Hardy-Weinberg principle 230319 220664952 2008-06-20T23:36:28Z Thorwald 74279 /* External links */ rm "New" [[Image:Hardy-Weinberg.gif|thumb|320px|Hardy–Weinberg principle for two [[allele]]s: the horizontal axis shows the two [[allele frequency|allele frequencies]] ''p'' and ''q'', the vertical axis shows the [[genotype]] frequencies and the three possible genotypes are represented by the different glyphs]] In [[population genetics]], the '''Hardy–Weinberg principle''' states that the [[genotype]] frequencies in a population remain constant or are in equilibrium from generation to generation unless specific disturbing influences are introduced. Those disturbing influences include non-random mating, mutations, selection, limited population size, random genetic drift and gene flow. Genetic equilibrium is a basic principle of [[population genetics]]. The Hardy-Weinberg principle is like a [[Punnett square]] for populations, instead of individuals. A Punnett square can predict the probability of offspring's genotype based on parents' genotype or the offsprings' genotype can be used to reveal the parents' genotype. Likewise, the Hardy-Weinberg principle can be used to calculate the frequency of particular alleles based on frequency of, say, an [[autosomal recessive]] disease. In the simplest case of a single locus with two [[allele]]s: the dominant allele is denoted '''A''' and the recessive '''a''' and their frequencies are denoted by ''p'' and ''q''; freq('''A''')=''p''; freq('''a''')=''q''; ''p'' + ''q'' = 1. If the population is in equilibrium, then we will have freq('''AA''')=''p''<sup>2</sup> for the '''AA''' [[homozygote]]s in the population, freq('''aa''')=''q''<sup>2</sup> for the '''aa''' homozygotes, and freq('''Aa''')=2''pq'' for the [[heterozygote]]s. Based on these equations, we can determine useful, difficult-to-measure facts about a population. For example, a patient's child is a carrier of a recessive mutation that causes [[cystic fibrosis]] in homozygous recessive children. The parent wants to know the probability of her grandchildren inheriting the disease. In order to answer this question, the genetic counselor must know the chance that the child will reproduce with a carrier of the recessive mutation. This fact may not be known, but disease frequency is known. We know that the disease is caused by the homozygous recessive genotype; we can use the Hardy-Weinberg principle to work backward from disease occurrence to the frequency of heterozygous recessive individuals. This concept is also known by a variety of names: '''HWP''', '''Hardy–Weinberg equilibrium''', '''HWE''', or '''Hardy–Weinberg law'''. It was named after [[G. H. Hardy]] and [[Wilhelm Weinberg]]. ==Derivation== A better, but equivalent, [[probability|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 is equal to the product of the row and column probabilities. {| class="wikitable" |+ Table 1: Punnett square for Hardy–Weinberg equilibrium |- !colspan="2" rowspan="2"| !colspan="2"|Females |- ! A (''p'') ! a (''q'') |- !rowspan="2"| Males !A (''p'') |AA (''p''²) |Aa (''pq'') |- !a (''q'') |Aa (''pq'') |aa (''q''²) |} The final three possible genotypic frequencies in the offspring become: * <math>f(\mathbf{AA}) = p^2\,</math> * <math>f(\mathbf{Aa}) = 2pq\,</math> * <math>f(\mathbf{aa}) = q^2\,</math> 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 [[expected value|expectation]]. How this affects the population depends on the assumptions that are violated. Generally, deviation from the Hardy-Weinberg equilibrium denotes the evolution of a species. * [[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 [[Zygosity|homozygosity]] for all genes. ** [[Assortative mating]], which causes an increase in [[Zygosity|homozygosity]] only for those genes involved in 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. * [[Natural selection|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<sup>−4</sup> to 10<sup>−8</sup>, 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 linkage|sex-linked]], the heterogametic sex (''e.g.'', mammalian males; avian females) 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 <math>p</math> and <math>q</math> for the heterogametic sex but <math>p^2</math>, <math>2pq</math> and <math>q^2</math> 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, (<math>q = 0.083</math>) whereas it affects about 1 in 200 females (<math>0.005</math>, compared to <math> q^2 = 0.007</math>), 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, <math>r</math>. The two-allele case is the [[binomial theorem|binomial expansion]] of <math>(p+q)^2</math>, and thus the three-allele case is the trinomial expansion of <math>(p+q+r)^2</math>. :<math>(p+q+r)^2=p^2 + r^2 + q^2 + 2pq +2pr + 2qr</math> More generally, consider the alleles '''A'''<sub>1</sub>, ... '''A'''<sub>''i''</sub> given by the allele frequencies <math>p_1</math> to <math>p_i</math>; :<math>(p_1 + \cdots + p_i)^2</math> giving for all [[Zygosity|homozygote]]s: :<math>f(A_i A_i) = p_i^2</math> and for all [[Zygosity|heterozygote]]s: :<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, for organisms that have more than two copies of each chromosome. Consider again only two alleles. The diploid case is the [[binomial theorem|binomial expansion]] of: :<math>(p + q)^2</math> and therefore the polyploid case is the [[polynomial expansion]] of: :<math>(p + q)^c</math> where ''c'' is the [[ploidy]], for example with tetraploid (''c'' = 4): {| class="wikitable" |+ Table 2: Expected genotype frequencies for tetraploidy ! Genotype ! Frequency |- |<math> \mathbf A \mathbf A \mathbf A \mathbf A </math> |<math>p^4</math> |- |<math> \mathbf A \mathbf A \mathbf A \mathbf a</math> |<math>4p^3 q</math> |- |<math> \mathbf A \mathbf A \mathbf a \mathbf a</math> |<math>6p^2q^2</math> |- |<math> \mathbf A \mathbf a \mathbf a \mathbf a</math> |<math>4pq^3</math> |- |<math> \mathbf a \mathbf a \mathbf a \mathbf a</math> |<math>q^4</math> |} 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 theorem|multinomial expansion]] of <math>(p_1 + \cdots + p_n)^c</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 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 relationship|dominance]]. Assuming that the Hardy–Weinberg principle applies to the population, then <math>q</math> can still be calculated from ''f''('''aa'''): :<math>q = \sqrt {f(aa)}</math> and <math>p</math> can be calculated from <math>q</math>. And thus an estimate of ''f''('''AA''') and ''f''('''Aa''') derived from <math>p^2</math> and <math>2pq</math> respectively. Note however, such a population cannot be tested for equilibrium using the significance tests below because it is assumed ''[[A priori and a posteriori (philosophy)|a priori]]''. == Significance tests for deviation == Testing deviation from the HWP is generally performed using [[Karl Pearson|Pearson's]] [[Pearson's chi-square test|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 [[Ronald Fisher|Fisher's]] [[Fisher's exact test|exact test]], which requires a [[computer]] to solve. More recently a number of [[Markov chain Monte Carlo|MCMC]] methods of testing for deviations from HWP have been proposed (Guo & Thompson, 1992; Wigginton ''et al'' 2005) === Example <math> \chi^2</math> test for deviation === These data are from [[E.B. Ford]] (1971) on the [[Scarlet tiger moth]], for which the [[phenotype]]s 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. {| class="wikitable" |+ Table 3: Example Hardy–Weinberg principle calculation |- !Genotype |White-spotted ('''AA''') |Intermediate ('''Aa''') |Little spotting ('''aa''') !Total |- !Number |1469 |138 |5 |1612 |} From which allele frequencies can be calculated: :{| |<math>p</math> |<math>= {2 \times \mathrm{obs}(AA) + \mathrm{obs}(Aa) \over 2 \times (\mathrm{obs}(AA) + \mathrm{obs}(Aa) + \mathrm{obs}(aa))}</math> |- | |<math>= {1469 \times 2 + 138 \over 2 \times (1469+138+5)}</math> |- | |<math>= { 3076 \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 [[expected value|expectation]] is: :{| |<math>\mathrm{Exp}(AA) = p^2n = 0.954^2 \times 1612 = 1467.4</math> |- |} :{| |<math>\mathrm{Exp}(Aa) = 2pqn = 2 \times 0.954 \times 0.046 \times 1612 = 141.2</math> |- |} :{| |<math>\mathrm{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 [[degrees of freedom (statistics)|degree of freedom]] (degrees of freedom for test for Hardy-Weinberg proportions are # phenotypes - # alleles). 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. ===Fisher's exact test (probability test)=== [[Fisher's exact test]] can be applied to testing for Hardy-Weinberg proportions. Because 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 :<math>prob[n_{12} | n_1] = \frac{{{n}\choose{n_{11}, n_{12}, n_{22}}}} {{{2n}\choose{n_1}}} 2^{n_{12}}, </math> where n<sub>11</sub>, n<sub>12</sub>, n<sub>22</sub> are the observed numbers of the three genotypes, '''AA''', '''Aa''', and '''aa''', respectively, and n<sub>1</sub> is the number of '''A''' alleles, where <math>n_1 = 2 n_{11} + n_{12}</math>. '''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. {| class="wikitable" |+ Table 4: Example of Fisher's Exact Test for ''n''=100, ''p''=0.34.<ref>Example data from Emigh(1980).</ref> |- !Number of Heterozygotes !Significance Level |- |0 |0.000 |- |2 |0.000 |- |4 |0.000 |- |6 |0.000 |- |8 |0.000 |- |10 |0.000 |- |12 |0.000 |- |14 |0.000 |- |16 |0.000 |- |18 |0.001 |- |20 |0.007 |- |22 |0.034 |- |34 |0.067 |- |24 |0.151 |- |32 |0.291 |- |26 |0.474 |- |30 |0.730 |- |28 |1.000 |} 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''. ==Analysis Software== *[http://www.goldenhelix.com/SNP_Variation/HelixTree/hardy_weinberg_equilibrium.html HelixTree] ==Inbreeding coefficient== The inbreeding coefficient, ''F'' (see also [[F-statistics|''F''-statistics]]), is one minus the observed frequency of heterozygotes over that expected from Hardy–Weinberg equilibrium. :<math> 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}))} , \!</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 = 1 - {138 \over 141.2}</math> ::<math>= 0.023.\,</math> For two alleles, the chi square goodness of fit test for Hardy-Weinberg proportions is equivalent to the test for inbreeding, ''F'' = 0. == History == <!-- Deleted image removed: [[Image:Wilhelm Weinberg.jpg|thumb|right|Wilhelm Weinberg|{{deletable image-caption|1=Saturday, 8 March 2008}}]] --> [[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 characteristics. [[Udny Yule]] (1902) argued against Mendelism because he thought that dominant alleles would increase in the population. The [[United States|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''&nbsp;=&nbsp;''q''&nbsp;=&nbsp;0.5. [[Reginald Punnett]], unable to counter Yule's point, introduced the problem to [[Godfrey Harold Hardy|G. H. Hardy]], a [[United Kingdom|British]] [[mathematician]], with whom he played [[cricket]]. Hardy was a [[pure mathematics|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)<sup>2</sup>:2(p+q)(q+r):(q+r)<sup>2</sup>, or as p<sub>1</sub>:2q<sub>1</sub>:r<sub>1</sub>, 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<sup>2</sup> = pr. And since q<sub>1</sub><sup>2</sup> = p<sub>1</sub>r<sub>1</sub>, 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 language|English]]-speaking world until [[Curt Stern]] ([[1943]]) pointed out that it had first been formulated independently in [[1908]] by the [[Germany|German]] [[physician]] [[Wilhelm Weinberg]] (see Crow 1999). Others have tried to associate [[William E. Castle|Castle's]] name with the Law because of his work in 1903, but it is only rarely seen as the Hardy-Weinberg-Castle Law. </br></br> ==Graphical representation== It is possible to represent the distribution of genotype frequencies for a bi-allelic locus within a population graphically using a [[de Finetti diagram]]. This uses a triangular plot (also known as trilinear, triaxial or [[ternary plot]]) to represent the distribution of the three genotype frequencies in relation to each other. Although it differs from many other such plots in that the direction of one of the axes has been reversed. [[Image:De finetti diagram.png]] The curved line in the above diagram is the Hardy-Weinberg parabola and represents the state where alleles are in Hardy-Weinberg equilibrium. It is possible to represent the effects of [[Natural Selection]] and its effect on allele frequency on such graphs (e.g. Ineichen & Batschelet 1975) The [[De Finetti diagram]] has been developed and used extensively by [[A.W.F. Edwards]] in his book ''Foundations of Mathematical Genetics''. ==References and notes== ===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 (journal)|Genetics]]'' '''152''': 821-825. [http://www.genetics.org/cgi/content/full/152/3/821 link] *Edwards, A.W.F. 1977. ''Foundations of Mathematical Genetics.'' Cambridge University Press, Cambridge (2nd ed., 2000). ISBN 0-521-77544-2 * Emigh, T.H. (1980). A comparison of tests for Hardy-Weinberg equilibrium. ''Biometrics'' '''36''': 627 &ndash; 642. * Ford, E.B. (1971). ''Ecological Genetics'', London. * Guo, S.W., Thompson, E.A. (1992). Performing the exact test of Hardy-Weinberg proportion for multiple alleles. ''Biometrics'' '''48''': 361 &ndash; 372. * Hardy, G. H. ([[1908]]). "Mendelian proportions in a mixed population". ''[[Science (journal)|Science]]'' '''28''': 49 &ndash; 50. [http://www.esp.org/foundations/genetics/classical/hardy.pdf ESP copy] * Ineichen,R., Batschelet, E. (1975) Genetic selection and de Finetti diagrams. [http://www.springerlink.com/content/m213l5v11nmp025v/ ''Journal of Mathematical Biology'' '''2''' 33 &ndash;39] * Pearson, K. ([[1903]]). 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 law". ''[[Science (journal)|Science]]'' '''97''': 137–138. [http://links.jstor.org/sici?sici=0036-8075%2819430205%293%3A97%3A2510%3C137%3ATHL%3E2.0.CO%3B2-8 JSTOR stable url] * Weinberg, W. ([[1908]]). "Über den Nachweis der Vererbung beim Menschen". ''Jahreshefte des Vereins für vaterländische Naturkunde in Württemberg'' '''64''': 368–382. * Wigginton, J.E., Cutler, D.J., Agbecasis, G.R. (2005). A note on exact tests of Hardy-Weinberg equilibrium. ''American Journal of Human Genetics'' '''76''': 887 &ndash; 893. * Yule, G. U. ([[1902]]). Mendel's laws and their probable relation to intra-racial heredity. ''New Phytol.'' '''1''': 193–207, 222–238. ===Notes=== <references /> == External links == *[http://www.changbioscience.com/genetics/hardy.html Hardy-Weinberg Equilibrium Calculator] *[http://www.jmp.com/software/genomics/ Hardy-Weinberg Equilibrium Analysis Software: JMP Genomics] *[http://www.goldenhelix.com/SNP_Variation/HelixTree/hardy_weinberg_equilibrium.html Hardy-Weinberg Equilibrium Analysis Software: HelixTree] *[http://www.radford.edu/~rsheehy/Gen_flash/popgen Population Genetics Simulator] * [http://www.stat.washington.edu/thompson/Genepi/Hardy.shtml HARDY (C) implementation of Guo & Thompson 1992] * [http://www.sph.umich.edu/csg/abecasis/Exact/ Source code (C/C++/Fortran/R) for Wigginton ''et al'' 2005] * [http://finetti.meb.uni-bonn.de Online De Finetti Diagram Generator and Hardy-Weinberg equilibrium tests] * [http://ihg.gsf.de/cgi-bin/hw/hwa1.pl Online Hardy-Weinberg equilibrium tests and drawing of de Finetti diagrams] {{popgen}} [[Category:Population genetics]] [[Category:classical genetics]] [[Category:Statistical genetics]] [[bg:Закон на Харди-Вайнберг]] [[ca:Equilibri de Hardy-Weinberg]] [[de:Hardy-Weinberg-Gleichgewicht]] [[es:Ley de Hardy-Weinberg]] [[fr:Principe de Hardy-Weinberg]] [[lt:Hardžio-Veinbergo dėsnis]] [[hu:Hardy–Weinberg-törvény]] [[nl:Wet van Hardy-Weinberg]] [[ja:ハーディー・ワインベルクの法則]] [[pl:Prawo Hardy'ego-Weinberga]] [[pt:Equilíbrio de Hardy-Weinberg]] [[ru:Закон Харди — Вайнберга]] [[sr:Генетичка равнотежа популације]] [[fi:Hardyn–Weinbergin laki]] [[sv:Hardy-Weinbergs lag]] [[tr:Hardy-Weinberg Kuralı]] [[uk:Закон Гарді-Вайнберга]] [[zh:哈代-温伯格定律]]