# Additive genetic variance $V_A= 2pqa^2$ ... what is $a$?

Reading Falconer and Mackay looking at equation $V_A = 2pqa^2$ (equation 8.5, page 126).

I am trying to write a clear explanation of how additive genetic variance is defined and calculated, but I keep getting twisted up in the books I am working from.

Just what is the term $a$ and how is it calculated?

• can you add a little context or provide a link to that article? Jun 11, 2015 at 17:22
• there's not much more context one can add, but I've linked to the book Jun 11, 2015 at 17:23
• Actually I could not access the book. Though you got an answer, I would suggest that you put up a snapshot of the page so that other users may find the question useful. Jun 11, 2015 at 20:32

This information is given in p.118.

In essence $a$ represents the deviation from the population mean due additive genetic effects.

Say a trait is controlled solely by locus X with the effect alleles X impacting the phenotype Y in an additive manner (no dominance and epistasis effect), then we can calculate the effect size of carrying the X allele, called $a$, and representing the distance from the population phenotypic mean of homozygote XX. Based on this definition, the distance from the mean for individual with the following genotype is given by:

• XX: $a$
• Xx : 0
• xx: $-a$

This makes more sense using the genetic model:

$P=G+E=BV+I+E=A+D+I+E$ (parameters listed at the bottom)

In the example, the locus has only an additive effect then D=0 and assuming there is no epistasis then I=0 as well which gives $P=BV+E=A+E$ therefore $BV=A$

In this case $a$ can be computed directly from the BV, with p=f(X) and q=f(x), as:

• $BV(XX)=2qa$
• $BV(Xx)=(q-p)a$
• $BV(xx)=-2pa$

Now for the complete case of both additive and dominant effects, i.e. $BV=A+D$, you can refer to table 7.3 in p.118 which describe that XX individuals are given the value $a$, Xx -> $d$ and xx -> $-a$ with the corresponding relationship to the BV given by (care here it's not $a$ but alpha $\alpha$):

• $BV(XX)=2q\alpha$
• $BV(Xx)=(q-p)\alpha$
• $BV(xx)=-2p\alpha$

with $\alpha=a+d(q-p)$

• P=phenotype
• BV=A+D=Breeding value
• G=Genotypic effect
• E=Environmental effect
• A=Additive genetic effect quantified by $a$
• D=Dominant genetic effect quantified by $d$
• I=Epistasis effect
• Thanks @cagliari that is a really good way of explaining it - are you studyin quantitative genetics? (I am in the chat room) Jun 11, 2015 at 22:25
• @rg255 Thanks! Yes and no, just finished a pop gen class where we used Falconer & Mackay intensively so this is still very fresh in my mind but my main research interest is actually in RNA/DNA structure prediction. Jun 12, 2015 at 0:28
• Cool the I'm stuck firmly in a popgen group - Hartl & Clark only, I had to do my own Q.G book clubs... I've realised where I was getting stuck too, it was $a$ becoming $\alpha$ that was throwing me off, but seeing equation 7.5 I see that $a = \alpha$ when $d = 0$. Jun 12, 2015 at 8:37