6/23/2006

The Hardy-Weinberg law

(Population Genetics)

One of the earliest, and simplest, discoveries of population genetics was the Hardy-Weinberg law. This discusses the behaviour of a population under the following circumstances:
  • The population is effectively infinite
  • Mating is completely random
  • All the organisms have the same survival chance
  • Chromosomes come in pairs, both elements of which behave identically
  • There are two alleles competing for the same spot, both of which behave identically
Call these alleles A1 and A2. Then the three possible genotypes are A1A1, A1A2 and A2A2, where for example A1A2 means you have an A1 allele on one chromosome and an A2 allele on the other. What we're aiming to do is to figure out how the ratios of these genotypes to each other varies over time.

So, say the starting genotype frequencies are x11 for A1A1, x12 for A1A2 and x22 for A2A2, with x11+x12+x22=1. We now have to do a lengthy calculation to figure out the probability of a randomly-chosen member of the next generation having each genotype. Call these frequencies y11, y12 and y22.

For a child having genotype A1A1, the only way to produce it is with an A1 allele from both parents. The chance of getting an A1 allele from an A1A1 parent is 1, the chance of getting it from an A1A2 parent is 1/2 and the chance from an A2A2 parent is 0. The probabilities work out as follows:
    Parent 1 is A1A1, parent 2 is A1A1: child is alway A1A1
    Parent 1 is A1A1, parent 2 is A1A2: child is A1A1 half the time
    Parent 1 is A1A2, parent 2 is A1A1: child is A1A1 half the time
    Parent 1 is A1A2, parent 2 is A1A2: child is A1A1 a quarter of the time
    Either parent is A2A2: child is never A1A1

So, totalling these up, the probability that a given child is A1A1 is going to be:
    y11 = 1.x11.x11 + 1/2.x11.x12 + 1/2.x11.x12 + 1/4.x12.x12
      = (x11 + 1/2.x12)2


Now, one important thing to note is that x11 + 1/2*x12 is actually just the probability of getting an A1 allele if you select randomly from the population of alleles. So if you call this probability p, you get the result: y11 = p2. Similarly, y22 = q2 where q=1-p. And since (p+q)2 = p2 + 2.p.q + q2 = 1, we have y12 = 2.p.q.

So we have our new frequencies p2, 2.p.q and q2. The obvious question is: what happens in the next generation? Well, that's the cool bit. These probabilities are only dependent on the frequencies of the alleles in the previous generation, not of the genotypes. And since the frequency of each allele hasn't changed a whit (check if you like), every subsequent generation will also have this exact same distribution of genotypes.

That is the Hardy-Weinberg law.




Of course, the Hardy-Weinberg law is talking about a very idealised situation. It's interesting to look at what happens if we discard each assumption in turn:
  • If the population is not infinite, random fluctuations will start to come into play. The result is the neutral theory of evolution, which I'll hopefully cover next time.
  • If the mating is not completely random, the equilibrium is generally still reached - but it takes longer. For example, if there are two sexes with different initial genotype frequencies, the HW frequencies take two generations to reach (proof left as an exercise to the reader).
  • If not all the organisms have the same survival/reproduction chance, and especially if that chance is linked to the organism's choice of alleles, an allele can be selected out of existence. This is the basis of evolutionary theory, and will be covered at some point.
  • If the chromosomes come in threes or above, or there are more than two alleles to choose from, equations similar to the HW law can easily be formed (again, an exercise for the reader).

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