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Hardy-weinberg principle of equilibrium

In the early twentieth century, English mathematician Godfrey Hardy and German physician Wilhelm Weinberg stated the principle of equilibrium to describe the genetic makeup of a population. The theory, which later became known as the Hardy-Weinberg principle of equilibrium, states that a population’s allele and genotype frequencies are inherently stable— unless some kind of evolutionary force is acting upon the population, neither the allele nor the genotypic frequencies would change. The Hardy-Weinberg principle assumes conditions with no mutations, migration, emigration, or selective pressure for or against genotype, plus an infinite population; while no population can satisfy those conditions, the principle offers a useful model against which to compare real population changes.

Working under this theory, population geneticists represent different alleles as different variables in their mathematical models. The variable p, for example, often represents the frequency of a particular allele, say Y for the trait of yellow in Mendel’s peas, while the variable q represents the frequency of y alleles that confer the color green. If these are the only two possible alleles for a given locus in the population, p + q = 1. In other words, all the p alleles and all the q alleles make up all of the alleles for that locus that are found in the population.

But what ultimately interests most biologists is not the frequencies of different alleles, but the frequencies of the resulting genotypes, known as the population’s genetic structure    , from which scientists can surmise the distribution of phenotypes. If the phenotype is observed, only the genotype of the homozygous recessive alleles can be known; the calculations provide an estimate of the remaining genotypes. Since each individual carries two alleles per gene, if the allele frequencies (p and q) are known, predicting the frequencies of these genotypes is a simple mathematical calculation to determine the probability of getting these genotypes if two alleles are drawn at random from the gene pool. So in the above scenario, an individual pea plant could be pp (YY), and thus produce yellow peas; pq (Yy), also yellow; or qq (yy), and thus producing green peas ( [link] ). In other words, the frequency of pp individuals is simply p 2 ; the frequency of pq individuals is 2pq; and the frequency of qq individuals is q 2 . And, again, if p and q are the only two possible alleles for a given trait in the population, these genotypes frequencies will sum to one: p 2 + 2pq + q 2 = 1.

Art connection

The Hardy-Weinberg principle is used to predict the genotypic distribution of offspring in a given population. In the example given, pea plants have two different alleles for pea color. The dominant capital Y allele results in yellow pea color, and the recessive small y allele results in green pea color. The distribution of individuals in a population of 500 is given. Of the 500 individuals, 245 are homozygous dominant (capital Y capital Y) and produce yellow peas. 210 are heterozygous (capital Y small y) and also produce yellow peas. 45 are homozygous recessive (small y small y) and produce green peas. The frequencies of homozygous dominant, heterozygous, and homozygous recessive individuals are 0.49, 0.42, and 0.09, respectively.  Each of the 500 individuals provides two alleles to the gene pool, or 1000 total. The 245 homozygous dominant individuals provide two capital Y alleles to the gene pool, or 490 total. The 210 heterozygous individuals provide 210 capital Y and 210 small y alleles to the gene pool. The 45 homozygous recessive individuals provide two small y alleles to the gene pool, or 90 total. The number of capital Y alleles is 490 from homozygous dominant individuals plus 210 from homozygous recessive individuals, or 700 total. The number of small y alleles is 210 from heterozygous individuals plus 90 from homozygous recessive individuals, or 300 total.  The allelic frequency is calculated by dividing the number of each allele by the total number of alleles in the gene pool. For the capital Y allele, the allelic frequency is 700 divided by 1000, or 0.7; this allelic frequency is called p. For the small y allele the allelic frequency is 300 divided by 1000, or 0.3; the allelic frequency is called q.  Hardy-Weinberg analysis is used to determine the genotypic frequency in the offspring. The Hardy-Wienberg equation is p-squared plus 2pq plus q-squared equals 1. For the population given, the frequency is 0.7-squared plus 2 times .7 times .3 plus .3-squared equals one. The value for p-squared, 0.49, is the predicted frequency of homozygous dominant (capital Y capital Y) individuals. The value for 2pq, 0.42, is the predicted frequency of heterozygous (capital Y small y) individuals. The value for q-squared, .09, is the predicted frequency of homozygous recessive individuals. Note that the predicted frequency of genotypes in the offspring is the same as the frequency of genotypes in the parent population. If all the genotypic frequencies, .49 plus .42 plus .09, are added together, the result is one
When populations are in the Hardy-Weinberg equilibrium, the allelic frequency is stable from generation to generation and the distribution of alleles can be determined from the Hardy-Weinberg equation. If the allelic frequency measured in the field differs from the predicted value, scientists can make inferences about what evolutionary forces are at play.

In plants, violet flower color (V) is dominant over white (v). If p = 0.8 and q = 0.2 in a population of 500 plants, how many individuals would you expect to be homozygous dominant (VV), heterozygous (Vv), and homozygous recessive (vv)? How many plants would you expect to have violet flowers, and how many would have white flowers?

In theory, if a population is at equilibrium—that is, there are no evolutionary forces acting upon it—generation after generation would have the same gene pool and genetic structure, and these equations would all hold true all of the time. Of course, even Hardy and Weinberg recognized that no natural population is immune to evolution. Populations in nature are constantly changing in genetic makeup due to drift, mutation, possibly migration, and selection. As a result, the only way to determine the exact distribution of phenotypes in a population is to go out and count them. But the Hardy-Weinberg principle gives scientists a mathematical baseline of a non-evolving population to which they can compare evolving populations and thereby infer what evolutionary forces might be at play. If the frequencies of alleles or genotypes deviate from the value expected from the Hardy-Weinberg equation, then the population is evolving.

Use this online calculator to determine the genetic structure of a population.

Section summary

The modern synthesis of evolutionary theory grew out of the cohesion of Darwin’s, Wallace’s, and Mendel’s thoughts on evolution and heredity, along with the more modern study of population genetics. It describes the evolution of populations and species, from small-scale changes among individuals to large-scale changes over paleontological time periods. To understand how organisms evolve, scientists can track populations’ allele frequencies over time. If they differ from generation to generation, scientists can conclude that the population is not in Hardy-Weinberg equilibrium, and is thus evolving.

Art connections

[link] In plants, violet flower color (V) is dominant over white (v). If p=.8 and q = 0.2 in a population of 500 plants, how many individuals would you expect to be homozygous dominant (VV), heterozygous (Vv), and homozygous recessive (vv)? How many plants would you expect to have violet flowers, and how many would have white flowers?

[link] The expected distribution is 320 VV, 160Vv, and 20 vv plants. Plants with VV or Vv genotypes would have violet flowers, and plants with the vv genotype would have white flowers, so a total of 480 plants would be expected to have violet flowers, and 20 plants would have white flowers.

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Source:  OpenStax, Biology. OpenStax CNX. Feb 29, 2016 Download for free at http://cnx.org/content/col11448/1.10
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