expectations for autosomes, while genotype frequencies are equal to allele frequencies in the homogametic or haploid sex.
| Homozygotic or diploid sex | |||||
| Genotype | ZZ | ||||
| Gamete | Z‐A Z‐a | ||||
| Heterozygotic or haploid sex | Frequency | p q | |||
| Genotype | Gamete | Frequency | |||
| Z‐A | p | Z‐A Z‐A | Z‐A Z‐a | ||
| p 2 | pq | ||||
| ZW | Z‐a | q | Z‐A Z‐a | Z‐a Z‐a | |
| Pq | q 2 | ||||
| W | Z‐A W | Z‐a W | |||
| p | q | ||||
| Expected genotype frequencies under random mating | |||||
| Homogametic sex | Homogametic sex | ||||
| Z‐A Z‐A | p 2 | Z‐A W | p | ||
| Z‐A Z‐a |
|
||||
| Z‐a Z‐a | q 2 | Z‐a W | q | ||
Later, in Section 2.4, we will examine two categories of applications of Hardy–Weinberg expected genotype frequencies. The first set of applications arises when we assume (often with supporting evidence) that the assumptions of Hardy–Weinberg are true. We can then compare several expectations for genotype frequencies with actual genotype frequencies to distinguish between several alternative hypotheses. The second type of application is where we examine what results when assumptions of Hardy–Weinberg are not met. There are many cases where population genotype frequencies can be used to reveal the action of various population genetic processes. Before that, the next section builds a proof of the Hardy–Weinberg prediction that inheritance per se will not alter allele frequencies.
2.3 Why does Hardy–Weinberg work?
A proof of Hardy–Weinberg.
Hardy–Weinberg with more than two alleles.
The Hardy–Weinberg equation is one of the most basic expectations we have in population genetics. It is very likely that you were already familiar with the Hardy–Weinberg equation before you picked up this book. But where does Hardy–Weinberg actually come from? What is the logic behind it? Let's develop a simple proof that Hardy–Weinberg is actually true. This will also be our first real foray into the type of the algebraic argument that much of population genetics in built on. Given that you start out knowing the conclusion of the Hardy–Weinberg tale, this gives you the opportunity to focus on the style in which it is told. Algebraic or quantitative arguments are a central part of the language and vocabulary of population genetics, so part of the task of learning population genetics is becoming accustomed to this mode of discourse.
We would like to prove that p2 + 2pq + q2 = 1 accurately predicts genotype frequencies given the values of allele frequencies. Let's start off by making some explicit assumptions to bound the problem. The assumptions, in no particular order, are:
1 mating is random (parents meet and mate according to their frequencies);
2 all parents have the same number of offspring (equivalent to no natural selection on fecundity);
3 all progeny are equally fit (equivalent to no natural selection on viability);
4 there is no mutation that could act to change an A to a or an a to A;
5 it is a single population that is very large;
6 there are two and only two mating types.
Now, let's define the variables we will need for a case with one locus that has two alleles (A and a).
N = Population size of individuals (N diploid individuals have 2N alleles)
| Allele frequencies | |
| p = frequency(A allele) = (total number of A alleles)/2N | |
| q = frequency(a allele) = (total number of a alleles)/2N | |
| p + q = 1 | |
| Genotype frequencies | |
| X = frequency(AA genotype) = (total number of AA genotypes)/N | |
|
Y
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