ADVERTISEMENTS:

In this article we will discuss about the Hardy-Weinberg law with its applications.

In 1908, the mathematician G. H. Hardy in England and the physician W. Weinberg in Germany independently developed a quantitative theory for defining the genetic structure of populations. The Hardy-Weinberg Law provides a basic algebraic formula for describing the expected frequencies of various genotypes in a population.

The similarity of their work however, remained unnoticed until Stern (1943) drew attention to both papers and recommended that names of both discoverers be attached to the population formula. The Law states that gene frequencies in a population remain constant from generation to generation if no evolutionary processes like migration, mutation, selection and drift are operating.

ADVERTISEMENTS:

Thus if matings are random, and no other factors disturb the reproductive abilities of any genotype, the equilibrium genotypic frequencies are given by the square of the allelic frequencies.

**If there are only two alleles A and a with frequencies p and q respectively, the frequencies of the three possible genotypes are:**

(p + q)^{2} =p^{2} + 2pq + q^{2 }

If there are 3 alleles say A1, A2 and A3 with frequencies p, q and r, the genotypic frequencies would be;

ADVERTISEMENTS:

(p + q + r)^{2} = p^{2} + q^{2} + r^{2} + 2pq + 2pr + 2qr

This square expansion can be used to obtain the equilibrium genotypic frequencies for any number of alleles.

It must also be noted that the sum of all the allelic frequencies, and of all the genotypic frequencies must always be 1. If there are only two alleles p and q, then p + q = 1, and therefore p^{2} + 2pq + q^{2} = (p + q)^{2} = 1. If there are 3 alleles with frequencies p, q, and r, then p + q + r = 1, as well as (p + q + r)^{2} = 1.

The time required for attaining equilibrium frequencies has been determined. If a certain population of individuals with one set of allele frequencies mixes with another set and complete panmixis occurs (that is, random mating), then the genotypes of the next generation will be found in the proportion p^{2} + 2pq + q^{2} where p and q are allele frequencies in the new mixed populations.

Thus it takes only one generation to reach Hardy-Weinberg equilibrium provided the allelic frequencies are the same in males and females. If the allelic frequencies are different in the two sexes, then they will become the same in one generation in the case of alleles on autosomes, and genotypic frequencies will reach equilibrium in two generations.

In general equilibrium is arrived at within one or at the most a few generations. Once equilibrium is attained it will be repeated in each subsequent generation with the same frequencies of alleles and of genotypes.

The Hardy-Weinberg law is applicable when there is random mating. Random mating occurs in a population when the probability of mating between individuals is independent of their genetic constitution. Such a population is said to be panmictic or to undergo panmixis. The matings between the genotypes occur according to the proportions in which the genotypes are present.

The probability of a given type of mating can be found out by multiplying the frequencies of the two genotypes that are involved in the mating. Matings are not random for instance when a population consists of different races such as blacks and whites in the U.S., or different communities as in India as there are preferred matings between members of the same racial or communal group.

**Applications of the Hardy-Weinberg Law****:**

**(a) **

**Complete Dominance:**

ADVERTISEMENTS:

When Hardy-Weinberg equilibrium exists, allele frequencies can even be found out in presence of complete dominance where two genotypes cannot be distinguished. If two genotypes AA and Aa have the same phenotype due to complete dominance of A over a the allele frequencies can be determined from the frequencies of individuals showing the recessive phenotype aa.

The frequency of aa individuals must be equal to the square of the frequency of the recessive allele q. Let us suppose q = 0.5, then q^{2} – (0.5)^{2} = 0.25. In other words when aa phenotype is 0.25 in the population, then it follows that the frequency of the recessive allele a is √0.25 – 0.5. The frequency of the dominant allele A would be 1 – q or 1 – 0.25 = 0.75.

**(b) **

**Frequencies of Harmful Recessive Alleles:**

ADVERTISEMENTS:

ADVERTISEMENTS:

The Hardy-Weinberg Law can also be used to calculate the frequency of heterozygous carriers of harmful recessive genes. If there are two alleles A and a at an autosomal locus with frequencies p and q in the population and p + q = 1, then the frequency of AA, Aa, and aa genotypes would be p^{2} + 2pq + p^{2}.

If the aa genotype expresses a harmful phenotype such as cystic fibrosis, then the proportion of affected individuals in the population would be q^{2}, and the frequency of the heterozygous carriers of the recessive allele would be 2pq.

To illustrate with figures, suppose one out of 1,000 children is affected with cystic fibrosis, then the frequency q^{2} = 0.001, so that q = √0.001 which is about 0.032, then 2pq = 2 x 0.032 x 0.968 = 0.062. This means that about 62 individuals out of 1000 or one out of 16 is a carrier of the allele for cystic fibrosis.

As already mentioned the number of individuals (aa) who are actually affected is one out of 1000. This implies that the frequency of heterozygous carriers is much higher than that of affected homozygotes.

ADVERTISEMENTS:

Similar calculation shows that when an allele is very rare in the population the proportion of carriers is still much higher and of affected homozygotes much lower. Thus, lower the frequency of an allele, greater the proportion of that allele that exists in the heterozygotes.

**(c) **

**Multiple Alleles:**

The Hardy-Weinberg Law permits calculation of genotypic frequencies at loci with more than two alleles, such as the ABO blood groups. There are 3 alleles I^{A}, I^{B} and I° with frequencies p, q and r. Here p + q + r = 1. The genotypes of a population with random mating would be (p + q + r)^{2}.

**(d) **

**Sex-Linked Loci:**

ADVERTISEMENTS:

It is possible to apply Hardy-Weinberg Law for calculating gene frequencies in case of sex-linked loci in males and females. Red green color blindness is a sex- linked recessive trait. Let r denote the recessive allele which produces affected individuals, and R the normal allele. The frequency of R is p and of r is q where p + q = 1. The frequencies of females having RR, Rr, rr genotypes would be p^{2}, 2pq, q^{2} respectively.

Males are different as they are hemizygous, have only one X chromosome derived from the mother with a single allele either R or r. The frequency of affected r males would be the same as the frequency of the r allele among the eggs that is q. The frequency of normal R males would be p. Suppose the frequency of r alleles is 0.08, then the incidence of affected males would be 0.08 or about 8%.

The frequency of affected rr females would be (0.08)^{2} = 0.0064 or 0.64%. Thus the Hardy-Weinberg Law explains that males would be affected a hundred times more frequently than females. This is actually what is observed. Males are more affected by sex-linked recessive traits than females.

The difference between the sexes is even more pronounced if the recessive allele is still more rare. The incidence of a common form of haemophilia is one in a thousand males; thus q = 0.001. However, only one in 1000,000 females will be affected. Thus males could have haemophilia one thousand times more often than females.

**(e) **

**Linkage Disequilibrium:**

Consider two or more alleles at one locus and another locus on the same chromosome with two or more alleles. Due to genetic exchange by recombination occurring regularly over a period of time, the frequencies of the allelic combinations at the two syntenic loci will reach equilibrium.

ADVERTISEMENTS:

If equilibrium is not reached, the alleles are said to be in linkage disequilibrium. The effect is due to tendency of two or more linked alleles to be inherited together more often than expected. Such groups of genes have also been referred to as supergenes.