Bayesian coalescent analysis

Transcription

1 Bayesian coalescent analysis Alexei J Drummond Department of Computer Science, University of Auckland, Auckland, New Zealand August 4, 2011

2 Overview 1 Coalescent theory 2 Hepatitis C in Egypt 4 Phylodynamics 3 Bayesian skyline plot 5 Multi-species coalescent

3 The coalescent Data: a small genetic sample from a large background population. The coalescent is a model of the ancestral relationships of a sample of individuals taken from a larger population. describes a probability distribution on ancestral genealogies (trees) given a population history, N(t). Therefore the coalescent can convert information from ancestral genealogies into information about population history and vice versa. a model of ancestral genealogies, not sequences, and its simplest form assumes neutral evolution. can be thought of as a prior on the tree, in a Bayesian setting.

4 Theoretical population genetics Most of theoretical population genetics is based on the idealized Wright-Fisher model of population which assumes Constant population size N Discrete generations Complete mixing For the purposes of this presentation the population will be assumed to be haploid, as is the case for many pathogens.

5 Kingman s n-coalescent Consider tracing the ancestry of a sample of k individuals from the present, back into the past. This process eventually coalesces to a single common ancestor (concestor) of the sample of individuals. Kingman s n-coalescent describes the statistical properties of such an ancestry when k is small compared to the total population size N.

6 The coalescence of two ancestral lineages First, consider two random members from a population of fixed size N. By perfect mixing, the probability they share a concestor in the previous generation is 1/N. The probability the concestor is t generations back is Pr{t} = 1 N (1 1 N )t 1. It follows that g = t 1, has a geometric distribution with a success rate of λ = 1/N, and so has mean N and variance of N 3 /(N 1).

7 The coalescence of k lineages With k lineages the time to the first coalescence is derived in the same way, only now there are ( k 2) possible pairs that may coalesce, resulting in a success rate of λ = ( k 2) /N and mean time to first coalescence (t k ) of E[t k ] = N ( k 2). This implicitly assumes that N is much larger than O(k 2 ), so that the probability of two coalescent events in the same generation is small.

8 The coalescent likelihood for a genealogy For a genealogy with known coalescent times t = {t 2, t 3,..., t n } we can write the likelihood: ( f (t N) = 1 n k ) ) 2 tk N n 1 exp (. N k=2

9 Constant size Exponential growth The coalescent with serial samples Many epidemiological agents, like RNA viruses, evolve very rapidly, so that the effect of sampling the population at different times becomes important.

10 Bayesian integration of uncertainty in genealogies How similar are these two trees? Both of them are plausible given the data. We can use Bayesian Markov-chain Monte Carlo to average the coalescent over all plausible trees.

11 Hepatitis C in Egypt A case study of the coalescent approach to molecular epidemiology Hepatitis C (HCV) Identified in kb single-stranded RNA genome Polyprotein cleaved by proteases Tissue culture system only recently developed

12 How important is Hepatitis C? 170m+ infected 80% infections are chronic Liver cirrhosis and cancer risk 10,000 deaths per year in USA No protective immunity?

13 HCV Transmission By percutaneous exposure to infected blood Blood transfusion / blood products Injecting and nasal drug use Sexual and vertical transmission Unsafe injections Unidentified routes

14 Estimating demographic history of HCV using the coalescent Pybus et al (2003) Molecular Biology and Evolution Egyptian HCV gene sequences n=61 E1 gene, 411bp All sequence contemporaneous Egypt has highest prevalence of HCV worldwide (10-20%) But low prevalence in neighbouring states Why is Egypt so seriously affected? Parenteral antischistosomal therapy (PAT)?

15 Demographic model for Hepatitis C in Egypt The coalescent can be extended to model any integrable function of varying population size. The model we used was a const-exp-const model. A Bayesian MCMC method was developed to sample the gene genealogy, the substitution model and demographic function simultaneously. N c N(x) = N c e r(t x) N A if t x if x < t < y if t y

16 Estimated demographic history of Hepatitis C in Egypt

17 Uncertainty in parameter estimates Demographic parameters Mutational parameters Growth rate of the growth phase Grey box is the prior Rates at different codon positions, All significantly different

18 Full Bayesian Estimation Marginalized over uncertainty in genealogy and mutational processes Yellow band represents time over which PAT was employed in Egypt

19 Virus Phylodynamics

20 Skyline coalescent model

21 The generalized skyline plot Introduced by Pybus and Strimmer (2001). Visual framework for exploring the demographic history of sampled DNA sequences Input: a single estimated ancestral genealogy (a tree) Output: nonparametric plot of the population size through time Groups adjacent coalescent intervals Converts information within these intervals to estimates of population size for a single coalescent interval: ( ) k ˆN k = t k 2 for j adjacent intervals: ˆN k,j = k(k j) 2j k i=k j+1 t i

22 The generalized skyline plot - simulated data Constant population size, N(t) = N 0 Exponential growth, N(t) = N 0 e rt

23 The generalized skyline plot - HIV-1 group M The tree used here was estimated in Yusim et al (2001) Phil. Trans. Roy. Soc. Lond. B 356: The black curve is a parametric coalescent estimate obtained from the same data under the expansion model, N(t) = (N 0 N A )e rt + N A

24 The Bayesian skyline plot (Drummond et al, 2005) The Bayesian skyline plot estimates a demographic function that has a certain fixed number of steps (in this example 15) and then integrates over all possible positions of the break points, and population sizes within each epoch.

25 Extending the BSP with Stochastic Variable Selection Heled and Drummond (2008)

26 Comparison of EBSP to BSP on Egypt Hepatitis C

27 Detecting evolutionary bottlenecks using EBSP 480 contemporaneous samples from a single locus

28 Detecting evolutionary bottlenecks using EBSP 16 contemporaneous samples from each of 32 loci

29 Detecting evolutionary bottlenecks using EBSP 480 samples sampled through time from a single locus

30 The population dynamics of genetic diversity in Influenza A Rambaut et al (2008) Nature 453:

31 Validating the Bayesian skyline plot

32 Comparison of BSP to parametric coalescent model Hepatitis C in Egypt

33 Modeling complex demographic history Dengue 4 in Puerto Rico N(t) = N 0 exp( rt) log marginal likelihood = N(t) = scaled-translated case data log marginal likelihood =

34 Comparing BSP to incidence data Dengue 4 in Puerto Rico

35 The murky boundary between population genetics and phylogenetics There has been increased interest in analyses of closely related species, where the effect of population genetic processes, such as the coalescent can t be ignored. Different gene trees can have different topologies due to incomplete lineage sorting Divergence times of species can be overestimated due to ancestral polymorphism Sometimes the exact species identities of individuals are not known Sometimes researchers identify species based on a split in a single gene tree. Enter the multi-species coalescent.

36 All these patterns are the consequence of Òlineage driftó within a single population

37 Gene trees and species trees

38 Why Ancestral Polymorphisms?

39 Probabilities of Monophyly, Paraphyly and Polyphyly

40 Problems with estimation of divergence times Typically we use gene phylogenies to estimate species phylogenies But divergence time for genes will be longer than that of species.

41 Problems with estimation of divergence times From Edwards and Beerli, 2000, Evolution 54:

42 The multispecies coalescent

43 Population sizes on a species tree Define N i to be the effective population size at the present for the species i {1, 2,..., n}, and A i the ancestral effective population of species i at the time of the species origin. Then for all ancestral species j {n + 1, n + 2,..., 2n} (represented by internal branches in the species tree) define N j = A left(j) + A right(j), where left(i) is the left descendent of species i and right(i) is the right descendent species. A k Exp(Θ), 1 k 2n N i Gamma(2, Θ), 1 i n N j = A left(j) + A right(j), n < j < 2n

44 Population sizes prior

45 Species divergence times prior For a species tree of n species, define T i to be the time at which the species tree goes from having i to i 1 species, back in time. Additionally define τ i = T i T i+1, i {2,..., n 1} and τ n = T n. The Yule speciation prior supposes a uniform rate species birth (λ) on all lineages, implying a prior of: τ i Exp(1/iλ) More complex species tree priors that admit species extinction (Birth-death prior; Gernhard, 2008) and incomplete sampling (Birth-death-sampling; Stadler, 2009) are also possible. All of these species tree priors imply a uniform prior on labelled histories.

46 Coalescent prior for gene trees Consider a single species in the species tree, spanned by k = u v coalescent intervals (and a final interval without a coalescent event). t k is the time during which there are k lineages. Define N(s) as the population size of this species at time s. Define s i = ik=u t k. The prior density for each interval ending in a coalescent is: f (t k ) = 1 N(s k ) exp ˆs k s k 1 ( n 2) N(x) dx

47 Coalescent prior for gene trees The coalescent prior density for the final interval that does not end in a coalescent event is: ˆs v ( n f (t v ) = exp 2) N(x) dx s v 1 Define f g (g S) to be the total coalescent density for gene tree g, obtained from the product of all the intervals over all species in the species tree (S).

48 Posterior probability of multi-species coalescent If we additionally define Pr(D g g) as the likelihood of the sequence data for gene tree g, and f S (S λ, Θ) as the prior on the species tree times and population sizes, then the posterior distribution over gene trees and species trees looks like: h(g, S D) = 1 Z [Pr(D g g)f g (g S)] f S (S λ, Θ)f Λ (λ)f Θ (Θ) g G where f Λ (λ) is the prior for the speciation rate and f Θ (Θ) is the prior on the ancestral population size parameter.

49 *BEAST Coestimate Species tree, Gene trees, Population sizes and all other parameters. Any combination of number of individual and genes from each individual. Gene trees estimated using any BEAST models, any type of linkage between parameters. For example, all mutations rates may be equal or separate.

50 *BEAST Given data D we define the posterior distribution of the species tree S as follows, ˆ ( n ) P(S D) P(d i g i )P(g i S) P(S)dG (1) G i=1 The data D = d 1, d 2,..., d n is composed of n alignments, one per locus. G = (G 1 G 2 G n ) is the space of all gene trees over the respective alignments where g i G i is one specific gene tree on the i th alignment. The term P(d i g i ) is the Felsenstein likelihood of the data given a gene tree, P(g i S) is the multispecies coalescent and P(S) is a prior distribution on the space of species trees. Like most coalescent models this assumes no recombination within loci and free recombination between loci.

51 Population sizes prior

52 Four gene trees inside a 3-species tree

53 Simulating a rapid radiation of 7 species Repeat 100 times: 1 Simulate a random species tree of 7 extant species from Birth-death process 2 Simulate gene trees in species tree (1, 2, 3,... genes) 3 Simulate sequences down each gene tree (400, 800, 1600 nucleotides) Species tree Species tree and one gene tree

54 Gene concatenation - a terrible idea For each simulated data set we concatenated the gene alignments and estimated a (multi-individual) tree. Species tree Gene tree from concatenated genes The concatenated tree always had > 99% posterior support on all nodes. Despite this, in 72 out of 100 simulations (4 loci), there was no pruning (down to 1 individual per species) of the concatenated tree that was consistent with the true species tree. Concatenation is positively misleading!

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60 Comparing *BEAST and BEST and Supermatrix methods 100 simulated species trees (each with 7 species) with four individuals per species and four loci. topology inside 95% mean 95% size normalized branch score *BEAST BEST Supermatrix tree score speciation time inside 95% speciation time error/ci size *BEAST % 0.41/1.49 BEST % 1.32/2.06 Supermatrix N/A 0.6% 21.12/5.28 population size inside 95% population size error/ci size *BEAST 98% 0.33/1.11 BEST 56% 0.59/2.15 Supermatrix N/A N/A

61 Pocket Gophers Data from (Belfiore, 2008) 27 individuals, 7 loci (12 from T. bottae, 23 from others, 1 from outgroup)

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64 Open questions Are there better priors for the population sizes? What are efficient proposal distributions for MCMC on the multispecies coalescent? What about uncertain species identification? How do we characterize the prior distribution on species associations? (geography, morphology...) What about uncertain numbers of species?! How do we characterize the hypothesis space over a species trees with a random number of species, and gene tree tips with an uncertain species identity? Reversible-jump, yes, but what proprosal distributions? ;-) What about Bayesian stochastic variable selection?