally n &minus; 1, where n is the number of genotype classes. However, an extra degree of freedom is lost because the expected values were estimated 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.

Fisher's exact test (probability test)
Fisher's exact test can be applied to testing for Hardy-Weinberg proportions. Since the test is conditional on the allele frequencies, p and q, the problem can be viewed as testing for the proper number of heterozygotes. In this way, the hypothesis of Hardy-Weinberg proportions is rejected if the number of heterozygotes are too large or too small. The conditional probabilities for the heterozygote, given the allele frequencies are given in Emigh (1980) as
 * $$prob[n_{12} | n_1] = \frac  2^{n_{12}}, $$

where n11, n12, n22 are the observed numbers of the three genotypes, AA, Aa, and aa, respectively, and n1 is the number of A alleles, where $$n_1 = 2 n_{11} + n_{12}$$.

An Example Using one of the examples from Emigh (1980), we can consider the case where n = 100, and p = 0.34. The possible observed heterozygotes and their exact significance level is given in Table 4. Using this table, you look up the significance level of the test based on the observed number of heterozygotes. For example, if you observed 20 heterozygotes, the significance level for the test is 0.007. As is typical for Fisher's exact test for small samples, the gradation of significance levels is quite coarse.

Unfortunately, you have to create a table like this for every experiment, since the tables are dependent on both n and p.

Inbreeding coefficient
The inbreeding coefficient, F (see also F-statistics), is one minus the observed frequency of heterozygotes over that expected from Hardy–Weinberg equilibrium.


 * $$ F = \frac{\operatorname{E}{(f(\mathbf{Aa}))} - \operatorname{O}(f(\mathbf{Aa}))} {\operatorname{E}(f(\mathbf{Aa}))} = 1 - \frac{\operatorname{O}(f(\mathbf{Aa}))} {\operatorname{E}(f(\mathbf{Aa}))}, \!$$

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


 * $$ \operatorname{E}(f(\mathbf{Aa})) = 2\, p\, q, \!$$

For example, for Ford's data above;


 * $$F = 1 - {138 \over 141.2}$$


 * $$= 0.023.\,$$

For two alleles, the chi square goodness of fit test for Hardy-Weinberg proportions is equivalent to the test for inbreeding, F = 0.

History
Mendelian genetics were 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 q2 = pr. And since q12 = p1r1, 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 Curt Stern (1943) pointed out that it had first been formulated independently in 1908 by the German physician Wilhelm Weinberg (see Crow 1999). Others have tried to associate Castle's name with the Law because of his work in 1903, but it is only rarely seen as the Hardy-Weinberg-Castle Law.