Tree Searching Methods Exhaustive search (exact) Branch-and-bound search (exact) Heuristic search methods (approximate) –Stepwise addition –Branch swapping –Star decomposition

Exhaustive Search

Searching for trees Generation of all possible trees 1.Generate all 3 trees for first 4 taxa:

Searching for trees 2. Generate all 15 trees for first 5 taxa: (likewise for each of the other two 4-taxon trees)

Searching for trees 3. Full search tree:

Searching for trees Branch and bound algorithm: The search tree is the same as for exhaustive search, with tree lengths for a hypothetical data set shown in boldface type. If a tree lying at a node of this search tree has a length that exceeds the current lower bound on the optimal tree length, this path of the search tree is terminated (indicated by a cross-bar), and the algorithm backtracks and takes the next available path. When a tip of the search tree is reached (i.e., when we arrive at a tree containing the full set of taxa), the tree is either optimal (and hence retained) or suboptimal (and rejected). When all paths leading from the initial 3-taxon tree have been explored, the algorithm terminates, and all most-parsimonious trees will have been identified. Asterisks indicate points at which the current lower bound is reduced. Circled numbers represent the order in which phylogenetic trees are visited in the search tree.

Stepwise Addition (in a nutshell)

Searching for trees Stepwise addition A greedy stepwise-addition search applied to the example used for branch-and-bound. The best 4-taxon tree is determined by evaluating the lengths of the three trees obtained by joining taxon D to tree 1 containing only the first three taxa. Taxa E and F are then connected to the five and seven possible locations, respectively, on trees 4 and 9, with only the shortest trees found during each step being used for the next step. In this example, the 233-step tree obtained is not a global optimum. Circled numbers indicate the order in which phylogenetic trees are evaluated in the stepwise-addition search.

Stepwise Addition Variants As Is –add in order found in matrix Closest –add unplaced taxa that requires smallest increase Furthest –add unplaced taxa that requires largest increase Simple –Farris’s (1970) “simple algorithm” uses a set of pairwise reference distances Random –random permutation of taxa is used to select the order

Branch swapping Nearest Neighbor Interchange (NNI) E A C B D A D E C B D A C B E

Branch swapping Subtree Pruning and Regrafting (SPR) D A B C G F E  D G F E A B C G D E F B A C a

Branch swapping Tree Bisection and Reconnection (TBR) D A B C G F E D G F E A B C G D E F B C A G D E F B A C G D E F C A B 

Reconnection limits in TBR Reconnection distances:

In PAUP*, use “ReconLim” to set maximum reconnection distance Reconnection limits in TBR

Star-decomposition search

Overview of maximum likelihood as used in phylogenetics Overall goal: Find a tree topology (and associated parameter estimates) that maximizes the probability of obtaining the observed data, given a model of evolutionOverall goal: Find a tree topology (and associated parameter estimates) that maximizes the probability of obtaining the observed data, given a model of evolution Likelihood(hypothesis) Prob(data | hypothesis) Likelihood(hypothesis)  Prob(data | hypothesis) Likelihood(tree,model) = k Prob(observed sequences|tree,model) [not Prob(tree | data,model)]

Computing the likelihood of a single tree 1 j N (1) C…GGACA…C…GTTTA…C (2) C…AGACA…C…CTCTA…C (3) C…GGATA…A…GTTAA…C (4) C…GGATA…G…CCTAG…C 1 j N (1) C…GGACA…C…GTTTA…C (2) C…AGACA…C…CTCTA…C (3) C…GGATA…A…GTTAA…C (4) C…GGATA…G…CCTAG…C(1)(2)(3)(4) CCAG(6) (5)

Computing the likelihood of a single tree Prob CCAG A A Likelihood at site j = + Prob CCAG A C Prob CCAG T T + … + But use Felsenstein (1981) pruning algorithm

Computing the likelihood of a single tree Note: PAUP* reports -ln L, so lower -ln L implies higher likelihood

Finding the maximum-likelihood tree (in principle) Evaluate the likelihood of each possible tree for a given collection of taxa.Evaluate the likelihood of each possible tree for a given collection of taxa. Choose the tree topology which maximizes the likelihood over all possible trees.Choose the tree topology which maximizes the likelihood over all possible trees.

Probability calculations require… An explicit model of substitution that specifies change probabilities for a given branch length “Instantaneous rate matrix”An explicit model of substitution that specifies change probabilities for a given branch length “Instantaneous rate matrix” Jukes-Cantor Kimura 2-parameter Hasegawa-Kishino-Yano (HKY) Felsenstein 1981, 1984 General time-reversible An estimate of optimal branch lengths in units of expected amount of change ( = rate x time)An estimate of optimal branch lengths in units of expected amount of change ( = rate x time)

For example: Jukes-Cantor (1969) Kimura (1980) “2-parameter” Hasegawa-Kishino-Yano (1985) General-Time Reversible

E.g., transition probabilities for HKY and F84:

A Family of Reversible Substitution Models

The Relevance of Branch Lengths CCAAAAAAAA A C CCAAAAAAAA C A

When does maximum likelihood work better than parsimony? When you’re in the “Felsenstein Zone”When you’re in the “Felsenstein Zone”ACBD (Felsenstein, 1978)

In the Felsenstein Zone A C G T A C G T Substitution rates: Base frequencies: A=0.1 C=0.2 G=0.3 T=0.4ABCD

In the Felsenstein Zone Sequence Length parsimony ML-GTR Proportion correct

The long-branch attraction (LBA) problem Pattern type 14 AI = Uninformative (constant)A A A A A The true phylogeny of 1, 2, 3 and 4 (zero changes required on any tree)

The long-branch attraction (LBA) problem Pattern type 14 AI = Uninformative (constant)A AII = UninformativeG A A A A The true phylogeny of 1, 2, 3 and 4 (one change required on any tree)

The long-branch attraction (LBA) problem Pattern type 14 AI = Uninformative (constant)A AII = UninformativeG CIII = UninformativeG A A A A The true phylogeny of 1, 2, 3 and 4 (two changes required on any tree)

The long-branch attraction (LBA) problem Pattern type 14 AI = Uninformative (constant)A AII = UninformativeG CIII = UninformativeG G IV = MisinformativeG A A A A The true phylogeny of 1, 2, 3 and 4 (two changes required on true tree)

The long-branch attraction (LBA) problem G 4 A 2 A 3 G 1 … but this tree needs only one step

Concerns about statistical properties and suitability of models (assumptions) Consistency If an estimator converges to the true value of a parameter as the amount of data increases toward infinity, the estimator is consistent.

When do both methods fail? When there is insufficient phylogenetic signal...When there is insufficient phylogenetic signal

When does parsimony work “better” than maximum likelihood? When you’re in the Inverse-Felsenstein (“Farris”) zoneWhen you’re in the Inverse-Felsenstein (“Farris”) zone A B C D (Siddall, 1998)

Siddall (1998) parameter space a a b b b Both methods do poorly Parsimony has higher accuracy than likelihood Both methods do well p a p b 00.75

Parsimony vs. likelihood in the Inverse-Felsenstein Zone B B BBBBBBBBBB J J J J J J J J J J J J ,00010,000100,000 Sequence length B J Parsimony ML/JC 15% 67.5% (expected differences/site) Accuracy

Why does parsimony do so well in the Inverse-Felsenstein zone? A A C C AC A A C C A G A C G C A A C C A C A C True synapomorphy Apparent synapomorphies actually due to misinterpreted homoplasy

Parsimony vs. likelihood in the Felsenstein Zone B B B B BBBBBBBB J J J J J J J J J JJJ 15% 67.5% Accuracy ,00010,000100,000 B J Parsimony ML/JC (expected differences/site) Sequence length

From the Farris Zone to the Felsenstein Zone C D A B C D A B C D A B B C D A B D C A External branches = 0.5 or 0.05 substitutions/site, Jukes-Cantor model of nucleotide substitution

Parsimony Likelihood Simulationresults:

Maximum likelihood models are oversimplifications of reality. If I assume the wrong model, won’t my results be meaningless? Not necessarily (maximum likelihood is pretty robust)Not necessarily (maximum likelihood is pretty robust)

Model used for simulation... A C G T A C G T Substitution rates: Base frequencies: A=0.1 C=0.2 G=0.3 T=0.4ABCD

Performance of ML when its model is violated (one example)

Among site rate heterogeneity Proportion of invariable sites –Some sites don’t change do to strong functional or structural constraint (Hasegawa et al., 1985) Site-specific rates –Different relative rates assumed for pre-assigned subsets of sites Gamma-distributed rates –Rate variation assumed to follow a gamma distribution with shape parameter  Lemur AAGCTTCATAG TTGCATCATCCA …TTACATCATCCA Homo AAGCTTCACCG TTGCATCATCCA …TTACATCCTCAT Pan AAGCTTCACCG TTACGCCATCCA …TTACATCCTCAT Goril AAGCTTCACCG TTACGCCATCCA …CCCACGGACTTA Pongo AAGCTTCACCG TTACGCCATCCT …GCAACCACCCTC Hylo AAGCTTTACAG TTACATTATCCG …TGCAACCGTCCT Maca AAGCTTTTCCG TTACATTATCCG …CGCAACCATCCT equal rates?

Performance of ML when its model is violated (another example) Rate  =50  =200 Modeling among-site rate variation with a gamma distribution... …can also estimate a proportion of “invariable” sites (p inv )  =2  =0.5 Frequency

Performance of ML when its model is violated (another example)

“MODERATE”–Felsenstein zone

“MODERATE”–Inverse- Felsenstein zone

Bayesian Inference in Phylogenetics Uses Bayes formula: Pr(  |D) = Pr(D|  ) Pr(  ) Pr(D)  Pr(D|  ) Pr(  )  L(  ) Pr(  ) Calculation involves integrating over all tree topologies and model-parameter values, subject to assumed prior distribution on parameters (  =tree topology, branch-lengths, and substitution-model parameters)

Bayesian Inference in Phylogenetics To approximate this posterior density (complicated multidimensional integral) we use Markov chain Monte Carlo (MCMC) –Simulated Markov chain in which transition probabilities are assigned such that the stationary distribution of the chain is the posterior density of interest –E.g., Metropolis-Hastings algorithm: Accept a proposed move from one state  to another state  * with probability min(r,1) where r = Pr(  *|D) Pr(  |  *) Pr(  |D) Pr(  *|  ) –Sample chain at regular intervals to approximate posterior distribution MrBayes (by John Huelsenbeck and Fredrik Ronquist) is most popular Bayesian inference program

A B C D A B C D Likelihood Iterations A brief intro to Markov chain Monte Carlo (MCMC) A B C D... If the chain is run “long enough”, the stationary distribution of states in the chain will represent a good approximation to the target distribution (in this case, the Bayesian posterior) 1.Initialize the chain, e.g., by picking a random state X 0 (topology,branch lengths, substitution-model parameters) from the assumed prior distribution A B C D AB|CD A B C D A B C D BC|AD A B C D A B C D A B C D B C D A AC|BD AB|CD A B C D 2.For each time t, sample a new candidate state Y from some proposal distribution q(.|X t ) (e.g., change branch lengths or topology plus branch lengths) Calculate acceptance probability 3.If Y is accepted, let X t+1 = Y; otherwise let X t+1 = X t “burn in”

Model-based distances Can also calculate pairwise distances based on these modelsCan also calculate pairwise distances based on these models These distances estimate the number of substitutions per site that have accumulated since the two sequences shared a common ancestor, allowing for superimposed substitutions (“multiple hits”)These distances estimate the number of substitutions per site that have accumulated since the two sequences shared a common ancestor, allowing for superimposed substitutions (“multiple hits”) E.g.:E.g.: –Jukes-Cantor distance –Kimura 2-parameter distance –General maximum-likelihood distances available for other models

a d e c b p 12 = a+b p 13 = a+c+d p 14 = a+c+e p 23 = b+c+d p 24 = b+c+e p 34 = d+e p ij = d ij for all i and j if the tree topology is correct and distances are additive Distance-based optimality criteria “Additive trees”

Distance-based optimality criteria Minimum evolution and least-squares p ij d ij SS Least-Squares Minumum evolution (ME) LS branch lengths