Lab 6: Genetic Drift and Effective
population size
Goals
1. To calculate the probability of fixation or loss of an
allele.
2. To estimate mean time until fixation of an allele.
3. To estimate effective population size affected by
past cataclysms.
4. To learn how genetic drift and selection interact in
populations of various Ne.
Probability of fixation or loss
1. Genetic drift results from chance changes in allele
frequencies that result from sampling of gametes from
generation to generation in a finite population.
2. Exact probability of fixation of an allele is
equal to the initial frequency of that allele in
absence of selection.
u( p)  p0
3. Probability of fixation of an allele can be
calculated empirically by using Monte Carlo
simulations as implemented in Populus.
Problem 1. The frequencies of alleles A1 and A2 are p = 0.7
and q = 0.3, respectively. Use Populus to empirically estimate
the probabilities of fixation and loss for each of these alleles.
What do you think are the exact probabilities of fixation and loss
for each allele? Do these probabilities depend on the population
size?(15 minutes)
Sr.#
1
2
3
4
5
6
7
8
9
10
# of run
I
II
III
IV
V
VI
VII
VIII
IX
X
# of A1 loci lost
3
1
1
2
2
3
4
3
6
1
26
# of A1 loci fixed
7
9
9
8
8
7
6
7
4
9
74
Problem 2.
Consider a population with the following genotype counts ( 15
minutes).
Case
1
2
Genotype counts
A1A1
A1A2
18
4
0
7
A2A2
3
21
a.) Use Populus to empirically estimate the mean time (in
number of generations) until fixation for allele A1
b.) Show the mean time until fixation of A1 calculated using
the diffusion approximation
c.) Discuss the reasons for the differences (if any) between
the two types of estimates. What are some of the
assumptions underlying each method?
Mean time until fixation of an allele depends on
population size and initial frequency of that allele.
Kimura
and Ohta (1971) diffusion
T ( p) 
approximat
 4 N (1  p ) ln( 1  p )
ion :
.
p
A1A1 A1A2 A2A2
(N11) (N12) (N22)
N
p
q
T(p) in
terms
T(p) of N
16
2
2
20
0.85 0.15 26.78
1.34N
0
2
18
20
0.05 0.95 77.96
3.90N
Case - ?
Population for
allele A1
# generation when locus
fixed
1
2
3
4
5
6
7
8
9
10
3
8
4
13
15
44
56
57
68
89
357
357/10= 35.7
Problem 3. The census populations size of an isolated
population of finches on the Galapagos islands is as
follows. What is the effective population size in 2010?
Year
1930
1950
1970
1990
2010
Females
Males
120
250
15
290
250
350
1500
2500
3500
5000
Ne 
t
it

i 1
Ne 
1
Ni
4N f Nm
N
f
 Nm
When time is discontinuous, a transition matrix can be
used to determine the probability of fixation in the next
generation.
x ij  P (Y t 1  i | Y t  j ) 
( 2 N )!
( 2 N  i )! i!
p (1  s )  pq (1  hs )
2
p'
 p ' q '
1  2 pqhs  p s
2
i
2 N i
Fitness
Fitness in terms of s and h
(adaptive Darwinian
selection)
Fitness in terms of s and h
(purifying selection)
A1A1
ω11
Genotype
A1A2
ω12
A2A2
ω22
1+s
1 + hs
1
1
1 − hs
1−s
• Using traditional setup for adaptive Darwinian (positive) selection
• These can be easily converted to terms for purifying selection
Let s D  selection
coefficien
t under Darwinian
Let s P  selection
coefficien
t under purifying
Let h D  heterozygo
us effect under Darwinian
Let h P  heterozygo
us effect under purifying
sP 
sD
1  sD
h p  1  hD
selection
selection
selection
selection
Problem 4. If adaptive Darwinian selection
(characterized by h = 0.5 and s = 0.25) is operating on a
locus and the frequency of allele A1 at that locus is p =
0.1, predict whether A1 is more likely to get lost or to
become fixed:
sD
sP 
h p  1  hD
i.) In a population with N = 10.
e
ii.) In a population with Ne = 100.
1  sD
a.) For each of the cases, calculate the probability of A1 fixation empirically
b.) If Ne affects the probabilities of fixation and loss of A1, explain why. If not, explain why not.
Fitness
Fitness in terms of s and h
(adaptive Darwinian
selection)
Fitness in terms of s and h
(purifying selection)
A1A1
ω11
Genotype
A1A2
ω12
A2A2
ω22
1+s
1 + hs
1
1
1 − hs
1−s
Fitness
Fitness under adaptive
Darwinian selection
Fitness under
purifying selection
Genotype
A1A1
A1A2
ω11
ω12
A2A2
ω22
1.25
1.125
1
1
0.9
0.8
Problem 5. GRADUATE STUDENTS ONLY:
Starting with the conditions in Problem 4-a),
calculate the probability that:
a.) The frequency of A1 becomes 0.1 in the
next generation.
b.) A1 becomes fixed in the next generation.
c.) If the two transition probabilities differ
dramatically, explain why. If not, explain why
not.