Evolution is sampling error

by willbradshaw5 min read23rd May 20202 comments

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World Modeling
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This is a linkpost for a (somewhat old) post on my blog apomorphic.com. I'm posting this and other old posts here to get feedback and in hopeful preparation for some follow-on work[1].

See also: An addendum on effective population size

A common mistake people make about evolution is to think it's all about natural selection and adaptation. In fact, random non-adaptive changes often dominate the evolutionary process.

Today I'm going to lay out a useful framework that I hope makes this fact more intuitive, which might in turn help non-experts build better intuitive models of evolutionary processes. This will come in handy when I try to explain non-adaptive theories of ageing later on.

Sampling error and genetic drift

We can think of evolution as sampling error: deviation in the genetic composition of the offspring in a population relative to their parents. To illustrate this, let's imagine a simple, asexual population, evenly divided between two gene variants (alleles) which produce no difference in fitness[2]:

Ten dots representing individuals, stacked vertically and coloured to represent their genotype: five red and five blue

These individuals will reproduce, giving rise to the next generation. Since all the individuals are genetically identical and have the same chance of reproducing, we can think of these offspring being randomly sampled, with replacement, from the previous generation:

Two columns of ten dots, stacked vertically, with lines between the columns representing parentage. Some individuals produce no offspring, some one, and some more than one.

Since there is a great deal of randomness involved in who reproduces successfully and whose offspring survive, not all individuals will produce the same number of offspring in the next generation, even though they all had the same probability of reproducing to begin with. As a result, even in the absence of selection effects, the allele distribution of the new generation is likely to differ from that of the previous generation; this random, unbiased change in allele distribution is known as genetic drift.

As a result of genetic drift, the allele distribution will fluctuate up and down stochastically; sooner or later, one or the other will be eliminated from the population, resulting in fixation:

Twenty columns of ten dots, stacked vertically, with lines between the columns representing parentage. The red allele reaches fixation at generation 14.

The time to fixation depends on the population size[3] and some other population parameters; here's an example plot for a population with a carrying capacity of 200 instead of 10:

A plot of the allele frequency distribution of a larger population, reaching fixation at roughly generation 350.

Fairly dramatic genetic changes, then, can accumulate in a population based purely on genetic drift; there's not necessarily any need to invoke selection to explain why genetic differences between populations accumulate over time. That said, what happens when we add selection into the mix?

Natural selection is sampling bias

Suppose that one of the starting alleles in the population is less fit than the other: individuals with that allele are less likely to produce reproductively-successful offspring. What happens now?

If one allele is much less fit than the other, the individuals bearing it will probably die without issue, producing a very boring plot:

Another twenty-column plot, this time in green and purple. The single initial purple individual produces no offspring, so the purple lineage dies out immediately.

So far, so trivial. The interesting cases occur when the fitness of one genotype is close to (either a bit higher or a bit lower than) the old one. In this case, thanks to genetic drift, the less-fit allele (here in purple) can persist in the population for a surprisingly long time...

Another twenty-stage green-and-purple plot, this time representing two alleles with only a small difference in fitness. The less-fit purple allele persists until stage 18, then is lost.

...or even fix!

An independent run of the scenario from the previous figure. This time, the slightly-less-fit purple allele reaches fixation at stage 6.

Overall, under these conditions (carrying capacity = 10, relative fitness ~ 0.9) the less-fit allele will reach fixation about a quarter of the time; more than enough for 100% the population to be bearing many deleterious alleles.

These results are a pretty trivial application of statistics, but they have very important implications for how we should view evolution. Thanks to genetic drift, beneficial mutations will often die out and deleterious ones reach fixation. How often this occurs depends on various factors, the most obvious of which is the magnitude of the mutation's effect on fitness -- the more dramatic the effect, the greater selection's ability to overcome drift and eliminate the less-fit allele.

However, another crucial variable, underappreciated outside evolutionary biology, is population size.

Evolution and the law of large numbers

According to the law of large numbers, the average of a sample converges in probability towards its expected value as sample size increases: the larger the sample, the smaller the expected relative mean absolute difference between the sample mean and the expected value[4]. If you flip a coin ten times, the chance of deviating from the expected value (five heads) by at least 20% is more than 75%, whereas it's only 5% if you flip 100 times and virtually zero if you flip 1000 times. The larger the sample, the more likely you are to see roughly what you expect.

In our framework of evolution as sampling error, natural selection determines the expected value: the number of offspring of each genotype we expect to see in the next generation, given the distribution in the current generation. But the smaller the population, the more likely it is to deviate substantially from this expectation – that is, for random genetic drift to overwhelm the bias imposed by natural selection.

If you combine this sample-size-dependent variability with the absorbing nature of fixation and elimination (that is, once an allele has been eliminated, it isn't coming back), you obtain the result that the larger the population, the more likely it is that the fitter allele is actually the one that gets fixed, all else equal. We can see this in our toy model from earlier, where the green allele is 10% fitter than the purple allele and both start with 50% prevalence in the population:

A plot of population size vs probability of fixation for two competing alleles, one of which is 10% fitter than the other. The fitter allele fixes at just over 50% when population size is very small, rising to 100% at population sizes of 100 or larger.

When population size is very small, the chance that the fitter (green) allele is the one that eventually fixes is close to 50%; as population size increases, this probability increases, until for sufficiently-large populations it is virtually certain. Smaller differences in fitness would require larger population sizes to consistently fix the fitter allele[5].

Population size, then, is a crucial factor affecting the optimisation power of evolution: the larger the population size, the greater the capacity of natural selection to select for beneficial mutations and eliminate deleterious ones. This is why bottlenecks and founder effects are so important in evolution: by reducing the size of the population, they both increase the relative prevalence of rare mutations and decrease the relative strength of natural selection, resulting in very powerful drift. The results of this can be quite striking: on the tiny Micronesian island of Pingelap, for example, almost 10% of the population are completely colourblind, a condition that is extremely rare elsewhere[6]. This is believed to be the result of a typhoon in 1775 that left only 20 survivors, one of whom was a carrier of the condition[7].

Implications

What can we infer from all this? Firstly, when thinking about evolutionary processes it's vital not to neglect genetic drift. Just because something spread throughout a population and reached fixation does not mean it is adaptive. Secondly, this is especially true when populations are small, and we should always pay careful attention to population size when thinking about how a population might evolve. In general, we should expect larger populations[8] to be fitter than smaller ones, since (among other things) natural selection will be more effective at weeding out deleterious alleles and propagating beneficial ones.

Finally, it has not escaped my notice that this framework has obvious implications for thinking about analogous evolutionary processes that might occur outside of biology. More on this anon.


  1. Some of the images are hilariously huge here due to LessWrong's CSS defaults; there doesn't seem to be anything I can do about this within the markdown editor, though I'd love to be proven wrong. ↩︎

  2. I'm also assuming non-overlapping generations and a constant carrying capacity; relaxing these assumptions makes the maths more complicated but shouldn't alter the basic conclusion. Similarly, while new genetic variants are capable of spreading much more quickly through sexual populations, the same basic phenomena still apply. ↩︎

  3. Actually, the time to fixation (and many other aspects of the population's behaviour) depend on its effective population size, which depends not only on its actual population size but also on various demographic and genetic factors. This is an absolutely crucial distinction that I am eliding here for the sake of brevity (in my defence, population geneticists seem to also do the same thing when speaking casually). Effective population sizes are often much smaller than actual ("census") sizes; for example, the usual estimate that gets bandied about for global human effective population size is roughly 10,000. ↩︎

  4. In fact, the RMD looks like it might vary as a power law of sample size: A log-log plot of relative mean absolute difference vs sample size, showing a very linear-looking relationship I noticed this from simulations and haven't bothered to tease out the underlying mathematics here, but still, kinda cool. ↩︎

  5. See e.g. this plot for a 1% difference in fitness: A plot of population size vs probability of fixation for two competing alleles, one of which is 1% fitter than the other. The fitter allele fixes at roughly 50% when population size is very small, rising to about 95% at a population size of 300. The rise in fixation rate is much slower than when the fitness difference was 10%. ↩︎

  6. According to Wikipedia, the proportion of Americans with the same condition is 0.003%. ↩︎

  7. It probably didn't hurt that the suspected carrier was also the chief of the island. ↩︎

  8. Again, I'm actually talking about effective population size here, not census size. ↩︎

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2 comments, sorted by Highlighting new comments since Today at 2:17 PM
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I am new on this website, and thus don't feel experienced enough to give global remarks on your post. For the same reason, please correct me if I am going against guidelines or acting in a way which is unusual on LessWrong.

I think I might have an explanation for your note:

4. In fact, the RMD looks like it might vary as a power law of sample size

Your samplings can probably be modeled through a random 1D symmetric motion, as suggested by your coin tossing example. This one is known to present a standard deviation proportional to the square-root of the sample size. The RMD should behave in a similar way, namely:


This is perfectly consistent with your graph: a factor 10000 on the x-axis gives a factor 1/100 on the y-axis.

Nice.

Edited to add:

For the same reason, please correct me if I am going against guidelines or acting in a way which is unusual on LessWrong.

This is a great comment and I upvoted it.