Advances and Limitations
of Maximum Likelihood Phylogenetics
Olivier Gascuel
LIRMM-CNRS, Montpellier, France
Stéphane
Guindon
Wim
Hordijk
Quang
Le Si
Nicolas
Lartillot
Maria
Anisimova
Jean-François
Dufayard
Most of the talk will be about proteins
The data is a set of aligned sequences
Man
Frog
Zebrafish
Fly
Yeast
Amoeba
Paramecium
Blue algae
M
M
M
M
M
L
L
L
A
A
A
S
S
S
A
S
E
E
D
D
E
E
E
D
I
I
L
I
I
L
L
L
G
G
G
G
G
G
G
G
R
R
K
K
R
R
K
K
L
L
L
L
L
L
L
L
I
V
I
V
V
V
V
I
E
E
D
E
E
D
E
D
F
Y
Y
F
F
F
-
S
S
S
S
-
A
A
A
P
-
D the data
Di the data at site i
M
M
L
M
-
V
V
V
V
-
D
D
D
E
D
-
F
F
F
F
F
F
-
W
W
W
W
W
W
-
Q
Q
Q
Q
Q
N
-
N
N
N
Q
N
N
-
R
R
R
K
R
R
R
K
C
C
C
C
C
C
C
C
We aim to reconstruct
the phylogeny of the
sequences in the alignment
T a phylogeny with branch lengths
 We assume a substitution model, denoted as M
 The likelihood of data D, given M and T, is L T , M ; D 
 We search for the tree T* that maximizes data likelihood
T *  ArgMaxT L T , M ; D 
Algorithmics
 Simultaneous NNIs
 Fast SPRs
 Results
Statistical modeling
 An improved replacement matrix
 Accounting for the structure
 Results
Simulation data (40 taxa, random model trees)
Topological accuracy (RF)
N = NJ
M = FastME (distance)
D = DNAPARS
P = PHYML (ML)
Maximum pairwise divergence
Algorithmics
Algorithmics
NNI
Algorithmics
Algorithmics
Algorithmics
SPR
PHYML-NNI
a)
Start with a reasonnable tree with branch lengths (BIONJ)
b)
Compute all subtree partial likelihoods
c)
Independently compute all optimal branch-lengths and optimal
NNI configurations (i.e. local changes)
d)
When no local change significantly increases the likelihood,
return the current tree
e)
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)
p-value<0.01
 WAG+G4+I, with PHYML 3.0
A1
A2
LLK/site
A1>A2
A1<A2
A1=A2
SPR
NNI
0.004
28 (6)
8 (2)
24 (52)
SPR is about twice slower than NNI, ranging
from a few seconds to a few hours
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
A1
A2
LLK/site
A1>A2
A1<A2
A1=A2
SPR
NNI
0.004
28 (6)
8 (2)
24 (52)
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 M  M x  y

 M x  y is the instaneous rate of changes from x to y
 Key role in protein phylogenetics (and alignment)


 P  l   Px  y  l   eM l
 M is defined by: M x  y    y Rx  y
Global rate
 1 in estimation and
when using several
models
Equilibrium
frequency
Exchangeability
R  Rx  y


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
L  A    L  T a , M; D a 
a
a
 A D
 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

L  A    L M; D a , T a
a

 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 T a trees were inferred by PHYML with WAG+G4+I
 Each site i was categorized in the rate category c  i  with
maximum a posteriori probability, and rate c i 
 The LG replacement matrix was estimated using XRATE (Holmes
et al 06) EM-based software, with site likelihood

L c i   M; Dia , T a

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 T a trees were inferred by PHYML with WAG+G4+I
 Each site i was categorized in the rate category c  i  with
maximum a posteriori probability, and rate c i 
 The replacement matrix was estimated using XRATE (Holmes 06)
EM-based software, with site likelihood
Convergence problems
a a

L


M
;
D
,
T
 c c
i
c


LG/WAG matrices
 AA frequencies: relatively close, very low influence on likelihood
values when inferring trees
 Exchangeabilities: strongly correlated
~20 times slower with LG
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:
LG
0.69
ratio  0.6
AsparagineTyrosine
WAG
1.14
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:
LG
1.15
ratio  2.0
CysteinTyrosine
WAG
0.57
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.
M1
M2
Topology
M1M2
AIC/site
M1-M2
M1>M2
M1<M2
JTT
WAG
41/60
-0.17
15 (7)
45 (21)
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
M1
M2
Length
M1/M2
a
M1/M2
Topology
M1M2
AIC/site
M1-M2
M1>M2
M1<M2
LG
WAG
1.07
(58/60)
0.85
(46/60)
30/60
0.23
48 (39)
12 (2)
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 M i are known for
every site i
L T ,  Mi  ,  D    L T , Mi ,  Di 
i
No extra parameter,
regarding single-matrix
models
Extra parameters: gamma,
proportion of invariant sites, etc.
Mixture model
Site category is unknown. We have a set of replacement
matrices M corresponding to various categories with
probabilities M


L T , M ,  D      M L T , M,  Di  

i  M
M  1 extra parameters,
regarding single-matrix
models, or none when the M
are known (e.g. buried/exposed)
Confidence-based combination
Site category is “known”, but not fully reliable
c L T , M i ,  Di  


L T ,  M i  ,  D    
(1  c) M L T , M,  Di  
i 

M

Confidence coefficient, estimated
separately for each alignment;
c  1 useful site assignments,
c  0: useless site assignments
One more parameter
than mixture
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.
M1
M2
AIC/site
M1-M2
M1>M2
Topology
M1M2
LG
WAG
0.36
248/300
165/300
LG_EX
Partitioning
WAG
1.03
294/300
199/300
LG_EX
Confidence
WAG
1.15
297/300
201/300
LG_EX
Mixture
WAG
0.33
LG=0.23
49/60
LG=48
33/60
LG=30
HSSP
Treebase
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
M1
M2
AIC/site
PASSML
(-G-I)
WAG
(+G+I)
-0.6
M1>M2
database
HSSP
Elegant HMM model to account for
secondary structure and exposition, but
not incoporating any RAS (Lio et al, 98)
Discussion
M1
M2
AIC/site
PASSML
(-G-I)
WAG
-0.6
HSSP
JTT
WAG
-0.23
HSSP
Counting estimate
M1>M2
ML estimate
database
Discussion
M1
M2
AIC/site
PASSML
(-G-I)
WAG
-0.6
HSSP
JTT
WAG
-0.23
HSSP
LG
WAG
0.33
ML estimation
with RAS and
larger database
M1>M2
248/300
database
HSSP
Discussion
M1
M2
AIC/site
M1>M2
PASSML
(-G-I)
WAG
-0.6
HSSP
JTT
WAG
-0.23
HSSP
LG
WAG
0.33
248/300
HSSP
LG_EX
WAG
1.15
297/300
HSSP
Accounting for solvent
exposition of residues
database
Discussion
M1
M2
AIC/site
M1>M2
database
PASSML
(-G-I)
WAG
-0.6
HSSP
JTT
WAG
-0.23
HSSP
LG
WAG
0.33
248/300
HSSP
LG_EX
WAG
1.15
297/300
HSSP
SPR
NNI
0.009
28(6)/60
Treebase
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 …….
Number of AA per site (Lartillot et al 2004, 2007)
WAG
LG
M1500
Mean
3.33
3.25
2.69
Variance
8.13
7.53
4.59
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 …..
Thank you all, the organizers
and the Isaac Newton Institute
 Independence assumption: L T , M ; D    L T , M ; Di 
i
 Stationary distribution of AA:     x 
 The tree likelihood is recursively computed from the root:
L T , M ; Di  
  x L T , M , ai  x; Di 
xAA


   x   Px y  lU , M  L U , M , ui  y;U  Di  
 yAA

xAA


  ...V ...
 yAA

a
lU
u
U
lV
v
V
Probability
of change from
x to y
in time lU
Partial
likelihood of
rooted tree U
(L(U) for short)
 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).
U
u
l(u,v)
v
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 hillclimbing, exact way.
Silmutaneous NNIs : two (relatively) fast and easy operations
(when all partial likelihoods are known)
 Independently computing all optimal branch lengths
l(u,v)
U
u
v
V
 Independently computing all optimal NNI configurations
D
A
e
B
C
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.