Classical Population Genetics

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Classical Population Genetics Genetic Variation
at one Locus with 2 Alleles
Source Theory of Population Genetics and
Evolutionary Ecology, Jonathan Roughgarden,
Prentice Hall, Upper Saddle River, NJ, 1996
reprint of 1979 edition, Part One, pp17-100
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Consider a population with two Alleles, A and
a. Possible Genotypes AA, Aa, and aa Suppose
that we have a population of size, N (usually a
large number). Distribution of Genotypes NAA
Number of AA Homozygotes NAa Number of
Heterozygotes Naa Number of aa Homozygotes
N NAA NAa Naa
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Two important frequencies for us to
consider Genotype Frequencies
Gene Frequences
These are important relationships ? be sure that
you understand them.
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Hardy Weinberg Law If we assume no external
forces or processes, within one generation,
D ? p2 H ? 2pq R ?
q2 and these frequencies remain stable for all
future generations.
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• What assumptions are being made
• Individuals of different genotypes do not differ
  in fertility.
• Random union of gametes.

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3. All individuals, regardless of genotype, have
an equal likelihood of survival from gamete to
adulthood.
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An example to illustrate what is being said by
the law Suppose an aquarium owner purchases a
variety of fish with two alleles that determine
their fin color. A red
fin a blue fin In the shipment the owner
receives, 75 of the fish have red fins, 25 have
blue fins, and none have purple fins. What will
be the eventual distribution of fin colors in the
aquarium? After one generation
Note Aa purple fin
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Proof of the Law Because of random union of
gametes prob(AA) pp p2
prob(Aa) prob(aA) pq
or prob(Heterozygote) 2pq
prob(aa) qq q2 Note gamete frequencies at
start are p and q. At this point we use the
third assumption that equal ratios of gametes
survive, mate, and the zygotes survive until the
adult stage to produce gametes for the next
generation. Thus, D p2 H 2pq R
q2 Gametes are haploid and previous
information about previous diploids population
is lost.
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• What is missing?
• Natural selection
• Differential fertility and/or survival
• Mutation
• Immigration from other populations
• Genetic drift

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• Are any assumptions unnecessary?
• Random mating also produces the same results.
  Just slightly more complex to show than the
  random union case.
• The requirement of distinct generations is not
  necessary. However, this assumption makes the
  algebra easier.
• If there is a different distribution of genotypes
  among the sexes, the stable position does not
  emerge for two generations (assuming that all
  other assumptions hold in particular the
  survival one)

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Enter Natural Selection
Consider Survival Rates lAA , lAa ,
Iaa Fertility Rates mAA, mAa, maa Let
WAA lAAmAA WAa lAamAa Waa Iaamaa
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Go Back to slides 2 and 3 and we can derive the
number of gametes in the population at time, t
1 from AA adults 2WAApt2 Nt
from Aa adults
2WAa2ptqtNt from aa adults 2Waa
qt2Nt The total population size at time, t1, is
one half the sum of these three quantities.
Nt1 (WAApt2 WAa2ptqt WAA
qt2)Nt An equation such as this is called a
difference equation.
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This is an example of a fast evolutionary
change (lt 40 years). It was caused by industrial
pollution in the area of Birmingham, England.
Before pollution these moths had majority
coloration (light) that was difficult to see
against the lichen of trees growing in the area.
After pollution the bark became black and the
lichen died. This meant that the light colored
insects became easy prey. So selection
pressure favored the dark colored moths.
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The difference equation for the population size
leads to these two absolutely essential
difference equations for the gene frequencies
So what? These equations coupled with the
difference equation for the population size allow
us to assign different fertility and survival
rates to the existing three genotypes and model
how the gene pool and population size change as a
result. Question Is this absolutely the way
things will turn out?
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One last notational adjustment to make matters a
little more simple. We will work to eliminate the
preponderance of Ws from the equation by
multiplying them by a suitable constant. We
normalize by selecting one of the Ws to be 1.
Say WAA1. Then we must divide the remaining two
Ws by WAA. Thus,
wAA 1 (WAA/WAA)
wAa WAa/WAA waa
Waa/WAA Note that we denoted these normalized
values with a small, italicized w.
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And, FINALLY, we define the selectivity
coefficients sAA 1 wAA sAa 1
wAa saa 1 waa Notice that, in general,
these are selectivity against. That means that a
value of 0 is good and positive decreases the
gene pool. Example mAA 100
mAa 50 maa 25 lAA ¾
lAa ½ laa 1/5 Then, WAA
(100)(3/4) 75 WAa (50)(1/2) 25
Waa (25)(1/5) 5 wAA 75/75 1
wAa 25/75 1/3 waa 5/75
1/15 sAA 0 sAa
2/3 saa 14/15
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With all of these substitutions we finally have
an expression for pt1 that is manageable.

• The simulations that follow all used the first
  form of the difference equation.
• We will consider
• Selection against a dominant allele
• Selection against a recessive allele
• Heterozygote superiority

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Writing a program to implement this model is a
quite straight forward process. This program is
written in a functional programming language used
in the Derive Computer Algebra System.
p(pwdd qwdr)
dp(p, q, wdd, wdr, wrr) ? ?????????????????????
?????? 2
2 p wdd
2pqwdr q wrr
HWApprox(p, wdd, wdr, wrr, n, q, i, pp, pn, qp,
qn, hw) ? Prog
q ? 1 - p
i ? 0
pp ? p

qp ? q
pn ? p
qn ? q
hw ?
Loop
If
i gt n
RETURN hw
hw ? APPEND(hw, i, pn)
pn ? dp(pp, qp, wdd,
wdr, wrr) qn ? 1 - pn
pp
? pn
qp ? qn
i ? i 1
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Selection Against the Dominant Allele p0 .9
wAA .8 wAa .8 waa 1
Note that even though the recessive allele made
up only 10 of the gene pool, in approximately 70
generations it makes up the entire gene pool.
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Selection Against the Recessive Allele p0 .1
wAA 1 wAa 1 waa .8
The end result is expected, but there is a
qualitative difference. In the former case the
decline of the majority gene started slowly and
then accelerated. Here the initial decline is
rapid and then the rate slows down.
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Selection in Favor of Heterozygote Selection
against the recessive is four times that against
dominant p0 .9 .5 wAA .9 wAa 1
waa .6
Note that in each of the cases (in fact, all
cases except p0 0 or 1) The dominant allele
will eventually make up 80 of the gene pool and
the recessive will make up 20. This result is
called a stable equilibrium.
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Finally a Highly Unusual Result Selection against
the Heterozygote p0 .55 .5 .45 wAA 1
wAa .8 waa 1
Notice that if both populations start out with
50 of the gene pool then that percentage will
persist. However, if the percentage wanders off
of 50, the majority gene will become the entire
gene pool and the other will become extinct.
Thus 50 is called an unstable equilibrium.
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Of the four scenarios that we considered, three
resulted in the elimination of one of the
Alleles. Only the case of selection in favor of
the Heterozygote resulted in a mixed gene pool.
Thus, in the presence of natural selection (We
will see later in this lecture what a powerful
force this can be.), this is the only case where
genetic variation is maintained.
Polymorphism Other cases fix on one or the other
of the alleles.
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Selection in Favor of Heterozygote Selection
against the recessive is four times that against
dominant p0 .9 .5 wAA .9 wAa 1
waa .6
Note that in each of the cases (in fact, all
cases except p0 0 or 1) The dominant allele
will eventually make up 80 of the gene pool and
the recessive will make up 20. This result is
called a stable equilibrium. Can we determine
what this equilibrium will be?
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More notation (Mathematicians love it!!)
What do we mean by equilibrium? When equilibrium
is achieved then the frequency of the alleles
stays stable. pt1 pt for all t gt some t0
And of course, qt1 1 pt1 1 pt qt
On the previous slide this happens around
generation 50. So, t0 50. Lets see if we can
predict . Recall, .
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We start with the definition of
equilibrium pt1 pt Earlier we saw that in
the presence of natural selection,
Since pt ? 0, this means that
For all t gt t0 . Or at the equilibrium value,
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Some simple, but messy, algebra gives us the
following result.
Or,
In our example wAA .9 wAa
1 waa .6 So,
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Experimental evidence ST and CH are names of
blocks of genes in Drosophilia pseudo-obscura
because of a chromosomal feature called inversion
the genes in each block are held together and
function as two alleles at a single locus.
Solid line simulated path for p.
Dashed lines are 95 confidence limits
Vertical bars experimental data
Results correctly predicted the equilibrium and
the dynamics of the approach to equilibrium.
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But, what about mutation? Ordinarily it works
this way A a

We are going to stack the deck in favor of
mutation and assume
A a i.e. we assume v 0 In the
absence of any selection our difference equation
becomes pt1 (1 u) pt This is just
the difference equation for exponential decay
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Look at the time axis! This process is much
slower than our simulations of natural selection
that was anywhere from 1 generation (pure
Hardy-Weinberg) to about 15,000 generations to
drop from p.9 to p.1.
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To actually calculate the predicted time to move
from p0 to pt . Begin with
Rearrange bottom line as
Take log of both sides and solve for t. This
yields,
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Lets calculate the time to move from p0 .9 to
pt .1 for the first curve on the graph shown
two slides previously, i.e. u 10-5 .00001
Mathematical note Since this quantity involves
the quotient of two logarithms, any base
logarithms will give the same numerical result.
i.e. We can use either the log10 or ln button on
our calculator or even log2 if we care to do
this. Extra Credit Project Use a spreadsheet or
write a computer program to generate the graphs
that were shown two slides previously.
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• In general, mutation has little effect if
  selection is at work.
• If selection is virtually neutral, say s lt .001,
  then mutation can have an effect, but it is slow.
• However, recurrent mutation can not be totally
  disregarded.
• Recurrent mutation tends to maintain a supply of
  genetic variation for mutation to act upon
• Even if selection is tending to eliminate one
  allele, recurrent mutation tends to maintain its
  presence in the gene pool. Thus, if the
  environment changes to a situation that is more
  favorable to the allele that was being selected
  against, that allele is still available.
• Mutation is the ultimate source of genetic
  variation.

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Sometimes mutation may oppose selection. Suppose
selection is against A wAA 1 s
wAa 1 s (A is dominant) waa 1 However,
also assume v gt 0, i.e. There is recurrent
mutation of a to A at a rate, v. Then, it can be
shown
On the other hand if A is recessive, wAA 1
s wAa 1 waa 1 We have,
If A is recessive, mutation maintains a much
higher frequency than if it is dominant.
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• Genetic Drift
• So far every model we have considered has been a
  deterministic model, i.e. everything is set in
  motion on a predetermined path. Chance has been
  ignored.
• But, chance does play a role!
• In the sea urchin model, gametes can wash out
  to sea.
• Some types of individual may produce more
  offspring than others
• Survival rates may vary
• A theory involving chance is called a stochastic
  theory.
• Instead of getting a single number, we get a
  distribution between several states

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• Two sources for chance occurrences
• Changing environment
• Internal to the population they would occur
  even in a fixed environment.
• Genetic Drift refers to all chance events
  internal to the population

Example Suppose we start with a large
population and p ½ . From the gamete pool
draw 4 individuals (small sample) Could be 2 2
relative to the alleles Could also be 3 1, or
1 3, or 0 4, or 4 0. Suppose 3 1 is the
distribution in our sample, then p has moved from
½ to ¾ without any selective pressures. This is
called sampling error. NOTE Sampling error is
more likely to occur as the population size
decreases.
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Experimental evidence of Genetic Drift Kerr and
Wright (1954) sampled a population of Drosophilia
melanogaster heterozygotes. They constructed 96
groups of 4 males and 4 females. At each
generation they randomly extracted 4 males and 4
females from that generation, etc. The following
is their data.
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Note the U shape of the later histograms of the
frequency distributions. This is characteristic
of this type of situation