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using an accurate beta approximation
PAULA TATARU
THOMAS BATAILLON
ASGER HOBOLTH
AARHUS
UNIVERSITY
Bioinformatics
Research Centre
Vienna, July 17th 2015
Inference under the Wright-Fisher model

An accurate Beta approximationPaula Tataru paula@birc.au.dk
›Inference of population
history from DNA data
› Divergence times
› (Variable) population size
› Migration / admixture
› Selection coefficients
Inference problems

›Inference of population
history from DNA data
› Divergence times
› (Variable) population size
› Migration / admixture
› Selection coefficients
›Model
› Wrigth-Fisher
Inference problems

Inference problems: populations divergence
M. Gautier and R. Vitalis. Inferring population histories using genome-wide allele frequency data.
Molecular biology and evolution, 30(3):654–668, 2013
Kim Tree

Inference problems: populations admixture
J. K. Pickrell and J. K. Pritchard. Inference of population splits and mixtures from genome-wide allele
frequency data. PLOS Genetics, 8(11):e1002967, 2012
TreeMix

Inference problems: loci under selection
Steinrücken M. et al. S. A novel spectral method for inferring general selection from time series
genetic data. The Annals of Applied Statistics 8(4):2203–2222, 2014
spectralHMM

Inference problems: loci under selection
Terhorst J. et al.. S. Multi-locus analysis of genomic time series data from experimental evolution.
PLoS Genetics 11(4):e1005069, 2015

Population genetics: the Wright-Fisher model
individuals
generations(time)

Population genetics: the Wright-Fisher model
› Evolution of
allele frequency
forward in time
at a bi-allelic locus
individuals
generations(time)
3
2
3
3
4
5
5
allele count

Allele frequency distribution

›Diffusion ›Moment-based
Approximations to the Wright-Fisher

›Diffusion
› Large population size
› Infinitesimal change
›Moment-based

›Diffusion
›Moment-based
› Convenient distributions
› Normal distribution
› Beta distribution

›Diffusion
› No closed solution
› Cumbersome to evaluate
›Moment-based

›Diffusion
›Moment-based
› Closed analytical forms
› Fast to evaluate

›Diffusion
›Moment-based
› Closed analytical forms
› Fast to evaluate
› Problematic at boundaries

›Normal distribution
› Support: real line
›Beta distribution
› Support: [0, 1]
Behavior at the boundaries

› Truncation
› Incorrect variance
› Support: [0, 1]

› Truncation
› Incorrect variance
› Intermediary frequencies
› Support: [0, 1]
› Intermediary frequencies

The Beta with spikes
›Use of Wright-Fisher
› Scalable
›Use of moments
› Simple mathematical calculations
›Improve behavior at boundaries
› Preserve mean and variance

The Wright Fisher model
› Zt allele count
› Xt = Zt /2N
› Zt+1 follows a binomial
distribution
individuals
generations(time)
3
2
3
3
4
5
5
allele count

The Wright Fisher model
› Zt allele count
› Xt = Zt /2N
› Zt+1 follows a binomial
distribution
› g encodes the
evolutionary pressures
individuals
generations(time)
3
2
3
3
4
5
5
allele count

The Wright Fisher model: Drift only
individuals
generations(time)
3
2
3
3
4
5
5
allele count

The Wright Fisher model: Mutations
individuals
generations(time)
3
2
4
5
4
3
2
allele count
u v

The Wright Fisher model: Migration
individuals
generations(time)
3
2
3
5
4
2
3
allele count
m1 m2

The Wright Fisher model: Linear forces

›Mutations

›Mutations
›Migration

›Mutations
›Migration
›Mutations & Migration

›DAF: the density of Xt
The Beta approximation: Main idea

›Use recursive approach to calculate
› Mean and variance
The Beta approximation: Main idea

The Beta approximation: Drift only

The Beta with spikes: Main idea

The Beta with spikes: Main idea
›Use recursive approach to calculate
› Loss and fixation probabilities

The Beta with spikes: Drift only

Numerical accuracy: Drift only
Beta Beta with spikes

›Simulated data
› 5000 independent SNPs
› 100 samples
in each population
› 50 data sets (replicates)
Inference of divergence times: Drift only

›Simulated data
› 100 samples
in each population
›DAF is used for likelihood
calculation

›Simulated data
› 100 samples
in each population
calculation
›Likelihood is conditioned
to polymorphism

›Simulated data
› 100 samples
in each population
calculation
›Likelihood is conditioned
to polymorphism
›Likelihood is numerically
optimized

›Exome sequencing: 42,063 autosomal syn SNPs
6 western
12 central
11 eastern
Thomas Bataillon et al.
Genome Biol Evol 2015; 7:1122-1132

P. Tataru, T. Bataillon, A. Hobolth. Inference under a Wright-Fisher model using an accurate beta
approximation. bioRxiv, 2015

Conclusions: beta with spikes
›An extension built on the beta approximation

›Improves the quality of the approximation

›Simple mathematical formulation

›Works under linear evolutionary forces

›Comparable to state of the art methods
for inference of divergence times

›Comparable to state of the art methods
for inference of divergence times
›Recursive formulation enables incorporation
of variable population size

Future work
›Incorporate selection
› Non-linear evolutionary force

Future work
› Positive selection increases probability of fixation

Future work
› Positive selection increases probability of fixation
› Mean and variance are no longer available in closed form*
* Terhorst J. et al.. S. Multi-locus analysis of genomic time series data from experimental evolution.
PLoS Genetics 11(4):e1005069, 2015

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