Gene Expression

I know I’ve posted on this topic before, but I thought I’d revisit it again. You do know that sometimes population bottlenecks can actually result in more variation being freed up for selection? This may strike you as a bit strange; after all, the power of selection to effect phenotypic change is proportional to genetic variance, specifically, additive genetic variance. Population bottlenecks imply a reduction in effective population size, the increase of sample variance across generations, that is, random genetic drift. As population size drops the stochastic change in gene frequencies becames proportionately much greater and alleles rapidly go extinct, or fix, within populations (average time until fixations in generations is proportional to 4Ne, where Ne is effective population size). The homogenizing effect of this dynamic is similar to what might occur with inbreeding, where effective population size is reduced through population substructure, and individuals within the demes quickly become closely related over a few generations. Obviously you know that inbreeding leads to a loss of variation. So how exactly can we extract more additive genetic variance from this? In short, but converting other types of variance….

My exposition here is borrowed from Evolutionary Genetics: Case Studies and Concepts (an excellent book which has been much blog fodder over the past few years). We start with the idea of statistical epistasis; gene by gene interaction which varies the trait value. Additive genetic variance is pretty simple, you take a locus (gene) and substitue alleles (variants of genes) and see the effect it is has on the trait. Obviously you have variation on other genes, but for additive genetic variance you just assume the average of the genetic background; in other words, ceteris paribus is the order of the day. When it comes to epistasis this sort of averaging of the background won’t do because it’s the combinations across the genes which are relevant; think of epistatic effects as non-linear and additive ones as linear.

The example from Evolutionary Genetics uses additive-by-additive epistasis across two loci; the elements of the matrix are show the outcomes of genotype combinations.

  A1A1 A1A2 A2A2
B1B1 1 0 -1
B1B2 0 0 0
B2B2 -1 0 1

As you can see, the effect of the genotypes of A invert contingent upon the background of B. Now imagine that you start with a large panmictic population where A & B are at intermediate frequencies. Obviously there’s only a weak correlaton of phenotype with genotype for change on A in this case; no additive genetic variance. But if stochastic pressures result in a deviaton from a balance between the allelic variants of B, then A will contribute to additive genetic variance.

Here’s a figure which illustrates what I’m talking about….
i-c3568bb639e8a8a6f4ae100f430de81d-jasonwolf.jpg

R. A. Fisher famously believed that evolution via natural selection operated through mass action upon large panmictic populations. He might have been right, but in all cases? I’m not so sure anymore….

Related: The Shifting Balance.

Comments

  1. #1 agnostic
    May 15, 2008

    For people who haven’t seen this thing before, here’s what Razib is getting at –

    Let’s look at the genic fitness of each allele at the A locus, assuming random mating. We’ll only look across the top row for now. Take a single copy of A1 and see what else it could be paired with, and find the fitness of this combination. Set A1′s frequency to p. Then with probability p, A1 finds another A1, and fitness = 1; while with probability (1 – p) it finds an A2 and fitness = 0. In the top row, the weighted average of fitness of A1 = p.

    Repeating this for the middle row gives average fitness = p * 0 + (1 – p) * 0 = 0. And in the bottom row, average fitness = p * (-1) + (1 – p) * 0 = -p. In random mating, we are as likely to find ourselves in a B1 / B1 homozygote as in a B2 / B2 homozygote, so the average fitnesses of A1 are equally weighted for the top and bottom rows. Since these are p and -p, and since average fitness in the middle row is 0 no matter the weighting, average fitness of A1 across all possibilities at the B locus = 0.

    Do the exact same thing for the A2 allele, using (1 – p) for its frequency, and p for landing in the heterozygote, and we find the same thing. Average fitness of A2 across all possibilities at the B locus = 0. Therefore, selection cannot budge the frequency of either A1 or A2 since they have the exact same genic fitness.

    However, imagine that a population goes through a bottleneck, so that by chance the B2 allele is lost. Then the only genotype you’d find on the left is B1 / B1. If you look across the top row, you see that fitness increases by 1 unit as you go from right to left — that is, each copy of the A1 allele adds 1 unit of fitness. In this case, the A1 allele will sweep to fixation. (A1′s genic fitness is greater than that of A2: p vs. p – 1, respectively.)

    Had the bottleneck removed the B1 allele by chance, you would only find the bottom row, where fitness increases from left to right. So in this case, the A2 allele would fix. (A2′s genic fitness is greater than that of A1: 1 – p vs. -p, respectively.)

  2. #2 agnostic
    May 15, 2008

    To clarify, my comment assumes p = 1/2, so before the bottleneck, you aren’t more likely to find A1 or A2.

  3. #3 agnostic
    May 15, 2008

    Third try’s a charm. My original comment assumes the frequency of B1 and B2 — the background alleles — are both 1/2 before the bottleneck.

    [feel free to delete my second comment]

  4. #4 razib
    May 15, 2008

    i have no objection to the further exposition….

  5. #5 agnostic
    May 15, 2008

    i have no objection to the further exposition….

    Ha, yeah, why stop me when my pre-caffeine-high goofballery is providing free entertainment? Heheh. What I meant was, the frequency of A1 doesn’t matter, just that B1 and B2 are equally frequent.

    I have a feeling that when I’m a professor, I will chronically keep students a few minutes “after the bell” to fill in gaps in the lecture.

  6. #6 Caledonian
    May 15, 2008

    So by reducing variation, bottlenecks can eliminate gene combinations that are harmful as well as those that are beneficial?

    That makes sense. With some recessive genes that are only seriously bad in the homozygous state, it’s probably the only way to eliminate them from a population.

  7. #7 toto
    May 16, 2008

    For people who want to save a couple of brain cells, here’s what agnostic is getting at:

    “Epistasis” means that the effect of a gene is dependent on the presence or absence of other genes. For example, if you have gene B1, then having gene A1 is better than A2, but if instead you have gene B2, then the reverse is true. If this is the case, then natural selection has no reason to favour A1 over A2 across the whole population.

    But if, somehow, you eliminate a lot of individuals, and all remaining individuals have the same B-gene (say, B1), then there will be a clear advantage for one particular A-gene, namely A1. So in this new, decimated population, A1 will thrive and become fixed by natural selection.

    (For added precision, just replace “gene” with “allele”.)

    I have a feeling that when I’m a professor, I will chronically keep students a few minutes “after the bell” to fill in gaps in the lecture.

    Or you might do like one of my 1st year physics professors, who consistently ended his lectures with “OK, you’ll finish the rest at home”.

  8. #8 Fly
    May 16, 2008

    Continuing the thought…

    The fitness value for a trait often follows a “bathtub” curve. E.g., body temperatures that are too low or too high lower fitness.

    Ultimately traits are determined by the amount of specific proteins in specific cells. There are many ways that protein levels are regulated. E.g., gene transcription, mRNA splicing, mRNA translation, protein recycling. There are many biological processes that can affect each method of regulation. E.g., for gene transcription, promoters and inhibitors, copy number, etc.

    So for a specific trait there may be hundreds of genetic factors working together to produce that trait’s fitness “bathtub” curve. In an individual the fitness value of a particular genetic factor depends on all the other factors that are present. If due to genetic drift, the diversity of some genetic factor is reduced and the population average for the trait is no longer at a fitness optimum then the frequencies of the other genetic factors will change through selection to compensate. (The genetic factors are simultaneously affecting many different traits each having its own fitness curve. So there will be ripple effects throughout the biological system.)

    Over time, drift in sub populations “tests” many combinations of genetic factors. Robust genetic factor combinations that maintain homeostasis under a wide range of environments and genetic backgrounds evolve.

  9. #9 agnostic
    May 16, 2008

    For people who want to save a couple of brain cells,

    Hey, slogging through the math is good for their moral fiber.

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