Lecture 25 : Tests of Neutrality
April 14, 2014


 Out of Africa hypothesis
 Neanderthal and Denisovan genomes
 Introgression into humans


 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

Equilibrium Heterozygosity under IAM
H e
=
4 N e
4 N e m m +
1
= q q +
1
 Frequencies of individual alleles are constantly changing
 Balance between loss and gain is maintained
 4N e
μ>>1: mutation predominates, new mutants persist, H is high
 4N e
μ<<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 time?
Neutrality
Balancing
Selection/Overdominance

A.
10
8
6
4
2
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?
B.
C.
2 4 6 8 10
2 4 6 8 10
Number of Observations of Allele
2 4 6 8 10

Hartl and Clark 2007
Black: Predicted from Neutral
Theory
 Neutral theory allows a prediction of frequency distribution of alleles through process of birth and demise of alleles through time
White: Observed (hypothetical)
 Comparison of observed to expected distribution provides evidence of departure from
Infinite Alleles model
 Depends on f, effective population size, and mutation rate

Ewens Sampling Formula
.
Population mutation rate: index of variability of population:

Probability the i-th sampled allele is new given i alleles already sampled:

4 N e

Probability of sampling a new allele on the first sample:
Probability of observing a new allele after sampling one allele:




0

1




1

H e
Probability of sampling a new allele on the third and fourth samples:



2



 i



3
Expected number of different alleles (k) in a sample of 2N alleles is:
E ( k )

2 i
N 


1
0


 i

1




1




2

...




2 N

1
Example: Expected number of alleles in a sample of 4:
E ( k )
 i
N 2  
1

0


 i
 i
3 

0


 i

1




1




2




3

Ewens Sampling Formula
E ( n )

1



 i
N 2 

0

1






1
 i

2

...




2 N

1 where E(n) is the expected number of different alleles in a sample of N diploid individuals, and

= 4N e

.
f e

1
4 N e
 
1


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 (f e
) 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
 f e comes from infinite allele model simulations and can be found in tables for given sample sizes and observed allele numbers f
HW
  p i
2

Ewens-Watterson Test Example
Hartl and Clark 2007 f e
 Drosophila pseudobscura collected from winery
 Xanthine dehydrogenase alleles
 15 alleles observed in 89 chromosomes
 f
HW
= 0.366
 Generated f e mean 0.168
by simulation:
How would you interpret this result?

Most Loci Look Neutral According to Ewens-
Watterson Test
Hartl and Clark 2007

 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

 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
 
N
N

1
 ij p i p j
 ij where
N is number of sequences in sample, p i and p j are frequency of sequences i and j in the sample, and
π ij is the proportion of sites that differ between sequences i and j

5 10 15 20 25 30 35
A
B
C
A->B, 1 difference
A->C, 1 difference
B->C, 2 differences
 
N
N

1
 ij p i p j
 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

 where m is length of sequence, and

4 N e


  m
 Expected number of polymorphic sites in all sequences:
E ( S )
 a
1

S a
1
 i

1 n 

1
1 i

S

S a
1 where n is number of different sequences compared

A
B
C

S
5 10 15 20 25 30 a
1
 i

1 n 

1
1 i

1
1

1
2

1 .
5
 
0 .
01867
Two polymorphic sites
S=2

S

S a
1

2
1 .
5

1 .
33


  m

( 0 .
01867 )( 35 )

0 .
65
35

Tajima ’ s D Statistic
 Two different ways of estimating same parameter:


  m

S

S a
1
 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
 

 
S
 D>0
 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


‘ balanced ’ mutation
Neutral mutation d
 

 
S
Balancing selection

 Should increase nucleotide diversity (

)
 Decreases polymorphic sites (S) initially.
 D>0
Slide adapted from Yoav Gilad

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 recovery

S
Advantageous mutation
Neutral mutation d
 

 
S
 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
Slide adapted from Yoav Gilad

Rapid Population Growth will also result in an excess of rare alleles even for neutral loci
Standard neutral model
Rapid population size increase
 Most alleles are rare
 Nucleotide diversity (

) depressed
 Polymorphic sites (S) unchanged or even enhanced : 4N e
μ is large
 D<0
Often two main haplotypes, some rare alleles
Slide adapted from Yoav Gilad
Most alleles are rare
 
4 N e
 d
 

 
S

How do we distinguish these two forms of divergence
(selection vs demography)?

 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 Test Locus A
Polymorphism 8 3
Divergence
20
Polymorphism: Variation within species
Divergence: Variation between species
8
8/20 ≈ 3/8
Slide adapted from Yoav Gilad

Hudson-Kreitman-Aguade (HKA) test
Neutral Locus Test Locus B
Polymorphism 8 3
Divergence
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

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