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# Hardy-Weinberg Equilibrium

The Hardy-Weinberg equilibrium is the statement that allele frequencies in a population remain constant over time, in the absence of forces to change them. Its name derives from Godfrey Hardy, an English mathematician, and Wilhelm Weinberg, a German physician, who independently formulated it in the early twentieth century. The statement and the set of assumptions and mathematical tools that accompany it are used by population geneticists to analyze the occurrence of, and reasons for, changes in allele frequency. Evolution in a population is often defined as a change in allele frequency over time. The Hardy-Weinberg equilibrium, therefore, can be used to test whether evolution is occurring in populations.

## Basic Concepts

A population is a set of interbreeding individuals all belonging to the same species. In most sexually reproducing species, including humans, each organism contains two copies of virtually every geneone inherited from each parent. Any particular gene may occur in slightly different forms, called alleles. An organism with two identical alleles is called homozygous for that gene, and one with two different alleles is called heterozygous. During the formation of gametes , the two alleles separate into different gametes. Mating unites egg and sperm, so that the offspring obtains two alleles for each gene.

The two alleles for a gene typically have different effects on the phenotype, or characteristics, of the organism. For many genes, one allele will control the phenotype if it is present in either one or two copies; this allele, which is often represented by a single, uppercase letterB, for exampleis said to be dominant. The other allele will only exert a visible effect if the dominant allele is not present; it is said to be recessive and is often represented by a lowercase letterb, for example. The genotype of an organism specifies both alleles for a particular gene and is often symbolized by pairs of letters, such as BB, Bb, or bb, with each letter representing an allele.

It is important to understand that "dominant" does not mean an allele is more common in the populationlethal dominant alleles are very rare, for instance. Nor does dominant necessarily mean an allele will spread through the population. Likewise, "recessive" does not necessarily mean an allele will become less common. Indeed, the Hardy-Weinberg equilibrium shows conditions under which allele frequencies remain unaltered over generations.

## Assumptions of the Hardy-Weinberg Model

Before examining the mathematical model underlying the Hardy-Weinberg equilibrium, let us look at the assumptions under which it operates:

1. Organisms reproduce sexually.
2. Mating is random.
3. Population size is very large.
4. Migration in or out is negligible.
5. Mutation does not occur.
6. Natural selection does not act on the alleles under consideration.

While the list appears to be so restrictive that no population can meet its requirements, in fact many do, to a very good first approximation. Even more to the point, variation from the Hardy-Weinberg equilibrium tells a population geneticist that one or more of these assumptions is not being met, thereby providing a clue about the forces at work within the population. Perhaps surprisingly, populations need not be very big to meet the conditions abovepopulations with as few as one thousand to two thousand individuals can do so.

## Allele Frequencies Remain the Same Between Generations

Suppose we want to study the allele frequencies of the gene for coloration in a population of moths. The allele for the dark color pattern, B, is dominant to the allele for the light color pattern, b. In a certain population, the frequency of B is found to be 0.9, and that of b is 0.1 (we will see, below, how to determine these frequencies by studying the moths themselves). This means that 90 percent of all the alleles are B, and 10 percent are b.

The Hardy-Weinberg equilibrium states that, given the above conditions, allele frequencies will not change from one generation to the next. To show this is true, we need some algebra.

Random mating means each allele has an equal chance of being paired with each other allele. During random mating, the likelihood that a B allele from a mother will unite with a B allele from a father is given by

B × B = 0.9 × 0.9 = 0.81.

The genotype of this offspring will be BB.

Similarly, the likelihoods of other combinations:

B × b = 0.9 × 0.1 = 0.09 for genotype Bb ;

b × B = 0.1 × 0.9 = 0.09 for genotype Bb ; and

b × b = 0.1 × 0.1 = 0.01 for genotype bb.

Note that the two Bb genotypes are the same. Therefore the frequency of BB is 0.81, the frequency of Bb is 0.18, and the frequency of bb is 0.01. These add to 1, just as we would expect, since they represent all the members of the next generation.

Are the allele frequencies still 0.9 and 0.1? For simplicity, imagine we're looking at one hundred individuals, so that eighty-one are BB, eighteen are Bb, and one is bb. Since each individual has two alleles, there are 200 alleles in all.

The number of B alleles is given by (81 × 2) + (18 × 1) = 180.

The number of b alleles is given by (1 × 2) + (18 × 1) = 20.

By comparing 180 to 20, you can see the frequency of B is still 0.9 and that of b is still 0.1.

## Allele Frequencies Can Be Calculated from Phenotypes

This is all very interesting, but it requires knowing the allele frequencies in a population. The power of the Hardy-Weinberg equilibrium formulas is in their ability to allow us to determine these frequencies by simple observation of the population.

When we see an organism with the dominant phenotype, we do not know whether we are looking at a homozygote (BB ) or a heterozygote (Bb ). When we see one with the recessive phenotype, though, we know it has the genotype bb. If we determine the proportion of individuals in the population with the bb genotype, it is a simple step to calculate the frequency of b. Let's work backward through our example above. If we determine that 1 percent of the population is homozygous recessive,

bb = 0.01.

Since the frequency of bb organisms is the product b × b, we can take the square root of this number to get the frequency of b : and

b = (bb )1/2 = (0.01)1/2

b = 0.1.

Since B + b must equal 1 (assuming there are only two alleles), B must be 0.9. With these calculations in hand, we can predict what the frequency for each of the other genotypes should be: BB is 0.81, and Bb is 0.18.

## Departures from Equilibrium Indicate Evolutionary Forces at Work

With these simple tools, we can look at populations to see if they conform to these numerical patterns. If they differ, we seek the reasons for the difference in some violation of the Hardy-Weinberg assumptions. Two processes, natural selection and genetic drift, are the most common and important factors at work in most populations that are not at equilibrium.

For example, suppose we find a population in which the recessive allele frequency is declining over time. We might then investigate whether homozygous recessives are dying earlier. (Many genetic diseases, such as cystic fibrosis, are due to recessive alleles.) This could be due to natural selection, in which those that are better adapted to the environment survive longer and reproduce more frequently.

Or suppose we find a population in which there is a smaller-than-expected number of homozygotes of both types, and a larger number of heterozygotes. This could be due to heterozygote superioritywhere the heterozygote is more fit than either homozygote. In humans, this is the case for the allele causing sickle cell disease, a type of hemoglobinopathy.

Nonrandom mating is another potential source of departure from the Hardy-Weinberg equilibrium. Imagine that two alleles give rise to two very different appearances. Individuals may choose to mate with those whose appearance is closest to theirs. This may lead to divergence of the two groups over time into separate populations and perhaps ultimately separation into two species.

In very small populations, allele frequencies may change dramatically from one generation to the next, due to the vagaries of mate choice or other random events. For instance, half a dozen individuals with the dominant allele may, by chance, have fewer offspring than half a dozen with the recessive allele. This would have little effect in a population of one thousand, but it could have a dramatic effect in a population of twenty. Such changes are known as genetic drift.

see also Gene Flow; Genetic Drift; Inheritance Patterns; Mutation; Population Bottleneck; Population Genetics.

Richard Robinson

#### Bibliography

Hartl, D. L., and A. G. Clark. Principles of Population Genetics, 3rd ed. Sunderland, MA: Sinauer, 1997.

## CALCULATING ALLELE FREQUENCIES AND GENOTYPES FROM THE OBSERVED FREQUENCY OF HOMOZYGOUS RECESSIVES

B : dominant allele frequency.

b : recessive allele frequency.

b = (observed homozygote recessive frequency)1/2.

B = 1 b.

B × b = expected frequency of heterozygotes in the population.

B 2 = expected frequency of homozygous dominants in the population.

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# 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 + 2pq + 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 2pq 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.

## 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").

## 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.

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## Hardy–Weinberg law

Hardy–Weinberg law The law which states that in an infinitely large, interbreeding population, in which mating is random and in which there is no selection, migration, or mutation, gene and genotype frequencies will remain the same between the sexes and constant from generation to generation, with no overlap between generations. In practice these conditions are rarely strictly present, but unless any departure is a marked one, there is no statistically significant movement away from equilibrium. Consider a single pair of alleles, A and a, present in a diploid population with frequencies of p and q respectively. Three genotypes are possible, AA, Aa, and aa, and these will be present with frequencies of p2, 2pq, and q2 respectively. The law was established in 1908 by G. H. Hardy and W. Weinberg.

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## Hardy–Weinberg law

Hardy–Weinberg law In an infinitely large, interbreeding population in which mating is random and in which there is no selection, migration, or mutation, gene and genotype frequencies will remain constant from generation to generation. In practice these conditions are rarely strictly present, but unless any departure is a marked one, there is no statistically significant movement away from equilibrium. Consider a single pair of alleles, A and a, present in a diploid population with frequencies of p and q respectively. Three genotypes are possible, AA, Aa, and aa, and these will be present with frequencies of p2, 2pq, and q2 respectively. The law was established in 1908 by G. H. Hardy and W. Weinberg.

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## Hardy–Weinberg law

Hardy–Weinberg law The law that states that in an infinitely large, interbreeding population in which mating is random and in which there is no selection, migration, or mutation, gene and genotype frequencies will remain constant from generation to generation. In practice these conditions are rarely strictly present, but unless any departure is a marked one, there is no statistically significant movement away from equilibrium. Consider a single pair of alleles, A and a, present in a diploid population with frequencies of p and q respectively. Three genotypes are possible, AA, Aa, and aa, and these will be present with frequencies of p2, 2pq, and q2 respectively.

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