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# Hardy-Weinberg Principle: Understanding allele and genotype frequency in population in Hardy-Weinberg Equilibrium.

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- 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.

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