I am reading Charlesworth et al. 1997. They talk about diversity within and between allelic classes.
Nucleotide diversities ($π$) at each neutral site were estimated from the mean of $2 \sum z_t (1-z_t)$, over replicated introductions at the site of single variants, where zt is the frequency of the neutral variant at time t, and summation is over all times until either fixation or loss occurs.
The total genetic diversity at the neutral sites ($π_T$) was also decomposed into that within and between allelic classes at the polymorphic locus. Diversity within allelic classes, which will be written here as $π_A$, was estimated from the mean of $2 \sum \left( x_t(1-x_t)+y_t)(1-y_t) \right)$ where $x_t$ and $y_t$ are the frequencies of the neutral variant within the first and second allelic classes, respectively. Diversity between allelic classes with respect to the polymorphic locus was calculated as the difference between the total diversity values and $π_A$
Note that the parentheses don't match up but this this is what is written in the paper!
Why am I confused about this text?
I am confused about the term allelic class. I first think there is anything fancy in here and I think we can simply replace the term "allelic class" by "allele". but then when I saw the equation for $\pi_A$ I realize that the frequency of the two allelic classes does not necessarily adds to 1 (even though we consider only two allelic classes).
I also got also a little confused about the difference between $\pi$ and $\pi_T$ but I think that they just used two notations for the same think ($\pi = \pi_T$)
In population genetics's jargon, diversity just mean expected heterozygosity. $\pi_T$ makes sense to me. It is just the average heterozygosity $\left(2 z(1-z)\right)$ calculated over all time steps. Maybe a more intuitive to put it would to integrate rather than summing over time rather than time steps.
Question
I can read the equation for $\pi_A$ but I fail to get any intuition behind what it means. For example, I have no idea why it should be called within-allelic class diversity. Where does $2(x(1-x)+y)(1-y)$ come from? My whole issue might boil down to the definition of allelic class.
EDIT
The term allelic class
is defined in Innan and Tajima (1997)
Suppose that there are two nucleotides, say A and T, in a particular site. Then, we can divide DNA sequences into two classes: one class includes sequences with A and the other includes sequences with T in this site. We call such a class an allelic class
(Slatkin 1996 might help as well).
I am still not quite sure what the within allelic class variance
. Maybe it is: Take the most common sequence in the considered allelic class. For each, sequence, calculate the number of pairwise differences to the most common sequence and square this value. Sum over all sequence and divide by the number of sequences. In math form it would be: $\frac{1}{2N}\sum_i^{2N} (D_i)^2$, where $N$ is the population size and $D_i$ is the number of pairwise difference between the sequence $i$ and the most common sequence in the considered allelic class. Does it sound right to you?