Effective population size for advantageous mutations

For neutral genetic diversity, the effective population size is the harmonic mean of the population over time. That is, you take the inverse of the population size, average that over the population history, and then invert it.  This harmonic mean is dominated  by  periods in which the population is very low – so, judging from the amount of neutral diversity, the effective human population size is about 20,0oo or so, rather than the census size of 7 billion.  Small population sizes back in the Old Stone Age, and possibly ancient bottlenecks associated with population crashes and expansions of small groups, probably explain this.

The story is very different for beneficial mutations. Mutations that confer much advantage are rare – sometimes very rare, when only a specific molecular change will do the job. This means that in a small population, you may have to wait a long time for the right favorable mutation – as much as hundreds of thousand of years in ancient humans, when you consider than most favorable mutations are lost by chance.

Anyhow,  you improve the chances, shorten the waiting time,  with a larger population.  And if the population stays large, avoids a crash,  for very long –  then many of the favorable alleles generated during the period of large population size will have had time to spread and become common, so they’re not likely to be lost in later crashes.

So, with beneficial mutations, the effective population size is very different. Instead of being dominated by bottlenecks, it is more influenced by eras of large population size – more and more so as the selective advantage of the mutation increases. In the limit, if we imagine  mutations so advantageous that they spread  very rapidly, the effective population size approaches the population mean.

For example, -13910*T , the European lactase persistence mutation, occurred in the context of a fair-sized population that was using milk from domesticated animals (probably cows, maybe horses or goats).  Suppose that Europeans go through a bottleneck – let 100 adults of reproductive age  make it to Alpha Centauri and start a colony.

Mutations like -13910*T won’t be lost.  On the other hand, recent advantageous mutations that have not had much time to spread, like that apoliprotein mutation in Limone Sul Garda, probably would be.

Limone Sul Garda

Limone Sul Garda

So the effective population size for advantageous mutations is a function of the population size and the degree of selective advantage.  I’ve never seen this in the texts, although I’ve seen Nick Barton mention it in an article about the evolution of pesticide resistance.  We ran into this when we were thinking about how the post-agricultural population expansion must have greatly increased the rate of creation of advantageous mutations  (as well as putting strong new selective pressures on the farmers).

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34 Responses to Effective population size for advantageous mutations

  1. rightsaidfred says:

    “Apoliprotein”

    Wow. Where do I sign up? Maybe if I believe strongly enough that every human trait is a social construct, I can get some of that.

    • jorge videla says:

      actually athero is a social problem in the sense that it isn’t found in primitive peoples even when they live long enough.

      • gcochran9 says:

        Yes, it is. Mummy CT scans show preindustrial hunter gatherers had clogged arteries. Even if hunter-gatherers were immune, that would hardly guarantee that it is a “social problem” in the modern world, or that it would necessarily be easy to fix. For example, if atherosclerosis were caused by a high-carbohydrate diet, well, most food production is grains, and so it’s not possible for most of the world’s population to avoid a high-carbohydrate diet.

      • aisaac says:

        I thought Kitavans didn’t have that problem. While trying to look that up I found that the evidence for that proposition is not as strong as I thought, but I also came across an interesting but highly speculative abstract proposing that antibodies to yaws are cardioprotective:
        http://www.ncbi.nlm.nih.gov/pubmed/?term=kitava+atherosclerosis

        A quick look around shows that there’s more evidence that IgM antibodies against PC are good guys, even if it’s not so certain that yaws has anything to do with it. Are there any known cases where autoimmunity caused by bacteria is a good thing?

      • jorge videla says:

        right. it has been found in inuit/eskimo mummies. i knew that. humans do appear to be meat eaters more than chimps, but nothing but meat? and athero and athero-disease are two different things. it’s possible to have athero and never have any clinical manifestation, because it’s so limited. and athero has also been found absent from wild animals according to hayflick.
        here are two (popularized) reports:
        http://www.empr.com/hunter-gatherer-lifestyle-tied-to-low-atherosclerosis-risk/article/242392/
        http://www.npr.org/templates/story/story.php?storyId=111889498
        and in his happy isles theroux reports that bp decreases with age in the trobriands. he was quoting a doctor he met there.

        “Even if…”

        right. even if the paleo-diet and exercise regime were the best possible diet-exercise regime, the paleo diet is too expensive for most people. yet it costs nothing for the few primitves left and nothing for the very few brave civilized people who leave civilization. but then someone who understands this shouldn’t claim his hereditarianism is anything other than SOCIAL darwinism.

      • jorge videla says:

        “Overall, the researchers found probable or definite atherosclerosis in 34 percent of the mummies studied”

        YET

        “Atherosclerosis starts very early in life. In the United States, most kids have little bumps on their arteries.”

        so everyone in the us but only 34% of ancient mummies!!!

    • jorge videla says:

      and the bigger point is that despite the explosion in population with agriculture cultural change may be so rapid that beneficial alleles aren’t benefcicial for long.

  2. Cloudswrest says:

    Re: -13910*T (ApoA-1 Milano) – According to Wikipedia, the protein coded by this, when intravenously injected, has remarkable therapeutic value in regard to atherosclerosis. Pfizer owns the rights but they seem to not be pursuing it. Given that exogenous application of this mutation’s products are therapeutic, this would seem to be a perfect application for the, allegedly robust, “Crisper” gene editing technology. Extract a bunch of relevant stem cells, crispr in the new mutation, and re-inject them back into the body – Voila!

  3. dave chamberlin says:

    I am no longer a big fan of science fiction because it sells out, meaning the writer rarely builds his plot around the scientifically plausible. For example let me take the information provided by Greg in this Blog and try to build a future that is plausible.

    The time of the functional simpletons came to an end within two generations. The baton was handed off to their tweaked children and the era of ordinary genius began. Fertilized eggs of prospective parents were held in a suspended state until three kinds of adjustments could be made. First the mutational load was cleaned away. Second adjustments were made for optimum synchronicity, to use an analogy to the eye everyone was given 15/20 minds eye vision. This effectively erased from the population everyone with an IQ below 145. But the third adjustment was the collection from all of mankind beneficial mutations influencing intelligence.
    The fourth adjustment was temporarily made illegal to appease the fundys, the religious fundamentalists who fought tooth and nail against any genetic engineering at all. That fourth adjustment was to build genes that did not yet exist to increase human intelligence further. The fundys refused any tweaking of their children’s genes and of course didn’t think of themselves as fundys but as normal people and the tweaked humans as freaks.

    This could then lead in a thousand directions. Dramatic conflict between the fundys and the freaks, or cooperation so that the fundys and freaks lived together harmoniously because they need each other. I love to wonder about these things.

    • little spoon says:

      “. This effectively erased from the population everyone with an IQ below 145.”

      You think this will happen? I kind of think it will. But then idiocracy could also happen.

      I seem to be a lady of smart fraction, and I have no children. I feel kind of bad for having no children as my childless kind contributes to the continual decrease of real IQ over generations. However, I sometimes figure that within a couple of hundred years, we will understand genetic engineering or gene based therapies well enough to make everyone smart anyway. Thus, perhaps I am wise and harming no one by not putting my energy and resources into having a child.

      • dave chamberlin says:

        “You think this will happen?”
        Nope, but at least it is built on the possible from from what we now know. I don’t pretend to know the future, but I also don’t dismiss that we can’t speculate with increasing accuracy of this kind of scenario as we move through time. We live in fascinating times and we might as well look around and enjoy the spectacle. I think it likely we are fast approaching a singularity and something like my fictional story will happen not too long after our lifetimes. We won’t be playing cowboys and indians on rocket ships in space as the hollywood science fiction likes to tell us. Better brains got us here and better brains will get us to where we are going and we won’t be able to imagine our destination any more than a man living in the 1700’s could imagine the life we live today.

        I like to think we are apart of something greater than ourselves, that we are almost at the point when the slow cruel march of evolution ends, and we are able to lift ourselves up from being just extra smart self centered apes. Stay tuned.
        ,

      • Wanderer says:

        Well, sure we will likely have genetic engineering, and then the rich and powerful will use it to improve their lineages, and all the men and women that contributed genes to you will curse you from their graves for not passing on their legacy.

      • Richard Sharpe says:

        and all the men and women that contributed genes to you will curse you from their graves for not passing on their legacy

        Are you on drugs?

        Surely the living are more likely to be thankful that she has chosen not to compete with their offspring?

      • sinij says:

        This is interesting problem. Is your Darwinian fitness reduced by genetically engineering beneficial traits to your direct offspring? That is, you chose not to pass some of your own genes.

        If so, is inclination to genetically engineer going to be naturally selected against?

      • jorge videla says:

        “fundys” are jut a distraction.

        the elite is opposed to eugenics because with it would come the necessary end to liberal capitalism. there would no longer be the “you know those people. what d’you expect.” argument for it. of course already wealth and iq are poorly correlated, and the effect of iq on income is mediated entirely by education.

      • banned56 says:

        “However, I sometimes figure that within a couple of hundred years, we will understand genetic engineering or gene based therapies well enough to make everyone smart anyway. Thus, perhaps I am wise and harming no one by not putting my energy and resources into having a child.”

        Or, you may be an utter fool. Perhaps it’s your (nonexistent) progeny who would have made the critical breakthrough. Meanwhile the dullards continue to breed without restraint, making ever more likely the Idiocracy.

    • aisaac says:

      Current high-fertility groups wouldn’t go for genetic manipulation for several reasons –

      Many of them are “fundies,” who, as you point out, won’t go for it, and that number will only grow with time as fundies continue to breed like crazy. The way things are going, Israel will tip ulra-orthodox this century, and other first world countries are a few centuries behind.

      For the rest of us, genetic manipulation will be expensive and will require prior planning, which will effectively put it out of reach of the functional simpletons. Functional simpletons probably don’t want their kids to be a bunch of nerds who talk like fags anyway.

      That leaves the middle class and the tiger moms. They have low fertility now, and genetic engineering would only depress it further, because they would come to believe that their kids HAVE TO HAVE the right DNA, so if they can’t afford it they won’t have them, just as they won’t have kids now if they can’t afford decent schools and other mostly unnecessary status symbols that middle-class and higher people think they need.

      You would get a small number of really smart kids, more than you do now as the top few percent diverge from everyone else, and just as many idiots.

      • sinij says:

        “Just as many idiots” is not really a problem. As population increases, cost of meeting basic needs will eventually contain the population of unenhanced. Genetic manipulation will provide a way to create an optimal number of enhanced, as decided by existing enhanced population.

        The only problem is that if initial enhanced population decides to narrowly focus on their own betterment over humanity’s greater good. (e.g. Star Trek’s Khan). Otherwise, controlled evolution and singularity.

    • Very likely different nations will choose different policies. In that case, those countries that embrace genetic engineering of humans will get a big advantage, even though there may be occasional disasters when the engineering doesn’t work the way it’s supposed to do.

      Might the Chinese be among the winners?

  4. Patrick Boyle says:

    Ah yes, the harmonic mean! I used to ask the class a simple question when I taught statistics. Almost everyone got it wrong.

    What is the average speed of 60 mph and 30 mph?

    • Anonymous says:

      Depends if you’re averaging over distance or over time.

      • Patrick Boyle says:

        Dear me, I misstated my own anecdote. I should have said a car travels from point A to Point B at 60 mph and returns at 30 mph. Most students just respond 45 mph and are surprised to learn that the right answer is 40 mph. This little puzzle is an attempt to get students think as I introduced the idea of the Harmonic Mean, but I scotched it. I stepped on my own punch line. My bad.

        BTW it’s interesting that In music the geometric mean is more relevant – not the harmonic mean as you might think.

    • Bert Derpleson says:

      Average speed is still a distance/time question. It’s certainly not incorrect to assume that you travel at 60mph for an hour and 30 mph for an hour, go 90mi in 2hr and average 45mph.

  5. dearieme says:

    It seems to be the norm in the teaching of statistics that questions are routinely so badly phrased that they mean little more than “Tell me what I am thinking about”. It’s when I realised that, as a fresher, that I became much more relaxed about it all. Once you know that your job is to guess what the devil an inarticulate and imprecise lecturer means, it’s all much easier, especially as the maths itself is so trivially easy.

  6. Jim says:

    dearieme – Statistics is not a trivial subject.

    • dearieme says:

      No, but some of the teaching of it is. I’ll tell you an exception – I once attended lectures by A W F Edwards. Clear as a bell: top man. But look at some of the stuff on probability and statistics for the beginner: kids of 16-19, say. The imprecision with which situations are described, and the ambiguity of the questions that are asked about them, are all too often a bloody embarrassment. Yet books instructing the same age group on, say, calculus are conspicuously superior (in my experience). This is mad: calculus is a piece of piss; prob and stats can be subtle and therefore need better teaching, not worse.

  7. Gottlieb says:

    http://bjp.rcpsych.org/content/184/4/327.full.pdf

    Correlation between ”psychosis” (schizophrenia and mood disorders) and high-achievers students (top 10), specially on maths, ”find” (not so recent study) in Iceland.
    Very interesting!

  8. Pingback: Minimum Viable Population vs. Effective Population Size | al fin next level

  9. RCB says:

    Old post, maybe no one will see this…

    My understanding is that effective population size is the size of a Wright-Fisher population (non-overlapping generations, random gamete union) that “behaves like” the population in question. “Behaves like” either refers to the rate of coalescence (Inbreeding effective population size) or the variance in allele frequency change across generations (Variance effective population size; measures the strength of drift for segregating loci). The harmonic-mean calculation applies to the former. The two needn’t be the same size; humans have a relatively small inbreeding effective size (not much neutral diversity), but I think we have a large variance effective population size (we are currently very large, so selection is much stronger than drift).

    I haven’t seen any effective population size that accounts for the rate of fixation of NEW mutations, e.g. the size of a selected Wright-Fisher population for which the rate of adaptive substitutions is equal to that of the population in question. This sounds like what Greg is getting at; we might call it the Adaptive effective population size. Have people published on this? I guess probably yes… but if not, a good opportunity.

    • gcochran9 says:

      I’ve never seen it in the literature but it might exist somewhere. Personally, I just simulated it for the population histories I wanted and for different values of selective advantage.

      • RCB says:

        Here’s some back-of-the-envelope calculations:

        Rate at which beneficial mutations arise: 2Nu (u = mutation rate)
        Rate at which beneficial mutations fix: approximately 2sN_e/N (N_e = variance effective population size; in a Wright-Fisher pop, N_e=N, yielding the familiar 2s. I think this assumes fully dominant mutation, but can’t recall at the moment.) There are more accurate calculations for this, but I’m going for linearity here. Actually, the rate is higher for growing populations, because more mutations survive the drift threshold. A model I did a while back gave me something like 2(s+r)N_e/N, where r is the intrinsic rate of increase for the population.

        Hence, the rate of adaptive evolution for a population that is far from equilibrium (and therefore will undergo many adaptive substitutions) is about 4u(s+r)*N_e. So, at least to a decent approximation, the rate of adaptive evolution is proportional to variance effective population size.

        An estimate of the total number of fixations over a long time period would be the average of this quantity times the total time period. This would imply that the arithmetic mean of the breeding population size (and growth rate) over time, not the harmonic mean, is what matters. Arithmetic means are not as affected by small numbers.

        About the variance effective population size is: It is not strongly affected (or at all?) by population history, because it only predicts the strength of drift at the time of calculation. It’s determined mostly by the size of the breeding population and reproductive skew, I think. So while recent growth means that our current inbreeding effective population size is something like 20k (based on neutral diversity estimates), our variance effective population size might be >1 billion, and has grown enormously since Out of Africa and agriculture.

        • RCB says:

          So, yes, a population that goes through a very small bottleneck to leave Africa will show low neutral diversity, because that bottleneck will weigh hugely in the harmonic mean population size. But if the population quickly grows to large size after that, and stays large, the bottleneck will have relatively little effect on adaptation to new environments. The arithmetic mean isn’t so sensitive.

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