Hardy-Weinberg Principle

What is the Hardy-Weinberg principle?

The Hardy-Weinberg principle, named after English mathematician G. H. Hardy and German doctor Wilhelm Weinburg, can calculate the frequency of an allele in a population at equilibrium. It is a null model in genetics. According to the Hardy-Weinberg principle, at equilibrium, the allele frequencies of a gene within a population will not differ from one generation to the next. Unless an allele leads to a phenotype with a significant advantage or disadvantage over other alleles, its frequency in a population is unlikely to change. A population in Hardy-Weinburg equilibrium is not evolving means it is described as ‘stable’.

As we will see below, the Hardy-Weinberg principle provides us with a mathematical equation to calculate the expected frequency of an allele in a population.

${\mathrm{P}}^2+2\mathrm{PQ}+{\mathrm{Q}}^2=1$

What are the assumptions of the Hardy-Weinberg principle?

Imagine a population of diploid organisms that reproduce sexually. Let’s assume that there is no overlap between generations and that the frequencies of all alleles are equal in males and females.

There are five conditions for Hardy-Weinberg equilibrium. These are:

1. There is no selection: all alleles are equally likely to be passed on to the next generation.
2. No mutations arise.
3. There is no migration: the population is isolated.
4. The population size is infinitely large.
5. Mating is random.

These exact conditions are unlikely to be met in natural populations; however, the Hardy-Weinberg principle provides an essential and valuable null model which we can use to study gene frequencies. In biology, null models attempt to describe what would happen in a system not influenced by any biological processes of interest (such as the conditions above!).

If the predicted frequencies do not match the observed frequencies, we can conclude that at least one of the Hardy-Weinberg equilibrium conditions has not been met. For instance, that population might be undergoing directional selection, wherein an extreme phenotype is favoured over the mean or another phenotype.

The inheritance pattern for the gene in question might also be non-Mendelian, which affects its chances of being passed on to the next generation. (For more information, check out our articles on Selection and Inheritance!)

How can we calculate the allelic frequency?

Let’s take, for example, a population of 10,000 humans.

• The hypothetical gene B controls hair colour; the dominant allele B results in brown hair, while the recessive allele b results in blonde hair.
• Recall from the material on Genetics that every individual has two copies of this gene, and therefore, can have two of either allele (a homozygous genotype) or one of each allele (a heterozygous genotype).
• Therefore (because each individual has two copies of the gene), our population has (10,000 x 2 =) 20,000 alleles of this gene.
• If all individuals in this population had blonde hair (homozygous recessive), the probability of anyone being bb would be 1.0 and BB, 0.0. The allele b appears at a frequency of 100%, while B appears at a frequency of 0%.

• It gets trickier if you throw in heterozygous individuals. Take, for example, a cross of two heterozygous individuals (shown in the table below). To figure out the frequency of “b”, take the number of “b” alleles, and divide by the number of alleles for this gene. Since there are 10,000 individuals, there are a total of 20,000 alleles. Two individuals produce 4 B alleles and 4 b alleles. If there are 20,000 alleles, 10,000 would be B and 10,000 would be b, giving an allele frequency of 0.5 or 50%.

What is the Hardy-Weinberg equation?

${\mathrm{P}}^2+2\mathrm{PQ}+{\mathrm{Q}}^2=1.0$

Where p² = BB, 2pq = Bb, and b² = aa.

• Consider gene B. It has the dominant allele B and recessive allele b.
• Let the probability of allele B = p and allele b = q.
• There are only two alleles, so the probability of one plus the other must be 1.0/ Thus, p+q = 1.0

If you are told that the frequency of a dominant allele in a population is 70%, you have been given p directly, p = 0.7. Therefore q = 0.3, as p + q = 1.0

In the article on Inheritance, we learned that there are only four possible arrangements of these two alleles. It follows that all together, the probability of all four will equal 1.0.

How can we calculate the probability of a genotype in a population?

Now that we have the Hardy-Weinberg equation, we can calculate the frequency of an allele in a population.

Suppose that blondeness is the result of recessive allele b, and only one in 10 people display this phenotype. What is the probability that an individual in this population is a heterozygote?

1. Since this trait is recessive, it will only appear in individuals with the genotype bb.

2. Only 1 in 10 individuals have this phenotype; the probability of bb is $q^2$ = 0.10.

3. q is the square root of 0.10. q = 0.3162.

4. p + q = 1.0. Therefore, p = 1.0 - q = 1.0 - 0.3162 = 0.6838.

5. The probability of heterozygotes is 2pq. 2pq = 0. 4324