Last week I talked about how Fisher’s original definition of additive genetic variance failed when average effects don’t equal average excesses. Basically this happens when ever there is some form of linkage disequilibrium. This gives us the easiest framework in which to think about this situation, that is in a “metapopulation”. A metapopulation is a population of populations that are partially connected by some level of migration, although being derived from a single ancestral population also works.

So, in theory Fisher’s additive genetic variance would technically be measured across the entire metapopulation. This would, of course be useless at best, and more likely down right misleading. The obvious solution is to measure the additive genetic variance separately for each population, but that leaves us with the problem of what do we do with migrants? In particular, what is the fate of genes as they move from one population to other. In an additive world we know what will happen. The gene will have the same effect on the phenotype as it did before.

Because the average effect is a function of the population mean, it will change, but that shift will be consistent for all alleles. Thus, if we considered two alleles. In a particular sub-population of interest allele A1 on average subtracts 1.5 inches to the height of an individual relative to the mean, and allele A2 on average adds 2 inches to the height of an individual relative to the mean. In this case the difference between the two alleles is 3.5 inches. If we move both of those alleles into another deme and measure their average effects will change because the population mean will change, but the difference between them will remain a constant 3.5 inches.

What this means is that if we are selecting for taller stature the A2 allele is always better than the A1 allele, regardless of which population we measure it in.

Fisher’s average effects would technically be measured as a deviation from the metapopulation mean averaged across all of the subpopulations. This would of course be a useless measure. Instead, we want to measure a separate average effect for each subpopulation, but continue to measure it as a deviation from the metapopulation mean. To be consistent with Fisher I have re-named these as “local average effects”.

If we add gene interaction things become more complicated. I will save you the math, but if you are interested in Goodnight 2000 (Heredity 84: 587-598) I used a Markov chain two-locus two-allele approach to examine these local average effects for all of the possible two-locus genetic effects. In this I found that the best thing to measure was the “variance in local average effects”, which conceptually is the variance in what a single allele does to the phenotype after correcting for the population mean. As you might expect from the figure above the variance in the local average effect in an additive system is zero. In a non-additive system it is very much non-zero. For a system with dominance or epistasis the result would look more like this:

That is, the relative difference between our pair of alleles shifts as they are moved among subpopulations, and in some cases may even reverse. In other words, the variance in local average effects is decidedly not zero.

The beauty of the Markov chain approach is that ANY one or two locus interaction an be examined. The first step is to examine the 8 “pure” forms of genetic effects (additive A locus, additive B locus, dominance A locus, dominance B locus, additive by additive epistasis, additive by dominance epistasis, dominance by additive epistasis and dominance by dominance epistasis). To examine these I start at a gene frequency of 0.5 for both loci, and allow them to undergo drift, which eventually leads to fixation and loss of all variation.

The first conclusion is that all of the forms of gene interaction show a conversion to additive genetic variance. Strictly additive effects decline linearly as inbreeding coefficient increases (yes, the traditional (1-f)σ2 works in the additive world). The other effects show a maximal conversion at an intermediate inbreeding coefficient.

More interesting is the variance in the local average effects of alleles:

Importantly the variance in the local average effects for an additive system is zero, but it is non-zero for all other forms of genetic effects.

So, what this means in common terms is that the same allele is doing different things in different subpopulations. In some subpopulations it may increase the size of an individual, and in others it may decrease the size of an individual. An allele may be the “good gene” in one subpopulation, and the “bad gene” in another population.

In short, what this is saying is that there are two types of population differentiation. We are all familiar with populations differentiating for their mean. In fact, in introductory population genetics we are told that the genetic variance among demes is 2fσ2. This is simply the expected variance among demes in which only additive effects are acting. It is not surprising that we never learned about differentiation for average effects, if for no other reason than it does not occur in additive systems. With non-additive systems the differentiation for average effects is important, and often very uncoupled with the variance in population means. Compare the variance in population means below with the variance in local average effects above.

Note that there is little correlation between differentiation for population means and differentiation for local average effects. Additive effects cause substantial differentiation for population means, but none for local average effects. Conversely dominance by dominance epistasis causes virtually no differentiation for population means, but has a huge effect on local average effects.

There has always been the question about how differentiated do two populations have to be before they can be considered separate species. When that question is asked people are usually asking how different do the population means have to be before they can be considered different species. Note that this includes questions such as how great a genetic distance do you need for speciation to occur. It should be clear at this point that this is the wrong question to ask. It is rather like asking what color car can go the fastest. A better question would be how differentiated for local average effects do two populations have to be before they can be considered different species. I don’t have the answer for that, but this is a question that has a meaningful answer and ends up being a substantial change in perspective.

Which color is fastest?

Finally, returning to my blog of two weeks ago, I suggested that one migrant every four generations was optimal. At the time I based it on the additive genetic variance:

but now I can add the second part of the story:

The variance in local average effects declines as the number of migrants increases. At one migrant every 4 generations the additive variance is maximized, and the variance in local average effects is still very large. Basically one migrant every four generations is the sweet spot between maximizing both the additive genetic variance and the variance in local average effects.