I've already started with the eye color example so I figured
no harm in continuing it.
Let's say we live on a planet or we're a species where
there's only two possible eye colors: either blue-- and the
genotype or the allele for blue I'll abbreviate with a
lowercase b --or you could have brown eyes.
I'm doing it in this salmon color just because I got tired
of the brown.
We'll do an uppercase B.
And I'll make a few other simplifying assumptions.
First of all, let's assume that the expression of this
gene is very simple, that brown is dominant to blue.
So if you have one of each, you're going
to see brown eyes.
The only way you're going to see blue eyes is if you have a
blue allele from each of your parents.
And the other assumptions I'm going to make is that this
population essentially has a stable gene pool with respect
to the eye color gene.
What do I mean by that?
No selection is taking place, no natural selection based on
this trait.
So, for example, you're not more or less likely to be able
to reproduce, or the number of children you have is not going
to be larger or greater dependent on what your eye
color is in this population.
I'm also going to assume that there's no mutations.
So for whatever reason, a chromosome that has a blue
allele can't randomly turn into a brown or vice versa or
turn into a third color.
I'm also going to assume that I'm going to have a large
population.
The reason why I'm making all of these assumptions is
because I essentially want to have a stable gene pool, at
least relative to this gene.
Relative to the eye color gene, I want to be stable.
I want to have a stable allele frequency.
What do I mean by allele frequency?
I said a large population, but let's take a very small
population.
Let's say that there's two people in the population, and
one guy right here, his genotype is big B and
lowercase B, and let's say that the other guy has blue
eyes, so he has to be homozygous recessive.
Let's say this is the entire population.
Of course, a population of two can't apply to what I'm about
to do, but I just want explain allele frequency.
The allele frequency here of the blue eyes, the blue-eyed
allele frequency here, is what?
Well, in this population, I have exactly four alleles.
I only have two individuals, but they
each have two alleles.
So it's going to be 75%.
75% of the alleles in this population are blue, right?
There's one, two, three, four alleles, and
three of them are blue.
The frequency of the brown allele is 25%.
And this is different than the actual expression.
If I asked you the frequency of brown eyes versus blue
eyes-- So this is allele frequency.
Let me write that down.
That's allele frequency right there.
If I were ask you phenotype frequency, if I said how
frequent are blue eyes seen in my population?
Well, it's only one out of the two people in my population,
so you'd say it's 50%, and then for brown eyes, you would
say it's 50%.
So I want to make this distinction very clear because
it can be very confusing when people talk
about allele frequency.
Allele frequency is literally if you were able to go and
look at the actual genotype, the actual chromosomes of
every person in the population, and count how many
of them had the blue allele versus the brown allele, you
would come up with this number.
This is different than the actual phenotype frequency.
Now, all of that is just as a background, because if these
set of assumptions are true and my population isn't in any
way evolving, then my allele frequency is going to be
roughly constant.
I have to assume a large population, because a small
population, just from random chance, my allele frequency
could start to change.
But let's say my allele frequency is constant, then
I'm in a Hardy-Weinberg equilibrium.
All this is a situation where the allele
frequency isn't changing.
The reason why we're making all of those assumptions is
because, if we can assume this, we can start to deduce
some things about what we observe, what must be the
genotypes in the population or the frequencies of different
phenotypes and whatever else.
So, for example, if p is the frequency of blue eyes, so let
me say p is equal to the frequency of blue eyes, and q
is equal to the frequency of brown eyes, what's p plus q
going to be?
What's p plus q?
Well, everything is either going to have blue eyes or
brown eyes.
If this is 20%-- And anyway, this is the allele.
Let me make that b allele.
Let me make that very clear.
This isn't necessarily what you observe.
This is the allele itself.
This is the allele frequency, if you were actually to count
the chromosomes and see what percentage have the blue
allele and what percentage have the
uppercase brown allele.
Well, every chromosome has to have either of those, so these
are going to add up to 100%, or equal to 1.
For example, if 30% of the alleles are blue, then 70% are
going to have to be brown because, I already told you at
the beginning of this video, that we're in a world where
you either have a blue allele or a brown allele.
Now, what if we don't care about alleles, we actually
care about the frequency of the genotypes?
So there's a couple of things we could do.
We could literally just square both sides of this and then
think about what that gives us.
If we square both sides as this equation, you get what? p
plus q squared is p squared plus 2pq plus q squared, and
you square the other side and it's equal to 1 as well.
What does this tell us?
What is p squared?
Well, it's the probability that I get two of the
lowercase blue alleles, right?
What's the probability that I get two, that I end up with
lowercase b, lowercase b as my genotype?
Well, I have to get a lowercase b from my first
parent, and that's with a probability of p, and then a
lowercase b from my second parent with a
probability of p.
So it's p times p, so that's p squared.
What's q squared?
Well, that's the probability that I get a big B, a
brown-eyed allele from each parent, because the
probability from parent 1 is q and then the probability from
parent 2 is also q.
So that's q squared right there.
Now, what's 2pq?
Well, essentially, this entire term is the
frequency that I'm a hybrid.
I have a heterozygous genotype.
And why is that?
Because there's two ways that I can be a heterozygote.
I could be like this.
I want to do that other blue.
I could get blue eyes from one parent and brown eyes from the
other or I could get brown eyes from the first parent and
blue eyes from the second.
I don't know anything about these parents, so if I'm
making no assumptions, then I just have to assume that the
probability that I get an allele from either one of them
is equal to the frequency in the population.
So there's two ways of getting these.
The probability of each of these ways, the probability of
this combination, is p times q.
The probability of this is q times p, which is the same
thing, and you add them together and you get 2pq.
So this is essentially the frequency of the hybrids in
the population.
I've been kind of abstract so far.
Let's see if we can apply this to a real world problem, and
obviously, I'm making some simplifying assumptions.
Let's say that we go into a population.
Let's say it's a population of a million people, reasonably
large, and we observe that 9% have blue eyes.
This is their phenotype.
This is the frequency of the genotype lowercase b,
lowercase b, right?
I started with this phenotype blue eyes, but I know, since
blue is a recessive trait, that they must have two copies
of the recessive allele.
So this is the frequency.
The frequency of having two lowercase b's is 9%.
Well, if you just look here, I just showed you that that's
also equal to p squared.
p squared is equal to 9%.
And how do you think about that?
Well, p is the probability of getting a blue allele.
It's the frequency of the blue allele in the population.
In order to get two of them, you have to multiply p by
itself twice.
So what's the frequency of the blue allele in the population?
p is equal to the square root of 0.09, which is 0.3.
So if you went and actually counted all of the alleles in
the population, you would actually find that 30% of them
are the lowercase blue.
Now, a much smaller percentage of people, only 9%, showed the
blue eyes because you need two of them.
You have a 30% chance of getting it from your mom and a
30% chance of getting it from your dad.
So what's the frequency of a brown-eyed allele?
Well, we know that the frequency of the blue eyes
plus the brown eyes is 100%, so that's going to be 70%.
If 30% of all of the chromosomes or of all of the
alleles in the population are blue, the other 70% are going
to have to be brown.
So what percentage of my population are going to be--
Well, there's a couple of things we can do.
What percentage of my population are going to have--
Well, I'll do an easy one.
What percentage of my population are going to have
brown eyes, so if I just look at the phenotype brown eyes?
I don't even have to use any of these
formulas for this one.
I already told you 9% have blue eyes, so the rest must
have brown eyes.
So 91% percent have brown eyes.
Now, we just said we have 9% who have blue eyes, 91% have
brown eyes.
What percentage of the population are going to be
homozygous for brown eyes?
So they need to have the capital B from both parents.
Well, we know that the frequency of the capital B
allele is 70%, so what percent are going to
be homozygous dominant?
Let me draw that.
Homozygous dominant.
Well, the frequency is going to be q squared.
They're going to have to get a q.
They're going to have to get an uppercase B from each
parent, so that's going to be 0.7 squared, which is equal to
0.49, or 49%.
So just starting already from that one idea, from that one
idea that 9% have blue eyes, we've already been able deduce
that 91% must have brown eyes.
And of the 91% of the brown eyes of the whole population,
49%-- this isn't 49% of the 91%.
This is 49% of the population.
49% percent are homozygous dominant.
And then what percentage of the population
are going to be hybrids?
Well, hybrids also have brown eyes, but they're not
homozygous dominant, so the remainder here.
So what is that?
That's 42%.
42% are going to be hybrids.
And if we go back to the Hardy-Weinberg
equation, we see that.
So we get p plus q has to be equal to 100%, or I could just
write equal to 1.
And we figured out that p was 30%.
That's the frequency of the blue-eyed allele, the actual
trait, not the observation of it or the genotype.
And the frequency of the brown eyes was 70%.
And then this is actually a breakdown when you square
that, so we also know that p squared plus 2pq plus q
squared is equal to 1.
This is the percentage of the population that has blue eyes.
They have two blue alleles, so that's 9%.
This is the percentage of the population that has two brown
alleles, homozygous dominant, so that is 49%.
And then the remainder, this 2pq, this whole term right
here is what's left over.
So 9 plus this 58, this is going to be 42%.
42% are hybrids.
And if I just talked about phenotypes, I'd say 9% have
blue eyes and then the remainder,
91%, have brown eyes.
Hopefully, you found that reasonably useful.
Just from very simple deductions, it's almost kind
of silly that this is almost a separate principle, because
you can kind of deduce this from very basic ideas, and you
can square both sides of that.
But you can come up with some fairly fascinating results
about what's going on in a population because we can
observe maybe the frequency of, let's say, some disease
that only happens when someone is homozygous recessive.
So you can go and see, OK, what percentage of the
population has that disease, but then doing the math that
we just did in this video, you can figure out what percentage
of the population have the allele for the disease.
They're carriers for the disease, but they don't
actually show it, and that would be the
hybrids right there.
So this is actually a pretty powerful
tool you've just learned.