Estimating Ancient Population Sizes using the Coalescent with Recombination

Transcription

1 Estimating Ancient Population Sizes using the Coalescent with Recombination Sara Sheehan joint work with Kelley Harris and Yun S. Song May 26, 2012 Sheehan, Harris, Song May 26,

2 Motivation Introduction Out of Africa bottleneck 1 Humans 2 Neanderthals 3 Early Hominids Sheehan, Harris, Song May 26,

3 Introduction Effects of population size variation Sheehan, Harris, Song May 26,

4 Introduction Effects of population size variation Sheehan, Harris, Song May 26,

5 Introduction How can we infer past population sizes? The population size at time t is inversely proportional to the rate of coalescence at time t Sheehan, Harris, Song May 26,

6 Introduction How can we infer past population sizes? The population size at time t is inversely proportional to the rate of coalescence at time t Coalescence times are unknown, but are correlated with mutation frequencies: Sheehan, Harris, Song May 26,

7 Adding recombination Introduction Model Genealogies of nearby loci are correlated with each other Use mutations to learn about the ancestral recombination graph Sheehan, Harris, Song May 26, 2012 Sheehan, Harris, Song May 26,

8 Introduction TMRCA changes throughout the genome Previous work: Pairwise Sequentially Markovian Coalescent (PSMC), Li and Durbin 2011 uses 2 haplotypes Sheehan, Harris, Song May 26,

9 Model Conditional Sampling Distribution Data: multiple sequence alignment of haplotypes h 1 = CCATACCTGGTCAATTTTTGTATTTGAAGTAGAGACG h 2 = CCAGACCTAGTCAATTTTTGTATTTGAAGTAGAGACG h 3 = CCAGACCTGGTCAATTTCTGTATTTTTAGTAGAGACG h 4 = CCATACCTAGTCAATTTTTGTATTTTTAGTAGCCACG Sheehan, Harris, Song May 26,

10 Model Conditional Sampling Distribution Data: multiple sequence alignment of haplotypes h 1 = CCATACCTGGTCAATTTTTGTATTTGAAGTAGAGACG h 2 = CCAGACCTAGTCAATTTTTGTATTTGAAGTAGAGACG h 3 = CCAGACCTGGTCAATTTCTGTATTTTTAGTAGAGACG h 4 = CCATACCTAGTCAATTTTTGTATTTTTAGTAGCCACG Want to compute: likelihood of our data given past population size function N(t) P(h 1, h 2, h 3,, h n, h n+1 N(t)) Sheehan, Harris, Song May 26,

11 Model Conditional Sampling Distribution Data: multiple sequence alignment of haplotypes h 1 = CCATACCTGGTCAATTTTTGTATTTGAAGTAGAGACG h 2 = CCAGACCTAGTCAATTTTTGTATTTGAAGTAGAGACG h 3 = CCAGACCTGGTCAATTTCTGTATTTTTAGTAGAGACG h 4 = CCATACCTAGTCAATTTTTGTATTTTTAGTAGCCACG Want to compute: likelihood of our data given past population size function N(t) P(h 1, h 2, h 3,, h n, h n+1 N(t)) Can obtain via the CSD: P(h n+1 h 1, h 2,, h n, N(t)) Sheehan, Harris, Song May 26,

12 roduction Background Sequential Model interpretation Discretization Resu Conditional Sampling Conditional Distribution genealogy C G n Suppose conditional we observe genealogy the C(hidden) genealogy G n for h n. Construct a conditional (hidden) genealogy genealogy G C for a single haplotype: n for h 1,, h n Coalescence, mutation, and recombination as usual. Paul, Steinrücken, and Song (2011) Absorption of lineage in C into a lineage in G n at rate 1. Sheehan, Harris, Song May 26,

13 equential interpretation Discretization Model Results Introduction Background Sequential interpretation Discretization Model Conditional Sampling Distribution ional genealogy A realization of ˆπ PS G n G n (h n ) den) Simplification genealogy An example I: Greplace n for hrealization the n. Construct distributionaof coalescent genealogies with a a single trunk haplotype: Assume genealogy G n = Gn (h n). and recombination Construct as usual. conditional Paul, Steinrücken, genealogy and Song (2011) C, with absorption at rate 1/2 C into a lineage Sheehan, Harris, in GSong n at rate 1. May 26,

14 Model Introduction Background Sequential interpretation Discretization Results Conditional Sampling Distribution Marginal conditional genealogies t (a) h (a) t (b) h (b) C G n(h n ) Simplification II: (Sequentially Markov assumption) If t 1,..., t l is the sequence of times at which segments of h n+1 coalesce with the trunk genealogy, then The marginal conditional genealogy S l at locus l is described by: 1 the absorption P(t l time t l 1 T, t l 2 l,,..., t 1 ) = P(t l t l 1 ) 2 the absorption haplotype H l. Paul, Steinrücken, and Song (2011) Sheehan, Harris, Song May 26,

15 Model Introduction Background Sequential interpretation Discretization Results Conditional Sampling Distribution Marginal conditional genealogies t (a) h (a) S 1 t (b) h (b) C G n(h n ) C G n(h n ) Marginal conditional S 1 =(T 1, H 1 )= ( genealogy t (1), h (1)) at locus 1 (red lines): absorption time = t (a) The absorption marginal haplotype conditional = h genealogy (a) S l at locus l is described by: 1 the absorption time T l, Paul, Steinrücken, and Song (2011) 2 the absorption haplotype H l. Sheehan, Harris, Song May 26,

16 Model Introduction Background Sequential interpretation Discretization Results Conditional Sampling Distribution Marginal conditional genealogies t (a) h (a) S 1 t (b) h (b) S 2 C G n(h n ) C G n(h n ) Marginal conditional S 1 =(T 1, H 1 )= ( genealogy t (1), h (1)) at locus 2 (red lines): S 2 =(T 2, H 2 )= ( t (2), h (2)) absorption time = t (b) The absorption marginal haplotype conditional = h genealogy (b) S l at locus l is described by: 1 the absorption time T l, Paul, Steinrücken, and Song (2011) 2 the absorption haplotype H l. Sheehan, Harris, Song May 26,

17 Model Introduction Background Sequential interpretation Discretization Results Conditional Sampling Distribution Marginal conditional genealogies t (a) h (a) S 1 S 3 t (b) h (b) S 2 C G n(h n ) C G n(h n ) Marginal conditional S 1 =(T 1, H 1 )= ( genealogy t (1), h (1)) at locus 3 (red lines): S 2 =(T 2, H 2 )= ( t (2), h (2)) S 3 = S 1 absorption time = t (a) The absorption marginal haplotype conditional = h genealogy (a) S l at locus l is described by: 1 the absorption time T l, Paul, Steinrücken, and Song (2011) 2 the absorption haplotype H l. Sheehan, Harris, Song May 26,

18 Model Hidden Markov Model framework 2D State space: S l = (t l, h l ) t l = absorption time h l = absorption haplotype Sheehan, Harris, Song May 26,

19 Model Hidden Markov Model framework 2D State space: S l = (t l, h l ) t l = absorption time h l = absorption haplotype ( ) Initial absorption time distribution: ζ(t) = n N(t) exp n t 0 N(τ)dτ where N(t) is the population size change function Sheehan, Harris, Song May 26,

20 Model Hidden Markov Model framework 2D State space: S l = (t l, h l ) t l = absorption time h l = absorption haplotype ( ) Initial absorption time distribution: ζ(t) = n N(t) exp n t 0 N(τ)dτ where N(t) is the population size change function Transitions: Given (t l, h l ), { (t (t l+1, h l+1 ) = l, h l ), if no recombination new time and haplotype, if recombination Sheehan, Harris, Song May 26,

21 Model Hidden Markov Model framework 2D State space: S l = (t l, h l ) t l = absorption time h l = absorption haplotype ( ) Initial absorption time distribution: ζ(t) = n N(t) exp n t 0 N(τ)dτ where N(t) is the population size change function Transitions: Given (t l, h l ), { (t (t l+1, h l+1 ) = l, h l ), if no recombination new time and haplotype, if recombination Emissions: mutations occur on the lineage h n+1 only to create the emitted allele (Poisson process with rate θ) Sheehan, Harris, Song May 26,

22 Model Improved accuracy: the wedding cake genealogy Too many lineages in the past under estimation of population sizes Replace the trunk genealogy with wedding cake genealogy where n(t) = the expected number of remaining lineages at time t Sheehan, Harris, Song May 26,

23 EM framework Inferring past population sizes Discretization: Discretize time into d intervals 0 = t 0 < t 2 < < t d = Constant population size N i during interval [t i 1, t i ) Sheehan, Harris, Song May 26,

24 EM procedure EM framework Fix all sizes N i = 1. E-step: Given population size estimates, use a leave-one-out likelihood approach. P(h 1,, h n N(t)) n P(h k h 1,, h k 1, h k+1,, h n, N(t)) k=1 M-step: Minimize the difference between E-step transitions and expected transitions to obtain approximate maximum likelihood estimates for population sizes. Terminate when likelihood has plateaued. Sheehan, Harris, Song May 26,

25 Results and Future Work Inference of TMRCA Red line = psmc meantmrca, Blue line = our meantmrca, Green line = our mediantmrca TMRCA 3 4 Black line = True TMRCA, 0e+00 2e+04 4e+04 6e+04 8e+04 1e+05 Position Sheehan, Harris, Song May 26,

26 Results and Future Work Drosophila melanogaster data haplotypes from the Drosophila Population Genomics Project 37 from North America (Raleigh, USA) 22 from Africa (Gikongoro, Rwanda) used trimmed intergenic regions longer than 70 kb and with < 10% missing data 9 such regions: two from chromosome 2 six from chromosome 3 one from chromosome X 650 kb in total Sheehan, Harris, Song May 26,

27 Data analysis goal Results and Future Work [Figure: Karasov, Messer, and Petrov (2010)] Sheehan, Harris, Song May 26,

28 Results and Future Work Simulated Drosophila history # Population size results, dros orig, hist1 True 3.0 population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

29 Results and Future Work Simulated Drosophila history # 1 1.3x Population size results, dros orig, hist1 PSMC True population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

30 Results and Future Work Simulated Drosophila history # 1 1.3x Population size results, dros orig, hist1 PSMC n=4 True population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

31 Results and Future Work Simulated Drosophila history # 1 1.3x Population size results, dros orig, hist1 PSMC n=6 True population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

32 Results and Future Work Simulated Drosophila history # 1 1.3x Population size results, dros orig, hist1 PSMC n=8 True population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

33 Results and Future Work Simulated Drosophila history # 1 1.3x Population size results, dros orig, hist1 PSMC n=10 True population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

34 Results and Future Work Simulated Drosophila history # Population size results, dros orig, hist2 True population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

35 Results and Future Work Simulated Drosophila history # Population size results, dros orig, hist2 PSMC True population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

36 Results and Future Work Simulated Drosophila history # Population size results, dros orig, hist2 PSMC n=4 True population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

37 Results and Future Work Simulated Drosophila history # Population size results, dros orig, hist2 PSMC n=6 True population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

38 Results and Future Work Simulated Drosophila history # Population size results, dros orig, hist2 PSMC n=8 True population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

39 Results and Future Work Simulated Drosophila history # Population size results, dros orig, hist2 PSMC n=10 True population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

40 Results and Future Work Simulated constant population size results, N(t) = Population size results, null, L500kb PSMC n=10 True population size (scaling factor) time (2N generations) Sheehan, Harris, Song May 26,

41 Results and Future Work Simulated constant population size results, N(t) = 0.1 Population size results, null, size0.1 population size (scaling factor) 1.1x PSMC n=10 True time (2N generations) Sheehan, Harris, Song May 26,

42 Results and Future Work Impact of using wedding cake genealogy Population size results, n vs. n, macs data 1.0 population size (scaling factor) n=6 n=6, no n True time (2N generations) Sheehan, Harris, Song May 26,

43 Results and Future Work Real Drosophila data: Raleigh, USA 5.9e e+06 Population size results, real data, drosophila, RAL PSMC n=10 effective population size 3.0e e e e e e e time (years) Sheehan, Harris, Song May 26,

44 Results and Future Work Real Drosophila data: Gikongoro, Rwanda 5.2e e+06 Population size results, real data, drosophila, RG PSMC n=10 effective population size 3.0e e e e e e e time (years) Sheehan, Harris, Song May 26,

45 Future work Results and Future Work Accounting for selection in the Drosophila genome Incorporating variable recombination rate Incorporating migration Use our model for human data What is the right data for certain time periods Sheehan, Harris, Song May 26,

46 Thank you! Results and Future Work Kelley Harris Yun S. Song The Song Group Sheehan, Harris, Song May 26,

47 Results and Future Work Parameter values for Drosophila data θ = 4N 0 µ = ( ) = ρ = (rescaled based on θ) mutation matrix (order ACGT): P = Sheehan, Harris, Song May 26,