0:11

And the important thing I emphasized at the very end of the last video was,

Â with these sorts of assumptions that gametes just come together at random

Â based on their proportions, we end up with a stable equilibrium.

Â And, in essence, the allele frequencies don't change over time.

Â The genotype frequencies don't change over time.

Â That is, by definition, an equilibrium.

Â Now, this pattern was first described by these three gentlemen here.

Â This is Godfrey Hardy over here.

Â Wilhelm Weinberg over here.

Â So they're the ones who typically are named for this,

Â as the Hardy-Weinberg equilibrium.

Â There's also American, William Castle, who introduced a similar idea around 1903.

Â But, the important thing was the idea of self perpetuation.

Â This is what I showed you in the end of the last video that you had an allele,

Â a big A allele frequency of 0.6 in gametes.

Â They created the seven genotypes with a little allele, 0.36 with big A,

Â big A, 0.48 with big A, little A, 0.16 with little a, little a.

Â The following generation you would again have this .6 frequency of the big

Â A gametes and correspondingly the .4 for the little a.

Â Now, up until 1902, several people thought that it was possible that

Â the dominate alleles would intrinsically increase in the population.

Â They are after all called dominant.

Â And some people also assume that rare alleles would just tend to get lost and

Â have this inherent drive towards loss.

Â Now it was in 1908, and also probably from the 1903, contribution.

Â Hardy and Weinberg independently showed that both these assumptions are not true.

Â That if you can just use this purely probabilistic approach, allele and

Â genotype frequency will stay completely stable.

Â Now there are assumptions underlying this and

Â we're gonna come back to those assumptions in just a moment.

Â Before we come back to that, let's change this into a more mathematical notation.

Â So let's formalize the math.

Â Now let's say these are typical notations people use.

Â Let's say the frequency of big A is referred to as p.

Â Also the frequency of little a, is referred to as q.

Â Again we're assuming there's only two alleles in this population.

Â Because there are only two types, p plus q must necessarily equal one.

Â Plus if p is one, then q must necessarily be zero.

Â You can't have negative numbers because these are frequencies.

Â What would the frequency of big A, big A be?

Â Well, it would be the probability of a big A encountering another big A.

Â So necessarily, it would be p times p or p squared.

Â Right, we can push this in P squared for big A big A.

Â Q squared for little a,little a.

Â 2pq for big A, little a.

Â Why is it 2pq?

Â Why isn't it just pq?

Â Doesn't it just mean a big A and a little a?

Â Again there are two different ways you can have it.

Â You can have a big A, sperm and a little a, egg or a little a, sperm and

Â a big A egg so there's two different ways you can get it.

Â Again, these genotype frequencies must necessarily add up to one.

Â So P squared plus two p q plus q squared equals one.

Â You note that this quantity squared will come out to that.

Â Right? P plus Q squared is 2pq plus q squared.

Â So, it all comes out very elegantly.

Â Let me show you how the frequencies would look if you were to plot

Â these things together with these assumptions.

Â 3:45

And as you can see the abundance of the heterozygote

Â peaks out around 50% and as you get to the two

Â ends you end up having one of the two homozygotes being completely abundant.

Â Obviously at the extreme end here,

Â if there are no big A's then everybody in the population must be aa.

Â If there are no little a's, then everybody in the population must be big A,big A.

Â Now this allows you to infer genotype frequencies from allele

Â frequencies because you can say, well I know the allele frequency is .8 for

Â big A and .2 for little a.

Â So you immediately say,

Â oh here's the abundance of big A,big A, Big A little a, little a, little a.

Â However some assumptions have to be met and we haven't gotten into that yet.

Â Now let me give you three points.

Â The third is going to come back to these assumptions.

Â 5:11

Let's say for example we have these three genotype frequencies.

Â We can use this little trick I showed you before.

Â All the homozygote plus half of the heterozygote gives you

Â the particular allele frequency.

Â So for example for big A, all the homozygotes 0.4.

Â Plus half the heterozygote, which is half of 0.32 or 0.16.

Â It comes out to 0.20.

Â All of this one is 0.64 + half of 0.32 or 0.16 = 0.80.

Â And again they add up to 100%.

Â The third one though, the third point is you cannot

Â always know genotype frequencies from allele frequencies.

Â That's what it seemed like we could do with that figure I showed you earlier, but

Â let me show you why.

Â Let's say, for example,

Â that the frequency of big A's 0.5, frequency of little a is 0.5.

Â Simple case, right?

Â Well, you could have 0.25, 0.5, and 0.25, right?

Â That adds up to 100%.

Â If you do all of these plus half of these, that is 0.5.

Â All of these plus half of these, that's 0.5.

Â That's actually at the Hardy-Weinberg expectation.

Â But you could also have other possibilities that don't

Â match the Hardy-Weinberg expected abundances.

Â You can have this, 0.45 plus half of .10 is still 0.5.

Â This .45 plus half of .10 is still point.

Â You can even have this, what if you had heterozgotes?

Â Again the little frequencies here are 0.5 and 0.5.

Â So, the bottom line is you can always calculate

Â allele frequencies from genotype frequencies.

Â The reason for this is that alleles are the ingredients of the genotypes.

Â 6:44

But you cannot always calculate genotype frequencies from the allele frequencies,

Â because genotypes are specific combinations of alleles.

Â Again, just like ingredients.

Â If somebody gave you a pancake, you can know it's made from flour and sugar and

Â baking powder, and things like that.

Â In contrast, if somebody gave you flour and sugar and baking powder,

Â there's a lot of different things you can make with that.

Â Alright many combinations are possible.

Â So coming back to Hardy-Weinberg.

Â Hardy-Weinberg allows predictions of genotype frequencies from

Â allele frequencies under certain conditions.

Â And this is when that sort of simple multiplicative rule applies.

Â Those conditions include random mating.

Â This is where the multiplying probabilities rule works very well.

Â We don't assume that big A's are more likely to encounter other big A's than

Â little a's relative to the proportions in the population.

Â Assumes no selection, migration, or mutation at that locus.

Â At terms of selection we're assuming that it's not like the big A,

Â big A individuals tend to die more readily and therefore they're not constant.

Â Assume an infinite population size and this is why statistic, or

Â probabilistically, everything works out very nicely.

Â This is referring to the absence of what's called genetic drift.

Â We'll come back to this at a later lecture.

Â But overall, Hardy-Weinberg is predicting a completely

Â boring population that probably could never exist.

Â 8:20

So we'll come back to this actually over the course of the next couple of lectures

Â about some specific evolutionary processes that could be operating.

Â But let's look a little bit at testing for Hardy-Weinberg.

Â So here's an example.

Â So here we have big A ,big A is 245 individuals.

Â Big A,little a, is 210 individuals.

Â Little a, little a is 45 individuals.

Â Let's walk through this one to see if this population is at Hardy-Weinberg and

Â then I'll have you do one later.

Â So the four steps you want to do is figure out the true genotype frequencies,

Â figure out the true allele frequencies,

Â and then figure out the Hardy-Weinberg expected genotype frequencies,

Â which may or may not be the same as the first one, and then ask yourself,

Â do the true frequencies match the expected frequencies, or close enough to them?

Â So let's do this now.

Â Again, from the genotype counts we can always get the genotype frequency,

Â you just get the totals 500 divided by the total and

Â here are the genotype types so that's five.

Â These are the true genotype frequencies and these true frequencies we can get

Â the true allele frequencies so all of these and half of these for big A.

Â All of these plus half of these for little a.

Â Very simple 0.7, 0.3.

Â Now the question is, if we put these together in the Hardy-Weinberg expected

Â proportions, does it actually work out as we expect?

Â Is it p squared for big A,big A, 2pq for big A, little a and q squared for

Â little a?

Â The answer is p2, 0.7 squared, 0.49.

Â That matches beautifully.

Â [SOUND] 0.42, 0.42, matches beautifully.

Â 0.09 to 0.09, matches beautifully.

Â So, yes, this population is at Hardy-Weinberg.

Â Now, let me ask you to try this next one.

Â 10:09

Well thank you, I hope that wasn't too hard.

Â So again what we do is, we do the first step is figure out the total.

Â So in this case the total adds up to 400 + 200 + 400 that is 1,000,

Â nice easy number there for you to start with.

Â So the genotype frequencies, the true genotype frequencies in this case, so

Â it would be 400 divided by 1,000 so that would be 0.40.

Â 200 divided by 1000.

Â 0.20. 400 divided by 1000, 0.40.

Â Okay so these are our true genotype frequencies.

Â What about our allele frequencies?

Â The allele frequency for big A would be all of these plus half of these.

Â So that would be 0.4 plus one half of 0.2 equals 0.5.

Â Q sub little a is equal to same thing exactly, 0.5.

Â You can check here at this point, yes, they still add up to 100%.

Â These still add up to 100%.

Â So, the question is,

Â then, do the Hardy-Weinberg expected genotype frequencies match these?

Â Well, Our expectation for big A, big A, p squared which would be 0.25.

Â For big A, little a, it should be 2 pq is equal to 0.5.

Â Little a, little a would be q squared.

Â Q is 0.5. So that would be 0.25 Uh-oh.

Â These don't match up at all.

Â 0.25 is not the same at 0.40.

Â 0.50 is not the same at 0.20.

Â 0.25 is not the same as 0.40.

Â So, people often tend to think that this is circular, but

Â as you can see in this case, it was not actually circular.

Â We actually are seeing a deviation from the idea that

Â these gametes just come together at random.

Â Everybody survives it all moves on.

Â So there is a deviation from Hardy-Weinberg in this case.

Â Now seeing how natural populations deviate from Hardy-Weinberg expected frequencies.

Â We can actually infer what evolutionary forces are operating.

Â In that previous slide we saw a deficit of heterozygotes.

Â There were fewer heterozygotes than we expected from Hardy-Weinberg.

Â What does that mean in particular.

Â Well we'll look at that as one possible deviation in the next video.

Â Thank you.

Â