1. Lecture 20 : Tests of Neutrality
November 6, 2015

2. Last Time
Mutation and selection
Infinite alleles and stepwise mutation models
Introduction to neutral theory

3. 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
nov6_neutraltest

4. The main power of neutral theory is it provides a theoretical expectation for genetic variation in the absence of selection.

5. Equilibrium Heterozygosity under IAM
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

6. 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

7. 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

8. Fate of Alleles in Mutation-Drift-Selection Balance
Purifying Selection
Which case will have the most alleles on average at any given time?
Highest HE?
What will this depend upon?
Neutrality
Balancing Selection/Overdominance

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

10. Allele Frequency Distributions
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
Black: Predicted from Neutral Theory
White: Observed (hypothetical)

11. Ewens Sampling Formula
Population mutation rate: index of variability of population:
Probability the i-th sampled allele is new given i alleles already sampled:
Probability of sampling a new allele on the first sample:
Probability of observing a new allele after sampling one allele:
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:
Example: Expected number of alleles in a sample of 4:

12. Ewens Sampling Formula
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
where E(n) is the expected number of different alleles in a sample of N diploid individuals, and  = 4Ne.

13. 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

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

15. Most Loci Look Neutral According to Ewens-Watterson Test
Expected Homozygosity fe
Hartl and Clark 2007

16. 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

17. Sequence Alignment
Necessary first step for comparing sequences within and between species
Many different algorithms
Tradeoff of speed and accuracy

18. Quantifying Divergence of Sequences
Nucleotide diversity (π) is average number of pairwise differences between sequences
where
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

19. Sample Calculation of π5 10 15 20 25 30 35 A B C
A->B, 1 difference
A->C, 1 difference
B->C, 2 differences
On average, there are polymorphisms per kb between pairs of haplotypes in the population

20. Tajima’s D Statistic
Infinite Sites Model: each new mutation affects a new site in a sequence
Expected number of polymorphic sites in all sequences:
where m is length of sequence, and
where n is number of different sequences compared

21. Sample Calculation of S5 10 15 20 25 30 35 A B C
Two polymorphic sites
S=2

22. where V(d) is variance of d
Tajima’s D Statistic
Two different ways of estimating same parameter:
Deviation of these two indicates deviation from neutral expectations
where V(d) is variance of d

23. Tajima’s D Expectations
D=0: Neutrality
D>0
Balancing Selection: Divergence of alleles (π) increases OR
Bottleneck: S decreases
D<0
Purifying or Positive Selection: Divergence of alleles decreases
Population expansion: Many low frequency alleles cause low average divergence

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

25. Standard neutral model
Recent Bottleneck
Rare alleles are lost
Polymorphic sites (S) more severely affected than nucleotide nucleotide diversity ()
D>0
Standard neutral model

26. Positive Selection and Purifying Selection
sweep
recovery
 S
 s
Time
Advantageous mutation
Neutral mutation
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

27. Standard neutral model
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
Most alleles are rare
Nucleotide diversity () depressed
Polymorphic sites (S) unchanged or even enhanced : 4Neμ is large
D<0
Time
Often two main haplotypes, some rare alleles
Slide adapted from Yoav Gilad

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