So, at least my twitterverse has been on fire suddenly with the appearance of a new article in PLoS by Patrick Monnahan and John Kelly “Epistasis Is a Major Determinant of the Additive Genetic Variance in Mimulus guttatus” .

It really is a nice study in which they identified 11 quantitative trait loci (QTL) in a single population of monkey flower, then used these to estimate the functional (also known as physiological) direct effects, and all of the two locus epistatic interactions. They then used these estimates to estimate additive genetic variances and total genetic variances in the population.

What is nice about this study is that they use actual data from a QTL analysis of a natural population, and then use the resulting analyses to estimate bi-allelic functional epistasis for each of the pairs of QTL. In fact it would be a great teaching tool to have access to some of those two locus genotypic values for teaching purposes! I would also love to have the actual allele frequencies, so that we could in fact estimate the standing statistical variance components in the natural populations. This also brings up a very important point: all of the models to date have put in fixed values for the genotypic values (or avoided the issue entirely using inbreeding coefficients). In the real world we collect organisms, identify genes, and phenotype them. There is ample room for error at every step. So the one thing we know for sure is that any QTL measures or assignment of phenotype to genotype is an estimate. This really is the first attempt to couple field estimates of genotypic values to variance components.

One other thing that is nice about this paper is that they bring up both the Kempthorne/Cokerham variance components and the more recent terminology of “positive”, “negative” and “sign” epistasis. Nicely, Hanson (2013 Evolution 67: 3501-3511) provided two locus examples of these types of epistasis. It turns out that if we set the gene frequencies to 0.5, and do the appropriate regressions we can directly relate these molecular concepts of epistasis to the quantitative genetic components. It also turns out that this is critical, for while functional epistasis is loads of fun, it is only the quantitative genetic variance components that tell us how phenotypic evolution works.

Anyway, from Hanson (2013) these different types of functional epistasis are:

Using the JMP program shown below it is easy to show that positive epistasis is a hodgepodge of variance components (89% additive variance, 3.6% AXA epistasis, 3.6% AXD epistasis, and 3.6% DXD epistasis), negative and sign epistasis is additive variance and AXA epistasis (negative epistasis: 80% additive variance, 20% AXA epistasis, sign epistasis: 50% additive variance, 50% AXA epistasis). Maybe its because I am a curmudgeon, but I am happier with the old fart Kempthorne partitioning, because it relates directly to variance components, and can be much more easily converted to statistical genetic components.

Now here is the critical point. These variance components are a function of gene frequency, thus the variance components will change as gene frequencies change. Using the example of positive epistasis above I can now tell you the additive genetic variance for any two locus gene frequency:

*Graph of the additive genetic variance for two locus two allele positive epistasis as described by Hansen (2013). A JMP program to calculate VA for a single gene frequency is listed below. Note that I rotated the graph to best show the shape of the surface. The highest additive genetic variance occurs when both the A2 and B2 alleles are at low frequency (around 0.2).*

Finally, I know it is impolite to promote your own work, but well, it’s my blog and I will do what I want. My ego was a bit hurt by the fact that that my work on epistasis and additive genetic variance was not cited, in particular, my paper on average effects and additive variance (Goodnight. 2000. Heredity 84: 587-598.), which was quite relevant. That and my earlier paper using breeding values (Goodnight,1988. Evolution 42: 441-454) were the first papers to describe the conversion of epistasis in to VA, and they have historical significance if nothing else. I have long been fighting a bit of a rear-guard action to keep those papers from falling into the obscurity of common knowledge. There is actually another reason that they could have benefited from citing those papers. One of the things that comes out of those papers is that if you can write down the functional values for the 9 genotypes of a pair of interacting two allele loci you can use regression to calculate the additive genetic variance for any given gene frequency. I do actually know why they might have missed my paper. They use the Falconer partitioning that was first pioneered by Cheverud and Routman (1995. 139: p. 1455–1461) which is enough different that my paper really didn’t need to be cited, so it is hard to get too mad at them.

*Its my blog and I will whine if I want to. You would whine to if it happened to you. (picture from (http://www.amazon.com/Its-My-Party-Mercury-Anthology/dp/B000VHKHZA )*

If you have JMP and are savvy in its use, the files that I use for calculating the additive genetic variance can be found here (variance regressions). I fixed it by changing the file extension to .txt. It is still a .jmp, so after you download it please change the txt to jmp, then it should work.

Basically you add your own dependent variables, add the allele frequencies of your choice (I put it in as a formula, so use the get column info route to change those), and the linkage disequilibrium. Then run the script in the upper left hand corner. Finally, if the gene frequencies are other than 0.5 and in linkage equilibrium use sequential (type 1) sums of squares. Type 3 sums of squares will give you the wrong answer. If you have any questions feel free to ask me. OK, if you want the program I need to send it to you under a separate cover, so email me if you would like it. If I ever figure it out I will fix tings.

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