Hi. It's Mr. Andersen and in this video I am going to talk about the Hardy-Weinberg
Punnett square which is a mouthful but it is also a very powerful tool you can use to
look at changes in the gene pool over time. And if you want to call it a population Punnett
square or a gene pool Punnett square that is maybe a better way to look at it. But the
reason I wanted to talk about it as Hardy-Weinberg is these are two fundamental concepts that
we deal with in biology. And so let's start with the work of Reginald Punnett. You know
him because he came up with this idea of the Punnett square which is applying the work
of Mendel. And it is a really neat model you can use to look at genetics. So imagine we
have this cross right here. We have heterozygous purple flower crossed with another heterozygous
purple. So what do these two letters represent? The two genes that that flower could give
off. And so we could figure out all of the possibilities that we could get. And so since
purple is dominant we are going to get a three to one of purple to white flowers. And so
most students like doing Punnett squares. And it is a really easy way to apply basic
genetics in story problems. Now when we get to Hardy and Weinberg, this is really a confusing
kind of a concept and students struggle with it all the time. It is got a couple of equations.
But they never know what is the allele frequency, what is the phenotype. And so by combining
these two the work of Hardy and Weinberg, and also these Punnett squares, you get a
more powerful model. And so we have to start with a little bit of history. Remember Mendel
had come up with all of these laws, basic laws, of Mendelian genetics and they were
lost. And then they were discovered about 100 years ago. And so Reginald Punnett was
one of those who found them. And he started developing these Punnett squares as an easy
way to work out problems. But there were a lot of doubters at that time. So for example
Udney Yule was a statistician and he said that your Punnett square might work in one
generation but over a series of generations we are going to see an increase in the dominant
allele. And so the recessive alleles not going to stick around. And so Punnett was not much
of a mathematician so he asked a friend of his, G.H. Hardy, to take a look at the problem,
and at least the contradiction that Yule was putting forward. He knew G.H Hardy because
the played cricket together. Hardy did not really enjoy doing applied mathematics, but
when he looked at the problem he realized immediately that Yule was wrong and the answer
was very, very simple. And that is where we get these Hardy-Weinberg equations. And so
let's start with Punnett squares and then work into a Hardy-Weinberg Punnett square.
And so let's start with a simple thing that we would find in humans. And that is red hair.
Red hair is caused by a recessive allele. And so just like the purple flowers, if we
set up a Punnett square, we could say let's cross somebody who is heterozygous for red
hair with one who is heterozygous for red hair. So again both of these parents are not
going to have red hair since it is a recessive trait. So Punnett would allow us to set it
up like this. And we would find a three to one ratio of non-red hair to red hair. So
that is a Punnett square. But again we are just looking at one generation. But what if
we could apply this to a whole gene pool over a long period of time? We are going to need
a different model? And so this is this Hardy-Weinberg or population Punnett square. What do we have
on the top? This is going to be the two genes that could be drawn from the gene pool itself.
And the reason I put a grid in here is that these right here, these five squares represent
the 5 in 10 probability that we could give the big R and the 5 in 10 probability that
we could give the little r. Now you might say this looks just like that last Punnett
Square. It just has more squares in it. But just hold on tight because we are going to
get there. So on the other side this would another gene that we could pull from the gene
pool. Again it is going to be 5 out of 10 and 5 out of 10. And so what I can work out
is I can work out what is the probability in this population that I am going to get
somebody who has red hair. Well how do you figure that out? It is going to be all of
those in this square right here. In other words it is going to be a 5 by 5 or 25 out
of 100, 25 percent or 0.25 is going to be the amount of people that have red hair. But
you might say that was just like the Punnett square we had a second ago. And so let's look
at a population where it is not constant. And so if we were to go to Scotland we would
find that we have a high level of red hair allele in the gene pool itself. It is not
exactly 5 and 5. And so what I have to do is I have to move this line over to here.
Why is that? Well there is a 3 in 10 probability that we are going to get a red gene, that
recessive gene. And notice as I move this one, this one also moved. Because the other
gene, since it is part of that gene pool is also going to be a 3 in 10 probability. Now
what are the odds that I am going to have somebody with red hair? Well it is going to
be in this box right here. So it is going to be a 3 in 10 times 3 in 10, or you could
just count the boxes. It is going to be a 9 out of 100 probability. Or we would call
that 9 percent. So thatss going to be right here. Well, how many people are heterozygous,
for example? Those would be in this box right here. They get the big R and the little r.
Or they would be in this box right. So I could just count that up as well. So that is going
to be 7x3 which would be 21 right here, 21 over here. So there would be a 42 percent
probability that you are going to be heterozygous for the trait. And what about homozygous dominant?
You just count all of these boxes. So it is going to be 7 by 7. And so there would be
49 percent. And so hopefully you kind of understand how we could apply this. And so this is that
Hardy-Weinberg Punnett square. What do we put on the top and what do we put on the side
now? It is the frequency of the dominant and recessive gene in the gene pool itself. And
so if I were to give you these p and q values, 0.5 and 0.5, you should be able to figure
out what is in each of those big boxes. In other word what is going to be the p squared
value, the 2pq value. Notice I have to multiply this times two. And then the q squared. Now
you should start to see where those equations are coming from. And so if we can apply a
model like this, it is going to be more efficient or it will make more sense to hopefully as
a student. And so let's say I give you this p and 1 value. Again what is my p value at
this point? It is going to be 8 out of 10 or 0.8 for my p value. And then my q value
is going to be 0.2. Likewise, notice as I move this line I had to move this line as
well, so my p over here is also going to be 0.8 and my q is going to be 0.2. And so what
is going to be my frequency of those who are q squared or homozygous recessive? That is
going to be 0.04 or 4 out of 100. What would be out here? It is just simply going to be
8x8 and that is going to be 0.64. And then I could add these up. And that is just going
to be the remainder which is going to be 0.32. And so as I move this to a different p value
or q value you can see that my q squared is going to change. My p squared is going to
change. And then remember we have to add those two that are in the p and q values. And so
a quick way to work Hardy-Weinberg problems is to just do a Punnett square. You do not
have to draw all the grids, but I could simply write the number here. So this is 0.4 and
this is 0.6. What is going to be in this box? If I put 0.4 here as well I am going to multiply
those two together so I get 0.16. What would be down here? It is 0.6 times 0.6. So I would
expect this to be 0.36 down here. And so for my students this is a really easy way to tie
together the work of Punnett looking at one generation with Hardy-Weinberg equilibrium.
And what Hardy had found is that over a long period of time these remain stable. The only
thing that changes them are things like small sample size. Non-random mating. All those
things that are accompanied with Hardy-Weinberg equilibrium. So again that is our Hardy-Weinberg
Punnett square. And you should start to understand really where this equation is coming from.
Why does p+q=1? Because the gene can either be a p or a q. In this case it would be 0.5
and 0.5. So that obviously has to add up to 1. And then where does this equation come
from? Again it is those homozygous dominant, the two heterozygous and the homozygous recessive.
And so what would my dream be? I would love for somebody to write an app where you could
put up a Hardy-Weinberg Punnett square like this and you could simply drag this line back
and forth. What is it going to do? It is going to move this line along the bottom. And if
I were able to write an app I would put numbers. Then as you drag it you are going to see your
p and q values. And you are going to see that reflected in the phenotypes of that population.
So I hope that was helpful.