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Molecular phylogenetics continued…
Outline
1. Models of evolution
2. Phylogenetic tree reconstruction methods:
-- distance based methods
-- maximum parsimony (MP)
-- maximum likelihood (ML)
3. Bootstrapping: evaluating the significance of a tree
The simplest model of evolution:
pairwise distance
The simplest approach to measuring distances
between sequences is to align pairs of sequences, and
then to count the number of differences.
The degree of divergence is called the p-distance.
For an alignment of length N with n sites at which there
are differences, the degree of divergence D is:
D=n/N
Consider an alignment where 3/60 aligned residues differ.
The p-distance is 3/60 = 0.05.
Common assumptions of simple
evolutionary models
Simple models of the evolutionary process make several incorrect
assumptions:
1) equal base or amino acid substitution rates
2) an equal frequency of all bases or amino acids
3) an equal evolutionary rate at all sites of an alignment
4) independent evolution between sites of an alignment.
Observations of DNA/protein alignments demonstrates these
assumptions are often not met in nature. Therefore much more
realistic models of DNA/protein evolution have been devised (more
on this to follow).
Evolutionary models: The Poisson distance correction
-- A simple correction of the p-distance can be derived
by assuming the probability of mutation at a site follows
a Poisson distribution (with a uniform mutation rate)
-- Correction takes account of multiple mutations at the
same site
Evolutionary models: The Poisson distance correction
-- A simple correction of the p-distance can be derived
by assuming the probability of mutation at a site follows
a Poisson distribution (with a uniform mutation rate)
-- Correction takes account of multiple mutations at the
same site
Poisson corrected distance:
dp = -ln(1-p)
-- The corrected distance starts to deviate
noticeably from the p-distance for p > 0.25
Assumption: equal rate of mutation at all sites
Figure 8.1
Evolutionary models: the Gamma distance correction
-- The Gamma distance correction takes account of
mutation rate variation at different sites
-- A Gamma distribution (Γ) can effectively model
realistic variation in mutation rates
Evolutionary models: the Gamma distance correction
-- The Gamma distance correction takes account of
mutation rate variation at different sites
-- A Gamma distribution (Γ) can effectively model
realistic variation in mutation rates
DΓ = a[(1-p)-1/a – 1]
-- The parameter a determines the rate variation
-- Values of a estimated from real protein sequence
data vary between 0.2 (high variation) and 3.5 (lower
variation)
Figure 8.2
Evolutionary models
p-distance, Poisson model, Gamma distance
correction:
These mutation models do not include any information
relating to the chemical nature of the sequences, which
means they can be applied to both nucleotide and
protein sequences.
So, it follows that there are a whole series of more
complex evolutionary models specific for nucleotide
sequence or protein sequence evolution
Jukes and Cantor (JC) one-parameter
model of nucleotide substitution: all
substitutions occur with equal probability
a
A
G
a
a
a
a
T
a
Substitution rate matrix
A
C
G
T
A
-3α
α
α
α
C
α
-3α
α
α
G
α
α
-3α
α
T
α
α
α
-3α
C
P. 271
Kimura two-parameter model (K2P) of nucleotide
substitutions: the probability of transitions and
transversion occurring are different
a
A
G
b
b
b
b
T
a
Substitution rate matrix
A
C
G
T
A
-2β-α
β
α
β
C
β
-2β-α
β
α
G
α
α
-2β-α
α
T
β
α
β
-2β-α
C
P. 272
Incorporation of unequal base frequencies
HKY85 substitution rate matrix: this is a K2P model, but rate
matrix has been modified to account for differences in base
composition (πA:πC:πG:πT)
A
C
G
T
A
(-2β-α)πA
βπA
απA
βπA
C
βπC
(-2β-α)πC
βπC
απC
G
απG
απG
(-2β-α)πG
απG
T
βπT
απT
βπT
(-2β-α)πT
P. 273
Different models of molecular evolution
(nucleotides)
Model name
Base
composition
Different
transition and
transversion
rates
All transition
rates identical
All
transversion
rates identical
JC 1:1:1:1
No
Yes
Yes
F81 Variable
No
Yes
Yes
K2P Variable
Yes
Yes
Yes
HKY85 Variable
Yes
No
No
Tamura-Nei (TN) Variable
Yes
No
Yes
K3P Variable
Yes
No
Yes
Yes
No
No
Yes
No
No
SYM 1:1:1:1
REV (GTR) Variable
Table 7.2
Evolutionary models: amino acid
substitution matrices
There are empirically based models of amino acid
substitution, which consist of a 20 x 20 rate matrix that
estimates the probabilities for each amino acid being
replaced by each alternative amino acid.
The Jones-Taylor-Thornton model (JTT) is the same as the
Dayhoff models but based on a more up to date substitution
matrix constructed from a larger database of sequence
The PMB model is derived from the BLOCKS database of
conserved protein motifs and is therefore related to BLOSUM
Common assumptions of simple
evolutionary models
Simple models of the evolutionary process make several incorrect
assumptions:
1) equal base or amino acid substitution rates
2) an equal frequency of all bases or amino acids
3) an equal evolutionary rate at all sites of an alignment
4) independent evolution between sites of an alignment.
Observations of DNA/protein alignments demonstrates these
assumptions are often not met in nature. Therefore much more
realistic models of DNA/protein evolution have been devised
Common assumptions of simple
evolutionary models
Simple models of the evolutionary process make several incorrect
assumptions:
1)equal base or amino acid substitution rates  solution: use a more
complex substitution matrix
2)an equal frequency of all bases or amino acids  solution: estimate
from sequence alignment data
3)an equal evolutionary rate at all sites of an alignment  solution:
model among site rate variation (ASRV) with a Gamma distribution
4) independent evolution between sites of an alignment  solution:
yikes! No easy solution here…
How to select an appropriate
evolutionary model:
While it is easy to identify models that are formally more realistic,
these are not necessarily more effective in representing the real data
(i.e. the MSA)
Figure 7.18
How to select an appropriate
evolutionary model:
While it is easy to identify models that are formally more realistic,
these are not necessarily more effective in representing the real data
(i.e. the MSA)
Figure 7.18
How to select an appropriate
evolutionary model:
While it is easy to identify models that are formally more realistic,
these are not necessarily more effective in representing the real data
(i.e. the MSA)
Example of model selection
Model
No. of
parameters
log-likelihood AIC
(lnL)
JC
17
-19864
39762
F81
20
-19859
39758
HKY85
21
-19779
39601
HKY85+Γ
22
-19462
38968
Akaike information criterion (AIC): measures the support
in the data for a given model. The model with the
smallest AIC value is regarded as the most suitable
Table 7.3
Outline
1. Models of evolution
2. Phylogenetic tree reconstruction methods:
-- distance based methods
-- maximum parsimony (MP)
-- maximum likelihood (ML)
3. Bootstrapping: evaluating the significance of a tree
Phylogenetic tree reconstruction
Phylogenetic inference is an hypothesis-generating
procedure, where an inferred tree represents the “best
hypothesis” of evolutionary relationships based on the
limited information contained in molecular sequence data
and the assumptions of the phylogenetic reconstruction
method.
Of the many possible evolutionary histories that could
produce the observed differences between homologous
sequences, we must have some method for choosing one
or more best trees from all possible trees.
Tree reconstruction methods
Algorithmic methods follow a fixed series of procedures (an
algorithm) to derive a tree from the data.
- computationally fast
- how well the tree fits the data relative to an alternative
tree is unknown.
- e.g. UPGMA or neighbor-joining methods
Tree reconstruction methods
Algorithmic methods follow a fixed series of procedures (an
algorithm) to derive a tree from the data.
- computationally fast
- how well the tree fits the data relative to an alternative
tree is unknown.
- e.g. UPGMA or neighbor-joining methods
Optimality criterion methods define a criterion for comparing
trees and then finds the tree that maximizes/minimizes the
criterion.
- can define how good or bad any one tree is compared to
other possibilities
- e.g. maximum parsimony and maximum likelihood
methods
Outline
1. Models of evolution
2. Phylogenetic tree reconstruction methods:
-- distance based methods
-- maximum parsimony (MP)
-- maximum likelihood (ML)
3. Bootstrapping: evaluating the significance of a tree
Distance matrix methods
Phylogenetic inference by distance matrix methods
involves two sequential steps:
1) the evolutionary distances (i.e. number of substitutions)
between all taxa in an alignment is estimated based on
a model of evolution.
1) the results are tabulated in a distance matrix and one of
a variety of approaches is used to reconstruct a
phylogenetic tree from the pairwise distance values.
The general flow
of a distance
matrix method for
phylogenetic
inference
Species A
Species B
Species C
Species D
Species E
Species
A
Species
B
Species
C
Species
D
Species A
Species B
Species C
Species D
Species E
Species D
Species C
Species B
Species E
Species A
Inferring a tree from a distance matrix
The simplest algorithm is Unweighted pair-group method
with arithmetic mean (UPGMA).
UPGMA uses a sequential clustering algorithm to group
taxa in order of decreasing similarity.
Ultrametric tree
The details of this algorithm are
presented in Chapter 8 (p 278-279)
Assumptions of UPGMA
UPGMA makes the assumption that there is a linear relationship
between evolutionary distance and divergence time, or, in other
words, that the rate of evolution is equal and constant among
taxa (i.e. ultrametric or clock-like).
This assumption is rarely, if ever, met and therefore it is advised
that UPGMA not be used to infer a best tree.
There are many other superior methods for tree reconstruction
that are as easy to implement and are computationally fast.
The neighbor joining (NJ) method
NJ does not assume all sequences have the same constant rate of
evolution
The basis of the method lies in the concept of minimum evolution,
specifically that the tree with the shortest total branch length is the
best tree
The first steps of NJ; start
with a star tree, identify the
first pair of nearest
neighbors… full details on
p 282- 285
NJ is a star decomposition algorithm that attempts to minimize the
overall branch length of the tree.
Figure 8.6
Neighbor joining method
Modified versions of the original neighbor-joining method
such as BioNJ and Weighbor have been formulated and they
tend to outperform the original neighbor-joining algorithm.
Because of fast run times neighbor-joining is particularly
useful for large studies or bootstrap resampling studies that
require analysis of multiple datasets (e.g. Bootstrap Analysis).
Outline
1. Models of evolution
2. Phylogenetic tree reconstruction methods:
-- distance based methods
-- maximum parsimony (MP)
-- maximum likelihood (ML)
3. Bootstrapping: evaluating the significance of a tree
Parsimony methods
Parsimony methods are based on the concept that the best
hypothesis is the one that requires the least amount of evolutionary
changes.
Objective: to find the tree (i.e. hypothesis) that requires the
minimum number of substitutions to explain the observed/inferred
difference between sequences.
Maximum parsimony (MP) is thus an optimality-criterion method in
which the criterion (i.e. number of substitutions) is to be minimized.
The tree that minimizes the number of substitutions required to
explain the data is called the maximum parsimony tree.
There are only 3 possible trees with 4 taxa
A
C
A
C
B
D
D
B
A
D
A
B
C
B
C
D
Which two trees are the same?
Parsimony methods
Parsimony begins with the classification of sites as either
informative or uninformative. A site is considered informative if
it favors a subset of trees over all possible trees.
Site 1 is uninformative because the
character states are all identical
Parsimony methods
Parsimony begins with the classification of sites as either
informative or uninformative. A site is considered informative if
it favors a subset of trees over all possible trees.
Site 2 is uninformative because
there are two mutations required for
all possible trees
Site 2 is
uninformative
(2 substitutions in
all trees)
Site 3 is informative
and tree 1 is most
parsimonious
Site 4 is informative
and tree 2 is most
parsimonious
Tree 2 is the
maximum
parsimony tree
Site 5 is informative
and tree 2 is most
parsimonious
Searching through the “forest” for the
“best tree”
As the number of taxa becomes large (10+), the number of possible
trees becomes enormous and searching this “tree space” for the
optimal tree can become computationally impossible.
Procedures exist for
reducing the search time
(e.g. heuristic search)
Searching tree space
Heuristic tree searches seek the optimal tree though the use of
iterative trial and error processes, which examine a subset of all
possible trees
Some common branch swapping algorithms:
Nearest neighbor interchange (NNI) a branch swapping
method that results in local rearrangements of a tree.
Subtree pruning and regrafting (SPR), all possible subtrees
are “pruned” from the reference tree and then “regrafted” at an
alternative location.
Problems with parsimony
Correct phylogeny
True convergent evolution
Incorrect reconstruction
This inconsistency in parsimony clusters long branches together
and is termed “long branch attraction”. It can be a problem in
all phylogenetic methods.
Outline
1. Models of evolution
2. Phylogenetic tree reconstruction methods:
-- distance based methods
-- maximum parsimony (MP)
-- maximum likelihood (ML)
3. Bootstrapping: evaluating the significance of a tree
Maximum likelihood methods
Maximum likelihood is an optimality based method, which
evaluates a hypothesized tree in terms of the probability that it
would lead to the observed sequence data under a proposed
model of evolution.
ML methods are among the most accurate at inferring
phylogenetic trees, but also some of the most time consuming
methods to run
The principle of maximum likelihood is to find the tree that
maximizes the likelihood of observing the data.
Maximum likelihood methods
Maximum likelihood is an optimality based method, which
evaluates a hypothesized tree in terms of the probability that it
would lead to the observed sequence data under a proposed
model of evolution. The principle of maximum likelihood is to
find the tree that maximizes the likelihood of observing the data.
Data
Hypothesis
A very brief overview of the maximum likelihood method
1) Calculate the likelihood
(L) of each site given the
tree
A very brief overview of the maximum likelihood method
1) Calculate the likelihood
(L) of each site given the
tree
2) Sum the ln (L) to get
the likelihood of the whole
alignment
This calculation must be
performed for each tree
during a heuristic search
Outline
1. Models of evolution
2. Phylogenetic tree reconstruction methods:
-- distance based methods
-- maximum parsimony (MP)
-- maximum likelihood (ML)
3. Bootstrapping: evaluating the significance of a tree
Error associated with inferred trees
Random error is the deviation from the true tree, because there is a
limited length of sequence data. Random error will therefore tend to
decrease with an increasing length of data, as the stochastic variation
associated with small sample size becomes less.
Systematic error is the deviation from the true tree due to incorrect
assumptions in the method or model used for phylogenetic inference.
Systematic error will introduce a bias that may support the wrong tree
and, unlike random error, the addition of more data will tend to
increase support for the incorrect tree.
Evaluating trees: bootstrapping
Bootstrapping is a commonly used approach to
measuring the robustness of a tree topology.
Given a branching order, how consistently does
A phylogenetic method find that branching order in a
randomly permuted version of the original data set?
IMPORTANT: Bootstrapping allows an assessment of
random error only, not systematic error due to
inaccurate assumptions in an evolutionary model.
Evaluating trees: bootstrapping
- To bootstrap, make an artificial dataset obtained by
randomly sampling columns from your multiple
sequence alignment. Make the dataset the same size
as the original.
- Do 100 (to 1,000) bootstrap replicates.
- Observe the percent of cases in which the assignment
of clades in the original tree is supported by the
bootstrap replicates.
- >70% is considered significant.
Evaluating trees: bootstrap analysis