Mixture Models in Phylogenetic Inference. Mark Pagel and Andrew Meade Reading University.

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1 Mixture Models in Phylogenetic Inference Mark Pagel and Andrew Meade Reading University

2 Mixture models in phylogenetic inference!some background statistics relevant to phylogenetic inference!mixture models!a mixture model for pattern heterogeneity Q!performance and results!real data!a mixture model for heterotachy t!4-taxon simulation study!performance and results!application to real data!mixtures of topologies T

3 The growth of GenBank

4 Growth in the use of molecular phylogenies in scientific research Cumulative publications: molecular AND phylogen*24000 PCR made available Nobel Prize to Kerry Mullis 2 years 3 years 4 years Year projected 2007 source: Web of Science, April, Search terms molecular AND phylogen*. Line of best fit is a quadratic spline or quadratic polynomial

5 Mixture models in phylogenetic inference L(Q)! P(D Q) conventional likelihood model mixture model in Q P(D Q,T) = " i P(D i Q,T) P(D Q 1,Q 2, Q J,T) = " i #w j P(D i Q j,t) Let T = (S, t) mixture model in t P(D Q,t 1,t 2 t J ) = " i #w j P(D i Q,t j )

6 Overview: Phylogenetic Inference from gene sequences 1. Homogeneous Process Model: A C G T Sp 1 AACGTTGTCCCTT Sp 2 AACGTTCCTTGCT.. Sp n CCGGTTGCAAGCT A --- q ac q ag q at C --- q cg q ct G --- q gt =1 T popular modifications to basic homogeneous model a) rate-heterogeneity (Yang, 1994): apply homogeneous model but allow rates to vary over sites according to a discrete gamma. Equivalent to fitting k rate matrices to each site, where the matrices differ from each other only by a proportional scalar b) partitioning data: apply a different rate matrix to different sites, chosen beforehand by the investigator

7 Limitations to homogeneous, rate-heterogeneity and partitioning approaches Homogeneous model: not all sites may evolve according to the same model Gamma model: rate variation may not be gamma or sites may vary in some other way Partitioning data: presumes investigator knows with certainty which model best applies to each site

8 Pattern-Heterogeneity Model of Gene-Sequence Evolution Allow for different sites to evolve in quantitatively or qualitatively different ways without prior partitioning by the investigator. Method: fit more than one model of evolution to each site, summing the likelihood at each site over all models. Allow the data to determine the best model for each site. Motivation for model: Pattern-heterogeneity model will always equal or better the performance of homogeneous or gamma rate heterogeneity models and normally improve on partitioning models. Frequently yields substantial improvements (100 s of log-units) Applications Phylogenetic inference Detecting regions within genes that evolve differently Detecting differences among genes

9 Applications of pattern-heterogeneity model Single gene alignment species 1 species n pattern 1 pattern 2 pattern 3 Concatenated gene alignment species 1 species 2 gene 1 gene 2 gene 3 gene k... species n Supermatrix alignment species 1 species 2 species n-k species n

10 Detecting pattern-heterogeneity in simulated data Method: I) generate simulated gene-sequence data on a random tree according to two different models of sequence evolution, creating two genes with different patterns of substitutions 2) analyse simulated data using homogeneous, gamma rates and pattern-heterogeneity model Tree used in simulations rate matrices A C G T A C G T --- Data: two genes of length 1200 and 800 sites

11 Posterior likelihoods for data simulated from two qualitatively distinct rate matrices log-likelihood Homogeneous Gamma (4) Pattern-Homogeneity Iteration of Markov Chain A <--> C A <--> G A <--> T C <--> G C <--> T G <--> T Q Q

12 Pattern-heterogeneity model: Simulated and estimated values of the rate parameters 5 A C G T Estimated values of rate parameters A C G T --- Q2 Q Actual values of rate parameters

13 Detecting pattern-heterogeneity in simulated data 15 log(q1 likelihood) - log(q2 likelihood) gene 1 gene Site in simulated sequence

14 Detecting rate-heterogeneity in simulated data Method: I) generate simulated gene-sequence data on a random tree according to a gamma rate heterogeneity model (continuous-gamma, $=1.0) 2) analyse data using homogeneous model, gamma rates (4 categories) and patternheterogeneity model (4 rate matrices) Tree used in simulations rate matrices A C G T A C G T --- Data: a gene of length 2000 sites

15 Posterior likelihoods for data simulated using four gamma-distributed rate categories log-likelihood Q(4) Q(1) Gamma(4) Iteration of Markov Chain

16 Average estimated transition rates from pattern-heterogeneity model applied to simulated gamma rate heterogeneity data: input values shown

17 Detecting secondary structure in Mitochondrial ribosomal 12s sequences n=57 mammals

18 Detecting secondary structure using the pattern-heterogeneity model Stem and loop structure leads to predictions about the pattern of nucleotide substitutions stems: dominated by Watson-Crick base pairings. Therefore expect compensatory substitutions to maintain Watson-Crick pairings. Non-compensatory changes have lower fitness. This predicts that transitional changes will occur at a far higher rate than transversional changes. Often observe Tr/Tv ratio of in mitochondrial DNA loops: no a priori substitutional pattern expected

19 Detecting secondary structure; likelihoods of pattern-heterogeneity and Gamma models log-likelihood Gamma (4) Pattern-heterogeneity (4) E7 2E7 3E7 4E7 5E7 Iteration number

20 Mammal 12S data: analysis of rate matrices A <-> C A <-> G A <-> T C <-> G C <-> T G <-> T Tr/Tv Q Q Q Q Best fit to loop or stem Q1 Q2 Q3 Q4 Stem Purines A G Pyrimidines C T Loop

21 Application of pattern heterogeneity to n=159 datasets (using a reversible-jump algorithm) RJ-algorithm allows GTR rate matirices to be gained and lost using split and merge moves Q split Q i merge Q Q j Jumps to higher or lower dimensions governed by: likelihood ratio % prior ratio % proposal ratio % jacobian Mean no. of taxa =39.9±32, median = 29 (range 7-178) Mean alignment size = 1983±2545 nucleotides, median = 1140 (range ) Mean no. of genes = 2.7±4.5, median = 1, (range 1-37) Allow RJ algorithm to run to apparent convergence Improvement over conventional modelling How many Qs?

22 Distribution of number of rate matrices (RJ-algorithm) and empirical weight 1 mean = 2.2±1.4 median =2 correlation with no. of taxa, r 2 =0.15 no. of sites in alignment, = r 2 =0.27 no. of genes, r 2 =0.25 no. of taxa has highest partial correlation number of rate matrices Weight Position

23 Improvement in log-likelihood over a single Q model 100 mean improvement = 265±608 log-units 75 mean improvement = 379± 698 for n=111 datasets with > 1 Q (light green) Increase in log-likelihood over single Q model

24 Improvement in log-likelihood versus no. of rate matrices 4000 log-likelihood difference No Q's note: curve is a quadratic fit

25 Conclusions Mixture model can detect pattern-heterogeneity in simulated and real data and recover the parameters of the model of sequence evolution Can lead in general to large improvements in likelihood over single Q model Can improve topology and reduce long-branch attraction Can reduce the node density artefact Can be used to investigate gene evolution (such as secondary structure) or be applied to concatenated data sets of multiple genes.

26 Heterotachy mixture model in t Mixture model in branch lengths, t P(D Q,t 1,t 2 t J ) = " i #w j P(D i Q,t j ) A B C A B C A B C What does the model deliver?!variation in rates across the tree (heterotachy): a non-parametric covarion? basic covarion model (Tuffley and Steel, 1998) Q (off or slow) - s ij I s ij I Questions!does it work?!can we estimate sets of branch lengths?!are there a small number of sets in real data? Q= s ji I Q (on or fast)- s ji I

27 Does the branch-lengths model work? Variable evolutionary rates over time: Where Parsimony beats ML site pattern 1 site pattern 2 A C A C p r B q D B D Site-specific variation in rates of evolution, causes maximum likelihood but not parsimony to find the wrong tree. One-half of the sites vary their rates as in the left diagram, and the remaining sites vary according to the right diagram. Maximum likelihood tends to put A and C and B and D together, whereas parsimony tends to find the true tree (Kolaczkowski and Thornton, 2004)

28 Simulation!four taxon tree A site pattern 1 site pattern 2 C A C!two sets of site patterns p r!750 sites per pattern B q D B D!generate 5000 random data sets of a hard case: p=0.75, q=0.05, r={0,0.4}!analyse with MP, JC-MCMC, Covarion, BL-MCMC (blind) Do we get back the true tree? A B C D

29 Information ratio = xxyy/(xyxy + xyyx) A r C e.g., aacc/(acac+acca) B D 6 5 information ratio r

30 A C 1 Probability of True Tree versus r B r D p 50 scores in r Parsimony = p 50 Y 0.5 JC MCMC = Covarion = BL MCMC = R r BL MCMC: recovered two branch length sets of correct lengths

31 Probability of recovering true tree vs. information ratio: BL MCMC model 1.8 probability correct information ratio

32 Can we estimate sets of branch lengths? Simulated branch-length sets data* Two branch length sets Analysis Likelihood Variance Weight BLS BLS BLS BLS Three branch length sets Analysis Likelihood Variance Weights BLS BLS BLS BLS Four branch length sets Analysis Likelihood Variance Weights BLS BLS BLS BLS *Data simulated on a tree of about 60 tips sites. Likelihood is mean from Markov chain; Variance is of the likelihood.

33 A reversible-jump algorithm for heterotachy -- allow each branch to choose one or more length -- need mechanism for splitting and merging branches -- test on simulated data 0.2 branch length set branch length set 1

34 Data simulated to have two branch-length sets log-likelihoods returned by one and two branch-length sets models and RJ algorithm Y Y log-likelihood BLS 1 log-likelihood BLS 2 log-likelihood RJ +925 log-units over BLS extra branches +870 log-units 73±2.46 branches 94% of improvement in log-likelihood with 65 fewer parameters iteration

35 Time series plot of no. of extra branches inferred from RJ algorithm No. branches iteration

36 Eukaryote phylogeny of 48 species based on four genes (EF1-$, EF2- $, and two trna synthetases) One branch-length set (homogeneous model) log-likelihood= two branch-length set log-likelihood= log-units for 94 extra branches RJ-algorithm log-likelihood = log-units(96%) for 45± 1.4 extra branches

37 Summary Branch-lengths model seems to work Can estimate branch length sets RJ algorithm can identify in which branches the heterotachy is most pronounced Uses Improve phylogenetic inference (e.g., should reduce long branch attraction) Identify where in tree and for which sites in the alignment the heterotachy is occurring Sp 1 AACGTTGTCCCTT Sp 2 AACGTTCCTTGCT.. Sp n CCGGTTGCAAGCT

38 A mixture of topologies L(Q)! P(D Q) conventional likelihood model mixture model in Q (Pagel and Meade, 2004) P(D Q,T) = "P(D i Q,T) i P(D Q 1,Q 2, Q J,T) = "#w j P(D i Q j,t) i j mixture model in T P(D Q,T 1,T 2 T J ) = "#w j P(D i Q,T j ) i j

39 Some causes of conflicting phylogenetic signals -- gene trees vs species trees (coalescent events) -- convergent evolution (e.g., lysozyme in cows and monkeys) -- gene duplication -- lateral gene transfer gene-duplication lateral gene transfer

40 How does the multiple-topologies mixture model work? conventional (Bayesian) inference alignment T likelihood iteration Li=1=P(D T1) Li=2=P(D T2) Li=3=P(D T3) n multiple-topologies mixture model alignment.. Li=n=P(D Tn) k distinct Markov chains T1 T2..Tk combined likelihood iteration n Converged combined likelihood: Li=n=w1L1+w2L2+..wkLk combined likelihood allows tree to diverge if this increases L calculate posterior probabilities of each tree separately Li=1=w1L1+w2L2+..wkLk Li=2=w1L1+w2L2+..wkLk Li=3=w1L1+w2L2+..wkLk.. Li=n=w1L1+w2L2+..wkLk

41 Mixture model in T mixture model in T P(D Q,T 1,T 2 T J ) = "#w j P(D i Q,T j ) i j A B C A B C A C A C B B What does the model deliver?!variation in the topology!variation in rates across the tree: a non-parametric covarion? Questions!does it work?!can we estimate sets of topologies/branch lengths?!are there a small number of sets in real data?

42 Does the multiple-topology model work? Simulation of two-topologies A topology 1 topology 2 C A B!two four taxon trees!two topologies p B q r D p C q r D!generate 20,000 random alignments of 1000 sites sets. Vary the proportion of sites per topology!analyse with Bayesian MCMC using: one-tree model (conventional), two-tree model Can we detect two trees?

43 Converged Markov chains for one and two-topology models Data: 1000 sites simulated up two different trees (80/20 split) two-topology model log-likelihood one-topology model Iteration of Markov chain

44 Proportion of sites favouring topology 1 topology 2 Outcome one-topology model two-topology model A B C D A B C D A D A C A B B C B D C D A B A B C D C D

45 Performance of one and two-topology models given conflicting phylogenetic signal A B (1) C A (2) D C B D Posterior support for topology two-topology model Analysis one-tree model two-tree model topology 1 two-tree model topology one-topology model Proportion of sites favouring topology 1

46 significant differences detected with ~ 10% of sites Posterior support for topology Proportion of sites favouring topology 1 likelihood difference Proportion of sites favouring topology 1

47 Estimating the number of topologies Data: 1000 sites simulated up two different trees of 50 tips, 60/40 split Model Log-likelihood s.d W 1 W 2 W 3 One-topology Two topologies Three topologies

48 Conflicting signal for ecdysozoa/coelomata phylogenies Nematodes (e.g., C. elegans) Arthropods (e.g., Drosophila) Chordates (e.g., humans) Nematodes (e.g., C. elegans) Arthropods (e.g., Drosophila) Chordates (e.g., humans) ecdysozoa coelomata dataset: 13,946 BP alignment of fourteen genes from four eukaryotic species, Homologene (NCBI) database Number Name Length 1 Beta tubulin EEF1A HSP40-4B HSP HSP HSP70-9B HSP HSP90-1A s s s RNA-Bind-Motif TUBA TUBB. 900

49 one-topology model: likelihood = ±2.58 data from Homologene database (NCBI) posterior support for coelomata = 87 tree length 1.01±0.03

50 One-topology model versus two-topology results likelihood = ±2.58 likelihood = ±3.01 weight = 0.57 weight = 0.43 coelomata coelomata ecdysozoa tree length 1.01±0.03 tree length 0.93±0.13 tree length 1.33±0.3 note: two recent papers support ecdysozoa!log-l = 53.1 log units

51 Simulation of null hypothesis distribution of differences in likelihood n=1000 simulated data sets of 13,946 sites. Fit one and two topology model to each. Find difference in likelihoods. n=1000 < 53, p<0.001 log(l2)-log(l1)

52 Prokaryote phylogenetics: placement of hyperthermophiles Log likelihood =-199,345

53 Prokaryotes: two topologies showing that hyperthermophiles become derived (longer tree) w 1 =0.55 w 2 =0.45 Log likelihood =-194,022!L = 5323

54 Summary Multiple topologies model seems to work Can estimate more than one topology and identify conflicting signal directly Some questions How to test? How many distinct trees in real data?

55 Acknowledgement BBSRC and NERC (UK)

56 Phylogeny of the Ascomycota Fungi showing the evolution of lichen-formation lichen forming ambiguous not lichen forming

57 log-likelihoods for combined LSU/SSU nrdna data set: 54 Ascomycota species n=800 sites log-likelihood Q(2) Gamma(4) Q(4) iteration number

58 Site by site analysis fitting two independent rate matrices: LSU/SSU nrdna Ascomycota combined data set SSU/LSU divide log-likelihood Q1 - log-likelihood Q Note: data modelled with two independent rate matrices site in the alignment

60 Compensatory substitutions transitional changes transitional changes tranversinal changes low fitness Purines A G Pyrimidines C U

61 Detecting secondary structure: application of pattern-heterogeneity model to 12s mitochondrial DNA data on 57 mammals Tree here

62 Mixture models in phylogenetic inference!mixture models!a mixture model in T!simulation study!performance at estimating T s!application to ecdysozoa/coelomata and prokaryotes

63 Detecting Conflicting Phylogenetic Signals a mixture-model approach

64 Average estimated transition rates from gamma and pattern-heterogeneity models applied to simulated gamma rate heterogeneity data: input values shown A C G T A C G T --- average rate = Average Q1 Gamma1 Q2 Gamma2 Q3 Gamma3 Q4 Gamma4

66 Results for branch length sets RJ model for branch lengths RJ for each branch. How many branches? identifying where and when

67 Is the number of branch lengths sets small in real data? Two prokaryote data sets 1) rrna 751 sites 2) ribosomal + proteins ~ 6000 sites Conventional likelihood model Beta proteobacteria Gamma proteobacteria Alpha proteobacteria Epsilon proteobacteria Chlamydiae Spirochaete Cyanobacteria DeinococcusThermus Actinobacteria Firmicutes Fusobacteria Thermotoga Aquificae Nanoarchaeota Crenarchaeota Euryarchaeota Branch-lengths mixture model Beta Proteobacteria Gamma Proteobacteria Alpha Proteobacteria Epsilon Proteobacteria Chlamydiae Spirochaetes Actinobacteria Aquifex Thermotoga Fusobacterium Firmicutes DeinococcusThermus Cyanobacteria Nanoarchaeota Crenarchaeota Euryarchaeota

Elements of Bioinformatics 14F01 TP5 -Phylogenetic analysis

Elements of Bioinformatics 14F01 TP5 -Phylogenetic analysis 10 December 2012 - Corrections - Exercise 1 Non-vertebrate chordates generally possess 2 homologs, vertebrates 3 or more gene copies; a Drosophila