Transcript ppt

Lecture 25 : Tests of Neutrality
April 14, 2014
Last Time
Human origins
 Out of Africa hypothesis
 Neanderthal and Denisovan genomes
 Introgression into humans
Signatures of selection
Today
Sequence data and quantification of variation
 Infinite sites model
 Nucleotide diversity (π)
Sequence-based tests of neutrality
 Ewens-Watterson Test
 Tajima’s D
 Hudson-Kreitman-Aguade
 Synonymous versus Nonsynonymous substitutions
 McDonald-Kreitman
The main power of neutral theory is it
provides a theoretical expectation for genetic
variation in the absence of selection.
Equilibrium Heterozygosity under IAM
4N em
q
He =
=
4N em +1 q +1
 Frequencies of individual
alleles are constantly
changing
 Balance between loss and
gain is maintained
 4Neμ>>1: mutation
predominates, new
mutants persist, H is high
 4Neμ<<1: drift dominates:
new mutants quickly
eliminated, H is low
Effects of Population Size on Expected Heterozgyosity
Under Infinite Alleles Model (μ=10-5)
 Rapid approach to equilibrium in small populations
 Higher heterozygosity with less drift
Fate of Alleles in Mutation-Drift Balance
Generations from
birth to fixation
Time between
fixation events
 Time to fixation of a new mutation is much longer than time to
loss
Fate of Alleles in Mutation-Drift-Selection Balance
Purifying Selection
Which case will have the most
alleles onHighest
averageHat
E?any given
What will this depend
upon?
time?
Neutrality
Balancing
Selection/Overdominance
Assume you take a sample of 100 alleles from a
large (but finite) population in mutation-drift
equilibrium.
What is the expected distribution of allele frequencies in
your sample under neutrality and the Infinite Alleles
Model?
Number of Alleles
A.
B.
C.
10
8
6
4
2
2
4
6
8
10
2
4
6
8
10
Number of Observations of Allele
2
4
6
8
10
Allele Frequency Distributions
Black: Predicted from Neutral
Theory
White: Observed (hypothetical)
 Neutral theory allows a
prediction of frequency
distribution of alleles through
process of birth and demise of
alleles through time
 Comparison of observed to
expected distribution provides
evidence of departure from
Infinite Alleles model
 Depends on f, effective
population size, and mutation
rate
Hartl and Clark 2007
Ewens Sampling Formula
  4 Ne 
Population mutation rate: index of variability of population:


 i
Probability the i-th sampled allele is new given i alleles already sampled:


Probability of sampling a new allele on the first sample:
 0

Probability of observing a new allele after sampling one allele:
.
1

 1
Probability of sampling a new allele on the third and fourth samples:
Expected number of different alleles (k) in a sample of 2N alleles is:
E (k ) 
2 N 1

  i
 1
i 0



 1   2
 ... 

  2N 1
Example: Expected number of alleles in a sample of 4:
E (k ) 
2 N 1

  i
i 0
3

i 0

 i
1





 1   2   3
 He


 2
 3
Ewens Sampling Formula
E ( n) 
2 N 1

  i
i 0
 1



 1   2
 ... 

  2N 1
where E(n) is the expected
number of different alleles in a
sample of N diploid individuals,
and  = 4Ne.
1
1
fe 

4Ne   1   1
 Predicts number of different
alleles that should be observed
in a given sample size if
neutrality prevails under
Infinite Alleles Model
 Small , E(n) approaches 1
 Large , E(n) approaches 2N
  can be predicted from
number of observed alleles for
given sample size
 Can also predict expected
homozygosity (fe) under this
model
Ewens-Watterson Test
 Compares expected homozygosity under the neutral model to
expected homozygosity under Hardy-Weinberg equilibrium
using observed allele frequencies
 Comparison of allele frequency distributions
 fe comes from infinite allele model simulations and can be
found in tables for given sample sizes and observed allele
numbers
f HW   p
2
i
Ewens-Watterson Test Example
 Drosophila pseudobscura
collected from winery
 Xanthine dehydrogenase
alleles
 15 alleles observed in 89
chromosomes
 fHW = 0.366
Hartl and Clark 2007
fe
 Generated fe by simulation:
mean 0.168
How would you interpret this result?
Expected Homozygosity fe
Most Loci Look Neutral According to EwensWatterson Test
Hartl and Clark 2007
DNA Sequence Polymorphisms
 DNA sequence is ultimate view of standing genetic variation:
no hidden alleles
 Is this really true?
 What about back mutation?
 Signatures of past evolution are contained in DNA sequence
 Neutral theory presents null model
 Departures due to:
 Selection
 Demographic events
- Bottlenecks, founder effects
- Population admixture
Sequence Alignment
 Necessary first step for comparing sequences within and
between species
 Many different algorithms
 Tradeoff of speed and accuracy
Quantifying Divergence of Sequences
Nucleotide diversity (π) is average number of pairwise
differences between sequences
where
N

pi p j ij

N  1 ij
N is number of sequences in
sample,
pi and pj are frequency of
sequences i and j in the sample,
and
πij is the proportion of sites that
differ between sequences i and j
Sample Calculation of π
5
10
15
20
25
30
35
A
B
C
A->B, 1 difference
A->C, 1 difference
B->C, 2 differences
N

pi p j ij

N  1 ij
3
  (0.33)(0.33)(1 / 35)  (0.33)(0.33)(1 / 35)  (0.33)(0.33)(2 / 35)
2
 0.01867
On average, there are 18.67 polymorphisms per kb between pairs of haplotypes
in the population
Tajima’s D Statistic
 Infinite Sites Model: each new mutation affects a new
site in a sequence
E ( ) 

m
  m
where m is length of sequence, and
  4 Ne 
 Expected number of polymorphic sites in all sequences:
E(S )  a1 S
1   S
S
a1  
a1
i 1 i
n 1
where n is number of different sequences compared
Sample Calculation of S
5
10
15
20
25
30
A
B
C
Two polymorphic sites
S=2
n 1
1 1 1
a1      1.5
1 2
i 1 i
S
2
S  
 1.33
a1 1.5
  0.01867
  m  (0.01867)(35)  0.65
35
Tajima’s D Statistic
 Two different ways of estimating same parameter:
S
  m  S 
a1
 Deviation of these two indicates deviation from neutral
expectations
d    S
d
D
V (d )
where V(d) is variance of d
Tajima’s D Expectations
 D=0: Neutrality
 D>0
d    S
 Balancing Selection: Divergence of alleles (π) increases
OR
 Bottleneck: S decreases
 D<0
 Purifying or Positive Selection: Divergence of alleles
decreases
OR
 Population expansion: Many low frequency alleles cause low
average divergence
Balancing Selection
Balancing
selection

‘balanced’ mutation
Neutral mutation
d    S
Slide adapted from Yoav Gilad

 Should increase nucleotide diversity ()
 Decreases polymorphic sites (S)
initially.
 D>0
Recent Bottleneck
d    S
 Rare alleles are lost
 Polymorphic sites (S) more severely affected than
nucleotide nucleotide diversity ()
 D>0
Standard neutral model
Positive Selection and Purifying Selection
sweep
S
Advantageous mutation
Neutral mutation
d    S
Slide adapted from Yoav Gilad
recovery

s
s
Time
 Should decrease both nucleotide
diversity () and polymorphic sites
(S) initially.
 S recovers due to mutation
  recovers slowly: insensitive to
rare alleles
 D<0
Rapid Population Growth will also result in an
excess of rare alleles even for neutral loci
Rapid population
size increase
Time
Standard neutral
model
Often two main
haplotypes, some
rare alleles
Slide adapted from Yoav Gilad
Most alleles are rare
  4 Ne 
 Most alleles are rare
 Nucleotide diversity ()
depressed
 Polymorphic sites (S)
unchanged or even
enhanced : 4Neμ is large
 D<0
d    S
How do we distinguish these two forms of divergence
(selection vs demography)?
Hudson-Kreitman-Aguade Test
Divergence between species should be of same
magnitude as variation within species
Provides a correction factor for mutation rates at
different sites
Complex goodness of fit test
Perform test for loci under selection and
supposedly neutral loci
Hudson-Kreitman-Aguade (HKA) test
Neutral Locus
Polymorphism
Divergence
8
3
20
8
Polymorphism: Variation within species
Divergence: Variation between species
Slide adapted from Yoav Gilad
Test Locus A
8/20 ≈ 3/8
Hudson-Kreitman-Aguade (HKA) test
Neutral Locus
Polymorphism
Divergence
Test Locus B
8
3
20
19
8/20 >> 3/19
Conclusion: polymorphism lower than
expected in Test Locus B: Selective sweep?
Slide adapted from Yoav Gilad
http://www.nsf.gov/news/mmg/media/images/corn-and-teosinte_h1.jpg
Teosinte
Maize
Maize w/TBR mutation
http://www.nsf.gov/news/mmg/media/images/corn-and-teosinte_h1.jpg
Mauricio
2001; Nature Reviews Genetics 2, 376
HKA Example: Teosinte Branched
 Lab exercise: test Teosinte-Branched Gene for signature of purifying
selection in maize compared to Teosinte relative
 Compare to patterns of polymorphism and diversity in Alchohol
Dehydrogenase gene