Lecture 9: Introduction to Genetic Drift
September 25, 2015
Last Time
◆ Overdominance and Underdominance
◆ Overview of advanced topics in
selection
◆ Introduction to Genetic Drift
Today
◆ First in-class simulation of
population genetics processes:
drift
◆ Fisher-Wright model of genetic
drift
What Controls Genetic Diversity Within
Populations?
4 major evolutionary forces
Mutation
+
Drift
-
Diversity
+/Selection
+
Migration
Genetic Drift
◆  Relaxing another assumption: infinite populations
◆  Genetic drift is a consequence of having small
populations
◆  Definition: chance changes in allele frequency that
result from the sampling of gametes from generation to
generation in a finite population
◆  Assume (for now) Hardy-Weinberg conditions
Ø  Random mating
Ø  No selection, mutation, or gene flow
Drift Simulation
Parent 1
m
m
Parent 2
m
heads
m
m
m
m
m
m
m
m
m
m
m
m
m
m
u Form two diploid ‘genotypes’ as
you wish
u Flip a coin to make 2 offspring
tails
m
m
u Pick 1 red and 3 other m&m’s so
that all 4 have different colors
m
Ø  Draw allele from Parent 1: if
‘heads’ get another m&m with the
same color as the left ‘allele’, if
‘tails’ get one with the color of the
right ‘allele’
Ø  Draw allele from Parent 2 in the
same way
u ‘Mate’ offspring and repeat for
3 more generations
u Report frequency of red ‘allele’
in last generation
Genetic Drift
A sampling problem: some alleles lost by random
chance due to sampling "error" during reproduction
Simple Model of Genetic Drift
◆  Many independent subpopulations
◆  Subpopulations are of constant size
◆  Random mating within subpopulations
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
N=16
Key Points about Genetic Drift
◆ Effects within subpopulations vs effects in
overall population (combining subpopulations)
◆ Average outcome of drift within subpopulations
depends on initial allele frequencies
◆ Drift affects the efficiency of selection
◆ Drift is one of the primary driving forces in
evolution
Effects of Drift
Simulation of 4 subpopulations with 20 individuals, 2 alleles
◆  Random
changes
through time
◆  Fixation or loss
of alleles
◆  Little change in
mean
frequency
◆  Increased
variance among
subpopulations
How Does Drift Affect the Variance of Allele
Frequencies Within Subpopulations?
p(1 − p)
Varp =
2N
Drift Strongest in Small Populations
Effects of Drift
http://www.cas.vanderbilt.edu/bsci111b/drosophila/flies-eyes-phenotypes.jpg
◆  Buri (1956) followed
change in eye color
allele (bw75)
◆  Codominant, neutral
◆  107 populations
◆  16 flies per
subpopulation
◆  Followed for 19
generations
Modeling Drift as a Markov Chain
⎛ n ⎞ y n − y
P(Y = y ) = ⎜⎜ ⎟⎟ s f ,
⎝ y ⎠
◆  Like the m & m
simulation, but
analytical rather
than empirical
◆  Simulate large
number of
populations with two
diploid individuals,
p=0.5
◆  Simulate transition
to next generation
based on binomial
sampling probability
(see text and lab
manual)
Modeled versus
Observed Drift in
Buri’s Flies
Effects of Drift Across Subpopulations
◆  Frequency of eye color
allele did not change
much
◆  Variance among
subpopulations
increased markedly
Fixation or Loss of Alleles
◆  Once an allele is lost
or fixed, the population
does not change (what
are the assumptions?)
◆  This is called an
“absorbing state”
◆  Long-term
consequences for
genetic diversity
44
Probability of Fixation of an allele within a
subpopulation Depends upon Initial Allele Frequency
u (q) = q0
q0=0.5
N=20
where u(q) is probability of a subpopulation to be fixed for allele A2
N=20
Effects of Drift on Heterozygosity
◆  Can think of genetic drift as random selection of alleles from a
group of FINITE populations
◆  Example: One locus and two alleles in a forest of 20 trees
determines color of fruit
◆  Probability of homozygotes for alleles IBD in next generation?
Prior Inbreeding
2
PIBD
1
⎛ 1 ⎞
= 2 N ⎜
⎟ =
2N
⎝ 2 N ⎠
1 ⎛
1 ⎞
f t +1 =
+ ⎜1 −
⎟ f t
2 N ⎝ 2 N ⎠
Drift and Heterozygosity
◆  Expressing previous equation in terms of heterozygosity:
1 ⎛
1 ⎞
f t +1 =
+ ⎜1 −
⎟ f t
2 N ⎝ 2 N ⎠
1 ⎞
⎛
1 − f t +1 = ⎜1 −
⎟(1 − f )t
⎝ 2 N ⎠
H
1− f =
◆  Remembering: H = 2 pq(1− f ) and
2 pq
t
Ht "
1 % H0
= $1−
'
2 pq # 2N & 2 pq
p and q are stable through
time across subpopulations,
so 2pq is the same on both
sides of equation: cancels
◆  Heterozygosity declines over time in subpopulations
◆  Change is inversely proportional to population size
Diffusion Approximation
◆  Greatly simplifies the problem of simulating drift
◆  Readily extended to incorporate other factors
Time for an Allele to Become Fixed
◆  Using the Diffusion Approximation to model drift
Ø  Assume ‘random walk’ of allele frequencies behaves like
directional diffusion: heat through a metal rod
Ø  Yields simple and intuitive equation for predicting time to
fixation:
4 N (1 − p) ln(1 − p)
T ( p) = −
p
◆  Time to fixation is linear function of population size
and inversely associated with allele frequency
Time for a New Mutant to Become Fixed
4 N (1 − p) ln(1 − p)
T ( p) = −
p
◆  Assume new mutant occurs at frequency of 1/2N
◆  ln(1-p) ≈ -p for small p
◆  1-p ≈ 1 for small p
T ( p) ≅ 4 N
◆  Expected time to fixation for a new mutant is 4
times the population size!
Effects of Drift
◆  Within subpopulations
Ø  Changes allele frequencies
Ø  Degrades diversity
Ø  Reduces variance of allele frequencies (makes frequencies more
unequal)
Ø  Does not cause deviations from HWE
◆  Among subpopulations (if there are many)
Ø  Does NOT change allele frequencies
Ø  Does NOT degrade diversity
Ø  Increases variance in allele frequencies
Ø  Causes a deficiency of heterozygotes compared to HardyWeinberg expectations (if the existence of subpopulations is
ignored = Wahlund Effect)