The Hardy-Weinberg Equilibrium Model

The Hardy-Weinberg equilibrium model (or law, or principle) states that the frequencies of alleles and genotypes in a population will remain constant over time in the absence of other evolutionary forces. These forces are:

  1. Genetic drift
  2. Mutation
  3. Migration
  4. Selection (natural or otherwise)

Assumptions of the Model

  • None of the the four forces listed above are brought to bear on the model: in effect, evolution does not take place.
  • There is random sexual reproduction (no sexual selection).
  • There is an infinite population size (therefore it is not susceptible to genetic drift).
  • Generations are non-overlapping, i.e. when a new generation is born, the old generation dies out.
  • Body cells are diploid.

Clearly, this is not a realistic scenario, so of what use is the Hardy-Weinberg equilibrium model?

The answer is that it establishes a null model, against which changes in a system can be measured after the application of one or all of the four forces listed above.

Possible Genotypes in a Diploid Cell

To keep things trackable, we focus on one gene, e.g. the one for eye colour *, where the alleles are:

  • B = brown eyes (dominant)
  • b = blue eyes (recessive)

By convention a capital letter represents the dominant allele, and a lower-case letter the recessive allele.

* This is simplified for the sake of the example. In reality, eye-colour is determined by multiple alleles.

Father
B b
Mother B BB bB
b Bb bb

The possible genotypes for the B and b alleles are:

  • BB: homozygous dominant, brown eyes.
  • Bb/bB: heterozygous, brown eyes (the dominant allele will always trump the recessive allele).
  • bb: homozygous recessive, blue eyes.

Calculating Genotype Frequencies

We can calculate the genotype frequencies for a population only if we know the percentage of that population in which the homozygous recessive genotype (bb) has been expressed.

That is, in any population we could, in theory, count all the people who had the recessive genotype (bb, blue eyes).

The same could not be done for people with brown eyes: they might have either the homozygous dominant genotype (BB) or the heterozygous genotype (Bb/bB).

The Hardy-Weinberg Equation

The Hardy-Weinberg equation is used to measure genotype frequencies in a population that is in equilibrium, i.e. not evolving.

As can be seen from the following graphic,

The Hardy-Weinberg equation: P squared + 2Pq + q squared = 1
  • P2 is the frequency of the homozygous dominant genotype (BB)
  • 2Pq is the frequency of the heterozygous genotype (Bb/bB) and
  • q2 is the frequency of the homozygous recessive genotype (bb).
  • The total value is always 1.

Furthermore:

P + q = 1
  • P is the frequency of the dominant allele (B)
  • q is the frequency of the recessive allele (b)
  • The total value is always 1.

Example Problem

Twenty individuals, out of a population of a hundred, have blue eyes. The rest have brown eyes. For the population as a whole, what is the frequency of:

  1. the dominant B allele?
  2. the recessive b allele?
  3. the homozygous recessive bb genotype?
  4. the homozygous dominant BB genotype?
  5. the heterozygous Bb/bB genotype?

Steps to the Solution

We have already been supplied with the information that twenty out of a hundred individuals have blue eyes, i.e. the homozygous recessive genotype (bb) is carried by 20% of the population.

The corresponding genotype frequency, q2 is: 1. q2 = 20 ÷ 100 = 0.20

From this we can calculate the q allele frequency: 2. q = √q2 = 0.45

From P + q = 1, we can derive the value of P: 3. P + 0.45 = 1 ⇒ 1.00 − 0.45 = P ⇒ P = 0.55

By squaring P we get the frequency of the BB genotype: 4. P2 = 0.55 × 0.55 = 0.30

Finally, we can get the frequency of the Bb/bB genotype by using the Hardy-Weinberg equation: 5. 2Pq = 1 − P2 − q2 = 1.00 − 0.30 − 0.20 = 0.50

Note: we can confirm that the genotype frequencies add up correctly in the Hardy-Weinberg equation: P2 + 2Pq + q2= 0.30 + 0.50 + 0.20 = 1

Answers

In summary, the frequency of the:

  1. dominant B allele is 0.55,
  2. recessive b allele is 0.45,
  3. homozygous recessive bb genotype is 0.20,
  4. homozygous dominant BB genotype is 0.30, and
  5. heterozygous Bb/bB genotype is 0.50.

Calculator

To use the calculator you must already know the percentage of the population expressing the homozygous recessive (bb) genotype.

Enter a number between 1 and 100

Note: both genotype and phenotype totals do not always equal 1 or 100% respectively, due to rounding. This is expected.

Glossary

Evolution
The change in allele frequencies in a population over time.
Natural Selection
The change in allele frequencies in a population over time as a result of their fitness for reproduction.
Genetic Drift
The change in allele frequencies in a population over time due to random events.
Allele
Version of a gene, e.g. the gene for eye colour.
Genotype
The combination of two related alleles in a cell.
Phenotype
The expression of the gene in the organism, e.g. brown eyes.
Allele frequency
The proportion of alleles in a gene pool that are either dominant, e.g. code for brown eyes, or recessive, e.g. code for blue eyes.
Genotype frequency
The proportion of allele combinations in a cell.
Homozygous dominant genotype
Both the allele inherited from the mother and the one inherited from the father are dominant.
Homozygous recessive genotype
Both the allele inherited from the mother and the one inherited from the father are recessive.
Heterozygous genotype
One allele is dominant and one is recessive.

Further information

I found the following Bozeman Science videos on YouTube very helpful: