Inference under the model using an accurate beta...

using an accurate beta approximation

PAULA TATARU

THOMAS BATAILLON

ASGER HOBOLTH

AARHUS

UNIVERSITY

Bioinformatics

Research Centre

CSHL, April 15th 2015

Inference under the Wright-Fisher model

An accurate Beta approximation

Paula Tataru [email protected]

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Theoretical population genetics

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›Mathematical models formalize the evolution of

genetic variation within and between populations

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›Mathematical models formalize the evolution of

genetic variation within and between populations

›Provide a framework for inferring evolutionary paths

from observed data to

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Inference problems

› Inference of population history from DNA data

› (Variable) population size

› Migration / admixture

› Divergence times

› Selection coefficients

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Inference problems: population size

4

H. Li and R. Durbin. Inference of human population history from individual whole-genome

sequences. Nature, 475:493–496, 2011

PSMC



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Inference problems: populations divergence

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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



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Inference problems: populations admixture

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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



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Inference problems: populations admixture

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Gronau I., Hubisz M. J., Gulko B., Danko C. G., Siepel A. Bayesian inference of ancient human

demography from individual genome sequences. Nature genetics 43(10): 1031-1034, 2011

G-PhoCS



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Inference problems: loci under selection

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Steinrücken M., Bhaskar A. and Song Y. 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



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Population genetics: the Wright-Fisher model

› Evolution of a population

forward in time

› Follow one locus (region

in the DNA)

›Different variants at the

locus are called alleles

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individuals

ge

ne

rati

on

s (t

ime

)



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Population genetics: the Wright-Fisher model

›Basic model: only two

alleles per locus

› Follow the frequency of

one of the alleles

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individuals

ge

ne

rati

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s (t

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)

3

2

3

3

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5

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allele count



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Allele frequency distribution

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Population genetics: the coalescent model

› Trace the genealogy of

sampled individuals

backward in time

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individuals

ge

ne

rati

on

s (t

ime

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sampled individuals

backward in time

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individuals

ge

ne

rati

on

s (t

ime

)

MRCA



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sampled individuals

backward in time

›Coalescent process

terminates when

reaching MRCA

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individuals

ge

ne

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on

s (t

ime

)

MRCA



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›The Wright-Fisher

›The coalescent

Two dual models

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› Forward in time

›The coalescent

› Backward in time

Two dual models

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› Forward in time

› Follow allele frequency

›The coalescent


› Follow genealogy

Two dual models

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› Forward in time


› Selection

›The coalescent



› Recombination

Two dual models

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› Forward in time


› Selection

› Scalability

›Sample size decreases

uncertainty

›The coalescent



› Recombination

› Scalability

›Sample size increases

complexity

Two dual models

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›Diffusion

›Moment-based

Approximations to the Wright-Fisher

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›Diffusion

› Large population size

› Infinitesimal change

›Moment-based


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›Diffusion



›Moment-based

› Convenient distributions

› Normal distribution

› Beta distribution


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›Diffusion



› No closed solution

› Cumbersome to evaluate

›Moment-based




› Closed analytical forms

› Fast to evaluate


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›Diffusion



› No closed solution

› Cumbersome to evaluate

›Moment-based




› Closed analytical forms

› Fast to evaluate

› Problematic at boundaries


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›Normal distribution

›Beta distribution

Behavior at the boundaries

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› Support: real line


› Support: [0, 1]


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› Truncation

›Incorrect variance


› Support: [0, 1]


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› Truncation

›Incorrect variance

› Intermediary frequencies


› Support: [0, 1]

› Intermediary frequencies


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The Beta with spikes

›Use of Wright-Fisher

› Scalable

›Use of moments

› Simple mathematical calculations

› Improve behavior at boundaries

› Preserve mean and variance

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The Wright Fisher model

›Zt allele count

›Xt = Zt /2N

›Zt+1 follows a binomial

distribution

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individuals

ge

ne

rati

on

s (t

ime

)

3

2

3

3

4

5

5

allele count



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The Wright Fisher model

›Zt allele count

›Xt = Zt /2N

›Zt+1 follows a binomial

distribution

›g encodes the

evolutionary pressures

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individuals

ge

ne

rati

on

s (t

ime

)

3

2

3

3

4

5

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allele count



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The Wright Fisher model: Drift only

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individuals

ge

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rati

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s (t

ime

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3

2

3

3

4

5

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allele count



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The Wright Fisher model: Mutations

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individuals

ge

ne

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on

s (t

ime

)

3

2

4

5

4

3

2

allele count

u v



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The Wright Fisher model: Migration

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individuals

ge

ne

rati

on

s (t

ime

)

3

2

3

5

4

2

3

allele count

m1 m2



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The Wright Fisher model: Linear forces

›Mutations

›Migration

›Mutations & Migration

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The Beta approximation: Main idea

›The density of Xt



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The Beta approximation: Main idea


›Use recursive approach to calculate

› Mean and variance



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The Beta approximation: Drift only



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The Beta with spikes: Main idea


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The Beta with spikes: Main idea


›Use recursive approach to calculate

› Loss and fixation probabilities

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The Beta with spikes: loss probability

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The Beta with spikes: loss probability

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The Beta with spikes: fixation probability

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The Beta with spikes: Drift only



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Numerical accuracy: Drift only

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Beta Beta with spikes



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Inference of divergence times: Drift only

›Simulated data

› 5000 independent loci

› 100 samples in each population

› 50 data sets (replicates)



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›Simulated data

› 5000 independent loci

› 100 samples in each population

› 50 data sets (replicates)

›Allele frequency distribution is used to

calculate likelihood of data

› Likelihood is numerically optimized



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Conclusions

›Beta with spikes

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Conclusions

›Beta with spikes

› An extension built on the beta approximation

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Conclusions

›Beta with spikes


› Improves the quality of the approximation

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Conclusions

›Beta with spikes



› Simple mathematical formulation

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Conclusions

›Beta with spikes




› Works under linear evolutionary forces

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Conclusions

›Beta with spikes





› Comparable to state of the art methods

for inference of divergence times

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Conclusions

›Beta with spikes





› Comparable to state of the art methods

for inference of divergence times

› Recursive formulation enables incorporation

of variable population size

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Future work

› Incorporate selection

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Future work


› Non-linear evolutionary force

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Future work



› Positive selection increases probability of fixation

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Future work




› Mean and variance are no longer available in closed form

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Future work




› Mean and variance are no longer available in closed form

› Extend the approximation for loss/fixation probabilities to

mean and variance

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Inference under the model using an accurate beta...

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