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As I understand it, if a population is at Hardy-Weinberg equilibrium, then the genotype frequencies should be $$p^2 + 2pq + q^2 = 1,$$ given the allele frequencies of $p$ and $q$, which you can figure out if given the number of homozygous recessive individuals in a population (such as 1/10,000 babies are born with PKU, so $q = \sqrt{1/10000} = 0.01$).

What I don't understand is what does it mean if your expected genotype frequencies do not match your expected frequencies. For example, here is a textbook problem where there are two alleles, $V$ and $v$. There are 16 $VV$, 92 $Vv$, and 12 $vv$ individuals.

If I did this right, that means

  • $V = (16\times 2 + 92)/240 = 0.516$
  • $v = (92 + 12\times2)/240 = 0.483$

and then the expected frequencies are:

  • $VV = 0.516\times0.516 = 0.266$
  • $Vv = 2\times0.516\times0.483 = 0.498$
  • $vv = 0.483\times0.483 = 0.233$

which are different from the observed frequencies:

  • $VV = 16/(120) = 0.133$
  • $Vv = 92/(120) = 0.766$
  • $vv = 12/(120) = 0.1$

The book says that the population is evolving because the observed frequencies are different from the expected frequencies. What exactly does that mean though? How can you tell if the population is evolving if you don't even know the allele frequencies or the genotype frequencies of the next generation? The reason I ask is because I thought to disrupt the Hardy-Weinberg equilibrium means that the allele frequencies would change over generations, so could you say the population is evolving is you're only given the data for one single generation? I mean, you can get the same allele frequencies from both the expected and the observed genotype frequencies, so I don't know if the population can be said to be evolving. Maybe I'm understanding this wrong. Please help. Thank you all.

My related question is: Can you have no change in allele frequencies across generations, while in each generation, the observed genotype frequency is different from the expected genotype frequency. If so, does this still mean the population is evolving?

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From Wikipedia, bold added by me:

In population genetics, the Hardy–Weinberg principle, also known as the Hardy–Weinberg equilibrium, model, theorem, or law, states that allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. These influences include genetic drift, mate choice, assortative mating, natural selection, sexual selection, mutation, gene flow, meiotic drive, genetic hitchhiking, population bottleneck, founder effect, inbreeding and outbreeding depression.

Therefore, if you observe allele frequencies that do not match what Hardy-Weinberg would suggest, that means one of those other things is happening. Think of Hardy-Weinberg a bit like the "null hypothesis". You can't tell directly which of these is occurring, and multiple evolutionary influences can occur simultaneously.

We can think about a particular example, though. Let's say you have a case with a recessive trait where the phenotype of vv individuals makes them more likely to die. You would expect when you look at genotype frequencies to find them out of equilibrium with respect to Hardy-Weinberg: you'll find fewer vv individuals than you'd otherwise expect based on the overall rate of v alleles in the population.

It's important to consider sampling effects in your measurement, too. Your estimates will rarely match the Hardy-Weinberg frequencies perfectly just from noise in your sampling. You'll want some estimate of how likely it is that, assuming the population actually does follow Hardy-Weinberg, you'd get a deviation as large as the one you observe. That is achieved by performing a chi-squared test.

How can you tell if the population is evolving if you don't even know the allele frequencies or the genotype frequencies of the next generation?

You can tell because if the population isn't evolving (this statement means exactly that the allele frequencies stay the same) and mating is random then the allele frequencies would be equal to the Hardy-Weinberg frequencies. A bit like if you park your bicycle, go inside, and come back outside and see no bicycle there, you can infer that your bicycle has moved even though you didn't actually observe it moving. You may not be able to conclude whether it was ridden away, was blown by the wind, or picked up and placed in a truck and driven off, but just the fact that it's not where you left it provides information that something is happening besides the "null hypothesis" that bicycles left in a place stay there.

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