# 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) expression of some traits is based on a simple dominant-recessive relationship between the alleles coding for the trait. In Mendel's original work for instance, 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:

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

Mathematically, since
*
p + q
*
1
*
, (p + q)
*
^{
2
}
must also equal 1, and so:

The usefulness of this final form is that
*
q
*
^{
2
}
*
,
*
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,
*
q
*
2
*
0.01,
*
then
*
q
*
0.1
*
, p 0.9, p
*
^{
2
}
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.

**
SEE ALSO
**
Adaptation
;
Evolution
;
Genetic Diseases
;
Natural Selection
;
Population Genetics

*
William P. Wall
*

## Bibliography

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.

## DOBZHANSKY, THEODOSIUS (1900–1975)

Dobzhansky is a Ukrainian-born U.S. biologist and author who showed that ongoing change in gene frequencies in natural populations was the rule, not the exception. He also showed that individuals with two different versions of the same gene ("heterozygotes") could be better adapted than individuals with identical copies of a gene ("homozygotes").

2pq = 0.18 in this example. 2 * 0.1 * 0.9 = 2 * 0.09 = 0.18

or: 1-(0.81 + 0.01) = 0.18