1. Betaspikes
The Beta distribution approach
PAULA TATARU
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
Aarhus, October 23rd 2014
Modelling allele frequency data under the
Wright Fisher model of drift, mutation and selection
Joint work with Thomas Bataillon and Asger Hobolth

2. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
Motivation
›Inference population parameters from DNA data
› mutation rates
› selection coefficients
› split times
› variable population size back in time
›Backward in time (coalescent)
›Forward in time (Wright Fisher)
2

3. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
3
The Wright Fisher model: Drift only

4. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
4
The Wright Fisher model: Mutations

5. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
5
The Wright Fisher model: Selection

6. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
Allele frequency distribution: Drift only
6

7. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
›Diffusion
› Kimura 1964
› Gautier & Vitalis 2013
› Malaspinas et al. 2012
› Steinrucken et al. 2013
› Zhao et al. 2013
›Moment based
› Normal distribution
› Nicholson et al. 2002
› Prickrell & Pritchard 2012
› Beta distribution
› Balding & Nichols 1995
› Siren et al. 2011
› Beta with spikes
7
Approximations to the Wright Fisher

8. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
8
The Beta approximation: Main idea
›The density of Xt
›Use recursive approach to calculate
› mean and variance

9. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
9
The Beta approximation: Drift only

10. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
10
The Beta approximation: Drift only

11. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
11
The Beta approximation: Drift only

12. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
The Beta with spikes: Main idea
›The density of Xt
›Use recursive approach to calculate
› mean and variance
› loss and fixation probabilities
› mean and variance conditional on polymorphism
12

13. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
Approximations: Drift only
13

14. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
14
Approximations: Drift only

15. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
15
The Beta with spikes: Drift only / Selection

16. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
16
The Beta with spikes: Drift only / Selection

17. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
17
The Beta with spikes: Drift only / Selection

18. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
18
Inference of split times: Drift only
›Felsenstein’s peeling algorithm
›Numerically optimized likelihood
›5000 independent loci
›100 samples in each population
›40 data sets

19. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
Inference of split times: Drift only
19

20. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
Conclusions
›Beta with spikes: new approximation to the WF
› Quality of approximation
› Consistent
› Diffusion > Beta with spikes > Beta
› Simple mathematical formulation -> decrease in speed
› Inference of split times
› Beta with spikes ~ Kim Tree
20

21. Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
21
Loss and fixation probabilities