Evolution · Math

Drunken Walks 2: Neutral Drift Boogaloo

In our last episode, our heroes saw that just by taking lots of steps in random directions, you could wind up arbitrarily far from where you started. This time, we’ll take a look at what it means for population genetics and evolution.

As we’ve all learned, evolution is “allele frequency change in a population over generations”. And the way we all visualize this, I think, is to think of a population in which one member has a slight advantage over the others, such as sharper teeth, better resistance to poison, slightly more efficient camouflage, etc. As generations pass, a greater and greater percentage of the population has this advantageous trait, until eventually it’s the norm. Either that, or detrimental traits get removed from the gene pool.

We can ask how quickly a beneficial or detrimental mutation spreads through a population. But that won’t tell us much unless we compare it to something. If a beneficial mutation spreads throughout the population in 200 generations, is that fast? Slow? How do we know?

To answer this, we should know how quickly a neutral mutation, one that makes no difference, behaves. This could be a mutation in which GCT becomes GCA: they both code for the amino acid alanine, so this mutation makes no difference. Neither variant gives an organism any advantage or disadvantage. Or it could be a gene that controls fur color, or or wing length, or seed shape, as long as (for our example) it provides no advantage or disadvantage.

I’ve written a Perl script to simulate this: it starts with a population with one pair of chromosomes, with one gene: variant (or allele) A on one chromosome, and allele B on the other. It then chooses two parents at random, picks one of the chromosomes at random from each parent to create the child’s genome. It repeats this until it reaches the desired population size. It also counts the number of organisms in each generation that are homozygous (have two copies of either allele A or allele B) or are heterozygous (have one copy of each). It does this for 1000 generations and prints the results (plotting the results is left as an exercise for the reader).

Here are some typical results for a population of 100,000 organisms, with time, in generations, along the horizontal axis:

As you can see, the number of heterozygotes tends to remain fairly constant, while the number of homozygotes goes up and down. At some points, these populations have had 10% more organisms with two copies of one allele than of the other, even though neither of the three possible variants gives any advantage whatsoever. This is just a random walk.

But what happens if, instead of 100,000 individuals, we look at a population of just 10,000? Here are a few such results:

Already, we see a lot more variation. This makes sense, since while the numbers are changing at the same rate, as absolute numbers, any difference constitutes a larger percentage of the population.

Although the number of heterozygotes remains roughly constant more often than not, this is by no means universal. In two of the examples above, one of the homozygotic variants has nearly disappeared, for no particularly good reason.

Let’s try with an even smaller population: 1000 individuals:

Here, the effect is even more pronounced, to the point where one allele can disappear entirely in less than 1000 generations, simply by a random walk.

And finally, let’s try with a tiny population: just 100 individuals:

Here, the effect is so extreme that one allele always ends up dominating, while the other disappears entirely, sometimes in less than 100 generations.

So what does it all mean?

For one thing, this shows how genetic diversity can be reduced in a small inbred population. The smaller the population, the more likely that some variant of a gene will disappear purely by chance.

This also touches on speciation: imagine that part of a population becomes isolated from the rest. Over the course of generations, the two groups will drift. Since there’s no reason for them to drift the same way, they’ll tend to drift apart. After a sufficient number of generations, they may become quite different from each other, and may even be unable or unwilling to interbreed even if the two groups are brought back together.

3 thoughts on “Drunken Walks 2: Neutral Drift Boogaloo

  1. This test seems to prove, or at the very least is consistent, with the story of Babel in the Bible. Each group separated and formed “common traits” for Chinese or African or Indian or any other. We see this happening in dogs, for example, where breeders breed common traits of dogs together to push out dominate traits. Unfortunately these extreme methods used by breeders like breeding siblings bad traits come out also as dominate.

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