Gene Expression

Why overdominance ain’t all that

A few days ago I posted about how overdominance, the fitness advantage of a heterozygote (an Aa genotype instead of an AA or aa genotype), can maintain polymorphism (genetic variation) within a population at a locus. Roughly, the equilibrium ratio between the two alleles is determined by their respective fitnesses in the homozygote state. For example, if AA & aa are of equal fitness and of lower fitness than the heterozygote then the final equilibrium frequencies will be 0.5.

Heterozygote advantage is intuitively comprehensible to many people, after all we’ve all heard of “hybrid vigor.” During the genesis of theoretical evolutionary genetics some thinkers promoted the idea that balancing selection of various kinds, including heterozygote advantage, would result in a non-trivial proportion of extant genetic variation within a population. Coalescing around Sewall Wright and Theodosius Dobzhansky this was the “Balance School,” which stood in contrast with the “Classical School” which took its lead from R.A. Fisher, who emphasized directional selection constraining polymorphism and either fixing or eliminating mutant alleles within a population. In 1966 Richard Lewontin and J.L. Hubby published A molecular approach to the study of genic heterozygosity in natural populations. II. Amount of variation and degree of heterozygosity in natural populations of Drosophila pseudoobscura (full paper at link), which showed that the extant of genetic variation as inferred from allozyme diversity was far greater than either the Balance or Classical School has predicted. At first Lewontin & Hubby offered the idea that selection for heterozyogosity might be maintaining the variation that they observed (Lewontin was a student of Dobzhansky), but soon enough they rejected this on theoretical grounds. The logic is simple, so I will first go through the formal motions.

Consider a population whose fitness can modeled as such:

mean fitness = 1 – (selection against p1)*(p12) – (selection against p2)*(p22)

Roughly, this model shows the decrement in fitness from the heterozygote maximum which is generated by resegregation of homozogyotes each generation within a population at equilibrium.1 The fitness of homozygotes is weighted by their frequency within the population, and this is removed from the ideal maximum, normalized to 1. But this is only at one locus. The reality is that organisms exhibit a complex genetic architecture and innumerable loci. So let’s assume that fitness is multiplicative, that is, the sum total fitness of an organism is the product of fitness across individual loci. Ergo:

mean fitness across all loci = product of mean fitnesses across all loci

Let’s assume that the fitness of the two homozygotes across a number of loci is the same so that the proportion of alleles 1 & 2 at each diallelic locus is 0.5. Our relation simplifies to2:

mean fitness across all loci = (1 – 0.5*(homozygote fitness))(number of loci)

Now, by definition we know that homozygote fitness is going to be less than 1 because we’re modeling overdominance. Let’s assume that the homozygotes are all of fitness 0.85 (selection against = 0.15), and assume that there are 100 such loci. We obtain a mean fitness of:

0.0004 = (1 – 0.5*0.150)100

So the you see the problem, if we assume widespread heterozygote advantage very few individuals within a population would resegregate across independent loci so as to attain the maximum fitness. In fact, if you assume that some of the homozygotes are lethal you are trapped by the same problem that you encounter in inbreeding: it becomes difficult for a zygote to run the gauntlet of the myriad possibilities of a fatal genetic combination and be viable.

Then why variation? Neutral Theory and its heirs!

1 – First, homozygote matings produce only homozygote matings. Second, matings of heterozygotes with homozygotes and other heterozygotes also produce homozygotes.

2 – The formalism would be mean(w) = 1 – s1p12 – s2p22, where s1 = s2 and p12 = p22. This means you really have 1 – sp2 – sp2, where p = 0.5, so algebraically it is equivalent to 1 – 0.5(s).

Comments

  1. #1 J Thomas
    April 18, 2007

    While there’s every reason to think many alternative alleles are neutral, the 0.0004 number is not a great argument against balance theory. It’s only an argument against 100 loci with heterozygote advantages being multiplicative.

  2. #2 razib
    April 18, 2007

    which would be why i said:

    At first Lewontin & Hubby offered the idea that selection for heterozyogosity might be maintaining the variation that they observed (Lewontin was a student of Dobzhansky), but soon enough they rejected this on theoretical grounds. The logic is simple, so I will first go through the formal motions.

    but thanks for rebutting an argument i didn’t make!

  3. #3 J Thomas
    April 19, 2007

    At first Lewontin & Hubby offered the idea that selection for heterozyogosity might be maintaining the variation that they observed (Lewontin was a student of Dobzhansky), but soon enough they rejected this on theoretical grounds. The logic is simple, so I will…

    I didn’t intend to claim you made the argument. I merely wanted to point out that the argument only works with multiplicative fitnesses.

    So if there is selection for heterozygosity across a large number of genes, it must be that the selection among those genes is not multiplicative. And of course the multiplicative assumption is convenient but there’s no particular reason to expect it would happen all that often….

  4. #4 razib
    April 19, 2007

    And of course the multiplicative assumption is convenient but there’s no particular reason to expect it would happen all that often….

    it doesn’t need to. even if it was a very small proportion of the genome it would cause problems. there are, after all, more than 100 loci on a “typical” genome.

  5. #5 J Thomas
    April 20, 2007

    Sure, but that means the alleles with multiplicative fitness wouldn’t be the ones that showed the variation.

    All the others (at least all the others that didn’t fail for some other reason yet to be determined) could perhaps be selectively maintained.

  6. #6 razib
    April 20, 2007

    All the others (at least all the others that didn’t fail for some other reason yet to be determined) could perhaps be selectively maintained.

    do you mean selectively maintained via heterozygote advantage, or other evo dynamics (e.g.,freq. dep., statistical epistasis, spatial selec., etc.?).

  7. #7 J Thomas
    April 20, 2007

    We’ve established that it’s hard for selection to maintain a whole lot of multiplicative alleles with heterozygote advantage. If the average individual has two offspring that each share half its genes, then the fantasticly rare individual that’s heterozygous for a hundred loci might on average have thousands of surviving offspring.

    I’m supposing that there are various ways that the selection could be arranged that would select for a lot of heterozygotes at a time. But it seems implausible the selection would be multiplicative, as you have shown.

  8. #8 razib
    April 20, 2007

    I’m supposing that there are various ways that the selection could be arranged that would select for a lot of heterozygotes at a time.

    say more.

  9. #9 J Thomas
    April 21, 2007

    OK, first I want to warn you that I’m not a professional geneticist. And I’m going to do a lot of handwaving.

    I think selection is very often frequency-dependent. For example, if you grow E Coli in continuous culture with lactose as the sole carbon source, there’s a big advantage to better transporting and using lactose. Get a mutant that does that, and if it gets past the time it can be washed out at random, it will increase. But when it increases enough that the concentration of lactose in the medium is low enough the wild-type can’t keep up, then the wild-type will get washed out faster than they can maintain themselves. The population has increased, everything grows slower with the reduced lactose concentration, and the wildtype can’t even maintain its numbers — because the environment has changed. The mutants changed it. There’s no reason to think that even the relative fitnesses will stay unchanged while the environment changes.

    Now I’m all ready to outline an example showing that it’s *possible* for heterozygote advantage to preserve a hundred pairs of alleles at once. Something along the lines of a whopping selective advantage for individuals that have x number more heterozygotes than the norm. Each individual heterozygote will on average be in an invidual that has n+1 total heterozygote sites where the comparable homozygotes will be in individuals that have only n heterozygote sites, which will be enough to provide some selection against the homozygotes, and more against the one that’s more common. But it occurs to me that you probably already accept that it’s *possible*, and the question is how likely it is. So I should ask if you agree that far before I go to the trouble of laying out the details.

    I would make an example that shows it’s possible, without any particular reason to think that exact example was at all plausible. Is some version of density-dependent multi-heterozygote selection plausible? How about all of them put together? I think this is a much harder question and I don’t know how we’d look for data about it. I could do a lot of handwaving. Various songbirds and elephant whales collect territories; the ones that are best at holding territories do much more of the reproduction. All that matters is being best against the current field. Maybe being best might sometimes involve being heterozygous at some particular sites? Probably I could make it seem plausible that some selection can work to maintain heterozygosity in a lot of sites at once, for real. Or maybe not.

    And of course neutral mutations do spread. Neutral mutations will on average increase slowly whenever there is no selection at all within their linkage group, and a particular neutral mutation will increase fast when a selected mutant happens to get linked to it, but the variability as a whole within the linkage group will drop fast — all the neutral mutations that aren’t linked to the selected mutation will decline as fast as it increases.

  10. #10 razib
    April 22, 2007

    Something along the lines of a whopping selective advantage for individuals that have x number more heterozygotes than the norm.

    first, thanks for the comment! this is what i like the comment boards to turn into. second, negative frequency dependent selection can maintain heterozygosity, but as a byproduct. that is, at MHC it looks like balancing selection has been maintained by strong frequency dependent effects (i.e., host-pathogen coevolution where rare morphs always have selective advantage against disease). but there is less confidence that it is heterosis itself which is elevating the fitness then that heterozygotes by their nature are more likely to have multiple rare morph alleles and/or populations with many extant rare allelic morphs will naturally more be heterozygous. do you see what i’m getting at? the problem with widespread heterozygote advantage in the way postulated above where homozygotes increase genetic load is because the variance in fitness in the population is empirically untenable (assuming appropriate boundaries of n). on the other hand, if frequency dependence effects selective for rare morphs in an additive & independent fashion then heterozygosity will be maintained as a matter of course (because homozygosity requires matching at the locus of alleles). individuals who are homozygotes though are not necessarily less fit in this scenario than heterozygotes if they carry rare alleles. that is, homozy. that is (rare1)(rare1) is more fit than heterozy. that is (common1)(common2).

    in any case, there are frequency dependent models as well as ecological balancing selection (spatial and temporal variation in fitness) which emerged in response to the lewontin & hubby paper. i will explore some of the ecological genetic models soon, or at least outline the bare outline of the formalism. it seems that on the molecular level the preponderance of the researchers would contend that neutral/nearly neutral dynamics are generating most of the polymorphism. but, MHC is a counterargument.

    note: there is some data which implies that heterozygosity qua heterozyogsity might increase fitness at MHC loci above & apart from the negative frequency dependent effects.

  11. #11 J Thomas
    April 22, 2007

    Razib, thank you.

    Let me see if we’re on the same wavelength. You’re saying that it’s untenable to have a whopping heterozygote advantage because that gives the common individuals too much of a genetic load — their fitness is too low, they can’t survive.

    I think this doesn’t have to be right, but what I say is not so right may not be what you’re saying.

    There is a big difference between fitness and survival. Say that every female lays 20 eggs, the environment will sort things out so that on average 10 of them survive. If a rare mutation or a collection of rare mutations result in all 20 eggs surviving, that’s a big deal — but it’s only twice as many offspring, and it doesn’t reduce the survival of other eggs much while it’s rare. If that doubling is spread over a hundred sites, that’s still room for each of them to get about a 1% advantage. The individuals that don’t have *any* of these advantages might do perfectly well when the more-fit individuals aren’t present. It’s only in competition with the others that they would tend to lose out.

    And when it’s heterozygotes that get that advantage they tend to balance. Whichever allele is in lower frequency has relatively fewer homozygotes and more heterozygotes, and it will tend to be selected.

    What I would expect from that would be a lot of alleles with >10% frequency, and that isn’t what people usually see. So I think the math doesn’t show it’s impossible — unless the increased fitness of the individuals with many heterozygotes requires absurdly low survival of the norm. It could happen. But it isn’t what’s usually there to explain.

    If you encourage me even a little I will suggest a wild idea for selection producing heterozygosity that tends toward one very high-frequency allele and one or many low frequency alleles. I came up with it over 20 years ago and I haven’t kept up with the literature to see whether someone has published it, so it may be old news.

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