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Beta spikes
The Beta distribution approach
PAULA TATARU
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
Oxford, July 19th 2014
Modelling allele frequency data under the Wright Fisher model of drift, mutation and selection
Joint work with Asger Hobolth and Thomas Bataillon
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
Motivation
› Infer 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
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 3
The Wright Fisher model
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 4
The Wright Fisher model
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 5
The Wright Fisher model
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
6
Approximations to the WF
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 WF
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 8
The Beta approximation
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 9
The Beta approximation
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 10
The Beta approximation
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
The Beta with spikes approximation
› The density of Xt
› Use recursive approach to calculate
› mean and variance
› loss and fixation probabilities
› mean and variance conditional on polymorphism
11
Allele frequencies: the Beta distribution approach AARHUS
UNIVERSITY
Bioinformatics
Research Centre Paula Tataru paula@birc.au.dk 12
› Hellinger distance
› true vs approximated distributions
› between 0 and 1
› Stationary: Beta distribution
› Diffusion > Beta with spikes > Beta
Allele frequencies: the Beta distribution approach AARHUS
UNIVERSITY
Bioinformatics
Research Centre Paula Tataru paula@birc.au.dk 13
Allele frequencies: the Beta distribution approach AARHUS
UNIVERSITY
Bioinformatics
Research Centre Paula Tataru paula@birc.au.dk 14
Allele frequencies: the Beta distribution approach AARHUS
UNIVERSITY
Bioinformatics
Research Centre Paula Tataru paula@birc.au.dk 15
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 16
The Beta with spikes: worst fit
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 17
The Beta with spikes: worst fit
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 18
The Beta with spikes: worst fit
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 19
Inference of split times
› Felsenstein’s peeling algorithm
› Numerically optimized likelihood
› 5000 loci
› 100 samples in each population
› 40 data sets
Allele frequencies: the Beta distribution approach AARHUS
UNIVERSITY
Bioinformatics
Research Centre Paula Tataru paula@birc.au.dk 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
› Inference of split times
› Beta with spikes ~ Kim Tree
› Diffusion ?
21
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
Future work
› Inference of
› mutation rates
› selection coefficients
› variable population size
22
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 23
The Beta approximation
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 24
Mean and variance
Allele frequencies: the Beta distribution approach
Paula Tataru paula@birc.au.dk
AARHUS
UNIVERSITY
Bioinformatics
Research Centre 25
Loss and fixation probabilities
Allele frequencies: the Beta distribution approach AARHUS
UNIVERSITY
Bioinformatics
Research Centre Paula Tataru paula@birc.au.dk 26
Allele frequencies: the Beta distribution approach AARHUS
UNIVERSITY
Bioinformatics
Research Centre Paula Tataru paula@birc.au.dk 27
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