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Using a Beta Approx for Wright-Fisher Model

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Paula Tataru
Paula Tataru

This document describes using a Beta approximation to model the Wright-Fisher model of genetic drift in population genetics. It discusses using a moment-based approach to calculate the mean and variance of allele frequencies over time, allowing the distribution to be approximated by a Beta distribution. It also describes adding "spikes" to the Beta distribution to better model loss and fixation probabilities at the boundaries of 0 and 1.

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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 paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Theoretical population genetics 2
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Theoretical population genetics ›Mathematical models formalize the evolution of genetic variation within and between populations 2
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Theoretical population genetics ›Mathematical models formalize the evolution of genetic variation within and between populations ›Provide a framework for inferring evolutionary paths from observed data to 2
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Inference problems ›Inference of population history from DNA data › (Variable) population size › Migration / admixture › Divergence times › Selection coefficients 3
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 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
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Inference problems: populations divergence 5 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
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Inference problems: populations admixture 6 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
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Inference problems: populations admixture 7 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
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Inference problems: loci under selection 8 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
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 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 9 individuals generations(time)
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Population genetics: the Wright-Fisher model › Basic model: only two alleles per locus › Follow the frequency of one of the alleles 10 individuals generations(time) 3 2 3 3 4 5 5 allele count
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Allele frequency distribution 11
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Population genetics: the coalescent model › Trace the genealogy of sampled individuals backward in time 12 individuals generations(time)
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Population genetics: the coalescent model › Trace the genealogy of sampled individuals backward in time 12 individuals generations(time)
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Population genetics: the coalescent model › Trace the genealogy of sampled individuals backward in time 12 individuals generations(time) MRCA
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Population genetics: the coalescent model › Trace the genealogy of sampled individuals backward in time › Coalescent process terminates when reaching MRCA 12 individuals generations(time) MRCA
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›The Wright-Fisher ›The coalescent Two dual models 13
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›The Wright-Fisher › Forward in time ›The coalescent › Backward in time Two dual models 13
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›The Wright-Fisher › Forward in time › Follow allele frequency ›The coalescent › Backward in time › Follow genealogy Two dual models 13
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›The Wright-Fisher › Forward in time › Follow allele frequency › Selection ›The coalescent › Backward in time › Follow genealogy › Recombination Two dual models 13
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›The Wright-Fisher › Forward in time › Follow allele frequency › Selection › Scalability ›Sample size decreases uncertainty ›The coalescent › Backward in time › Follow genealogy › Recombination › Scalability ›Sample size increases complexity Two dual models 13
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Diffusion ›Moment-based Approximations to the Wright-Fisher 14
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Diffusion › Large population size › Infinitesimal change ›Moment-based Approximations to the Wright-Fisher 14
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Diffusion › Large population size › Infinitesimal change ›Moment-based › Convenient distributions › Normal distribution › Beta distribution Approximations to the Wright-Fisher 14
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Diffusion › Large population size › Infinitesimal change › No closed solution › Cumbersome to evaluate ›Moment-based › Convenient distributions › Normal distribution › Beta distribution › Closed analytical forms › Fast to evaluate Approximations to the Wright-Fisher 14
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Diffusion › Large population size › Infinitesimal change › No closed solution › Cumbersome to evaluate ›Moment-based › Convenient distributions › Normal distribution › Beta distribution › Closed analytical forms › Fast to evaluate › Problematic at boundaries Approximations to the Wright-Fisher 14
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Normal distribution ›Beta distribution Behavior at the boundaries 15
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Normal distribution › Support: real line ›Beta distribution › Support: [0, 1] Behavior at the boundaries 15
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Normal distribution › Support: real line › Truncation ›Incorrect variance ›Beta distribution › Support: [0, 1] Behavior at the boundaries 15
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Normal distribution › Support: real line › Truncation ›Incorrect variance › Intermediary frequencies ›Beta distribution › Support: [0, 1] › Intermediary frequencies Behavior at the boundaries 15
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes ›Use of Wright-Fisher › Scalable ›Use of moments › Simple mathematical calculations ›Improve behavior at boundaries › Preserve mean and variance 16
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model › Zt allele count › Xt = Zt /2N › Zt+1 follows a binomial distribution 17 individuals generations(time) 3 2 3 3 4 5 5 allele count
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model › Zt allele count › Xt = Zt /2N › Zt+1 follows a binomial distribution 17 individuals generations(time) 3 2 3 3 4 5 5 allele count
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model › Zt allele count › Xt = Zt /2N › Zt+1 follows a binomial distribution › g encodes the evolutionary pressures 17 individuals generations(time) 3 2 3 3 4 5 5 allele count
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Drift only 18 individuals generations(time) 3 2 3 3 4 5 5 allele count
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Mutations 19 individuals generations(time) 3 2 4 5 4 3 2 allele count u v
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Mutations 19 individuals generations(time) 3 2 4 5 4 3 2 allele count u v
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Migration 20 individuals generations(time) 3 2 3 5 4 2 3 allele count m1 m2
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Migration 20 individuals generations(time) 3 2 3 5 4 2 3 allele count m1 m2
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Linear forces ›Mutations ›Migration ›Mutations & Migration 21
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Linear forces ›Mutations ›Migration ›Mutations & Migration 21
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 22 The Beta approximation: Main idea ›The density of Xt
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 22 The Beta approximation: Main idea ›The density of Xt ›Use recursive approach to calculate › Mean and variance
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 22 The Beta approximation: Main idea ›The density of Xt ›Use recursive approach to calculate › Mean and variance
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 23 The Beta approximation: Drift only
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 23 The Beta approximation: Drift only
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 24 The Beta approximation: Drift only
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 25 The Beta approximation: Drift only
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes: Main idea ›The density of Xt 26
An accurate Beta approximation 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 › Loss and fixation probabilities 26
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes: loss probability 27
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes: loss probability 28
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes: loss probability 28
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes: loss probability 28
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes: fixation probability 29
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 30 The Beta with spikes: Drift only
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 30 The Beta with spikes: Drift only
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 31 The Beta with spikes: Drift only
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 32 The Beta with spikes: Drift only
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Numerical accuracy: Drift only 33 Beta Beta with spikes
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 34 Inference of divergence times: Drift only ›Simulated data › 5000 independent loci › 100 samples in each population › 50 data sets (replicates)
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 34 Inference of divergence times: Drift only ›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
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Inference of divergence times: Drift only 35
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Conclusions ›Beta with spikes 36
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Conclusions ›Beta with spikes › An extension built on the beta approximation 36
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Conclusions ›Beta with spikes › An extension built on the beta approximation › Improves the quality of the approximation 36
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Conclusions ›Beta with spikes › An extension built on the beta approximation › Improves the quality of the approximation › Simple mathematical formulation 36
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 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 36
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 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 36
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 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 › Recursive formulation enables incorporation of variable population size 36
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Future work ›Incorporate selection 37
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Future work ›Incorporate selection › Non-linear evolutionary force 37
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Future work ›Incorporate selection › Non-linear evolutionary force › Positive selection increases probability of fixation 37
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Future work ›Incorporate selection › Non-linear evolutionary force › Positive selection increases probability of fixation › Mean and variance are no longer available in closed form 37
An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Future work ›Incorporate selection › Non-linear evolutionary force › Positive selection increases probability of fixation › Mean and variance are no longer available in closed form › Extend the approximation for loss/fixation probabilities to mean and variance 37

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Using a Beta Approx for Wright-Fisher Model

  • 1. 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
  • 2. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Theoretical population genetics 2
  • 3. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Theoretical population genetics ›Mathematical models formalize the evolution of genetic variation within and between populations 2
  • 4. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Theoretical population genetics ›Mathematical models formalize the evolution of genetic variation within and between populations ›Provide a framework for inferring evolutionary paths from observed data to 2
  • 5. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Inference problems ›Inference of population history from DNA data › (Variable) population size › Migration / admixture › Divergence times › Selection coefficients 3
  • 6. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 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
  • 7. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Inference problems: populations divergence 5 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
  • 8. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Inference problems: populations admixture 6 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
  • 9. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Inference problems: populations admixture 7 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
  • 10. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Inference problems: loci under selection 8 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
  • 11. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 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 9 individuals generations(time)
  • 12. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Population genetics: the Wright-Fisher model › Basic model: only two alleles per locus › Follow the frequency of one of the alleles 10 individuals generations(time) 3 2 3 3 4 5 5 allele count
  • 13. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Allele frequency distribution 11
  • 14. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Population genetics: the coalescent model › Trace the genealogy of sampled individuals backward in time 12 individuals generations(time)
  • 15. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Population genetics: the coalescent model › Trace the genealogy of sampled individuals backward in time 12 individuals generations(time)
  • 16. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Population genetics: the coalescent model › Trace the genealogy of sampled individuals backward in time 12 individuals generations(time) MRCA
  • 17. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Population genetics: the coalescent model › Trace the genealogy of sampled individuals backward in time › Coalescent process terminates when reaching MRCA 12 individuals generations(time) MRCA
  • 18. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›The Wright-Fisher ›The coalescent Two dual models 13
  • 19. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›The Wright-Fisher › Forward in time ›The coalescent › Backward in time Two dual models 13
  • 20. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›The Wright-Fisher › Forward in time › Follow allele frequency ›The coalescent › Backward in time › Follow genealogy Two dual models 13
  • 21. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›The Wright-Fisher › Forward in time › Follow allele frequency › Selection ›The coalescent › Backward in time › Follow genealogy › Recombination Two dual models 13
  • 22. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›The Wright-Fisher › Forward in time › Follow allele frequency › Selection › Scalability ›Sample size decreases uncertainty ›The coalescent › Backward in time › Follow genealogy › Recombination › Scalability ›Sample size increases complexity Two dual models 13
  • 23. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Diffusion ›Moment-based Approximations to the Wright-Fisher 14
  • 24. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Diffusion › Large population size › Infinitesimal change ›Moment-based Approximations to the Wright-Fisher 14
  • 25. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Diffusion › Large population size › Infinitesimal change ›Moment-based › Convenient distributions › Normal distribution › Beta distribution Approximations to the Wright-Fisher 14
  • 26. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Diffusion › Large population size › Infinitesimal change › No closed solution › Cumbersome to evaluate ›Moment-based › Convenient distributions › Normal distribution › Beta distribution › Closed analytical forms › Fast to evaluate Approximations to the Wright-Fisher 14
  • 27. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Diffusion › Large population size › Infinitesimal change › No closed solution › Cumbersome to evaluate ›Moment-based › Convenient distributions › Normal distribution › Beta distribution › Closed analytical forms › Fast to evaluate › Problematic at boundaries Approximations to the Wright-Fisher 14
  • 28. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Normal distribution ›Beta distribution Behavior at the boundaries 15
  • 29. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Normal distribution › Support: real line ›Beta distribution › Support: [0, 1] Behavior at the boundaries 15
  • 30. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Normal distribution › Support: real line › Truncation ›Incorrect variance ›Beta distribution › Support: [0, 1] Behavior at the boundaries 15
  • 31. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre ›Normal distribution › Support: real line › Truncation ›Incorrect variance › Intermediary frequencies ›Beta distribution › Support: [0, 1] › Intermediary frequencies Behavior at the boundaries 15
  • 32. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes ›Use of Wright-Fisher › Scalable ›Use of moments › Simple mathematical calculations ›Improve behavior at boundaries › Preserve mean and variance 16
  • 33. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model › Zt allele count › Xt = Zt /2N › Zt+1 follows a binomial distribution 17 individuals generations(time) 3 2 3 3 4 5 5 allele count
  • 34. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model › Zt allele count › Xt = Zt /2N › Zt+1 follows a binomial distribution 17 individuals generations(time) 3 2 3 3 4 5 5 allele count
  • 35. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model › Zt allele count › Xt = Zt /2N › Zt+1 follows a binomial distribution › g encodes the evolutionary pressures 17 individuals generations(time) 3 2 3 3 4 5 5 allele count
  • 36. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Drift only 18 individuals generations(time) 3 2 3 3 4 5 5 allele count
  • 37. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Mutations 19 individuals generations(time) 3 2 4 5 4 3 2 allele count u v
  • 38. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Mutations 19 individuals generations(time) 3 2 4 5 4 3 2 allele count u v
  • 39. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Migration 20 individuals generations(time) 3 2 3 5 4 2 3 allele count m1 m2
  • 40. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Migration 20 individuals generations(time) 3 2 3 5 4 2 3 allele count m1 m2
  • 41. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Linear forces ›Mutations ›Migration ›Mutations & Migration 21
  • 42. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Wright Fisher model: Linear forces ›Mutations ›Migration ›Mutations & Migration 21
  • 43. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 22 The Beta approximation: Main idea ›The density of Xt
  • 44. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 22 The Beta approximation: Main idea ›The density of Xt ›Use recursive approach to calculate › Mean and variance
  • 45. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 22 The Beta approximation: Main idea ›The density of Xt ›Use recursive approach to calculate › Mean and variance
  • 46. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 23 The Beta approximation: Drift only
  • 47. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 23 The Beta approximation: Drift only
  • 48. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 24 The Beta approximation: Drift only
  • 49. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 25 The Beta approximation: Drift only
  • 50. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes: Main idea ›The density of Xt 26
  • 51. An accurate Beta approximation 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 › Loss and fixation probabilities 26
  • 52. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes: loss probability 27
  • 53. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes: loss probability 28
  • 54. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes: loss probability 28
  • 55. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes: loss probability 28
  • 56. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre The Beta with spikes: fixation probability 29
  • 57. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 30 The Beta with spikes: Drift only
  • 58. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 30 The Beta with spikes: Drift only
  • 59. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 31 The Beta with spikes: Drift only
  • 60. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 32 The Beta with spikes: Drift only
  • 61. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Numerical accuracy: Drift only 33 Beta Beta with spikes
  • 62. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 34 Inference of divergence times: Drift only ›Simulated data › 5000 independent loci › 100 samples in each population › 50 data sets (replicates)
  • 63. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 34 Inference of divergence times: Drift only ›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
  • 64. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Inference of divergence times: Drift only 35
  • 65. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Conclusions ›Beta with spikes 36
  • 66. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Conclusions ›Beta with spikes › An extension built on the beta approximation 36
  • 67. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Conclusions ›Beta with spikes › An extension built on the beta approximation › Improves the quality of the approximation 36
  • 68. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Conclusions ›Beta with spikes › An extension built on the beta approximation › Improves the quality of the approximation › Simple mathematical formulation 36
  • 69. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 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 36
  • 70. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 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 36
  • 71. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre 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 › Recursive formulation enables incorporation of variable population size 36
  • 72. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Future work ›Incorporate selection 37
  • 73. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Future work ›Incorporate selection › Non-linear evolutionary force 37
  • 74. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Future work ›Incorporate selection › Non-linear evolutionary force › Positive selection increases probability of fixation 37
  • 75. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Future work ›Incorporate selection › Non-linear evolutionary force › Positive selection increases probability of fixation › Mean and variance are no longer available in closed form 37
  • 76. An accurate Beta approximation Paula Tataru paula@birc.au.dk AARHUS UNIVERSITY Bioinformatics Research Centre Future work ›Incorporate selection › Non-linear evolutionary force › Positive selection increases probability of fixation › Mean and variance are no longer available in closed form › Extend the approximation for loss/fixation probabilities to mean and variance 37