The Hardy-Weinberg Equilibrium

 

    The Englishman HARDY and the German WEINBERG could show that the frequency of homozygotes and heterozygotes in a population stays constant for generations if certain conditions are fulfilled. The HARDY-WEINBERG law permits the theoretical calculation of the frequency a certain genotype has in a given population independent of the number of existing alleles.

The Mendelian laws start out from two individuals (parents) and their offspring. Hereditary traits as they have been described till now can only be understood under controlled conditions. Ratios like 3:1 will hardly be discovered in nature since every species has to be regarded as a group of populations in which certain genotypes occur in certain amounts hard to capture. The frequency of an allele can be very low and genetic combinations where it has part in will inevitably be very rare.

The Englishman G. H. HARDY and the German W. WEINBERG showed independent of each other in 1908 and 1909 that the frequency of homozygotes and heterozygotes stays constant for generations, if

• the population is very large,
• the individuals can pair without limitations (if they belong to different sexes and live at the same place and at the same time, of course),
• there is no selection of certain alleles,
• no gene migration occurs and
• no mutations take place.

Derivative:

Given: two pairs of alleles, A and a

Assumed: the frequency of A shall be p = 0.9 (= 90%) that of a shall be q = 0.1 (= 10%)

From that follows: p + q = 1

In the population, the genotypes AA, Aa and aa will thus be found. The produced germ cells would either contain A or a. If they cross according to chance, it has to be taken into account that germ cells containing A have the frequency p and germ cells equipped with a the frequency q. These genotypes will accordingly occur in the following generation with the following frequencies:

AA = 0.9 x 0.9 = 0.81

Aa = 0.9 x 0.1 = 0.09

aA = 0.1 x 0.9 = 0.09

aa = 0.1 x 0.1 = 0.01

or, expressed mathematically: AA = p² ; Aa + aA = 2pq ; aa = q²

or: p² + 2pq + q² = (p + q)² = constant

Or, expressed in words: under the conditions mentioned above, the original ratio of the alleles A and a will be retained from generation to generation. There can be any number of alleles per gene in a population. The genome of an individual is therefore just a chance selection of the whole gene pool.

The Hardy-Weinberg law allows the calculation of the heterozygous individuals' frequency. When two alleles exist, it can never be larger than 0.5. The following picture shows the quantitative relation between the frequencies of the alleles and those of the respective genotypes (according to D. S. FALCOMER, 1960).

 

If an allele has a high frequency, the relation of the genotypes will shift strongly in favour of the respective homozygous genotype. But since the preconditions for Hardy-Weinberg are usually not given, plant populations being often very small and self-pollination being no exception, the law cannot be applied here. MENDEL himself tackled this problem in his classic study in 1866 and asked how the splitting ratios of subsequent generations would look, if the offspring of every new generation would always be crossed with each other.

A population with random mating results in an equilibrium distribution of genotypes after only one generation, so that the genetic variation is maintained f When the assumptions are met, the frequency of a genotype is equal to the product of the allele frequencies AA:Aa: aa:  p2 2pq q2

The H-W equilibrium states that sexual reproduction does not reduce genetic variation generation after generation; on the contrary, the amount of variation remains constant if there are no disturbing forces acting against it. It establishes the relationship for calculating genotype frequencies under random mating and, in doing so, provides the foundation for many studies in population genetics.

Hardy–Weinberg equilibrium is impossible in nature.

·         Static allele frequencies in a population across generations assume: random mating, no mutation (the alleles don't change), no migration or emigration (no exchange of alleles between populations), infinitely large population size, and no selective pressure for or against any traits.

·         The Hardy-Weinberg model, named after the two scientists that derived it in the early part of this century, describes and predicts genotype and allele frequencies in a non-evolving population.

·         The model has five basic assumptions

·         The population is large (i.e., there is no genetic drift);

·         There is no gene flow between populations, from migration or transfer of gametes;

·         Mutations are negligible;

·         Individuals are mating randomly; and

·         Natural selection is not operating on the population. Given these assumptions, a population's genotype and allele frequencies will remain unchanged over successive generations, and the population is said to be in Hardy-Weinberg equilibrium. The Hardy-Weinberg model can also be applied to the genotype frequency of a single gene

Importance

·         The Hardy-Weinberg model enables us to compare a population's actual genetic structure over time with the genetic structure we would expect if the population were in Hardy-Weinberg equilibrium (i.e., not evolving).

·         If genotype frequencies differ from those we would expect under equilibrium, we can assume that one or more of the model's assumptions are being violated, and attempt to determine which one(s).

·         How do we use the Hardy-Weinberg model to predict genotype and allele frequencies? What does the model tell us about the genetic structure of a population?

 

 

The starting point is generation 0. We have a gene with two alleles, A1 and A2. The frequency of allele A1 is p and the frequency of allele A2 is q. The genotype frequencies in generation 0 are for A1 A1 = p2, for A1 A2 = 2pq and for A2 A2 = q2. If random mating occurs, the probability of any allele from the female plant meeting any allele from the male plant will be the sameThe frequency of occurrence of each genotype is given by the product of the frequency of each allele in the genotype (e.g. for A1 A1 is p x p = p2).

	Genotypes of G0			G0		Genotypes of G1			G1
Popul.	A1A1	A1A2	A2A2	p	q	A1A1	A1A2	A2A2	p	q
Pop.1	0.6	0.2	0.2	0.7	0.3	0.49	0.42	0.09	0.7	0.3
Pop.2	0.49	0.42	0.09	0.7	0.3	0.49	0.42	0.09	0.7	0.3
Pop.3	0.4	0.6	0	0.7	0.3	0.49	0.42	0.09	0.7	0.3

An example of three population in Hardy Weinberg equillibrium

    Population genotype frequencies are given in rows. Generations (G0 and G1) are given in columns. Again, we have one gene with two alleles, A1 and A2. The frequency of allele A1 is p and the frequency of allele A2 is q. Genotype frequencies are different for each population in generation 0 (e.g. the frequency of A1 A1 in population 1 is 0.6; in population 2, 0.49; and in population 3, 0.4, and so on for the other genotypes). We note, though, that the allele frequencies in the three populations are similar in G0 (p = 0.7 and q = 0.3). In the next generation, G1, if all requirements of the H-W principle are met, the genotype frequencies in the three populations balance (now the frequency of A1 A1 is 0.49 in the three populations, and the same happens with the frequencies of A1 A2 and A2 A2 ). The allele frequencies are kept.