Last week, we discussed the use of the Hardy Weinberg equations to estimate the rate of change in population under conditions of extreme selection, i.e. total elimination of one phenotype. This is essentially the goal of any sort of eugenics program. As an example of a way that this kind of policy could creep into culture, we watched GATTACA. Besides, it’s just a good film.

The purpose of the Hardy Weinberg equations is to model conditions under which allele frequencies can NOT change from one generation to the next. Therefore, it is evident that these are exactly those conditions that are responsible for allele frequency changes.

These conditions are:

- No Mutation
- No Selection (survival)
- No Sexual Selection
- No Genetic Drift –due to occasional fluctuations occurring by chance
- No Gene Flow – immigration / emigration

In order to prevent the random changes in allele populations stipulated in #4, we also need a sufficiently large population, where sufficient is likely definable by someone with better probability-computing skills than my own. (I feel like going off half-cocked on notions of probability and finite vs infinite time, but I’ll spare you).

Anyway, if we know something about the population, we might be able to work out the allele frequencies and then compute our theoretical proportions for the next generation from the equations…

p+q = 1,

where p and q are the frequencies of the (only) two alleles we are calculating.

and

p^{2}+2pq+q^{2} = 1

where each unit above represents the proportion of that genotype.

Mathematically, these equations provide insight into how rapidly the rate of an allele in a population could be eliminated if reproduction was prevented in a specific group. (This sounds completely esoteric without using an example, so let’s come up with one…)

A Healthy Gnome Couple

Imagine a population of fictional creatures – Garden Gnomes.

These gnomes have a recessive allele that makes them susceptible to a fungal disease. We’ll call the two alleles for this trait *H* – hearty (resistant) and *h*– weak (susceptible)

There was recently a new law passed amongst the gnomes forbidding susceptible gnomes from breeding (let’s imagine that the *H* allele is apparent by a normal complexion and the *h* allele is apparent by a jaundiced complexion. Like susceptibility to disease, jaundice only appears in the homozygous recessive (*hh*) gnomes.)

Imagine a population starting with equal allele frequencies, p=q=0.5.

p^{2}+2pq+q^{2} = 1

will give us genotype frequencies of:

25% HH + 50% Hh + 25%hh = 1

for the present generation.

Now, if we start our draconian, anti-jaundiced gnome policy and prevent breeding of these individuals, then this generation‘s breeding population only consists of the *HH* and *Hh* gnomes, where only the heterozygotes will contribute the h allele to the next generation.

If we call the next generation q_{1}, we can estimate the new proportion of the q allele in the population as the frequency of the heterozygote over the total population excluding the *hh* gnomes:

No wonder they want to get rid of these guys

After one generation, the frequency of the *H* allele is now 67%.

Since the same process would occur generation after generation (as long as the law was in place – and followed), we can determine the frequency of q at any generation, where n is the generation number.

- From this information, try calculating the frequency of both alleles after the policy has been in place for 5 generations.
- How long will it take to completely eliminate the
*h* allele?
- How would this change if the susceptible (
*h*) allele is dominant?