### Hardy-Weinberg Theorem

 Hardy-Weinberg Theorem

The Hardy-Weinberg theorem is the principal that, in the absence of external pressures for change, the genetic makeup of an ideal population of randomly mating, sexually reproducing diploid organisms will remain the same, at what is called Hardy-Weinberg equilibrium.

Population genetics is the branch of genetics that studies the behavior of genes in populations. The two main subfields of population genetics are theoretical (or mathematical) population genetics, which uses formal analysis of the properties of ideal populations, and experimental population genetics,which examines the behavior of real genes in natural or laboratory populations.

Population genetics began as an attempt to extend Gregor Mendel’s laws of inheritance to populations. In 1908 Godfrey H. Hardy, an English mathematician, and Wilhelm Weinberg, a German physician, each independently derived a description of the behavior of allele and genotype frequencies in an ideal population of randomly mating, sexually reproducing diploid organisms.

Their results, now termed the Hardy-Weinberg theorem, showed that the pattern of allele and genotype frequencies in such a population followed simple rules. They also showed that, in the absence of external pressures for change, the genetic makeup of a population will remain at an equilibrium.

Because evolution is change in a population over time, such a population is not evolving. Modern evolutionary theory is an outgrowth of the “New Synthesis” of R. A. Fisher, J. B. S. Haldane, and Sewell Wright, which was developed in the 1930’s.

They examined the significance of various factors that cause evolution by examining the degree to which they cause deviations from the predictions of the Hardy-Weinberg theorem.

Predictions

The predictions of the Hardy-Weinberg theorem hold if the following assumptions are true:
1. The population is infinitely large.
2. There is no gene flow (movement of genes into or out of the population by migration of gametes or individuals).
3. There is no mutation (no new alleles are added to the population by mutation).
4. There is random mating (all genotypes have an equal chance of mating with all other genotypes).
5. All genotypes are equally fit (have an equal chance of surviving to reproduce).

Under this very restricted set of assumptions, the following two predictions are true:
1. Allele frequencies will not change from one generation to the next.
2. Genotype frequencies can be determined by a simple equation and will not change from one generation to the next.

The predictions of the Hardy-Weinberg theorem represent the working through of a simple set of algebraic equations and can be easily extended to more than two alleles of a gene. In fact, the results were so self-evident to the mathematician Hardy that he, at first, did not think the work was worth publishing.

If there are two alleles (A, a) for a gene present in the gene pool (all of the genes in all of the individuals of a population), let p = the frequency of the A allele andq= the frequency of the a allele.

As an example, if p = 0.4 (40 percent) and q = 0.6 (60 percent), then p + q = 1, since the two alleles are the only ones present, and the sum of the frequencies (or proportions) of all the alleles in a gene pool must equal one (or 100 percent).

The Hardy-Weinberg theorem states that at equilibrium the frequency of AA individuals will be p2 (equal to 0.16 in this example), the frequency of Aa individuals will be 2pq, or 0.48, and the frequency of aa individuals will be q2, or 0.36.

The basis of this equilibriumis that the individuals of one generation give rise to the next generation. Each diploid individual produces haploid gametes. An individual of genotype AA can make only a single type of gamete, carrying the A allele. Similarly, an individual of genotype aa can make only a gametes. An Aa individual, however, can make two types of gametes, A and a, with equal probability.

Each individual makes an equal contribution of gametes, as all individuals are equally fit, and there is random mating. Each AA individual will contribute twice as many A gametes as each Aa individual. The frequency of A gametes is equal to the frequency of A alleles in the gene pool of the parents.

The next generation is formed by gametes pairing at random (independent of the allele they carry). The likelihood of an egg joining with a sperm is the frequency of one multiplied by the frequency of the other.

AA individuals are formed when an A sperm joins an A egg; the likelihood of this occurrence is
p × p = p2
(that is, 0.4 × 0.4 = 0.16 in the first example). In the same fashion, the likelihood of forming an aa individual is
q2 = 0.36. The likelihood of an A egg joining an a sperm is pq, as is the likelihood of an a egg joining an A sperm; therefore, the total likelihood of forming an Aa individual is 2pq = 0.48.

If one now calculates the allele frequencies (and hence the frequencies of the gamete types) for this generation, they are the same as before: The frequency of the Aallele is
p = (2p2 + 2pq)/2
(in the example (0.32 + 0.48) ÷2 = 0.4), and the frequency of the a allele is q = (1– p) = 0.6.
The population remains at equilibrium, and neither allele nor genotype frequencies change from one generation to the next.

The Real World

The Hardy-Weinberg theorem is a mathematical model of the behavior of ideal organisms in an ideal world. The real world, however, does not approximate these conditions very well. It is important to examine each of the five assumptions made in the model to understand their consequences and how closely they approximate the real world.

The first assumption is infinitely large population size, which can never be true in the real world, as all real populations are finite. In a small population, chance effects on mating success over many generations can alter allele frequencies. This effect is called genetic drift.

If the number of breeding adults is small enough, some genotypes will not get a chance to mate with one another, even if mate choice does not depend on genotype. As a result, the genotype ratios of the off spring would be different from those of the parents. In this case, however, the gene pool of the next generation is determined by those genotypes, and the change in allele frequencies is perpetuated.

If it goes on long enough, it is likely that some alleles will be lost from the population, because a rare allele has a greater chance of not being included. Once an allele is lost, it cannot be regained.

How long this process takes is a function of population size. In general, the number of generations it would take to lose an allele by drift is about equal to the number of individuals in the population.

Many natural populations are quite large (thousands of individuals), so that the effects of drift are not significant. Some populations, however, especially of endangered species, are very small: A number of plant species are so rare that they consist of a single population with less than one hundred individuals.

The second assumption is that there is no gene flow, or movement of genotypes into or out of the population. Individuals that leave a population do not contribute to the next generation.

If one genotype leaves more frequently than another, the allele frequencies will not equal those of the previous generation. If incoming individuals come from a population with different allele frequencies, they also alter the allele frequencies of the gene pool.

The third assumption concerns mutations. A mutation is a change in the deoxyribo nucleic acid (DNA) sequence of a gene—that is, the creation of a new allele. This process occurs in all natural populations, but new mutations for a particular gene occur in about 1 of 10,000 to 100,000 individuals per generation.

Therefore, mutations do not, in themselves, play much part in determining allele or genotype frequencies. Mutation, however, is the ultimate source of all new alleles and provides the variability on which evolution depends.

The fourth assumption is that there is random mating among all individuals. A common limitation on random mating in plants is inbreeding, the tendency to mate with a relative.

Because plants have a limited ability to move, and pollinators may not carry pollen very far, the plants in a population tend to mate with nearby individuals, which are often relatives. Such individuals tend to share alleles more often than the population at large.

The final assumption is that all genotypes are equally fit. Considerable debate has focused on the question of whether two alleles or genotypes are ever equally fit. Many alleles do confer differences in fitness; it is through these variations in fitness that natural selection operates. Newer techniques of molecular biology have revealed many differences in DNA sequences that appear to have no discernible effects on fitness.

As the cornerstone of population genetics, the Hardy-Weinberg theorem pervades evolutionary thinking. The advent of techniques to examine genetic variation in natural populations has been responsible for a great resurgence of interest in evolutionary questions. One can now test directly many of the central aspects of evolutionary theory.

In some cases, notably the discovery of the large amount of genetic variation in most natural populations, evolutionary biologists have been forced to reassess the significance of natural selection compared with other forces for evolutionary change.