Adding up is hard to do (warning: boring technical post)

(this was meant to nicely complements a post at Nature’s Journal Club
, but it looks as if we’ll have to wait for that one to appear in public)
One of science’s great secrets is that the most influential scientist of the early 20^th^ century was not Einstein, but R.A. Fisher. Einstein did some cool stuff in physics, but confined his science to this area. On the other hand, Fisher managed to father both modern statistics and evolutionary biology. the latter work also had a huge effect on agriculture and breeding (over and above his working out how to do field trials properly).
One of his early contributions to evolutionary biology was his technique for decomposing the effects of many genes on a trait, by decomposing the variance in a population (as a side-product he invented the term Analysis of Variance). What he showed was the the genetic contribution to the correlation between phenotypes depended on the additive genetic variance, which is the average effect a gene has on the phenotype in the population. This is then important because the rate of evolution depends on this variance: the more additive genetic variance for a trait, the faster it can respond to selection. The genetic value of an individual also depends on its contribution to the additive genetic variance, this is called the breeding value, because of its importance for deciding which animals to breed stock from.
Of course, gene actions are not that simple: in diploids like humans, the effect of one gene may depend on the other gene in the individual, for example of one gene is dominant (i.e. masks the effect of the other). There can also be interactions between different genes, so the effect of one gene may depend on what alleles there are at another locus (epistasis).
Where things start to get confusing with Fisher’s approach is when we find out that even these non-additive effects contribute to the additive variance: they still affect the average effect of a gene. As if that is not bad enough, the effect changes depending on the frequency of an allele. For example, if a recessive allele (i.e. one whose effect is masked by any other allele) is rare in a population, then it contributes almost nothing to the variance, because it almost never appears as two copies in the same individual.
So, additive genetic variance is important, but there are other forms of genetic variance. If we want to understand the genetic variation in populations, we should really understand how important these other forms are. This is especially important now that we are starting to approach quantitative genetic variation from another angle, finding the genes (or parts of the genome close to a gene) that influence the traits. These genes can show the different forms of interaction, but does this still convert into additive variance?
This forms the background for my latest work as an agent of the AAAS. I wrote a short commentary on a paper by Hill et al1. In this they ask how important additive genetic variance is. Part of their investigations revolved around asking how much genetic variation was additive when different types of non-additivity were operating. Now, this depends on both the effects of the genes, but also the gene frequencies. But if a trait is affected by many genes, then we only need to know the distribution of the gene frequencies, and then we can average over the distribution. Hill et al. do this for several models with extreme non-additive gene actions, and find that most genetic variation is still additive.
This worried me slightly, initially because in only needs three or four genes to have an effect on a trait for it to look like it is affected by many genes (in other words, our gene counting process goes 1,2,3,infinity). But then the deviations from the average could be considerable. So, I started to play around with the models, to see how gene frequencies affected the results.
I started with the simple model of one locus, with two alleles, so that the only non-additive effect is dominance. Using complete dominance I plotted the additive genetic variance (_V~A~_), total genetic variance(_V~G~_), and the ratio of the two (_V~A~_/_V~G~_):

We can see that all three depend on the genotype frequency, p. V~A~ and V~G~ both have a maximum at intermediate values. But the ratio has a maximum at zero, and decreases. The realistic distributions Hill et al. averaged over mainly had genotype frequencies near 0 or 1. So, half the time they were near the maximum (the the other half near the minimum), but I think more importantly, most of the time they were averaging over situations with almost no genetic variance – precisely the cases when we might conclude that there is no genetic effect. It is not clear, then what this says about the traits we declare as having a genetic basis.
That wasn’t enough for me, so I moved on to models with more than one gene, and with interactions (i.e. epistasis). Firstly was an additive by additive model (where the effect of an allele at locus 1 is additive, but the amount by which it is additive is determined by locus 2, in an additive way). We need to work in three dimensions, which is as good an excuse as any to exploit the nifty plotting in R, and gives us these colourful figures:

The x- and y- axes are the allele frequencies, and the z-axis is the amount of variance (or the ratio of additive to total genetic variance, in the third figure). The additive genetic variance and total genetic variances are both smallest when allele frequencies are rare, and this is (a) when the ratio of additive to total genetic variance is greatest, and (b) the most common realistic frequencies examined by Hill et al. So, again additive variance dominates most when there is little genetic variance.
The third set of parameters I tried were the complementary model: if either locus only had the recessive allele, the trait had a value of 0, otherwise it was 1. This gave these figures:

We can see that the maximum variance occurs when both alleles are rare, but not extremely rare. Over most of the parameter space the genetic variance is very small. But for this type of epistasis, there is no simple relationship between the total genetic variance and the proportion of additive genetic variance.
What does all this show? My main point is that the argument by Hill et al. does not give the full answer. on average genetic variance is mainly genetic, but there can be considerable deviations from that. So, their results may not say a great deal about any particular case. And in some of these cases, when there is observable genetic variation for a trait, the proportion of additive variance can be relatively low. So, things are not as simple as the authors make out (they never are, are they?).
Another point worth bringing up is that the authors consider extreme forms of epistasis, so their results may well be biased away from additivity in that way. Overall, the answer to the question of how much genetic variance is additive must come from the data: theory can be twisted both ways.

1 Hill, W.G., Goddard, M.E., Visscher, P.M., Mackay, T.F. (2008). Data and Theory Point to Mainly Additive Genetic Variance for Complex Traits. PLoS Genetics, 4(2), e1000008. DOI: 10.1371/journal.pgen.1000008

About rpg

Scientist, poet, gadfly
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7 Responses to Adding up is hard to do (warning: boring technical post)

  1. Maxine Clarke says:

    I’ve followed up on the Nature journal club blog: there have been a few problems but it should all be up to date by the end of today, I’m told.

  2. Bronwen Dekker says:

    Am embarrassed to say that I had not heard of Ronald Fisher… I have just read the Wikipedia entry and it does indeed sound like he was a very influential thinker (and had quite an interesting life).

  3. Bob O'Hara says:

    Thanks, Maxine. We’ll see if any comments arrive there, other than me pointing to this post.
    Bronwen – we’ll excuse you this time.

  4. Lee Turnpenny says:

    You should edit your deprecation to remove the word ‘boring’, because you provide a very nice exposition of complex subject matter.

  5. Heather Etchevers says:

    One more proof of the inverse proportional relationship between the scientific content of a post and the number of comments it elicits.
    When I read such posts I think, “sounds convincing,” which is where I would have been had I tried to slog my way through the original article by Hill et al. That is, I don’t have the desire to become sufficiently savvy to have a critical mind, here.
    I’d love to hear an example of how this affects breeding choices. For instance, I had a vague memory about an article I read 9 years ago now about the phenotypic association of white forelocks and tameness in foxes – that brought me today here.
    The story of the Paint horses is suspiciously reminiscent of the variable effect of Sox10 mutations.
    I wonder if the associated nervousness has something to do with brain demyelination?
    But getting back to what I meant to: from an embryological standpoint, it’s not easy for me to explain the association of tameness with facial depigmentation. (Pigment with skull shape, yes.) So I’d like to say there are two genes involved; either physically close on a chromosome and therefore usually segregating together, or in functional epistasis.
    What would one do to tease these possibilities apart? (It’s probably fairly obvious but not to me!)

  6. Chen Guo-Bo says:

    It seems to me that Dr. Hill and Dr. Visscher are few guys working in a quit classical method developed by R.A.Fisher still, maybe the only group now…
    It’s a quit interesting question. I’m thinking whether the magnitude of effects will be an concern or not…
    One assumption underling their paper was “multiple rivaled small effect genes.”

  7. Bob O'Hara says:

    Chen – I don’t know about human genetics, but this framework is still used in animal breeding and in evolutionary quantitative genetics.
    The assumption of multiple genes isn’t for their analysis, except that it makes the effect of averaging stronger, so my concerns won’t be so problematic. BTW, human height seems to be affected by at least 54 genes
    Heather – sorry, I should have answered you earlier! Do you mean pleiotropy, rather than epistasis? The problem isn’t easy, of course. One possibility is to try to knock out the genes through mutation. The maize people had a really neat trick for this called t-tagging. They would prod their maize in the right way so that one type of transposon would jump around, and then screen the offspring for changes in phenotype. Of course, this is likely to happen when the transposon inserts itself into a gene, and knocks out its effect. So, if two genes are next to each other and it only knocks out one, you’ll only see one phenotype change. The really cool thing is that you also know the sequence of the transposon, so searching for it is much easier.
    Wouldn’t work in foxes, mind you. We’d have to take on walkies past Chernobyl to do the mutation instead.

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