Rooting Phylogenetic Trees with Non-reversible Substitution Models Von Bing Yap* and Terry Speed § *Statistics and Applied Probability, National University of Singapore § Statistics, University of California Reference: BMC Evolutionary Biology 5:1 (2005)

Molecular Phylogenetics From alignments to trees. Many methods: parsimony, distance, stochastic models.

Reversible Models Almost all substitution models are reversible: for example, Pr(anc=A, des=C) = Pr(anc=C, des=A). Rooted trees that give the same unrooted tree are indistinguishable.

Stationary Models Character states have the same frequencies everywhere on the tree. Root can be identified (Yang 1994, Huelsenbeck et al. 2001).

Nonstationary Models Yang and Roberts (1985) Galtier and Gouy (1998)

NON-STATIONARY STATIONARY REVERSIBLE SUBSTITUTION MODELS

The Simplest NSTA Model Parameters: rooted tree topology θ: root base frequency Q: rate matrix (calibrated) branch lengths No relationship between θ and Q.

Specialisations If θ is the equilibrium distribution of Q, get STA. If in addition, Q satisfies the detailed balance conditions, get REV.

Probability of alignment Felsenstein’s algorithm can be used to compute the probability of one site. Multiplying across sites gives probability of alignment.

Tree Inference Fix a rooted tree. Find the most likely parameter values. The maximum likelihood is the support of the tree. Choose tree with highest support.

Site Heterogeneity Codon positions, secondary structure. Deterministic or random relative rates can be accommodated in the model. Two deterministic models: codon position, and codon position + fast/slow.

Two deterministic models codon: 3 fixed unknown rates, corresponding to codon positions, with weighted average 1. codonsite: get two classes of amino acids (fast/slow) from CLUSTAL alignment output. Coupled with codon positions, get 6 unknown rates with weighted average 1.

Test Data Sets A: human, chimp, gorilla B: human, mouse, rat C: human, chimp, gorilla, orangutan D: human, chimp, mouse, rat E: human, mouse, chicken, frog 13 mitochondrial protein-coding genes

Method Unrooted tree is assumed known. For each rooted tree consistent with the unrooted tree, its support is the maximum loglikelihood upon finding the MLE of the process parameters and branch lengths.

Method (continued) Three processes: REV, STA, NSTA Three site models: novar (no variations), codon (3 classes), codonsite (6 classes).

Method (continued) Two outcomes (a) number of genes for which the correct rooted tree is the most likely (b) does the model get the right rooted tree when the loglikelihoods are summed over genes?

Number of successes AB C DE novar NSTA STA47344 codon NSTA STA36225 codon site NSTA STA35156

Combined genes: Does it get the right tree? ABCDE novar NSTANYYYN STAYNNYN codon NSTAYYYYY STAYNNYY codonsite NSTAYYYYY STAYNNNN

Discussion (1) In general, NSTA fits much better than STA, which fits much better than REV, by the likelihood ratio test criterion. Not only does NSTA get the right tree more often than STA, it is also more discriminative: the best tree has much larger support compared to the other trees.

Discussion (2) The codon+site model of site variation is very crude, and this may explain why the performance is worse than codon model. Need to use better methods. Also need to compare with random model, like discrete gamma.

Discussion (3) The NSTA only has 3 more parameters than STA, and 6 more than REV, so the extra computation is not heavy. Also, since it is possible to identify the root, perhaps NSTA should be used routinely.

Discussion (4) Constraint on NSTA: base compositions of sequences that are equally distant from the root are the same. This may not hold. Software freely available upon request.