Why not take the Hardy-Weinberg Equilibrium one step further?

[MonDie]

p⁴ + 4p³q + 2p²q² + 4p²q² + 4pq³ + q⁴ = 1

A Google search showed me that this equation is mentioned in some genetics works written in other languages. Here is my question: why aren't these values normally included with descriptions of or calculators for the HWE? You can read further to find out what these values represent. I've attached an example problem at the bottom to make all of this more understandable. Also, I accidentally used the word "paternity" with an incorrect understanding that it encompassed both maternity and paternity.

 

I'm new to Biology, and the way I came to that equation is very simply. I used a probability grid with the values for p², 2pq, and q² on each axis. The grid reveals the paternity patterns one would expect by chance under equilibrium. These paternity patterns would also lead to equilibrium between generations by chance. The values of the grid can be summed up this way:

(p² + 2pq + q²)² = 1

Apply the distributive property and simplify, but don't combine the values from (2pq)² with the values from 2(p²·q²):

p⁴ + 4p³q + 2p²q² + 4p²q² + 4pq³ + q⁴ = 1

 

p⁴ is the value for the chance frequency of AA AA paternity (both parents are homozygous dominant).

4p³q = AA Aa

2p²q² = AA aa

4p²q² = Aa Aa

4pq³ = Aa aa

q⁴ = aa aa

 

This further proves the HWE if one takes it a step further. They would draw a punnet square for each paternity pair, and record (as decimal values) the chances of each genotype of offspring for each paternity pair. Then, for each pair, multiply the decimal value for each genotype of offspring by the decimal value for that genotype of paternity. Hypothetically, these numbers, which I will refer to as the "composite values," are the chances of any randomly chosen offspring 1) being that genotype and 2) having parents with those genotypes.

For example, the Aa aa paternity pair would have 0.5 for heterozygous genotype offspring and 0.5 for homozygous recessive genotype offspring because those offspring genotypes each have a 50/50 chance of resulting from that paternity pair. Then, multiply both 0.5 and 0.5 by 4pq³. The resulting values are the composite values.

In the end, if you multiply all the composite values for each genotype of offspring, you will get the original values for p², 2pq, and q². This means that, if the paternity pairings were in line with the values of p⁴ + 4p³q + 2p²q² + 4p²q² + 4pq³ + q⁴ = 1, then the only reasons I can think of for a different distribution of alleles in the next generation would be a small population size or a disruption in gene recombination at the molecular level. Furthermore, if one had enough in-depth data about mating patterns, they could use these maths to determine whether one particular effect can account for the extent to which gene circulation is shifted from equilibrium. Of course, there are certain conditions where the biologist would need a lot of data, like with polygynous primate populations containing many transient males, but the equation is still potentially useful.

 

Now that I've explained the meaning and hypothetical application of these values, I am going to ask those with more experience the question from the beginning. Why aren't these values normally included with descriptions of or calculators for the HWE? Also, feel free to critique anything I said in this post, or discuss real or hypothetical applications of this math.

 

Edited July 7, 2012 by Mondays Assignment: Die

[MonDie]

In the end, if you multiply all the composite values for each genotype of offspring

I meant add, not multiply.