In previous lecture,

we considered the fate of an allele in absence of evolutionary factors.

And we have arrived to Hardy-Weinberg equilibrium.

Now we are going to extend our model with other evolutionary factors and see

what happens to frequency of allele under these more rich conditions.

So let's start with enriching our model with selection.

So selection is a process of differential reproduction.

And differential here means genotype-dependent.

To quantify it in a model,

in mathematical model, we need to introduce a coefficient.

We will call it parameter of selection which is defined for specific genotype,

and which tells what proportion of the individuals with

these genotype does not contribute to the gametic pool of the next generation.

This selection can act through different biological mechanisms.

For example, the presence of mutation may lead to diminished survival,

to death before reproductive age,

or it can damage reproductive system,

or actually it can increase the fitness.

For example, the individuals getting this mutation reproduce more offspring,

or the offspring is better surviving.

However, we can always scale

their selection or reproductive success to the most fittest class,

and then we still can describe how their selection works by this parameter of selection.

So let's start with very simple example where we will

consider the fate of a dominant lethal allele.

So let's denote the alleles N as normal and D as the mutant one,

and again this mutation is going to work in a dominant manner.

So it's going to lead to lethality.

Now if we start with some population with some frequency of these alleles say 10 percent,

what's going to happen?

Well, of course this allele is going to be eventually eliminated.

But how quick this happens?

When we consider this simple scenario of dominant lethal mutation,

of course the mutation is going to be eliminated directly in the next generation.

This example is somewhat stupid,

but it allows us already to see how this model can operate.

Now, let's consider something more interesting.

Let's consider the fate of recessive allele.

So here, we are going to assume that only individuals who are

homozygote carriers of the mutation are selected against.

Unless for simplicity, assume that

again in the homozygous form the mutation leads to lethality.

Now, if we can see there,

again we start with a population where this mutation is present at certain frequency.

And then if we try to guess what happens to

the frequency of this mutation in the roll of generations,

then, well, naturally, it's going to fall down.

The question is, "How quickly?"

Well first of all,

the mutation is visible to selection only when it's homozygous form.

And the frequency of homozygote is the square of allelic frequencies.

This already tells us that a rare mutation is going to be eliminated very slowly.

While if mutation is at a higher frequencies,

the elimination process is going to happen quicker.

You can give this model an exact mathematical treatment and arrive to recursive formula,

which defines the frequency of this allele in

the next generation depending on its frequency in the previous generation,

and the form of this relation is shown on the screen.

So again you can see that initially,

this allele is eliminated at higher rate,

but the more rare it becomes,

the slower is the rate of elimination.

If we extend our model and say is that,

not all homozygous are not contributing to the next generation,

but only a certain proportion is penalized

as characterized by the selection parameter, s.

Then of course you can see as s become smaller,

the way how their frequency change becomes slower and slower.

So for the alleles,

which have penalized in recessive form,

very little, the elimination process is very long.

Now if you think of this,

in infinitely large population,

even this very small selection parameter,

the mutant allele is going to be eventually eliminated.

Although this process may be very very slow especially at lower allele frequencies.

Now let's try to enrich our model even further and consider the effects of mutation,

which is a random event converting some proportion

of normal gametes into the mutant gametes.

And this happens at rate Mu per gamete per generation.

If we try to think of this model,

then very soon you will realize that there should be a balance point.

And this balance point is reached when the number of or the proportion of

a newly derived mutant gametes by the mutation process,

equal to the number or the proportion of the gametes which are penalized by selection.

So this equilibrium point is reached at the point of

square root of mutation rate divided by the parameter of selection.

Now let's try to consider a specific example.

We are going to consider example of Cystic fibrosis.

This is a rare monogenic autosomal recessive disease

about which we have been already talking in one of the previous presentations.

The homozygote carriers of the mutation in Cystic fibrosis gene are having Cystic fibrosis.

And their survival and chances to

reproduce have been severely affected up until recent.

Now the frequency of the mutant alleles in European population is 1/30.

Let's try to think whether mutation/selection balance can explain

this high frequency of the Cystic fibrosis gene mutation in the European populations.

So to do that, we also need to assume what is

the selection parameter associated with the homozygotes for mutations in CFTR gene,

and while exact,

it is very hard to tell exactly what its coefficient used to be or is now.

We can assume that 100 years ago,

say it was relatively high.

And then actually if you use a range of possible reasonable values,

you come up to the same conclusion.

If we take selection parameter to be 1/4,

we can see that the corresponding mutation rate to

justify this high frequency is 10 to the minus five.

And this is a bit too much for a mutation rate.

Therefore, we need to come up with some different explanation of

why this mutation reached such high frequency in European populations.

So one of the explanation and probably the most reasonable one is that, well,

human populations are finite and random drift is

an important player in shaping the frequency distribution of different alleles.

So it could be just random factors,

it could be genetic drift which by random drove

the frequency of alleles at CFTR locus at this high frequency.

However, there could be other explanation, a biological explanation.

It had been hypothesized that heterozygote carriers of the mutation

in CFTR gene might have been protected against cholera.

A disease, pretty bad disease,

which was widespread in Europe only 150 years ago.

This example brings us to the concept of balancing selection.

Is described by a possibility that a mutation which might be deleterious in

homozygous form may give some selective advantage when it is being heterozygous form.

To treat this concept in mathematical terms,

we need to associate two selection parameters to the homozygotes.

Let's denote them as S1 and S2.

And then it's possible to show that the balancing point, that issue

the population have stable genotypic distribution

is given by a rather simple expression,

where the first selection parameter comes in the nominator,

and the sum of the selection parameter is in the denominator.

One very remarkable example of

balancing selection in human populations is given by sickle-cell anemia locus.

Sickle-cell anemia is monogenic recessive autosomal disease

which is actually quite prevalent in some of the human populations.

This disease is caused by mutations in human beta-globin gene.

When this mutations are in homozygous form,

this leads to the phenotype of sickle-cell anemia.

And this is associated with high risk of death before 3 years old,

and shorter life expectancy by an adult.

At the same time, it was noted that the frequencies of

the mutant for beta-globin are very high in certain populations.

And these are actually the populations where malaria is endemic.

So what was discovered is that heterozygote carriers of mutation in

beta-globin are protected against malaria.

So again, we see a very clear example of balancing selection,

and a good explanation of the observed distribution of allelic frequencies.

An important lesson to learn from

these few examples is that fitness is context dependent.

So it's very hard to tell about harmful allele in general

because harmful allele may be harmful in homozygous form,

but protective in heterozygous form.

This harmful allele may be

actually advantages in the past of the population when the conditions are different,

and then it became harmful with changes in environment in current time.

It might be that the effects of allele are good when a person is young,

and they become harmful when the person ages.

Other very important factor which we should not

forget is the factor of random genetic drift,

which is actually explained in quite a bit of

what we observe on the genetics of population of humans.