Hardy-Weinberg Equilibrium

The Hardy-Weinberg equilibrium is the fundamental concept in population genetics (the study of genetics in a defined group). It is a mathematical equation describing the distribution and expression of alleles (forms of a gene) in a population, and it expresses the conditions under which allele frequencies are expected to change.

Mendelian genetics demonstrated that the phenotypic (observable) ex-pression of some traits is based on a simple dominant-recessive relationship between the alleles coding for the trait. In Mendel’s original work for in-stance, green pea pods were dominant to yellow pods, meaning that a heterozygote (an individual with one allele for green and one for yellow) would show the green trait. (A common misunderstanding is that a dominant allele should also be common. This is not the case. Frequency of an allele in a population is independent of its dominance or recessiveness. Either type of allele may be common or rare.

Allele Frequencies

A significant question in population genetics, therefore, is determining the frequency of the dominant and recessive alleles in a population (for exam­ple, the frequency of blood type O allele in the United States), given the frequency of the phenotypes. Note that phenotypic and allelic frequencies are related but are not equal. Heterozygotes show the dominant phenotype, but carry a recessive allele. Therefore, the frequency for the recessive allele is higher than the frequency of the recessive phenotype.

Early in the twentieth century mathematician Godfrey Hardy and physi­cian Wilhelm Weinberg independently developed a model describing the relationship between the frequency of the dominant and recessive alleles (hereafter, p and q) in a population. They reasoned that the combined fre­quencies of p and q must equal 1, since together they represent all the alle­les for that trait in the population:

p + q = ¹

Hardy and Weinberg represented random mating in the population as the product (p + q)(p + q),which can be expanded to p² + 2pq + q². This corresponds to the biological fact that, as a result of mating, some new in­dividuals have two p alleles, someone p and one q, and some two q alleles. P² then represents the fraction of the population that is homozygous dom­inant while 2pq and q² represent the heterozygous and homozygous reces­sive fractions, respectively.

Mathematically, since p + q = 1, (p + q)² must also equal 1, and so:

p² + 2pq + q² = 1

The usefulness of this final form is that q², the fraction of the popula­tion that is homozygous recessive, can be determined with relative ease, and from that value all of the other frequencies can be calculated. For instance, if 1 percent of the population is found to be homozygous recessive, q2 = 0.01, then q = 0.1, p = 0.9, p² = 0.81, and 2pq = 0.09.

One value of the Hardy-Weinberg equilibrium equation is that it al­lows population geneticists to determine the proportion of each genotype and phenotype in a population. This may be useful for genetic counseling in the case of a genetic disease, for example, or for measuring the genetic diversity in a population of endangered animals.

Implications for Evolution

A significant implication of the Hardy-Weinberg relationship is that the fre­quency of the dominant and recessive alleles will remain unchanged from one generation to the next, given certain conditions. These conditions are:

(1)a sufficiently large population to eliminate change due to chance alone

(2)random mating (the phenotypic trait being examined cannot play a role in mate selection); (3) no migration of individuals either into or out of the population under study; (4) the genes under consideration are not subject to mutational change; and (5) the dominant or recessive phenotype must not have an adaptive advantage; in other words natural selection must not be favoring one trait over another.

If any of these constraints are not satisfied then the Hardy-Weinberg equilibrium does not hold true. When a population geneticist finds a change in allele frequency over time, therefore, he or she may be confident that one or more of these factors is at work. In fact, one definition of evolution is a change in allele frequencies over time.

J. B. S. Haldane was the first person to adapt the Hardy-Weinberg re­lationship to model evolutionary change. He introduced a selection coeffi­cient to represent a disadvantage for the homozygous recessive. His equation was later shown to successfully model the impact of industrial pollution on peppered moths in England.

Reference

Gillespie, John H. Population Genetics: A Concise Guide. Baltimore, MD: Johns Hop­kins University Press, 1998.

Kingsland, Sharon E. Modeling Nature: Episodes in the History of Population Ecology. Chicago: University of Chicago Press, 1995.

Pianka, Eric R. Evolutionary Ecology. San Francisco, CA: Benjamin Cummings, 2000.

Stearns, Stephen C., and Rolf F. Hoekstra. Evolution: An Introduction. New York: Ox­ford University Press, 2000.

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