How could we keep track of the frequency of each allele for a given trait when we have a dominant-recessive interaction? More specifically, how could we account for the visible dominant traits as homozygous or heterozygous, since both look the same?

This is where the **Hardy-Weinberg principle** comes in. Firstly, there are criteria for when this principle may be applied to a population:

**1. Random mating** must take place.

**2. No migration** must occur either inwards or outwards of the population.

**3. No mutations** must arise in the population.

**4. No natural selection** must take place due to one trait being better or worse adapted to the environment.

It's apparent that this is simply rarely, if ever, the case in a real wild population. However, the Hardy-Weinberg principle is useful at predicting allele frequencies in a reliable mathematical model.

The *frequency* of the dominant allele is noted **p **while that of the recessive allele is noted **q**. Both must necessarily account for the whole population, therefore:

**p + q = 1**

The values are frequencies, so they are noted as *percentages*. 1 is 100% while 0.5 is 50% and 0.05 is 5%, etc.

**Worked exercise**

If we know that the frequency of the allele for dark fur in a population of koala bears is 0.2, and this allele is dominant over the one for light fur, work out the frequency of the allele for light fur in the population.

**p = 0.2**

**p + q = 1**

Therefore, **0.2 + q = 1 **so **q = 1 - 0.2**

**q = 0.8 **or 80%.

Now the *allele frequency* has been worked out, how could we work out the actual **phenotype** of the koala bears in the population. How many are actually dark-furred? How many of the dark-furred ones are homozygous?

For this we use the same equation as before, but **squared**: (p + q)2

This is equivalent to **p**^{2} + 2pq + q^{2} = 1

Where **2pq** is the frequency of **heterozygotes**, and **p**^{2} and **q**^{2} the frequencies of **homozygous dominant** and **homozygous recessive** respectively.

We want to know how many koala bears have dark fur. We know that the allele frequency for dark fur is 0.2, so 0.2^{2} is the percentage of homozygous dark fur individuals; = 0.04 (4%).

This trait being dominant, the heterozygotes must also have dark fur. The frequency of heterozygous dark fur is 2pq = 2*0.2*0.8 = 0.32 (32%).

**So overall, there are (0.4 + 0.32) 0.36 or 36% dark-furred koala bears in the population.**

This leaves the remaining 64% with light fur. Note the contrast between the light *phenotype* only being 64% while the *allele frequency* for light fur is 80%. If the allele were dominant over dark fur, the frequency would be *higher* rather than *lower*.