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)  p 0
3. Probability of fixation of an allele can be
calculated empirically by using Monte Carlo
simulations as implemented in Populus.
Problem 1 (15 minutes). The frequencies of alleles A1 and A2 are
p = 0.75 and q = 0.25, 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?
Problem 2 (15 minutes). Consider a population with the following
genotype counts:
Genotype counts
Case
A1A1
A1A2
A2A2
1
14
3
3
2
0
3
17
 4 N (1  p ) ln(1  p )
T ( p) 
.
p
a) Use Populus to empirically estimate the mean time (in number of
generations) until fixation for allele A1 for each case.
b) Show the mean time until fixation of A1 calculated using the
diffusion approximation (given above) for each case.
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 approximation :
 4 N (1  p) ln(1  p)
T ( p) 
.
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
Problem 3 (15 minutes). The census population size of an
isolated population of finches on the Galapagos Islands is as
follows. What is the effective population size in 2010?
Year
Females
Males
1930
1950
1970
1990
2010
120
15
250
1500
3500
250
290
350
2500
5000
Dr. Robert Rothman, Rochester Institute of Technology
http://people.rit.edu/rhrsbi/GalapagosPages/DarwinFinch.html
When time is discontinuous, a transition matrix can be
used to determine the probability of fixation in the next
generation.
xij  P(Yt 1
(2 N )!
i
2 N i
 p' q'
 i | Yt  j ) 
(2 N  i)!i!
p (1  s)  pq(1  hs)
p' 
2
1  2 pqhs p s
2
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  selectioncoefficient under Darwinian selection
Let s P  selectioncoefficient under purifyingselection
Let h D  heterozygous effect under Darwinian selection
Let h P  heterozygous effect under purifyingselection
sD
sP 
1  sD
hp 1  hD
Problem 4 (15 minutes). 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 = 0.25, predict whether
A1 is more likely to get lost or to become fixed:
(i) In a population with Ne = 10.
(ii) In a population with Ne = 30.
a) For each of the two cases, calculate the probability of fixation of
A1 empirically (i.e., using Populus).
b) If Ne affects the probabilities of fixation and loss of A1, explain
why. If not, explain why not.
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.