Reconstructing phylogenies: how? how well? why? Joe Felsenstein Department of Genome Sciences and Department of Biology University of Washington, Seattle Reconstructing phylogenies: how? how well? why? – p.1/29
Reconstructing phylogenies: how? how well? why?
Joe Felsenstein
Department of Genome Sciences and Department of Biology
University of Washington, Seattle
Reconstructing phylogenies: how? how well? why? – p.1/29
A review that asks these questions
What are some of the strengths and weaknesses of different ways of
reconstructing evolutionary trees (phylogenies)?
How can we find out how accurate we may have been inreconstructing the phylogeny?
Why do we want to reconstruct it? What are phylogenies used for?
Reconstructing phylogenies: how? how well? why? – p.2/29
A review that asks these questions
What are some of the strengths and weaknesses of different ways of
reconstructing evolutionary trees (phylogenies)?
How can we find out how accurate we may have been inreconstructing the phylogeny?
Why do we want to reconstruct it? What are phylogenies used for?
Reconstructing phylogenies: how? how well? why? – p.2/29
A review that asks these questions
What are some of the strengths and weaknesses of different ways of
reconstructing evolutionary trees (phylogenies)?
How can we find out how accurate we may have been inreconstructing the phylogeny?
Why do we want to reconstruct it? What are phylogenies used for?
Reconstructing phylogenies: how? how well? why? – p.2/29
What does “tree space” (with branch lengths) look like?
t1t2
t1t2
an example: three species with a clock
A B C
t 1
t 2
t 1
t 2
OK
not possible
trifurcation
etc.
when we consider all three possible topologies, the space looks like:
Reconstructing phylogenies: how? how well? why? – p.3/29
For one tree topology
The space of trees varying all 2n − 3 branch lengths, each a nonegativenumber, defines an “orthant" (open corner) of a 2n − 3-dimensional realspace:
A
B
C
D
E
F
v1
v
vv
v
v
v23
v4
5
6
78
v9
wall wall
floor v9
Reconstructing phylogenies: how? how well? why? – p.4/29
Through the looking-glass
Shrinking one of the n − 1 interior branches to 0, we arrive at atrifurcation:
A
B
C
D
E
F
v1
v
vv
v
v
v23
v4
5
6
78
v9
A
B
C
E
F
v1
v
vv
vv2
3
D
v
v4
56
78
A
B
C
D
E
F
v1
v
v
vv
v
v23
v4
5
6
7
8
v9
A
B
C
D
E
F
v1
v
vv
v
v
v23
v4
56
78
v9
Here, as we pass “through the looking glass" we are also touch the space
for two other tree topologies, and we could decide to enter either.
Reconstructing phylogenies: how? how well? why? – p.5/29
The graph of all trees of 5 speciesThe space of all these orthants, one for each topology, connecting ones
that share faces (looking glasses):
C DB EA
D BC EA
D BE CA
C ED AB D C
A EB
A CD EB
E BC DA B C
D EA
C BD EA
A BD EC
A BE CD
B CE DA
B DC EA E B
D CA
E CB DA
The Schoenberg graph (all 15 trees of size 5 connected by NNI’s)Reconstructing phylogenies: how? how well? why? – p.6/29
There are very large numbers of trees
For 21 species, the number of possible unrooted tree topologies exceeds
Avogadro’s Number: it is
3 × 5 × 7 × 9 × 11 × 13 × 15 × 17 × 19×21 × 23 × 25 × 27 × 29 × 31 × 33 × 35 × 37
= 8, 200, 794, 532, 637, 891, 559, 375
... and that’s not even asking about how hard it is to optimize the 39
branch lengths for each of these trees.
What this goes with is that most methods of finding the best tree are
NP-hard, and not easy to approximate either.
Reconstructing phylogenies: how? how well? why? – p.7/29
Parsimony methods
Alpha Delta Gamma Beta Epsilon
1
23
4 45 56
SitesSpecies 1 2 3 4 5 6
Alpha T A G C A TBeta C A A G C TGamma T C G G C TDelta T C G C A AEpsilon C A A C A T
Reconstructing phylogenies: how? how well? why? – p.8/29
Advantages and disadvantages of parsimony methods
Disadvantage: not model-based so people think it makes noassumptions.
Advantage: reasonably fast, no search of branch lengths needed
and quick to compute the criterion.
Advantage: good statistical properties when amounts of change are
small.
Disadvantage: statistical misbehavior (inconsistency) when some
nearby branches on the tree are long (Long Branch Attraction).
Disadvantage: likely to make you think you have William of
Ockham’s endorsement.
Disadvantage: may lead to the delusion that you know exactly whathappened in evolution, in detail.
Reconstructing phylogenies: how? how well? why? – p.9/29
Advantages and disadvantages of parsimony methods
Disadvantage: not model-based so people think it makes noassumptions.
Advantage: reasonably fast, no search of branch lengths needed
and quick to compute the criterion.
Advantage: good statistical properties when amounts of change are
small.
Disadvantage: statistical misbehavior (inconsistency) when some
nearby branches on the tree are long (Long Branch Attraction).
Disadvantage: likely to make you think you have William of
Ockham’s endorsement.
Disadvantage: may lead to the delusion that you know exactly whathappened in evolution, in detail.
Reconstructing phylogenies: how? how well? why? – p.9/29
Advantages and disadvantages of parsimony methods
Disadvantage: not model-based so people think it makes noassumptions.
Advantage: reasonably fast, no search of branch lengths needed
and quick to compute the criterion.
Advantage: good statistical properties when amounts of change are
small.
Disadvantage: statistical misbehavior (inconsistency) when some
nearby branches on the tree are long (Long Branch Attraction).
Disadvantage: likely to make you think you have William of
Ockham’s endorsement.
Disadvantage: may lead to the delusion that you know exactly whathappened in evolution, in detail.
Reconstructing phylogenies: how? how well? why? – p.9/29
Advantages and disadvantages of parsimony methods
Disadvantage: not model-based so people think it makes noassumptions.
Advantage: reasonably fast, no search of branch lengths needed
and quick to compute the criterion.
Advantage: good statistical properties when amounts of change are
small.
Disadvantage: statistical misbehavior (inconsistency) when some
nearby branches on the tree are long (Long Branch Attraction).
Disadvantage: likely to make you think you have William of
Ockham’s endorsement.
Disadvantage: may lead to the delusion that you know exactly whathappened in evolution, in detail.
Reconstructing phylogenies: how? how well? why? – p.9/29
Advantages and disadvantages of parsimony methods
Disadvantage: not model-based so people think it makes noassumptions.
Advantage: reasonably fast, no search of branch lengths needed
and quick to compute the criterion.
Advantage: good statistical properties when amounts of change are
small.
Disadvantage: statistical misbehavior (inconsistency) when some
nearby branches on the tree are long (Long Branch Attraction).
Disadvantage: likely to make you think you have William of
Ockham’s endorsement.
Disadvantage: may lead to the delusion that you know exactly whathappened in evolution, in detail.
Reconstructing phylogenies: how? how well? why? – p.9/29
Advantages and disadvantages of parsimony methods
Disadvantage: not model-based so people think it makes noassumptions.
Advantage: reasonably fast, no search of branch lengths needed
and quick to compute the criterion.
Advantage: good statistical properties when amounts of change are
small.
Disadvantage: statistical misbehavior (inconsistency) when some
nearby branches on the tree are long (Long Branch Attraction).
Disadvantage: likely to make you think you have William of
Ockham’s endorsement.
Disadvantage: may lead to the delusion that you know exactly whathappened in evolution, in detail.
Reconstructing phylogenies: how? how well? why? – p.9/29
Distance matrix methods
A B C D E
A
B
C
D
E
0
0
0
0
0
0.20
0.24
0.20
0.24
0.19
0.19
0.17
0.17
0.16
0.16
0.24
0.24
0.15
0.15
0.25
0.25
0.10
0.10
0.24
0.24
ABCDE
CCTAACCTCTGACCC ...CGTAACCTCCGGCCC ...CGTAACCTCTGGCCC ...CGCAACCTCTGGCTC ...CCTAACCTCTGGCCC ...
The sequences: yield distances:
compare:alter tree until predictions matchobserved distances as closely as possible
A B C D E
A
B
C
D
E
0
0
0
0
0
0.23 0.16 0.20 0.17
0.23 0.17 0.24
0.11
0.21
0.23
0.16
0.20
0.17
0.23
0.17
0.24 0.11 0.21
0.10
0.07
0.05
0.08
0.030.06
0.05
A B
CD
E0.20
0.20
A suggested tree: predicts:
Reconstructing phylogenies: how? how well? why? – p.10/29
Advantages and disadvantages of distance methods
Advantage: model-based so assumptions are clearer.
Advantage: it’s geometry so mathematical scientists love it.
Advantage: often fast (especially Neighbor-Joining method), canhandle large numbers of sequences.
Disadvantage: not using data fully statistically efficiently.
Advantage: when tested by simulation, found to be surprisinglyefficient anyway.
Disadvantage: cannot easily propagate some information about
local features in the sequences from one distance calculation toanother.
Disadvantage: it’s geometry so mathematical scientists hang onto
it beyond the point of reason.
Reconstructing phylogenies: how? how well? why? – p.11/29
Advantages and disadvantages of distance methods
Advantage: model-based so assumptions are clearer.
Advantage: it’s geometry so mathematical scientists love it.
Advantage: often fast (especially Neighbor-Joining method), canhandle large numbers of sequences.
Disadvantage: not using data fully statistically efficiently.
Advantage: when tested by simulation, found to be surprisinglyefficient anyway.
Disadvantage: cannot easily propagate some information about
local features in the sequences from one distance calculation toanother.
Disadvantage: it’s geometry so mathematical scientists hang onto
it beyond the point of reason.
Reconstructing phylogenies: how? how well? why? – p.11/29
Advantages and disadvantages of distance methods
Advantage: model-based so assumptions are clearer.
Advantage: it’s geometry so mathematical scientists love it.
Advantage: often fast (especially Neighbor-Joining method), canhandle large numbers of sequences.
Disadvantage: not using data fully statistically efficiently.
Advantage: when tested by simulation, found to be surprisinglyefficient anyway.
Disadvantage: cannot easily propagate some information about
local features in the sequences from one distance calculation toanother.
Disadvantage: it’s geometry so mathematical scientists hang onto
it beyond the point of reason.
Reconstructing phylogenies: how? how well? why? – p.11/29
Advantages and disadvantages of distance methods
Advantage: model-based so assumptions are clearer.
Advantage: it’s geometry so mathematical scientists love it.
Advantage: often fast (especially Neighbor-Joining method), canhandle large numbers of sequences.
Disadvantage: not using data fully statistically efficiently.
Advantage: when tested by simulation, found to be surprisinglyefficient anyway.
Disadvantage: cannot easily propagate some information about
local features in the sequences from one distance calculation toanother.
Disadvantage: it’s geometry so mathematical scientists hang onto
it beyond the point of reason.
Reconstructing phylogenies: how? how well? why? – p.11/29
Advantages and disadvantages of distance methods
Advantage: model-based so assumptions are clearer.
Advantage: it’s geometry so mathematical scientists love it.
Advantage: often fast (especially Neighbor-Joining method), canhandle large numbers of sequences.
Disadvantage: not using data fully statistically efficiently.
Advantage: when tested by simulation, found to be surprisinglyefficient anyway.
Disadvantage: cannot easily propagate some information about
local features in the sequences from one distance calculation toanother.
Disadvantage: it’s geometry so mathematical scientists hang onto
it beyond the point of reason.
Reconstructing phylogenies: how? how well? why? – p.11/29
Advantages and disadvantages of distance methods
Advantage: model-based so assumptions are clearer.
Advantage: it’s geometry so mathematical scientists love it.
Advantage: often fast (especially Neighbor-Joining method), canhandle large numbers of sequences.
Disadvantage: not using data fully statistically efficiently.
Advantage: when tested by simulation, found to be surprisinglyefficient anyway.
Disadvantage: cannot easily propagate some information about
local features in the sequences from one distance calculation toanother.
Disadvantage: it’s geometry so mathematical scientists hang onto
it beyond the point of reason.
Reconstructing phylogenies: how? how well? why? – p.11/29
Advantages and disadvantages of distance methods
Advantage: model-based so assumptions are clearer.
Advantage: it’s geometry so mathematical scientists love it.
Advantage: often fast (especially Neighbor-Joining method), canhandle large numbers of sequences.
Disadvantage: not using data fully statistically efficiently.
Advantage: when tested by simulation, found to be surprisinglyefficient anyway.
Disadvantage: cannot easily propagate some information about
local features in the sequences from one distance calculation toanother.
Disadvantage: it’s geometry so mathematical scientists hang onto
it beyond the point of reason.
Reconstructing phylogenies: how? how well? why? – p.11/29
Maximum likelihood
A C C C G
xy
z
w
t1 t t
t t
t
t
23
4 5
6t7
8
t i are
"branch lengths",
X (rate time)
To compute the likelihood for one site, sum over all possible states(bases) at interior nodes:
L(i) =∑
x
∑
y
∑
z
∑
w
Prob (w) Prob (x | w, t7)
× Prob (A | x, t1) Prob (C | x, t2) Prob (z | w, t8)
× Prob (C | z, t3) Prob (y | z, t6) Prob (C | y, t4) Prob (G | y, t5)
Reconstructing phylogenies: how? how well? why? – p.12/29
Advantages and disadvantages of likelihood
Advantage: uses a model, so assumptions are clear.
Advantage: fully statistically efficient.
Disadvantage: computationally slower.
Advantage: statistical testing by likelihood ratio tests available
Disadvantage: can’t use the LRT test to test tree topologies.
Reconstructing phylogenies: how? how well? why? – p.13/29
Advantages and disadvantages of likelihood
Advantage: uses a model, so assumptions are clear.
Advantage: fully statistically efficient.
Disadvantage: computationally slower.
Advantage: statistical testing by likelihood ratio tests available
Disadvantage: can’t use the LRT test to test tree topologies.
Reconstructing phylogenies: how? how well? why? – p.13/29
Advantages and disadvantages of likelihood
Advantage: uses a model, so assumptions are clear.
Advantage: fully statistically efficient.
Disadvantage: computationally slower.
Advantage: statistical testing by likelihood ratio tests available
Disadvantage: can’t use the LRT test to test tree topologies.
Reconstructing phylogenies: how? how well? why? – p.13/29
Advantages and disadvantages of likelihood
Advantage: uses a model, so assumptions are clear.
Advantage: fully statistically efficient.
Disadvantage: computationally slower.
Advantage: statistical testing by likelihood ratio tests available
Disadvantage: can’t use the LRT test to test tree topologies.
Reconstructing phylogenies: how? how well? why? – p.13/29
Advantages and disadvantages of likelihood
Advantage: uses a model, so assumptions are clear.
Advantage: fully statistically efficient.
Disadvantage: computationally slower.
Advantage: statistical testing by likelihood ratio tests available
Disadvantage: can’t use the LRT test to test tree topologies.
Reconstructing phylogenies: how? how well? why? – p.13/29
Bayesian inference methods
Basically uses the likelihood machinery, and adds priors on parameters
and on trees.
Implemented by Markov chain Monte Carlo methods to sample from theposterior on trees (or parameters, or both).
Very popular right now.
Advantage: interpretation is straightforward, once theassumptions are met.
Advantage: gives you what you want, the probability of the result.
Disadvantage: how long is long enough to run the MCMC?
Disadvantage: where do we get priors from, what effect do they
have?
Disadvantage: they keep chanting in unison “We are the
statisticians of Bayes – you will be assimilated.”
Reconstructing phylogenies: how? how well? why? – p.14/29
Bayesian inference methods
Basically uses the likelihood machinery, and adds priors on parameters
and on trees.
Implemented by Markov chain Monte Carlo methods to sample from theposterior on trees (or parameters, or both).
Very popular right now.
Advantage: interpretation is straightforward, once theassumptions are met.
Advantage: gives you what you want, the probability of the result.
Disadvantage: how long is long enough to run the MCMC?
Disadvantage: where do we get priors from, what effect do they
have?
Disadvantage: they keep chanting in unison “We are the
statisticians of Bayes – you will be assimilated.”
Reconstructing phylogenies: how? how well? why? – p.14/29
Bayesian inference methods
Basically uses the likelihood machinery, and adds priors on parameters
and on trees.
Implemented by Markov chain Monte Carlo methods to sample from theposterior on trees (or parameters, or both).
Very popular right now.
Advantage: interpretation is straightforward, once theassumptions are met.
Advantage: gives you what you want, the probability of the result.
Disadvantage: how long is long enough to run the MCMC?
Disadvantage: where do we get priors from, what effect do they
have?
Disadvantage: they keep chanting in unison “We are the
statisticians of Bayes – you will be assimilated.”
Reconstructing phylogenies: how? how well? why? – p.14/29
Bayesian inference methods
Basically uses the likelihood machinery, and adds priors on parameters
and on trees.
Implemented by Markov chain Monte Carlo methods to sample from theposterior on trees (or parameters, or both).
Very popular right now.
Advantage: interpretation is straightforward, once theassumptions are met.
Advantage: gives you what you want, the probability of the result.
Disadvantage: how long is long enough to run the MCMC?
Disadvantage: where do we get priors from, what effect do they
have?
Disadvantage: they keep chanting in unison “We are the
statisticians of Bayes – you will be assimilated.”
Reconstructing phylogenies: how? how well? why? – p.14/29
Bayesian inference methods
Basically uses the likelihood machinery, and adds priors on parameters
and on trees.
Implemented by Markov chain Monte Carlo methods to sample from theposterior on trees (or parameters, or both).
Very popular right now.
Advantage: interpretation is straightforward, once theassumptions are met.
Advantage: gives you what you want, the probability of the result.
Disadvantage: how long is long enough to run the MCMC?
Disadvantage: where do we get priors from, what effect do they
have?
Disadvantage: they keep chanting in unison “We are the
statisticians of Bayes – you will be assimilated.”
Reconstructing phylogenies: how? how well? why? – p.14/29
Aren’t these graphical models?
x1
x4
x3
x5
x6
x7
x8
x9
x0
x1
x4
x3
x5
x6
x7
x8
x9
x0
x1
x4
x3
x5
x6
x7
x8
x9
x0
x1
x4
x3
x5
x6
x7
x8
x9
x0
x1
x4
x3
x5
x6
x7
x8
x9
x0
v1
v4 v
3
v6
v5
v8
v9
v7
(You have to imaging it going back 500 layers or so). The problem is to
use the data, which is at the tips but not available for the interior nodes,to infer the topology and branch lengths of the tree that is shared by allsites.
Reconstructing phylogenies: how? how well? why? – p.15/29
Could we use graphical model machinery here?
Like Moliére’s character who is delighted to discover that he’s beenspeaking prose all his life, we found we had already been using the
relevant Graphical Model machinery since about 1973.
So alas there was nothing to gain.
The same thing is true for statistical genetics, where the graphical model
machinery reinvents the standard “peeling” algorithms for computing
likelihoods on pedigrees, in use since 1970.
Reconstructing phylogenies: how? how well? why? – p.16/29
Bootstrap sampling of phylogenies
OriginalData
sequences
sites
Reconstructing phylogenies: how? how well? why? – p.17/29
Draw columns randomly with replacement
OriginalData
sequences
sites
Bootstrapsample#1
Estimate of the tree
sample same numberof sites, with replacementsequences
sites
Reconstructing phylogenies: how? how well? why? – p.18/29
Make a tree from that resampled data set
OriginalData
sequences
sites
Bootstrapsample#1
Estimate of the tree
Bootstrap estimate ofthe tree, #1
sample same numberof sites, with replacementsequences
sites
Reconstructing phylogenies: how? how well? why? – p.19/29
Draw another bootstrap sample
OriginalData
sequences
sites
Bootstrapsample#1
Bootstrapsample
#2
Estimate of the tree
Bootstrap estimate ofthe tree, #1
sample same numberof sites, with replacement
sample same numberof sites, with replacement
sequences
sequences
sites
sites
(and so on)
Reconstructing phylogenies: how? how well? why? – p.20/29
... and get a tree for it too. And so on.
OriginalData
sequences
sites
Bootstrapsample#1
Bootstrapsample
#2
Estimate of the tree
Bootstrap estimate ofthe tree, #1
Bootstrap estimate of
sample same numberof sites, with replacement
sample same numberof sites, with replacement
sequences
sequences
sites
sites
(and so on)the tree, #2
Reconstructing phylogenies: how? how well? why? – p.21/29
Summarizing the cloud of trees by support for branches
Bovine
Mouse
Squir Monk
Chimp
Human
Gorilla
Orang
Gibbon
Rhesus Mac
Jpn Macaq
Crab−E.Mac
BarbMacaq
Tarsier
Lemur
80
72
74
9999
100
77
42
35
49
84
Reconstructing phylogenies: how? how well? why? – p.22/29
Some alternatives to bootstrapping
Parametric bootstrapping – same, but simulate data sets from our
best estimate of the tree instead of sampling sites.
Bayesian inference of course gets statistical support informationfrom the posterior.
The Kishino-Hasegawa-Templeton test (KHT test) which comparesprespecified trees to each other by paired sites tests.
Reconstructing phylogenies: how? how well? why? – p.23/29
Some alternatives to bootstrapping
Parametric bootstrapping – same, but simulate data sets from our
best estimate of the tree instead of sampling sites.
Bayesian inference of course gets statistical support informationfrom the posterior.
The Kishino-Hasegawa-Templeton test (KHT test) which comparesprespecified trees to each other by paired sites tests.
Reconstructing phylogenies: how? how well? why? – p.23/29
Some alternatives to bootstrapping
Parametric bootstrapping – same, but simulate data sets from our
best estimate of the tree instead of sampling sites.
Bayesian inference of course gets statistical support informationfrom the posterior.
The Kishino-Hasegawa-Templeton test (KHT test) which comparesprespecified trees to each other by paired sites tests.
Reconstructing phylogenies: how? how well? why? – p.23/29
Why want to know the tree?
It affects all parts of the genomes – it is the essential part of propagating
information about the evolution of one part of the genome to inquiriesabout another part.
The standard method for finding functional regions of the genome isnow using “PhyloHMMs” which use Hidden Markov Model machinery
together with phylogenies to find regions that have unusually low rates
of evolution.
Reconstructing phylogenies: how? how well? why? – p.24/29
Another kind of tree: the coalescentCoalescent trees are trees of ancestry of copies of a single gene locus
within a species. They are weakly inferrable as most have only a few sites
(SNPs) varying among individuals.
Since each coalescent tree applies to a very short region of genome,maybe as little as one gene, there is less interest in the tree.
But they do illuminate the values of parameters such as population
size, migration rates, recombination rates etc. This allows us to
accumulate information across different loci (genes).
To don this we have to sum over our uncertainty about the tree byusing MCMC methods, accumulating the information (as log
likelihood or using Bayesian machinery) to make inferences aboutthe parameters.
This is the interface between within-species population geneticsand between-species work on phylogenies.
It is also the statistical foundation of inferences frommitochondrial genealogies (“mitochondrial Eve”) and Y
chromosome genealogies, and of the samples from the rest of the
genome that are now being added to this.Reconstructing phylogenies: how? how well? why? – p.25/29
A coalescent
Time
Reconstructing phylogenies: how? how well? why? – p.26/29
Yet another kind of tree: trees of gene families
Gene duplications in evolution create new genes. Both the new gene andthe original one then evolve.
Frog Human Monkey Squirrel
gene duplication
a a ab b b
species
boundary
tree of genes
Some forks are gene duplications, leading to subtrees that are all
supposed to have the same phylogeny as they are in the same set ofspecies. Example: Hemoglobin proteins.
Reconstructing phylogenies: how? how well? why? – p.27/29
Yet another kind of tree: trees of gene families
Gene duplications in evolution create new genes. Both the new gene andthe original one then evolve.
Frog Human Monkey Squirrel
gene duplication
a a ab b b
species
boundary
tree of genes
FrogHuman Monkey Squirrel Human Monkey Squirrel
a a a b b b
These twotrees should beidentical
Some forks are gene duplications, leading to subtrees that are all
supposed to have the same phylogeny as they are in the same set ofspecies. Example: Hemoglobin proteins.
Reconstructing phylogenies: how? how well? why? – p.28/29
References
Felsenstein, J. 2004. Inferring Phylogenies. Sinauer Associates, Sunderland,Massachusetts. [Book you and all your friends must rush out andbuy]
Semple, C. and M. Steel. 2003. Phylogenetics. Oxford Lecture Series inMathematics and Its Applications, 24. Oxford University Press. [Morerigorous mathematical treatment]
Yang, Z. 2007. Computational Molecular Evolution. Oxford Series in Ecology
and Evolution. Oxford University Press, Oxford. [Careful survey ofmolecular phylogeny methods, from a leader]
For a list of 348 phylogeny programs, many available free, see
http://evolution.gs.washington.edu/phylip/software.html
Reconstructing phylogenies: how? how well? why? – p.29/29