Derivation of the Hardy-Weinberg principle

We saw last time using the data from Zoarces viviparus that we can describe empirically and algebraically how genotype frequencies in one generation are related to genotype frequencies in the next. Let's explore that a bit further. To do so we're going to use a technique that is broadly useful in population genetics, i.e., we're going to construct a mating table. A mating table consists of three components:

1. A list of all possible genotype pairings.

2. The frequency with which each genotype pairing occurs.

3. The genotypes produced by each pairing.

		Offsrping genotype
Mating	Frequency	$A_1A_1$	$A_1A_2$	$A_2A_2$
$A_1A_1 \times A_1A_1$	$x_{11}^2$	1	0	0
$A_1A_2$	$x_{11}x_{12}$	$\frac{1}{2}$	$\frac{1}{2}$	0
$A_2A_2$	$x_{11}x_{22}$	0	1	0
$A_1A_2 \times A_1A_1$	$x_{12}x_{11}$	$\frac{1}{2}$	$\frac{1}{2}$	0
$A_1A_2$	$x_{12}^2$	$\frac{1}{4}$	$\frac{1}{2}$	$\frac{1}{4}$
$A_2A_2$	$x_{12}x_{22}$	0	$\frac{1}{2}$	$\frac{1}{2}$
$A_2A_2 \times A_1A_1$	$x_{22}x_{11}$	0	1	0
$A_1A_2$	$x_{22}x_{12}$	0	$\frac{1}{2}$	$\frac{1}{2}$
$A_2A_2$	$x_{22}^2$	0	0	1

Believe it or not, in constructing this table we've already made three assumptions about the transmission of genetic variation from one generation to the next:

Assumption #1

Genotype frequencies are the same in males and females, e.g., $x_{11}$ is the frequency of the $A_1A_1$ genotype in both males and females.³

Assumption #2

Genotypes mate at random with respect to their genotype at this particular locus.

Assumption #3

Meiosis is fair. More specifically, we assume that there is no segregation distortion; no gamete competition; no differences in the developmental ability of eggs, or the fertilization ability of sperm.⁴

Now that we have this table we can use it to calculate the frequency of each genotype in newly formed zygotes in the population,⁵provided that we're willing to make three additional assumptions:

Assumption #4

There is no input of new genetic material, i.e., gametes are produced without mutation, and all offspring are produced from the union of gametes within this population, i.e., no migration from outside the population.

Assumption #5

The population is of infinite size so that the actual frequency of matings is equal to their expected frequency and the actual frequency of offspring from each mating is equal to the Mendelian expectations.

Assumption #6

All matings produce the same number of offspring, on average.

Taking these three assumptions together allows us to conclude that the frequency of a particular genotype in the pool of newly formed zygotes is

$$\sum(\hbox{frequency of mating})(\hbox{frequency of genotype produce from mating}) \quad .$$

So

$$\begin{eqnarray*} \hbox{freq.}(A_1A_1\hbox{ in zygotes}) &=& x_{11}^2 + \frac{1... ...) &=& 2pq \\ \hbox{freq.}(A_2A_2\hbox{ in zygotes}) &=& q^2 \\ \end{eqnarray*}$$

Those frequencies probably look pretty familiar to you. They are, of course, the familiar Hardy-Weinberg proportions. But we're not done yet. In order to say that these proportions will also be the genotype proportions of adults in the progeny generation, we have to make two more assumptions:

Assumption #7

Generations do not overlap.

Assumption #8

There are no differences among genotypes in the probability of survival.