Hardy-Weinberg Equilibrium

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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 gene—one 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 letter—B, for example—is 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 letter—b, 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 population—lethal 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 above—populations 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 superiority—where 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.

Bibliography

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

—Richard Robinson