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Equilibri de Hardy-Weinberg - Viquip??dia

Equilibri de Hardy-Weinberg

De Viquip??dia

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???Principi Hardy-Weinberg per a dos al??lels: l'eix horitzontal mostra la dues freq????ncies d'al??lels p i q, l'eix vertical mostra les freq????ncies de genotips i els tres genotips possibles s??n representats per les diferents linies.
???Principi Hardy-Weinberg per a dos al??lels: l'eix horitzontal mostra la dues freq????ncies d'al??lels p i q, l'eix vertical mostra les freq????ncies de genotips i els tres genotips possibles s??n representats per les diferents linies.

???En gen??tica de poblacions, el principi Hardy-Weinberg manifesta que les freq????ncies de genotip en una poblaci?? romanen constants o s??n en equilibri de generaci?? en generaci?? llevat que s'introdueixin influ??ncies pertorbadores espec??fiques. Aquelles influ??ncies pertorbadores inclouen l'aparellament no aleatori, mutacions, selecci?? natural, mida de poblaci?? limitada i constant en el temps, abs??ncia de deriva gen??tica. L'equilibri gen??tic ??s un principi b??sic de gen??tica de poblacions.

???El principi Hardy-Weinberg ??s com un quadrat Punnett per a poblacions, en comptes d'individus. Un quadrat Punnett pot pronosticar la probabilitat del genotip del descendent basat en el genotip dels pares o el genotip de la descend??ncia es pot utilitzar per revelar el genotip dels pares. De la mateixa manera, el principi de Hardy-Weinberg es pot utilitzar per comptar la freq????ncia d'al??lels particulars basats en freq????ncia d'una malaltia recessiva d'autos??mica. ???El pare vol saber la probabilitat dels seus n??ts que hereta la malaltia. Per contestar a aquesta q??esti??, el conseller gen??tic ha de saber la possibilitat que el nen es reproduir?? amb un transportista de la mutaci?? recessiva. Aquest fet pot no saber-se, per?? se sap freq????ncia de malaltia. Sabem que la malaltia sigui provocada pel genotip recessiu homozigot; podem utilitzar el principi Hardy-Weinberg per treballar endarrere de l'ocurr??ncia de malaltia a la freq????ncia d'individus recessius heterozigot.

???Aquest concepte tamb?? s'anomena de les seg??ents formes: HWP, equilibri Hardy-Weinberg, HWE, o llei de Hardy-Weinberg. Reb?? el nom de G. H. Hardy i Wilhelm Weinberg.

Taula de continguts

[edita] Derivaci??

???Una descripci?? millor, per?? equivalent, probabil??stica per l'equilibri de Hardy Weinberg ??s que els al??lels per a la pr??xima generaci?? per a qualsevol individu donat s'escullen de forma aleatoria i independent l'un de l'altre succ??s. Considereu dos al??lels, A i a, amb freq????ncies p i q respectivament, en la poblaci??. Les diferents maneres de formar genotips nous es poden obtenir utilitzant una quadrat Punnett o Quadrat Prout), on la fracci?? en cada un ??s igual al producte de les probabilitats de files i columnes.

Taula 1: Quadrat Punnet per a l'equilibri de Hardy???Weinberg.
Femelles
A (p) a (q)
Mascles A (p) AA (p??) Aa (pq)
a (q) Aa (pq) aa (q??)

Els tres possibles genotips s'obtenen de la seg??ent manera:

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

???Aquestes freq????ncies s'anomenen freq????ncies de Hardy-Weinberg (o proporcions Hardy-Weinberg). Aix?? s'aconsegueix en una generaci?? i nom??s exigeix la suposici?? de l'aparellament aleatori amb una mida de poblaci?? infinita.

A vegades, una poblaci?? es crea reunint mascles i femelles amb freq????ncies d'al??lels diferents. En aquest cas, la suposici?? d'una poblaci?? ??nica es viola fins que despr??s de la primera generaci??, aix?? la primera generaci?? no es trobar?? en equilibri de Hardy-Weinberg. Les successives generacions es trobaran en equilibri Hardy-Weinberg.

[edita] 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. 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 homozygosity for all genes.
    • Assortative mating, which causes an increase in 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.

  • 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).

[edita] Sex linkage

Where the A gene is 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 p and q for the heterogametic sex but p2, 2pq and q2 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 q2 = 0.007), 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.

[edita] Generalitzacions

???La seg??ent derivaci?? simple pot ser generalitzada per a m??s de dos al??lels i en poliploidia.

[edita] Generalitzacions per m??s de dos al??lels

Considereu un al??lel addicional amb freq????ncia, r. Els en el cas de dos al??lels ??s l'expansi?? binomial de (p + q)2 i alheshores en el cas de tres al??lels l'expansi?? del trinomi (p + q + r)2.

(p + q + r)2 = p2 + r2 + q2 + 2pq + 2pr + 2qr

De manera general es poden considerar els al??lels A1, ... Ai determinats per les freq????ncies al??l??liques p1 a pi;

(p_1 + \cdots + p_i)^2

obtenint per a tots els homozigots:

f(A_i A_i) = p_i^2

i per tots els heterozigots:

f(AiAj) = 2pipj

[edita] Generalitzacions per a poliploidies

???El principi Hardy-Weinberg tamb?? es pot generalitzar a sistemes poliploides, ??s a dir, per a organismes que tenen m??s de dues c??pies de cada cromosoma. Consideri una altra vegada nom??s dos al??lels. El cas diploide ??s l'expansi?? binomial de:

(p + q)2

???i per aix?? el cas poliploide ??s l'expansi?? polin??mica de:

(p + q)c

???on c ??s la ploidia, per exemple per a tetraploides (c = 4):

Table 2: Expected genotype frequencies for tetraploidy
Genotip Frequ??ncia
\mathbf A \mathbf A \mathbf A \mathbf A p4
\mathbf A \mathbf A \mathbf A \mathbf a 4p3q
\mathbf A \mathbf A \mathbf a \mathbf a 6p2q2
\mathbf A \mathbf a \mathbf a \mathbf a 4pq3
\mathbf a \mathbf a \mathbf a \mathbf a q4

???Depenent de si l'organisme ??s un "veritable" tetraploide o un amfidiploide determinar?? quant temps portar?? que la poblaci?? arribi a equilibri Hardy-Weinberg.

[edita] Complete generalization

La f??rmula totalment expandida coma multinomi 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}

[edita] 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.

[edita] 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 p2 and 2pq respectively. Note however, such a population cannot be tested for equilibrium using the significance tests below because it is assumed a priori.

[edita] 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. More recently a number of MCMC methods of testing for deviations from HWP have been proposed (Guo & Thompson, 1992; Wigginton et al 2005)

[edita] Example ??2 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.

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:

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

and

q = 1 ??? p
= 1 ??? 0.954
= 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 = 3.4

Pearson's chi-square test states:

??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

There is 1 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 ???? value is less than this, the null hypothesis that the population is in Hardy???Weinberg frequencies is not rejected.

[edita] 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

prob[n_{12} | n_1] = \frac{{{n}\choose{n_{11}, n_{12}, n_{22}}}} {{{2n}\choose{n_1}}} 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 n1 = 2n11 + n12.

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.

Table 4: Example of Fisher's Exact Test for n=100, p=0.34.[1]
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.

[edita] Analysis Software

[edita] 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.

[edita] Historia

La gen??tica mendeliana fou redescoberta el 1900. Tot i aix??, ???romania una mica controvertit durant uns quants anys com no se sabia llavors com podria provocar caracter??stiques de variaci?? continua. Udny Yule (1902) parlava en contra del mendelisme]] perqu?? pensava que els al??lels dominants augmentarien en la poblaci??. L'americ?? William E. Castle (1903) postul?? que sense selecci??, les freq????ncies de genotip romandrien estables. Karl Pearson (1903) trob?? la posici?? d'equilibri a un amb valors de p = q = 0,5. Reginald Punnett, incapa?? de contrarestar el punt de Yule, present?? el problema a G. H. Hardy, un matem??tic brit??nic, amb qui jugava a criquet. Hardy era un matem??tic pur i tenia menystenia la matem??tica aplicada; el seu punt de vista matem??tic sobre passa al seu article de 1908 on descriu aix?? com a "molt simple":

???A l'Editor de Science: Em reca immisquir-me en una discussi?? pel que fa a q??estions de la qual no tinc cap mena de coneixements experts i m'hauria d'haver esperat el punt molt simple que desitjo familiaritzar als bi??legs. Tanmateix, alguns comentaris del sr. Udny Yule, a qui el Sr. R. C. Punnett han cridat la meva atenci??, suggerir que encara pot ser valor que fa...
???Suposi que Aa siguin un parell de car??cters mendelians, sent A dominant, i all?? en qualsevol generaci?? donada el nombre de dominants purs (AA), heterozigots (Aa) i recessius purs (aa) s??n com P:2Q:R. Finalment, suposi que els nombres siguin bastant grans, de manera que l'aparellament es pugui veure com fortu??t, que els sexes es distribueixen uniformement entre les tres varietats i que tots s??n igualment f??rtils. Una mica de matem??tiques del tipus de taula de multiplicaci?? s??n prou per mostrar que en la pr??xima generaci?? els nombres seran com (p+q)2:2(p+q)(q+r):(q+r)2 o com p1:2q1:r1.
???La q??esti?? interessant ??s - en quines circumst??ncies aquesta distribuci?? ser?? la mateixa que en la generaci?? anterior? ??s f??cil de veure que la condici?? per aix?? sigui q2 = p??r. I des de q12 = p1r1, per qualsevol valor de P, Q i R; la distribuci?? continuar?? en qualsevol cas inalterat despr??s de la segona generaci??.

El principi es coneixia com la llei de Hardy al m??n de parla anglesa fins a Curt Stern (1943) assenyal?? que s'havia formulat independentment anteriorment el 1908 pel metge alemany Wilhelm Weinberg (vegeu, Corb 1999). Altres han intentat associar el nom de Willian E. Castel amb la llei a causa del seu treball del 1903, per?? rarament s'anomena Llei de Hardy-Weinberg-Castle.

[edita] 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.

Imatge: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.

[edita] Refer??ncies i notes

[edita] Bibliografia

  • Yule, G. U. (1902). Mendel's laws and their probable relation to intra-racial heredity. New Phytol. 1: 193???207, 222???238.
  • 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.
  • 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.
  • Hardy, G. H. (1908). Mendelian proportions in a mixed population. Science 28: 49 - 50. ESP copy
  • Weinberg, W. (1908). ??ber den Nachweis der Vererbung beim Menschen. Jahreshefte des Vereins f??r vaterl??ndische Naturkunde in W??rttemberg 64: 368???382.
  • Stern, C. (1943). The Hardy???Weinberg law. Science 97: 137???138. JSTOR stable url
  • Ford, E.B. (1971). Ecological Genetics, London.
  • Ineichen,R., Batschelet, E. (1975) Genetic selection and de Finetti diagrams. Journal of Mathematical Biology 2 33 - 39
  • Edwards, A.W.F. (1977). Foundations of Mathematical Genetics. Cambridge University Press, Cambridge (2a ed., 2000). ISBN 0-521-77544-2
  • Emigh, T.H. (1980). A comparison of tests for Hardy-Weinberg equilibrium. Biometrics 36: 627 ??? 642.
  • Guo, S.W., Thompson, E.A. (1992). Performing the exact test of Hardy-Weinberg proportion for multiple alleles. Biometrics 48: 361 ??? 372.
  • Crow, J.F. (1999). Hardy, Weinberg and language impediments. Genetics 152: 821-825. link
  • 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: 887-893.

[edita] External links

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