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Goal: Investigate the Hardy Weinberg law
of equilibrium and some simple models of natural selection used in population
genetics. After today you should be able to do the following:
Introduction.
The Hardy Weinberg law of equilibrium
is one of the most important concepts in population genetics. First of all
for many situations involving complete dominance it provides a way to infer
what proportion of the population consists of homozygous dominant vs. heterozygotes
simply by knowing the proportion of the population that is homozygous recessive.
This is particularly useful in answering questions about the frequency of
individuals that might be carrying rare recessive alleles for human genetic
disorders.
More importantly in population genetics,
Hardy Weinberg provides a framework from which population geneticists construct
models to study the evolutionary process, particularly at the level of microevolution
or genetic change in a population.
The Hardy Weinberg Law of Equilibrium.
Suppose we are studying a single locus
with two alleles A and a. Let p = the frequency of the A allele in the population
and q be the frequency of the a allele. The Hardy Weinberg says that in the
absence of evolution then the following relationship holds or rapidly become
true:
The frequency of the genotypes AA, Aa
and aa respectively are given by p^{2}, 2pq and q^{2}. Note
that these terms are simply the expansion of the following:
(p +q)^{
2}.
Question 1.
Phenylketonuria(PKU) is an autosomal recessive disorder. The frequency of
individuals who have PKU is about 1/12,000. Use the Hardy Weinberg law to
estimate the frequency of individuals that are heterozygous carriers of PKU.
Question 2
. Suppose you have three alleles in a population A_{1}, A_{2}
and A_{3} which are found with frequencies p, q, and r respectively.
A. List the possible genotypes
B. Note that Hardy Weinberg law can be
generalized to this situation as an expansion of
(p + q + r)^{2}^{}
. Use this fact to estimate the frequencies of the possible genotypes in a
population where p = 0.2 , q = 0.4.
Hint: why didn't I give you r ?
Question 3. A puzzler!
The ABO blood system is a wellknown example
of codominance. A certain population has the following frequencies for the
blood types:
A =0.40
B =0.27
AB =0.24
O = 0.09
A.
Estimate the frequency of the 'i' allele in the population.
B.
Let p be the frequency of the I^{A} allele in the population and q
be the frequency of the I^{B} allele in the population and r be the
frequency of the 'i' allele. Estimate the frequency of the alleles. Clearly
show all your work.
Hints:
·Assume
Hardy Weinberg can be extended to situations where you have multiple allele
systems.
·Assume
random mating, which means random combinations of gametes.
Write a general algebraic expression for
the frequency of blood type A in terms of a function of p and r. A certain
formula from algebra dealing with solving certain equations might be useful. J
Question 4.
Suppose I am a geneticist studying sickle cell anemia in a particular population
and I find the following frequencies genotype frequencies in 1000 infants
screened for the B^{A} and B^{S} alleles:
B^{A}B^{A} = 0.60
B^{A}B^{S} = 0.35
B^{S}B^{S} = 0. 05
A.
Estimate the frequency of the B^{A} and the B^{S} alleles
in the population. Hint: remember how we did the snapdragon example in lecture.
See page 98  99 for a discussion of the genetics of sickle cell.
B.
Use the resulting allele frequencies to predict what the genotype frequencies
should be under Hardy Weinberg equilibrium.
C.
Do the observed data suggest that the Hardy Weinberg law holds for this population?
Use the observed data in A and the (predicted or expected) genotype frequencies
to test this by means of a certain statistical test.
Simple models of natural selection.
The Hardy Weinberg law ideally applies
when 5 basic assumptions are met which you should review. Population geneticists
attempt to develop mathematical models of evolution by modeling what happens
when one or more of the assumptions of the Hardy Weinberg law are seriously
violated.
Natural selection refers
to the differential survival and reproductive success of different phenotypes
in a population in response to environmental conditions. Lets suppose for
instance that butterflies in a certain population come in three wing colors:
white, grey and black. Suppose that we let W_{11}, W_{ 12}
and W_{22} be the relative number of offspring left by each type of
butterfly. This relative number is often called Darwinian fitness.
See page 657 in your text.
So for example suppose the white butterflies
leave on average 15 offspring per butterfly that survives to adult hood, the
grey butterflies leave 18 offspring per butterfly and the black butterflies
leave 9 offspring per butterfly. We then use these to calculate the Darwinian
fitnesses per adult butterfly of each phenotype as:
W_{11} = 15/18; W_{12}
= 18/18; W_{22} = 9/18
Notice that the Darwinian fitnesses are
relative to the phenotype with the highest absolute reproductive success.
Mathematically this just normalizes the fitness values so that the maximum
Darwinian fitness is 1.0. One thing this allows is the expression of fitness
in terms of what is called a selection coefficient which is simply 1 Wij
for each of the Darwinian fitnesses.
For our example what are the selection
coefficients S1, S12 and S2 for each phenotype?
Often times it is easier to express population
genetics models in terms of the selection coefficients rather than the Darwinian
fitnesses.
Note that natural selection is simply
differential survival and reproduction of different phenotypes and that natural
selection will only lead to evolution if the phenotype is at least in part
tied to genetics. So lets suppose for simplicity that wing color in this butterfly
is controlled by two alleles A_{1} and A_{2} . Complete the
following table for a hypothetical butterfly population:
Phenotype  White  Grey  Black 
Genotype  A_{1} A_{1}  A_{1} A_{2}  A_{2} A_{2} 
Number of Adults in parent generation (time t)  1000  750  100 
Reproductive success per adult of each genotype  11  18  5 
Darwinian Fitnesses  W_{11} =  W_{12} =  W_{22} = 
Selection Coefficients  S_{11} =  S_{12} =  S_{22} = 
Now we have all the information necessary
to predict how the frequencies of the A_{1} and A_{2 }alleles
will change over for this population. We will do this by writing an expression
for p', the allele frequency of A_{1 }at time t+1 as a function of
the current value of p.
Question 5.
Use the data from the table to calculate
the frequencies p of the A_{1} allele and q for the A_{2 }allele
during the parent generation. This will also give you the frequency of each
type of gamete produced by the parents. Remember that p + q =1.0 so check
yourself.
Average fitness of the alleles
In simple selection models written in
terms of p and q we can calculate the average Darwinian fitness (W_{ 1}
and W2) of each allele:
This is
W_{1} = (W_{11}*1000 +
1/2 W_{12}*750)/ (1000 + 1/2*750) for A_{1} and:
W_{2} = (1/2 W_{12} *750
+ W_{22}*100)/ (1/2*750 + 100) for A_{2} .
The A_{1} allele will spread if
W_{1} > W2 and A_{2} the allele will spread if W2 > W_{1}.
Which allele has the higher average fitness
and will thus become more common in the very next generation?
Equations for change in allele frequencies
What we typically want is an expression
for p' in terms of p during the previous generation when natural selection
is operating but all the other assumptions of Hardy Weinberg apply. It turns
out that this is relatively easy and is given by:
p' = W_{1}p/(W_{1}p +
W_{2 }pq) and it turns out that this becomes:
(Equation 2A)
p' = [W_{11}p^{2} +
W_{12}pq]/W
where
(Equation 2B)
W = W_{11}p^{2} + 2W_{12}pq
+ W_{22}q^{2} is called the average population fitness.
p' = [W_{11}p^{2} + W_{12}pq]/(W_{11}p^{2} + 2W_{12} pq + W_{22}q^{2})
This expression is the same as given in
your text in tables 22.10 and 22.11
Now we have an expression that allows
us to follow the change in allele frequencies from generation to generation
simply by reevaluating this expression at each generation.
Generally population geneticists use this
general formula and derive algebraic expressions for specific cases in terms
of the selection coefficients. Some of these common expressions for different
types of natural selection are in table 22.12 of your text.
Notice these are written in terms of D
p's that is the change in allele frequency between time t and the next time,
The general form of the difference equation is simply:
(Equation 3A)
Dp = p'  p
= W_{11}p^{2} + W_{
12}pq/W  p
After some algebra this becomes:
= pq[p(W_{11}  W_{12}
) + q(W_{12}  W_{22})]/W
D p = pq[p(W_{11} W_{12} ) + q(W_{12}  W_{22})] /( W_{11} p^{2} + 2W_{12}pq + W_{22} q^{2})
Note that the corrsponding expression
in your text is written in terms of the other allele which occurs with frequency
q. It the same expression as equation 3, except multiplied by 1. The
trick to deriving equation 3B from equation 3A is to multiply the top and
bottom of equation 3A by W and writing W in terms of the Darwinian fitnesses.
Then this can be rearranged to yield equation 3B.
The details are here.
These difference equations are useful
because they allow one, as we shall see, to investigate equilibrium values
for allele frequencies, that is values of p where p does not change.
Question 6.
A. Looking at table 22.12, which type
of selection is shown in the butterfly example? You should refer to the Darwinian
fitnesses and selection coefficients you calculated earlier to help you.
B. Substitute the values for the selection
coefficients and write a specific expression for the butterfly example:
Stability analysis:
Often population geneticists can find
equilibrium values and determine whether or not the equilibria are stable.
This can be done using calculus, but often a graphical analysis is useful.
You will analyze the selection model for the butterfly example. Graphical
stability analysis involves plotting Dp
as a function of p and examining the slope of the function around any equilibrium
points.
Question 7. Use a calculator or spreadsheet and your formula from 6B or the general formula in equation 2c or equation 3B to complete the following table for p' as a function of p.
A simple Excell
worksheet.
A more complex worksheet for graphing.
p  D p = p'p 
0 

0.05 

0.10 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

0.60 

0.65 

0.70 

0.75 

0.80 

0.85 

0.90 

0.95 

1.00 

Next roughly sketch the graph showing
the equilibrium vales. If you have access to a programmable graphing calculator
or spreadsheet program this should be easy. The points where the graph crosses
the x axis ( Dp
= 0 ) are equilibrium points.
B.
How many equilibrium points are there for this model?
C.
Which equilibrium is stable? How do you know? Hint: for a stable equilibrium
point what happens to D
p for values of p smaller than the equilibrium or larger than the equilibrium?
Graphical analysis example:
Graph of D p as a function of p:
Notice there are three equilibrium points one each at p = 0.0 and p =1.0 and the third at p = 0.65. Equilibrium points are typically either stable or unstable. Lets examine the first equilibrium point at p = 0. This means that the population only has the A_{2} allele. If a very small number of A_{1} alleles are introduced then this makes p slightly grater than zero. Taking this arbitrarily small value above the equilibrium point as the new p, notice that Dp is greater than zero and hence p' is larger. This means that over time p should increase for this happens as long as D p is positive. The opposite will be true for the equilibrium at p = 1.0.
But what about the equilibrium at p = 0.65? Notice here is you pick a value of p slightly smaller than .65, say p = 0.64, then Dp is positive meaning that p will creep back up toward the equilibrium point. If p is say 0.66 then D p is negative and p should creep back to the equilibrium point. Hence, the thrird equilibrium point is stable.
This is a pretty traditional analysis. Some types of genetic systems in population genetics may exhilbit nonlinear behavior, for example complex models involving certain types of frequency dependent selection. These models are often best analysed by plotting p' as a function of p.
pgd created 04/18/03 revised 12/18/04