Populations: The Hardy-Weinberg principle

In a nutshell

The Hardy-Weinberg principle is a mathematical model that predicts the frequencies of alleles in a population won't change between generations. It is used to calculate allele frequency and genotypes in different populations. It is only true in large populations where there is no immigration, emigration, mutations or natural selection. It also assumes random mating. 

Definitions

The Hardy-Weinberg principle

The Hardy-Weinberg principle is a mathematical model that can be used to predict allele frequencies, predict the percentage of the population with a certain genotype and show if external factors are affecting allele frequency.

Calculating the frequency of one allele

When a gene has two alleles, the frequency can be calculated using this formula:

​$$p + q =1$$

Example

Calculate the frequency of the dominant allele in a population if the frequency of recessive is $$0.3$$​.

Rearrange formula: 

​$$p=1-q$$

​

Substitute in $$0.3$$:

$$p= 1-0.3 \newline\underline {p=0.7}$$

​

$$\underline {Therefore,\ the\ frequency\ of\ the\ dominant\ allele\ in\ the\ population\ is\ 0.7.}$$

Calculating the frequency of one genotype

You can also calculate the frequency of one genotype using this formula:

​$$p^2 +2pq+q^2 = 1$$​

Example

Calculate the frequency of the heterozygous genotype in a population of flowers with red petals and yellow petals. The frequency of homozygous dominant genotype (RR) is $$0.64$$​ and the frequency of yellow flowers (rr) is $$0.04$$. 

Rearrange formula: 

​$$2pq= 1 -p^2-q^2$$

Substitute in numbers:

$$2pq=1 - 0.64 -0.04\newline \underline{2pq=0.32}$$

​$$\underline{Therefore,\ the\ frequency\ of\ the\ heterozygous\ genotype\ in\ the\ flower\ population\ is\ 0.32.}$$​​

This can then be used to work our phenotype frequencies:

Number of red flowers:

​$$0.32+0.64=0.96$$

​

Number of yellow flowers:

$$0.04$$

​​

​$$\underline{Therefore\ in\ a\ population\ of\ 100\ flowers,\ 96\ would\ be\ red\ and\ 4\ would\ be\ yellow.}$$

This can also be used to work out the allele frequencies:

Frequency of dominant allele:

​$$\sqrt {0.64} = 0.8 \newline\underline{p=0.8}$$

​​

Frequency of recessive allele:

​$$\sqrt{0.04}=0.2\newline \underline{q= 0.2}$$​​

Calculating the genotype percentage 

The Hardy-Weinberg principle can also be used to predict the percentage of a population with a certain genotype.

Example

The frequency of a recessive characteristic in the UK is $$1$$ birth in every $$25000$$. Use this information to estimate the percentage of people in the UK who are heterozygous for this characteristic.

Calculate $$q$$: 

​$$q^2 = 1\ in\ 25000= 0.00004\newline q=\sqrt {0.00004} \approx 0.00063$$

Calculate $$p$$:

$$p= 1-0.00063\newline p=0.9937$$

​​

Calculate $$2pq$$:

​$$2\times 0.00063\times 0.9937=0.00125$$​​

​$$\underline{Therefore,\ the\ frequency\ of\ the\ heterozygous\ genotype\ in\ the\ population\ is} \newline \underline{0.00125\ or\ 0.125 \%.}$$

The Hardy-Weinberg principle can also show if external factors are affecting allele frequency. If the calculated allele frequency between generations has changed there must be some external factors such as immigration, emigration, mutations and natural selection affecting it.