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Advances and Limitations of Maximum Likelihood Phylogenetics

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### Advances and Limitationsof Maximum Likelihood Phylogenetics

Olivier Gascuel

LIRMM-CNRS, Montpellier, France

M A E I G R L I E F S A M V D F W Q N R C

Frog

M A E I G R L V E Y S A M V D F W Q N R C

Zebrafish

M A D L G K L I D Y S A L V D F W Q N R C

Fly

M S D I G K L V E F S P M V E F W Q Q K C

Yeast

M S E I G R L V E F - - - - - F W Q N R C

Amoeba

L S E L G R L V D F - - - - D F W N N R C

Paramecium

L A E L G K L V E - - - - - - - - - - R C

Blue algae

L S D L G K L I D - - - - - - - - - - K C

The data is a set of aligned sequences

We assume a substitution model, denoted as M

- The likelihood of data D, given M and T, is
- We search for the tree T* that maximizes data likelihood

Statistical modeling

- An improved replacement matrix
- Accounting for the structure
- Results

Algorithmics

- Simultaneous NNIs
- Fast SPRs
- Results

Simulation data (40 taxa, random model trees)

N = NJ

M = FastME (distance)

D = DNAPARS

P = PHYML (ML)

Topological accuracy (RF)

Maximum pairwise divergence

Algorithmics

NNI

Algorithmics

SPR

PHYML-NNI

- Start with a reasonnable tree with branch lengths (BIONJ)
- Compute all subtree partial likelihoods
- Independently compute all optimal branch-lengths and optimal NNI configurations (i.e. local changes)
- When no local change significantly increases the likelihood, return the current tree
- Else, apply to the current tree all local changes; if the tree likelihood increases go to (b), else (~5% of the cases) apply as many as possible of these changes and go to (b)

Comments

- Simultaneous NNIs can change the tree dramatically, and are not included in (single) SPR or TBR
- The algorithm is very fast and able to deal with large datasets (up to 500-1000 taxa with DNA sequences)
- High topological accuracy with simulated data
- But real data tend to be harder than simulated data, specially the multiple-gene, concatenated datasets

Fast SPRs

- SPRs are non-local moves
- We start from a phylogeny with ML branch length estimates
- The SPR procedure involves testing all (subtree, edge) pairs
- This cannot be achieved in an exact way (i.e. with optimal branch lengths), thus the game is to focus on the most promising pairs (PHYML 3.0 uses a parsimony approach) and to minimize the number of length optimizations and partial likelihod calculations.
- As soon as an improving SPR is found, we fully optimize all branch lengths, compute all partial likelihoods and iterate the procedure.

Results

- 60 Treebase protein alignments (i.e. all available datasets, only removing redundancies and incomplete data).
- average of ~25 sequences and ~1000 sites
- 2 genomic datasets (e.g. 12.000 sites and 64 sequences)
- WAG+G4+I, with PHYML 3.0

SPR is about twice slower than NNI, ranging from a few seconds to a few hours

p-value<0.01

Results

- 60 Treebase protein alignments (i.e. all available datasets, only removing redundancies and incomplete data).
- average of ~25 sequences and ~1000 sites
- 2 genomic datasets (e.g. 12.000 sites and 64 sequences)
- WAG+G4+I, with PHYML 3.0

RAXML is in between in LLK values, and 2-3 times slower than PHYML SPR

Comments

- Fast with this representative, relatively small alignments
- Output trees are not statistically different (in most cases, 52/60)
- SPR trees do not depend (much) on the starting trees
- Some more intensive search strategy could be envisaged, e.g. based on tabu
- Genetic algorithms (e.g. MetaPIGA, GARLI) also perform well.

I do not expect high gains from further algorithmic developments (with such datasets)

Statistical modeling

- An improved, general AA replacement matrix
- Accounting for structure and exposition to solvent
- Results

AA time-reversible replacement matrices

- is the instaneous rate of changes from x to y
- Key role in protein phylogenetics (and alignment)
- M is defined by:

Global rate

- 1 in estimation and

when using several models

Equilibrium frequency

Estimating replacement matrices

- Counting approach of Dayhoff et al. (1972), using pairwise alignments of closely related proteins (PAM, JTT, …).
- Logarithmic (Gonnet et al 1992) and resolvent (Muller et al 2000) counting approaches to deal with pairs of remote proteins
- A strong tendency is to estimate different matrices for different protein groups (mitochondrial, prokaryotic, viral, arthropoda …).
- But general matrices (e.g., JTT, WAG) are widely used, e.g. to build deep phylogenies or to analyze concatenated datasets.

ML estimation of replacement matrices

- Counting methods are not able to deal with multiple alignments, which contain much more information than protein pairs
- ML methods exploit multiple alignments and phylogenies
- a set of multiple alignments, we aim to maximize
- But we cannot simultaneously estimate a number of trees and M. This full maximization was only used with unique concatenated alignments (e.g. Adachi&Hasegawa 1996, with mitochondrial genes, ~3350 sites and 20 taxa).

ML estimation of replacement matrices, Whelan&Goldman 2001

- First step: approximate trees are inferred using NJ and ML branch length estimation
- Second step: M is estimated using an EM algorithm maximizing
- WAG was estimated using BRKALN (186 aligments, ~51.000 sites, ~900.000 AAs)
- WAG is much better than JTT (also estimated from BRKALN)

ML estimation of replacement matrices, Whelan&Goldman 2001

- Variability of rates across sites (RAS) was not incorporated in likelihood calculations.
- It is now recognized that RAS is essential. Some sites are slow (invariant) due to strong evolutionary constraints, while others are very fast.
- RAS is usually implemented with a discrete gamma distribution of rates and invariant sites (G4+I), and used to infer most of trees.
- Moreover, BRKALN is limited regarding current databases, and likely biased toward proteins being easy to cristallize, with well defined 3D structure.

Lee & G., 2007 (submission next week !)

- We used the seed alignments of Pfam, which are manually verified multiple alignments of representative sets of sequences, and selected 3,913 large enough alignments (~600.000 sites, ~6.5 millions AAs).
- The trees were inferred by PHYML with WAG+G4+I
- Each site i was categorized in the rate category with maximum a posteriori probability, and rate
- The LG replacement matrix was estimated using XRATE (Holmes et al 06) EM-based software, with site likelihood

Lee & G., 2007 (submission next week !)

- We used the seed alignments of Pfam, which are manually verified multiple alignments of representative sets of sequences, and selected 3,913 large enough alignments (~600.000 sites, ~6.5 millions AAs).
- The trees were inferred by PHYML with WAG+G4+I
- Each site i was categorized in the rate category with maximum a posteriori probability, and rate
- The replacement matrix was estimated using XRATE (Holmes 06) EM-based software, with site likelihood

Convergence problems

LG/WAG matrices

- AA frequencies: relatively close, very low influence on likelihood values when inferring trees
- Exchangeabilities: strongly correlated

require 3 DNA substitutions

LG/WAG matrices

- Our estimation procedure has better ability to distinguish among the substitution events that are very rare (likely occuring in fast sites only) and those being not so rare (possibly occuring in slow sites).
- LG exchangeabilities are much more contrasted than WAG’s
- But LG cannot be viewed as a constrasted version of WAG:

ratio 0.6

AsparagineTyrosine

0.69

LG

1.14

WAG

LG/WAG matrices

- Our estimation procedure has better ability to distinguish among the substitution events that are very rare (likely occuring in fast sites only) and those being not so rare (possibly occuring in slow sites).
- LG exchangeabilities are much more contrasted than WAG’s
- But LG cannot be viewed as a constrasted version of WAG:

ratio 2.0

CysteinTyrosine

1.15

LG

0.57

WAG

LG/WAG in tree inference

- We analyzed the 60 Treebase alignments using PHYML_SPR with WAG+G4+I, LG+G4+I, and JTT+G4+I.
- We measured the tree length, the gama parameter value (a) and the loglikelihood. We also compared the tree topologies.

p-value<0.01

LG/WAG in tree inference

- LG trees are longer than WAG trees
- Topologies of the inferred trees differ with half of the data sets.
- Clear improvement in likelihood values
- Similar results with Pfam test aligments

Accounting for exposition and secondary structure

- Substitutions clearly depend on secondary structure and exposition; e.g., buried sites are and remain hydrophobic.
- Overington et al.1990; Lüthy et al. 1991; Topham et al. 1993; Wako and Blundell 1994; Goldman et al. 1996 (to infer both the structure and the phylogeny).
- Not (or rarely) used today in phylogenetics, though the structure of dozens of thousands of proteins is now available.
- We revisited the question thanks to (1) our improved ML-based estimation procedure, (2) the huge, current databases.

Learning and testing data

- We extracted from HSSP ((homology-derived structures of proteins) 4,889 non-redundant (sub)alignments.
- 290,000 sequences, 1,250,000 sites and 71 billions AAs.
- Secondary structure (Helix, Sheet, Turn, Coil) and exposition (Exposed, Buried) are available for all the sites, but not fully reliable (80-90% of conservation).
- We randomly selected 500 alignments as a test set, leaving 4,389 alignments to learn substitution matrices for various site categories ( E, B; H, S, T, C; E&H, E&S, E&T …).

Computing the tree likelihood using site partition

Each category is associated to a replacement matrix; the category and corresponding matrix are known for every site i

No extra parameter,

regarding single-matrix models

Extra parameters: gamma, proportion of invariant sites, etc.

regarding single-matrix models, or none when the are known (e.g. buried/exposed)

Mixture modelSite category is unknown. We have a set of replacement matrices corresponding to various categories with probabilities

Confidence-based combination

Site category is “known”, but not fully reliable

One more parameter

than mixture

Confidence coefficient, estimated separately for each alignment;

c 1useful site assignments,

c 0:useless site assignments

Treebase

Results of buried/exposed model (LG_EX)

- We analyzed the 60 Treebase and 300 HSSP test alignments with various models, all using G4+I option.

Results

- Likelihood gain is lower when using the secondary structure (LG_SS, ~0.85) and higher when combining both secondary structure and exposition (LG_EX_SS, ~1.6).
- The difference between LG_EX_SS+G4+I and WAG+G4+I, is of the same range as the difference between WAG+G4+I and WAG (~2.0).

Discussion

We revisited questions and models which were proposed and explored by N. Goldman, Z. Yang, their collaborators, … others, using today

- concepts, e.g. RAS MUST be accounted for in tree inference AND replacement matrix estimation,
- tools (XRATE, PHYML),
- and databases (Pfam, HSSP).

Discussion

We revisited questions and models which were proposed and explored by N. Goldman, Z. Yang, their collaborators, … others, using today

- concepts, e.g. RAS MUST be accounted for in tree inference AND replacement matrix estimation,
- tools (XRATE, PHYML),
- and databases (Pfam, HSSP),
- and computers !

Discussion

Elegant HMM model to account for secondary structure and exposition, but not incoporating any RAS (Lio et al, 98)

Discussion

ML estimation with RAS and larger database

Discussion

Accounting for solvent exposition of residues

Warm up conclusions

Statistical modelling provides much higher gains than algorithmics !

Warm up conclusions

Statistical modelling provides much higher gains than algorithmics !

This should continue in the next years, as current models are still rejected for a number of alignments …….

Warm up conclusions

Statistical modelling provides much higher gains than algorithmics !

This should continue in the next years, as current models are still rejected for a number of alignments …..

Independence assumption:

- Stationary distribution of AA:

lU

lV

u

v

V

U

- The tree likelihood is recursively computed from the root:

Partial likelihood of rooted tree U

(L(U) for short)

Probability of change from x to y in time lU

u

v

U

V

- With time reversible models, the tree likelihood can be obtained from any branch, using partial likelihoods L(U) and L(V), and branch length l(u,v).

(Relatively) time consuming

- Computing the partial likelihood of all subtrees
- Optimizing the branch lengths and computing the likelihood of a given topology

Very time consuming

- Searching the topology space in an hill-climbing, exact way.

Efficient algorithms simultaneously modify the branch lengths and the tree topology, thus searching the space of phylogenies with branch-lengths.

u

v

U

V

D

A

e

B

C

Silmutaneous NNIs : two (relatively) fast and easy operations (when all partial likelihoods are known)

- Independently computing all optimal branch lengths
- Independently computing all optimal NNI configurations

Evaluate AC|BD and AD|BC, optimizing l(e)or all five branches

Orchestrating calculations (RAXML, PHYML ….)

Step0 - All partial likelihoods are available

Orchestrating calculations

Step1 – Pruning the subtree and estimating the branch being left

Orchestrating calculations

Step2 – Computing 1 partial likelihood, estimating the 3 new branch lengths and computing the tree likelihood

Orchestrating calculations

Step3 – Computing 1 partial likelihood, estimating the 3 new branch lengths and computing the tree likelihood … etc.

Progressive filtering strategy (PHYML)

- All possible SPRs are first filtered by a fast distance-based (or parsimony) algorithm; typically, we retain for every subtree the 20% most promising edges for regraphting.
- Previous scheme is run several times with increasingly sophisticated branch-length estimations; when an improving SPR is found, it is returned and the procedure restart from the beginning; else, results are used to rank and filter remaining SPRs.
- This strategy allows considerable gain in computing time, without loss on the resulting tree.

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