# Additive genetic variance with $n$ alleles

The genetic variance of a quantitative trait (the quantitative trait in question is fitness) can be express as the sum of two components, the dominance and additive variance:

$$\sigma_D^2 + \sigma_A^2 = \sigma^2$$

, where $\sigma$ is the genetic variance, $\sigma_D^2$ is the dominance variance and $\sigma_A^2$ is the additive variance. $\sigma_D^2$ and $\sigma_A^2$ are given by

$$\sigma_D^2 = x^2(1-x)^2(2\cdot W_{12} - W_{11} - W_{22})^2$$

$$\sigma_A^2 = 2x(1-x)(xW_{11}+(1-2x)W_{12} - (1-x)W_{22})^2$$

, where $W_{11}$, $W_{12}$ and $W_{22}$ are the fitness of the three possible genotypes and $x$ and $1-x$ give the allele frequencies.

Question

The above definition makes sense for one bi-allelic locus.

• How are $\sigma_D^2$, $\sigma_A^2$ and $\sigma^2$ defined for a locus that have $n$ alleles?

Here is a related question

• this might be a useful resource for you @Remi.b - I'll read the Falconer and Mackay book tomorrow at work too nitro.biosci.arizona.edu/zbook/NewVolume_2/newvol2.html Mar 31, 2014 at 19:07
• really like this question, spent some time at the whiteboard today trying to work it out! made good progress! Apr 1, 2014 at 20:06
• Chapter 4 of lynch and Walsh seems to deal with this - from around table 4.2 Apr 3, 2014 at 18:13
• @GriffinEvo At which page is it? I could find table 4.1 at page 11 and table 4.3 at page 22! table 4.2 is probably in between :D Apr 3, 2014 at 18:26
• In my copy it is page ~77, I think you should see equation 4.23a and some of the preceeding section on "average excess" - this is Lynch and Walsh Genetics and analysis of quantitative traits. I'm away for the next couple of days but will be trying to write this in an R code with data to test it out next week - perhaps we should move to the chat room to discuss more! Apr 4, 2014 at 9:39

Well, the total genetic variance is just, by the definition of the variance, $$\sigma^2 =\sum_{i,j} f_i f_j (w_{ij}-\bar{w})^2$$ (using $f_i$ and $w_{ij}$ for frequency and fitness, respectively), and $$\bar{w} = \sum_{i,j} f_i f_j w_{ij}$$ is just the average fitness.
$$\sigma^2_A =\sum_{i,j} f_i f_j (w_iw_j-\bar{w}')^2 = \sum_i f_i(w_i-\bar{w}')^2.$$ with $$f_i=\sum_jf_{ij}; w_i=\sum_j f_{ij}w_{ij}; \bar{w}' = \sum_i f_i w_i;$$
• Thanks a lot for your answer. Could you please help me to develop the formula for $\sigma_A^2$ for a diploid population? Apr 2, 2014 at 15:07
• this should be true for a diploid population. The additive genetic variance is defined as the variance in a population where there are no interaction effects between genotypes, so $w_{ij}$ can be written as $w_iw_j$ Apr 2, 2014 at 18:33
• I'm sorry I might be a bit slow to understand :)… Can you please show the development to find back $\sigma_A^2 = 2x(1-x)(xW_{11}+(1-2x)W_{12} - (1-x)W_{22})^2$ from your equations? Thank you. Apr 3, 2014 at 8:17