6/24/2006

Genetic Drift and the Neutral Theory

(Population Genetics)

You'll hopefully recall that, in the last post on Population Genetics, I provided a short script for simulation of genetic drift. Hopefully you've all been happily playing away with it, trying to figure out the rule for how likely it is that the lone B allele will be fixed instead of one of the A alleles.

The maths is actually depressingly easy for this one.

First, assume that one of the allele types in the population will eventually go to fixation. This is not a crazy assumption - you can show quite easily that the average heterozygosity decays exponentially. This of course is equivalent to the assumption that at some point one of the original alleles will be the great-granddaddy of the entire population.

Now note that each allele has an equal chance of granddaddy-hood. So the chance for any one of them must be 1/(population size). In the case with a population size of 100 and one B allele, the chance of the B type becoming fixed is therefore 0.01. QED.

I rigged a Python script [1] to estimate the fixation probabilities for certain population sizes (I'll publish it when I get round to sorting out webspace). It takes a ridiculous length of time for large populations, but its first three results were:
    Pop. Size: 10
    P(B fixed): 0.090000
    Pop. Size: 100
    P(B fixed): 0.010000
    Pop. Size: 1000
    P(B fixed): 0.001300

Magic.

We can see that, in small populations, genetic drift will mutate the group at a high rate [2], whereas, with larger populations, its pace is glacial. This pretty much matches our intuition - for the effects of too small a population size, just consider Cletus the Slack-Jawed Yokel :)

[1] See http://coalescent.freewebpage.org/popgen/gendrift2.py - I'd link to it except that the crappy free service I'm using dislikes what it sees as hotlinking.

[2] It turns out, as a corollary of the heterozygosity calculations alluded to earlier, that the time to fixation is also proportional to population size, further reinforcing the point.




The Neutral Theory

The Neutral Theory states that the majority of genetic change is caused by this genetic drift. Now, anyone who's ever debated a creationist will spot the obvious problem here - if this were the case, their accusation that "evolution is random" would actually be accurate. Obviously natural selection plays some role. However, the Neutral Theory acts as a brilliant null hypothesis for seeing whether natural selection has occurred.

And actually it turns out to be fairly accurate in a variety of circumstances. One particular case where it's easy to test the neutral hypothesis is that of silent mutations - changes in DNA that don't actually affect the expressed protein. Long story short, the predictions of the Neutral Theory regarding the number of mutations that will be accumulated in a given time period fit a wide number of cases pretty much perfectly.

One really interesting consequence of all this is that we can actually make good guesses as to which DNA sequences in, say, humans are likely to have some unknown purpose. We can do this by asking the palaeontologists when our last common ancestor was with, say, rats and figure out which chunks of DNA have been better conserved than we'd expect. See the paper "Ultraconserved elements in the human genome" for more detail.

Pretty damn cool.

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