Prof: Okay,
today we're going to talk about
the origin and maintenance of
genetic variations;
and this is continuing our
discussion of central themes in
the mechanisms of
microevolution.
The reason we're interested in
this is that there cannot be a
response to natural selection,
and there cannot be any history
recorded by drift,
unless there's genetic
variation in the population.
So we need to understand where
it, where it,
comes from, and whether or not
it sticks around.
If it happened to be the case
that every time a new mutation
popped up it was immediately
eliminated,
either for reasons that were
random or selective,
evolution couldn't occur.
If a lot of variation came into
the population,
and then persisted for a
tremendously long time without
any sorting,
we would see patterns on the
face of the earth that are
totally different from what we
see today.
So these issues are actually
central issues in the basic part
of evolutionary genetics that
makes a difference to evolution.
So the context basically is
this.
Since evolution is based on
genetic change,
we need to know where genetic
differences come from;
and the rate of evolution
depends on the amount of genetic
variation that's available in
the population,
so we need to know what
maintains the variation.
If you were to go back fifty,
sixty years,
which is what we now think of
as the classical view--
remember the classical view is
a moving window in time--
at that point it was thought
there wasn't very much genetic
variation out there and that
evolution was actually limited
by the rate at which genetic
variation was created.
Since 1965, with the discovery
of protein isozymes,
and especially now,
since the discovery of ways to
sequence DNA very cheaply,
we know that's not true.
There is a tremendous amount of
genetic variation in Nature,
and I'm going to show you some
of it this morning.
So since about 1975,1980,
due to a series of studies,
some of them on the Galapagos
finches,
some of them on the guppies in
Trinidad,
some of them on mosquitofish in
Hawaii,
some of them on the world's
fish populations responding to
being fished,
we know that evolution can be
very fast when there's strong
selection acting on large
populations that have lots of
genetic variation.
So really the rate of
evolution--and,
for example,
the issue of climate change and
global warming--
will all the species on earth
be able to adapt fast enough to
get--
to persist in the face of
anthropogenic change on the
planet?--
that issue is directly
addressed by the things we're
talking about this morning.
If there isn't enough genetic
change to adapt,
say, the grassland populations
of the world,
or things that are living on
mountains,
to the kinds of climatic
changes that they are going to
be encountering,
and currently are encountering,
they'll go extinct.
Ei--they have to either move to
a place which is like the one
they're in, or they have to
adapt to the changed conditions
that they're encountering.
So the outline of the lecture
today basically is this.
Mutations are the ultimate
origin of all genetic variation.
Recombination has a huge impact
on variation.
So what that means basically is
that sexual populations have the
potential to be much more
variable than asexual
populations--
there is lots of genetic
variation in natural
populations.
And then we will run through
four mechanisms that can
maintain variations in single
genes,
and briefly mention the
maintenance of variation in
quantitative traits.
So mutations are where these
genetic differences come from,
and they can be changes in the
DNA sequence or changes in the
chromosomes,
and in the chromosomes they can
be changes in how many
chromosomes there are in the
form of chromosomes or in
aspects of chromosome structure.
So there can be gene
duplications and so forth.
Most of the mutations that
occur naturally are mutations
that are occurring during DNA
replication.
For those of you who are
thinking of being doctors,
this is important because the
probability that a cancer will
emerge in a tissue is directly
proportional to the number of
times cells divide in that
tissue;
which is why cancers of
epithelial cells are much more
common than cancers of cells
that do not divide.
You never get a cancer in your
heart muscle,
and you frequently get cancers
on your skin,
and in your lungs,
and in the lining of your gut,
and that's because every
mitotic event is a potential
mutation event.
The kinds of DNA sequence
mutations are point mutations;
there can be duplications,
and in the chromosomes as well
there can be inversions and
transplacements that go on.
Genes can be moved around from
one chromosome to another.
They can actually be turned
around so that they are in the
opposite reading direction,
along the chromosome.
All those things are going on.
There's good reason to think
that an intermediate mutation
rate is optimal.
If the mutation rate is too
low, then the descendants of
that gene cannot adapt to
changed conditions.
If it's too high,
then all the accumulation of
information on what has worked
in the past will be destroyed by
mutation;
which is what happens to
pseudogenes that are not
expressed.
So some intermediate rate is
probably optimal.
Now a gene that controls the
mutation rate will evolve much
more easily in an asexual
organism than in a sexual
species because sexual
recombination uncouples the gene
for the benefits of the process.
Let me illustrate that.
Suppose that I am engaged in a
process that Greg wants to
control,
and we've got a certain period
of time we can do it in,
and so he decides that he's
going to do it,
with me, on a bus going to New
York.
We go down to the bus station
and, because of recombination,
he gets into one bus and I get
into another.
He loses his opportunity to
control me, simply because I am
now riding in a different bus.
That's the effect of
recombination on genes.
Recombination,
instead of keeping me on the
same chromosome that Greg and I
were on, will actually end up
putting me into a different
body.
Okay?
So in a sexual organism the
gene that's controlling the
mutation rate becomes
disassociated from the genes
whose mutations it might try to
control,
and therefore even though down
in my ride to New York I invent
some kind of great process that
would benefit Greg,
he is now dissociated from it
and he doesn't get to benefit
from my adaptations.
So it is much more plausible
that we will see genes that are
controlling mutation rates
evolving in organisms like
bacteria and viruses than it is
that we will see mutations that
control mutation rates evolving
in us.
There is some reason to think
that there is weak selection on
them, but it's not as strong as
it is in bacteria.
And in fact,
interestingly,
in bacteria you can do
experimental evolution and show
that the mutation rate will
evolve up or down,
depending on the circumstances
that you put the bacteria under.
These are some representative
mutation rates,
and it's good to have some
general framework to think
about--how frequent is a
mutation?
So the per nucleotide mutation
rate in RNA is about 10^(-5);
in DNA it's 10^(-9).
So if you start evolving in an
RNA world,
and you want to lower the
mutation rate because your
information is getting eroded
and you can somehow manage to
engineer DNA as your molecule
rather than RNA,
you can see that you would be
able to pick up four orders of
magnitude by doing so.
That's just because DNA is more
stable.
DNA is a remarkably stable
molecule.
It's possible to recover DNA
from fossil bones.
Svante Paabo is in the middle
of a project to sequence
Neanderthal's genome.
He's already got significant
chunks of Neanderthal sequence.
So DNA is just a remarkably
stable molecule.
The per gene rate of mutation
in DNA is about one in a
million;
so this is like per meiosis.
The per trait mutation rate is
about 10^(-3) to 10^(-5).
The rate per prokaryotic genome
is about 10^(-3),
and per eukaryotic genome it's
between .1 and 10.
I once saw a really great talk
by a guy named Drake,
Frank Drake,
from NIH--this was like at a
big international meeting--
Drake walks up to the
blackboard and he writes 10^(-3)
on the blackboard;
he's going to give a talk about
mutation rates in prokaryotes.
He talks for 45 minutes about
this number;
no PowerPoints,
nothing else,
he's just speaking very
animatedly about how it was that
just about all viruses and
bacteria appear to have
converged on roughly this per
generation mutation rate,
per genome, which is pretty
strong evidence that it's an
optimal rate;
thousands of species have
converged on this rate.
And I asked him how it was that
he gave this great talk without
any slides,
and he said that he had lost
them in the airplane,
and that had happened about ten
times before,
and it was such a great talk
without the slides that he just
switched completely.
So a couple of years ago,
actually early last year in
this course, I tried giving
talks without the PowerPoints.
Ninety percent of the class
didn't like it and it ten
percent of the class did.
So that's why you're still
getting PowerPoints.
Okay?
Now what is your mutation rate?
Well each of you has about four
mutations in you that your--new
things, your parents didn't
have, and about 1.6 of those are
deleterious.
So this is something that's
always going on.
And there are about 100 of us
in the room;
that means there are somewhere
around 150 new,
deleterious mutations,
unique in this generation,
sitting here in the classroom.
Where did they happen?
Well they happened fifty times
more in males than in females.
And there are good biological
reasons why.
There are many more cell
divisions between the formation
of a zygote and the production
of a sperm than there are
between the formation of a
zygote and the production of an
egg.
In human development,
and in mammal development,
egg production pretty much
stops in the third month of
embryonic development,
at which point all the women in
this room had about seven
million eggs in their ovaries.
Since then oocytic atresia,
which means the killing of
oocytes, has reduced the number
of eggs in your ovaries down by
nearly seven million.
When you began menstruating you
had about 1500 eggs in your
ovaries.
You've gone from seven million
down to 1500.
When you were born you had gone
from seven million down to one
million;
you'd lost six million of them
before you were even born.
It appears to be a quality
control mechanisms,
ensuring that the oocytes that
survive are genetically in
really good shape.
So there are very,
very different kinds of biology
affecting the production of eggs
and sperm;
females have a mutation screen
that males do not.
Well the result of that is that
there are more mutations in the
sperm of older males;
they've lived a longer time.
Anybody that wants to get in to
mate choice and what kinds of
reproductive strategies should
result from this simple fact is
welcome to write a paper on it;
there's literature out there.
Okay?
Not very PC,
but it's very biological.
Okay, recombination.
What does recombination do to
this mutational variation that
builds up in populations?
Suppose we had ten genes,
and each of those genes had two
alleles, and each of those was
on a different chromosome.
That would mean that just
looking at those ten genes,
on those ten chromosomes,
we could get 3^(10) different
zygotes.
Can anybody tell me why?
Student:
>
Prof: How many genotypes
are there for the first gene?
How many different combinations
of Aa are there?
Three: AA, Aa, aa.
So there's three things that
the first gene can do.
There are three things that the
second gene can do.
There are three things that the
third gene can do.
And there are ten genes.
So we multiply them to get the
number of different
combinations,
and if they are independently
sorting on different
chromosomes,
that will result in 59,000
different zygotes.
Now if we had a real eukaryotic
genome that had free
recombination--
which we don't have--and
unlimited crossing over--
which we don't have--then the
number of possible zygotes is
about 3^(15,000) or 3^(50,000),
somewhere along that,
that order of magnitude.
Well the number of fundamental
particles in the universe is
only 10^(131).
We're talking about numbers
which are just inconceivably
large.
That means that in the entire
course of evolution the number
of genetic possibilities that
are present, just sitting in
you, have never been realized.
There is a huge portion of
genetic space that remains
unexplored,
simply because there hasn't
been enough time on the planet
for that many organisms to have
lived.
Now, how--you can see that this
would be free recombination with
independent assortment of
chromosomes.
That makes it easier than if
it's crossing over,
because crossing over happens
more frequently the farther
genes are apart on a chromosome,
and it doesn't happen very
often when they're close
together.
So there's been an evolution of
the chromosome number of a lot
of species.
And I've previously told you
about ascaris.
Ascaris is a nematode that
lives in the gut of vertebrates.
There is an ascaris that lives
in dogs, there's an ascaris that
lives in us, and it just has one
chromosome.
So that's kind of one limit,
things with one chromosome.
There are species that have
hundreds of chromosomes.
Sugarcane has I think about 110
chromosomes, something like
that.
So the chromosome number of the
species itself evolves,
and it can evolve fairly
dynamically.
There are actually some
populations within a single
species that have a different
chromosome number than other
populations within that species,
and when individuals from those
two populations meet and mate
with each other,
the offspring often run into
developmental difficulties
because of this difference in
chromosome number.
There is such a,
uh, contrast in house mice in
Denmark.
There's a spot where there's
sort of a hybrid zone in
Denmark,
and the house mice on one side
of the hybrid zone have
difficulty--
uh, they're in the same
species, but they just have
different chromosome numbers--
and they have difficulty
dealing with the house mice on
the other side of that hybrid
zone.
The difference in chromosome
numbers appears to have arisen
in the house mice during the
last glaciation,
and they recolonized northern
Europe from different places.
Some of them came up from Spain.
Some of them came up from
Greece.
They got together in Denmark
and they ran into problems.
Okay, now crossing over also
generates a lot of genetic
diversity.
And the amount of crossing over
can be adjusted.
Inversions will block crossing
over.
You take a chunk of chromosome
and flip it around,
so that in the middle of the
chromosome the gene sequences
are reversed,
and in that section of the
chromosome the inversion causes
mechanical difficulties.
It actually changes the shape
of the chromosomes when they
line up next to each other,
and it inhibits crossing over
during meiosis.
This is one way of taking a
bunch of genes that happen to
have really helpful interactions
with each other,
and locking them up in a
combination,
so that they don't recombine.
That has happened,
and it's thought to be
important in the evolution of
quite a few insects,
for example.
Now we can play the mental game
of asking ourselves what would
happen in a sexual population if
we just shut off mutation?
We can't actually do it,
of course.
But how long would it take
before we would even notice that
evolution had been shut off,
if we were just observing the
rate at which that population
was evolving?
And the answer to that is kind
of interesting.
We could wave a magic wand over
a moderately large sexual
population,
completely shut off mutation,
and the impact of recombination
on the standing genetic
diversity in that population
would create so many new diverse
combinations of genes that it
would take about 1000
generations before we would even
notice that mutation has been
shut off.
So think back to the beginning
of the lecture.
I said mutation is the origin
of all genetic diversity;
and that's true.
But once mutation and evolution
have been going on for awhile,
so much genetic diversity
builds up in populations that
you can actually shut off
mutation and mutation--
and evolution will keep going
for quite a while.
After 1000 generations it'll
run out of steam and stop;
but it takes quite awhile.
Okay, so where genetic
came--where genetic variation
came from and how much there
was,
was a huge issue and caused a
lot of research and controversy
for about fifty years.
Before 1965,
there was the concept of a wild
type out there.
After 1965--so there was one
really good genome,
and then there were a few
mutations.
After 1965, with
electrophoresis,
the impact of Clement Markert's
work,
and Dick Lewontin,
and his colleague Hubby,
we've recognized that there's a
lot of molecular variation.
This concept that each species
has a certain genomic type is no
longer tenable.
There's just a tremendous
number of different kinds of
genomes out there.
Since 1995, we've had a lot of
DNA sequence variation and now
we've got genomics.
So I want to illustrate the
impact of genomics with
something that's just become
possible in about the last four
years.
The HapMap Project was done
after the human genome was
sequenced,
and the motivation of it was to
try to associate diseases with
common genetic variants.
By the way, the upshot of that
effort is genes don't normally
account for very much,
usually about two or three
percent of the variation;
but that's another story.
So basically once we had the
human genome,
it was clear that we could then
look for places in genomes that
had single nucleotides,
that were different,
between one person and another;
these are called single
nucleotide polymorphisms.
And to do this the HapMap
Project looked at regions of the
human genome that were about
10,500 kilobytes long,
for 269 individuals.
So that's 10,500,000 bases,
for each of 269 individuals.
And they did it on people from
Nigeria, Utah,
Beijing and Tokyo.
And they discovered that our
genome is arranged in blocks.
There are, within each block,
within each let's say rarely
recombining section of DNA,
there are about 30 to 70 single
nucleotide polymorphisms,
and that means that you could
design a genechip just to pick
up enough of these to tag a
person as having that particular
block of DNA.
Okay?
So now there are these
genechips, and we've discovered
that there are some SNPs that
are associated with disease.
We can see that there are
portions of the genome that show
signatures of recent selection.
This is an interesting
literature.
This is what a little section
of our chromosome 19 looks like.
Okay?
So this is the position along
the chromosome,
starting at 40,000,000,
and going up to 50,000,000 base
pairs.
The little black dots are all
the genes that are in this
section of the chromosome,
and using the single nucleotide
polymorphisms,
you can identify people as
having a segment of DNA that is
not recombining very frequently.
And you will notice that they
are actually lined up right over
places where the recombination
rate is pretty high.
So you can see breaks in this
upper diagram here,
showing places where the
recombination rate is pretty
high.
So remember,
this was done over the entire
genome, all of our 23
chromosomes.
I am only showing you one tiny
part of one chromosome here,
and there are actually 650,000
of those blocks that have been
identified now in our genome.
So three years later a group
then goes out and takes 928
people, from 51 populations,
and looks at how much haplotype
diversity there is.
Remember, a haplotype is a
block that's got some specific
nucleotide polymorphisms on it.
The Y axis here has 650,000
entries on it.
Of course they all blend
together, it's hard to see them.
The X axis has 928 people
arranged across it.
This is a sample of human
genetic diversity on the planet.
You can see there's quite a bit.
You can see different colors.
Okay?
Now if you take this and you
then use the tools of
phylogenetic analysis to ask
what kind of historical
structure is there in this data
set,
this is what you get.
You get a group in Africa.
You can see the emergence of
mankind from Africa--
this is thought to have
happened about 100,000 years
ago--
and then you get a very,
very nice genetic trace of our
expansion across the globe.
We paused for awhile in the
Middle East, before we broke
out.
We were in the Middle East up
until about 50,000 years ago,
and then there was a group that
went into Europe,
and other groups then split off
from that and set off into Asia.
And about probably 40,000 years
ago people went to Papua New
Guinea and Australia,
and probably somewhere around
between say 15 and 20,000 years
ago,
a group of people headed off
over the Bering Straight for
North America,
to become Native Americans,
and then another group
diversified in East Asia.
So there is a huge amount of
information in the history of
genetic variation.
So what I'd now like to do is
give you four general reasons
why this much genetic variation
could be maintained in any
population.
If you look in the textbook you
will see that there is also a
tremendous amount of genetic
variation in wild populations of
practically any species,
just as there is in humans.
In humans it happens to be
better analyzed than in almost
any other species.
But something like that can be
done for any species on earth
now, and it's getting cheaper
and cheaper and cheaper to do
so.
So selection and drift can both
explain the maintenance of
genetic variation.
And for a long time there was a
fight within evolutionary
genetics about whether what we
saw was being explained by
selection or drift.
It appears not to be a
productive question.
It's extremely difficult to
answer,
in any specific case,
whether the pattern you see is
because of a history of natural
selection or because of a
history of drift.
Both of them are capable of
generating quite a few patterns,
and those patterns overlap.
So if you take a very specific
case and study it in detail,
you can give a leading role to
selection or to drift.
For example,
you can find a signature of
selection in a portion of a
human chromosome,
indicating that there's a gene
there that perhaps was affected
by a specific disease;
that's been done.
But the general answer,
for all species across the
planet, about whether selection
or drift is more important,
probably is unrealistic.
It's probably not a fruitful
research effort to try to answer
this question.
So here are the situations that
can maintain genetic variation
in principle;
there are four of them.
There can be a balance between
mutation and drift;
a balance between mutation and
selection;
there can be heterosis or
over-dominance;
and there can be negative
frequency dependence.
So I'm going to step through
these now and give you some
feeling for how the thinking
works on each of them.
In so doing,
we're going to be dealing with
equilibria,
and really there are other ways
of approaching the analysis,
but the equilibrium approach is
the one that allows you do it
with simple algebra,
rather than with complicated
computer models.
We do it for mathematical
convenience.
We do it also because the
periods during which things are
in balance may be pretty long,
compared to those in which
they're dynamically changing--
that does appear to be a
message of evolution--
but with respect to this
particular question of the
maintenance of genetic
variation,
we don't really know too much
about those periods.
Selection can go back and forth;
populations can appear to be in
stasis when things are going on
inside of them.
This question is really
unresolved.
We do know that in terms of our
immune genes that we share
certain polymorphisms with
chimpanzees.
Those appear to have been
things that evolved in terms of
disease resistance before humans
and chimps speciated,
about five to six million years
ago.
So certainly that genetic
variation is five to six million
years old.
We don't have too many cases
where we know that,
but there may be many more out
there, just undiscovered.
A little terminology.
The fixation probability of a
mutation is the probability that
it will spread and be fixed in
the population.
That's equal to its frequency,
at any point in time.
The fixation time is how long
it takes to become fixed in
generations.
And I put these ideas up on the
board earlier,
and I'd like to go back to
that, because I'd like to have
reference to it in a minute.
So if this is frequency here,
it can go from 0 to 1,
on the Y axis,
and if this is time,
over here, this can be many
thousands of generations.
And the fate of most neutral
alleles, when they come into the
population, will be to increase
in frequency for a little while
and then drift out.
They have low probability of
being fixed because when they
first originate they're very
rare,
and the probability of eventual
fixation is just directly equal
to their frequency.
So in a big population most
mutations disappear.
But every once in awhile one
will drift through,
and when it reaches frequency
1.0, it's fixed.
Okay?
So the fixation probability is
the probability that out of all
of the mutations that might
arise, most of which drift out,
this one will be fixed;
and that's a small number.
And the fixation time,
how long it takes to be fixed,
is on average how long it takes
for this process to occur.
So that's the fixation time,
and that's an average of many
such events.
So this picture that you're
looking at on the blackboard is
really just supposed to be an
evocative picture,
not some kind of precise,
concrete state.
Because it's representing many,
many different genes,
they are occurring at all the
different possible places in the
genome.
Now for a neutral allele,
like the one that I've been
sketching there,
the fixation rate is just equal
to the mutation rate.
That doesn't depend on
population size.
The probability of fixation,
as I said, is equal to the
current frequency.
For a new mutation,
one of these guys down here,
right at the beginning,
that's 1/2N,
to be fixed,
and 1-1/2N to be lost.
That means that most of them
are lost.
N is the population size.
N is a big number.
Because there are 2N copies of
the gene in the population,
and if mu is mutation rate,
that means in each generation
there are 2mu new mutations,
and for each of them the
probability of fixation is 1/2N.
So the rate of fixation of new
mutations is about 2mu times
1/2N, which is equal to the
mutation rate.
That's about 10^(-5) to 10^(-6)
per gene,
and that means the molecular
clock is ticking once every
100,000 to once every 1,000,000
generations per neutral gene.
The fixation rate doesn't
depend on the population size,
and that's because the
probability that a mutation will
occur in a population depends
upon how many organisms are
there.
You can think of all of their
genomes out there as being a net
spread out to catch mutations--
the bigger then net,
the more the mutations are in
any given generation--
and that will just exactly
compensate for the fact that it
takes them longer to get fixed.
The bigger the population,
the longer this process takes.
But the bigger the population,
the more of these are actually
moving through to fixation.
Those two things exactly
compensate.
Okay?
In a small population most of
them are lost.
The few that do reach fixation,
reach it rapidly,
and in large populations more
new mutations are fixed,
but each one does it more
slowly.
Those things compensate,
and the fixation rate doesn't
depend on population size,
if you're looking at the whole
genome.
The number of differences fixed
over the whole genome doesn't
depend on the size of the
population.
Now there is a technical
concept in evolutionary genetics
called effective population
size,
and that is the size of a
random mating population,
that is not changing in time,
whose genetic dynamic would
match those of the real one
under consideration.
And so we know that there are
lots of violations of these
assumptions.
Okay?
Populations don't have random
mating.
They are changing in time;
ta-da ta-da.
How do we take a real
population and then transform it
into something that's really
easy to calculate?
Well, there are methods of
doing so.
The factors that will have to
come into consideration are
variation in family size,
inbreeding, variation in
population size,
and variation in the number of
each sex that is breeding.
And so just to illustrate one
of these, to give you some idea
of its impact,
look at cattle in North
America.
There are about 100,000,000
female cattle in North America.
They are fertilized by four
males, on average,
through artificial
insemination.
So there are four bulls that
are inseminating 100,000,000
cows.
Genetically speaking,
how big is the population?
It's just about 16.
Okay?
So by restricting one sex to a
very small number,
we have restricted one pathway
that the genes can go through to
get to the next generation.
And by making the male side of
it so small, we have biased the
probability that a gene will get
fixed according to some process
like this.
That male side is a really
small population.
So it completely outweighs the
fact that there are 100,000,000
females there.
Because if you think about it,
every time one of those genes
goes through a female and goes
into a baby and grows up the
next generation,
it's going to go back through
the male side of the
population--
right?--as you go through the
generations.
And these formulas that have
been developed give us the
opportunity to take that complex
situation and make a quick,
useful, back of the envelope
calculation of how we can expect
genetic drift to be going on in
cattle in North America.
Basically they are a small
population.
So that's the basis of a
mutation-drift balance.
The amount of genetic variation
in a population,
in a mutation-drift balance,
is just a snapshot of the genes
that are moving through it.
If I were to go back to this
diagram,
and I were to put more genes
into this process,
and I were to ask you to go out
and take a sample out of a
population at any given time,
you would take the sample at
some time and you would tell me
that's how many genes we have,
that's how many are moving
through.
Okay?
Now the second possibility for
a mechanism that will maintain
genetic variation is a balance
between mutation and selection.
Mutation brings things into the
population.
Selection takes them out.
So if we had a haploid
population, with N individuals,
and we have a mutation rate mu,
we're getting Nmu new mutations
each generation.
The key idea is that if there
is a mutation selection balance,
then the number going in equals
the number going out;
that's what would keep this
mechanism balancing the amount
of genetic variation in the
population.
And so if the mutant
individuals have a lower fitness
than the non-mutants,
and if q is the frequency of
the mutants,
then selection is taking out
NSq mutants per generation.
And at equilibrium,
with the number coming in equal
to the number going out,
the number coming in equals the
number going out,
and that gives us an
equilibrium frequency of the
mutation rate divided by the
selection coefficient.
It's a very simple result.
And if you do the same kind of
thinking for a diploid
population,
you get that the equilibrium
frequency will be the square
root of the mutation rate,
divided by selection for
recessives,
and the same as it is for
haploids for dominance.
Okay?
So there are some examples of
this.
There are rare human genetic
diseases, such as
phenylketonuria--that's the
inability to metabolize
phenylalanine.
It has a frequency of about 1
in 200,000, in Caucasians and
Chinese.
It is probably in selection
mutation balance.
It's at low frequency but it's
present in a population.
People with it suffer a
selective disadvantage.
It keeps mutating and coming
back in, and it keeps getting
selected out.
The result is balance,
okay, and it's pretty rare.
The third mechanism that will
maintain selection in natural
populations is a balance of
selective forces;
that is, where the heterozygote
is better than either
homozygote.
And there is a classic,
famous case,
and it's always discussed in
this context,
and it's interesting that it's
the one that's always discussed
in this context,
and the answer is it's been
hard to find more.
>
Okay?
That's sickle cell anemia.
Now this is the normal
heterozygote which is
susceptible to malaria.
The heterozygote is resistant
to malaria, and the sickle cell
homozygote is anemic and sick.
And it sets up this kind of
relative fitness.
And, in fact,
if--H here is actually going to
be a negative number.
Okay?
So the fitness of the
heterozygote is going to be
higher than the fitness of
either homozygote.
And you can then set--the
equilibrium frequency is going
to be the one where P prime is
equal to p;
in other words,
the frequency in the next
generation is just the same as
the frequency in this
generation.
At what frequency does that
happen?
Well it happens when these
little equations are satisfied.
And the interesting thing,
when you look at them,
is that the selection
coefficient has dropped out of
them.
The equilibrium frequency
doesn't depend on the selection
pressure, it depends on how
frequently the gene is expressed
in a heterozygote.
So it depends really on the
heterozygote advantage.
Now the real situation is more
complicated than this.
There are several such sickle
alleles.
They're changing frequency.
The equilibrium assumption
doesn't really apply out there
in Nature,
but it does give us a rough
rule of thumb for how much to
expect,
and as soon as people who have
sickle cell anemia move out of
areas with malaria,
it takes quite awhile for that
allele to disappear from the
population.
The fourth mechanism is a
balance of selection forces,
so that, for example,
for A2, when A2 is 0,
it has high fitness here,
and as it increases in
frequency its fitness drops,
according to this equation.
Now the frequencies of A1 are
just reversed along this axis.
A1 is 1.0 here, and it's 0 here.
A1 has low frequency--has low
fitness when it's at high
frequency, and high fitness at
low frequency.
A2 has high fitness at low
frequency;
low fitness at high frequency.
So both of them do better when
they are rare.
And I think that you can see
intuitively from this diagram
that at equilibrium they will
stop changing when their
fitnesses are exactly the same.
Now there are some interesting
examples of this sort of thing.
One is Ronald Fisher's
classical argument on why 50:50
sex ratios are so common;
why in many populations we see
half females and half males.
The deviations from that are
interesting.
This kind of thing happens with
evolutionary stable strategies,
and those are the solution to
many problems within
evolutionary game theory.
They are also called Nash
equilibria, under certain
circumstances,
and they are important in
economics and political science
as well.
And the tremendous amount of
genetic variation in the immune
system is thought to exist for
reasons of frequency dependent
selection;
basically pathogen resistant
genes gain advantage when they
are rare, because when they're
common, the pathogens evolve
onto them.
They are more or less sitting
ducks;
they're a stable evolutionary
target.
But as they become more common
and more and more pathogens
evolve onto them,
and those organisms get sicker
and sicker,
the ones that are rare have an
advantage.
And then as they start to
increase in frequency,
the same process occurs;
the same process,
it continues again,
and after awhile you've got
hundreds of genes,
each of which is advantageous
at low frequency,
and none of which are
advantageous at high frequency.
So this is a very important
kind of mechanism maintaining
genetic variation in natural
populations, including our own.
If we look at quantitative
traits, such as birth
weight--here's a classical
example.
This is for babies born in the
United States in the 1950s and
1960s, and this is the percent
mortality for babies of
different weights.
You can see that there's
stabilizing selection that's
operating to stabilize birth
weight right at about 7 pounds,
and there's variation around
it.
And you might wonder,
why is there any variation
around that?
Why don't all babies have the
optimal birth weight?
It's such an important thing.
And there are really two
answers to that.
One is that there are
evolutionary conflicts of
interest between mother and
infant,
and father and mother,
over how much should be
invested in the infant,
and these lead to some
variation.
And there's mutation selection
balance.
So that this is a trait which
is probably determined by
hundreds of genes,
and at each of those genes
mutations are coming into the
population,
and at each of those genes
there is a mutation selection
balance,
and when you add that up,
over hundreds of genes,
you get quite a range of
variation.
Of course, some of this
variation is also due to
developmental effects of the
environment;
variations in the mother's diet
and other parts of her
physiological condition during
pregnancy.
So to summarize.
The origin and maintenance of
genetic variation are key
issues;
mutations are the origin.
Recombination has huge impact.
There's a tremendous amount of
genetic variation in natural
populations.
Remember that data from the
HapMap Project on us,
on humans, and that all of the
differences that you have,
in single nucleotide
polymorphisms,
from the person sitting next to
you,
and how you share them with
people who have had a similar
history since we came out of
Africa.
We can explain the maintenance
of this variation by various
kinds of mechanisms,
principally for balance between
mutation and drift,
between mutation and selection,
and by some kind of balancing
selection,
either heterosis or frequency
dependent selection.
And we think that variation in
many quantitative traits--
human birth weight,
human body size,
athletic performance,
lots of other things--
is probably maintained by
mutation selection balance,
as well as by other factors.
So next time I'm going to talk
about the role of development in
evolution.