Parsimony method
Jarno Tuimala
Thanks to James McInerney for the
slides with a darker background!
Topics
• Theory of parsimony method
• Evolutionary models (optimality criterion)
in parsimony
• Finding optimal trees
• Consensus trees
• Checking reliability
Ideas behind parsimony
Development of parsimony method
• Willi Hennig 1950
– ”Assumes that the tree that gives the fewest number of character
state changes along the branches of a tree gives the best
estimate of phylogeny of the characters being examined” (Fitch
2003)
• Farris 1969
– Parsimony on ordered characters (morphology)
– ”Wagner trees”
• Fitch 1970-1971
– Parsimony on unordered characters (nucleotides)
• Sankoff 1973
– Generalized parsimony
• Felsenstein 1978
– Introduces maximum likelihood to phylogenetics, and shows that
parsimony could be inconsistent (”Felsenstein zone”)
Characters
• Organisms are a set of features (characters):
0123456789AB
ACGTAGCTGAGT
ACGTAGCTGAGT
CCGTAGCAGAGT
CCGTAGCAGAGT
CCGTAGCAGAGT
• When organisms differ with respect to a
feature, different feature forms are called
character states
Unique and unreversed
characters (apomorphy)
• Because hair evolved only once and is
unreversed (not subsequently lost) it is
homologous and provides unambiguous
evidence of relationships
Human
Lizard
HAIR
Frog
change
or step
Dog
absent
prese
nt
Homoplasy - misleading evidence of
phylogeny
• If misinterpreted as homology, the absence
of tails would be evidence for a wrong tree:
grouping humans with frogs and lizards
with dogs
Human
Lizard
TAIL
Frog
Dog
absent
present
Homoplasy - independent
evolution
• Loss of tails evolved independently in
humans and frogs - there are two steps
on the true tree
Lizard
Human
TAIL (adult)
Frog
Dog
absent
present
Homoplasy - reversal
• Reversals are evolutionary changes back to
an ancestral condition
• As with any homoplasy, reversals can provide
misleading evidence of relationships
True tree
1 2 3 4 5 6 7 8 9 10 1 2
Wrong tree
7 8 3 4 5 6
9 10
Homoplasy in molecular data
• Incongruence and therefore homoplasy
can be common in molecular sequence
data
– There are a limited number of alternative character
states ( e.g. Only A, G, C and T in DNA)
– Rates of evolution are sometimes high
• Character states are chemically identical
– homology and homoplasy are equally similar
– cannot be distinguished by detailed study of
similarity and differences
Incongruence or Incompatibility
Lizard
Human
HAIR
Frog
Dog
absent
present
• These trees and characters are incongruent - both
trees cannot be correct, at least one is wrong and at
least one character must be homoplastic
Human
Lizard
TAIL
Frog
Dog
absent
present
Congruence
• We prefer the ‘true’ tree because it is
supported by multiple congruent
characters
Lizard
Frog
Human
Dog
MAMMALIA
Hair
Single bone in lower jaw
Lactation
etc.
Parsimony analysis
• Parsimony methods provide one way of
choosing among alternative phylogenetic
hypotheses
• The parsimony criterion favours
hypotheses that maximise congruence
and minimise homoplasy
• It depends on the idea of the fit of a
character to a tree
Character Fit
• Initially, we can define the fit of a
character to a tree as the minimum
number of steps required to explain the
observed distribution of character states
among taxa
• This is determined by parsimonious
character optimization
• Characters differ in their fit to different
trees
Tree A
1 step
Hair
absent
present
Tree B
2 steps
Bird
Bat
Human
Cocodile
Kangeroo
Frog
Human
Bat
Kangeroo
Bird
Cocodile
Frog
Character Fit
Parsimony Analysis
• Given a set of characters, such as aligned
sequences, parsimony analysis works by determining
the fit (number of steps) of each character on a given
tree
• The sum over all characters is called Tree Length
• Most parsimonious trees (MPTs) have the minimum
tree length needed to explain the observed
distributions of all the characters
wings
-
-
-
-
-
-
Human
Bat
Kangeroo
6
antorbital
fenestra
5
placenta
4
lactation
3
hair
Frog
2
amnion
1
Cocodile
CHARACTERS
Bird
Frog
Parsimony in practice
6
6
4
5
Tree 1
3
2
+
-
-
-
+
+
Crocodile
+
-
-
-
+
-
Kangeroo
+
+
+
-
-
-
Bat
+
+
+
+
-
+
Human
+
+
+
+
-
-
TREE
LENGTH
Tree 1
1
1
1
1
1
2
7
Tree 2
1
2
2
2
2
1
10
FIT
4
2
5
3
6
Tree 2
Of these two trees, Tree 1 has the shortest length
and is the most parsimonious
Both trees require some homoplasy (extra steps)
3
2
1
Human
Bird
Bat
Cocodile
Kangeroo
1
Frog
TAX A
Bird
5
4
Results of parsimony analysis
• One or more most parsimonious trees
• Hypotheses of character evolution associated
with each tree (where and how changes have
occurred)
• Branch lengths (amounts of change
associated with branches)
• Various tree and character statistics
describing the fit between tree and data
• Suboptimal trees - optional
Evolutionary models in
parsimony analysis
Character types
• Characters may differ in the costs
(contribution to tree length) made by
different kinds of changes
• Wagner (ordered, additive)
0
1
2 (morphology, unequal costs)
• Fitch (unordered, non-additive)
one step
A G (morphology, molecules) two steps
T
C
(equal costs for all changes)
Character types
• Sankoff (generalised)
A
G
(morphology, molecules)
one step
five steps
T C (user specified costs)
• For example, differential weighting of
transitions and transversions
• Costs are specified in a stepmatrix
• Costs are usually symmetric but can be
asymmetric also (e.g. costs more to gain than
to loose a restriction site)
Stepmatrices
transversions
Pu
Py
• Stepmatrices specify the costs of changes
within a character
PURINES (Pu)
A
G
T
C
PYRIMIDINES (Py)
transitions
Py
Py
Pu
Pu
A
From C
G
T
A
0
5
1
5
To
C
5
0
5
1
G
1
5
0
5
T
5
1
5
0
Different characters (e.g 1st, 2nd and 3rd)
codon positions can also have different
weights
Different kinds of changes
differ in their frequencies
To
A
A
C
G
T
Transitions
Transversions
C
From
G
T
Unambiguous changes
on most parsimonious
tree of Ciliate SSUrDNA
Weighted parsimony
• If all kinds of steps of all characters
have equal weight then parsimony:
– Minimises homoplasy (extra steps)
– Maximises the amount of similarity due to
common ancestry
– Minimises tree length
• If steps are weighted unequally
parsimony minimises tree length - a
weighted sum of the cost of each
character
Why weight characters?
• Many systematists consider weighting unacceptable, but weighting is
unavoidable (unweighted = equal weights)
• Transitions may be more common than transversions
• Different kinds of transitions and transversions may be more or less
common
• Rates of change may vary with codon positions
• The fit of different characters on trees may indicate differences in their
reliabilities
Number of Characters
250
200
Ciliate SSUrDNA data
150
100
50
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21
Number of steps
• However, equal weighting is the commonest procedure and is the
simplest approach
•
•
Parsimony can be
inconsistent
Felsenstein (1978) developed a simple model phylogeny including four taxa
and a mixture of short and long branches
Under this model parsimony will give the wrong tree
A
p
p
q
C
•
•
•
B
Model tree
q
q
D
Parsimony tree
Rates or
Branch lengths
p >> q
C
A
Wrong
B
D
Long branches are
attracted but the
similarity is
homoplastic
With more data the certainty that parsimony will give the wrong tree increases - so
that parsimony is statistically inconsistent
Advocates of parsimony initially responded by claiming that Felsenstein’s result
showed only that his model was unrealistic
It is now recognised that the long-branch attraction (in the Felsenstein Zone) is
one of the most serious problems in phylogenetic inference
Quest for the best tree Finding optimal trees
Practical issues
• Analysis is usually done using some
software package on a computer.
• First, an initial tree is created fast, possibly
using Wagner method
• This initial tree is then rearranged in order
to find the shortest (MPT) tree.
– Exact solutions
– Heuristics!
Development of parsimony method
• Hill climbing methods (heuristics)
– NNI: Robinson 1971, Moore 1973
– Branch and bound: Hendy, 1982
– SPR: Swofford 1987, 1993
– TBR: Maddison 1991
– Ratchet: Nixon 1999
– TD, TF, SS: Goloboff 1999
Tree space may be populated by local
minima and islands of optimal trees
RANDOM ADDITION SEQUENCE REPLICATES (RAS or jumble)
FAILURE
SUCCESS
Branch
Swapping
Branch Swapping
FAILURE
Branch SwappingTree
Length
Local
Minimum
GLOBAL
MINIMUM
Local
Minima
Finding optimal trees - exact
solutions
• Exact solutions can only be used
for small numbers of taxa
• Exhaustive search examines all
possible trees
• Typically used for problems with
less than 10 taxa
Finding optimal trees - exhaustive search
B
C
Starting tree, any 3 taxa
1
A
Add fourth taxon (D) in each of three possible positions -> three
trees
E
B
D
D
B
C
2a
C
2b
A
B
C
E
D
2c
A
E
E
A
E
Add fifth taxon (E) in each of the five possible positions on each
of the three trees -> 15 trees, and so on ....
Finding optimal trees - exact
solutions
• Branch and bound saves time by discarding
families of trees during tree construction that
cannot be shorter than the shortest tree
found so far
• Can be enhanced by specifying an initial
upper bound for tree length
• Typically used only for problems with less
than 18 taxa
Finding optimal trees - branch and
bound
C2.1
C
D
C2.2
B
B
C
A1
B
C3.1
C
C3.2
D
C2.3
C3.3
A
C2.4
B2
C2.5
B
E
A
B
D
C
C
C1.5
A
A
B
D
D
E
E
C
C1.2
D E
B
B1
A
B
C3.5
C
D
C1.1
C3.4
B3
A
E
A
C1.3
D
B
C
A
C1.4
C
A
Finding optimal trees heuristics
• The number of possible trees increases
exponentially with the number of taxa making
exhaustive searches impractical for many data
sets (an NP complete problem)
• Heuristic methods are used to search tree space
for most parsimonious trees by building or
selecting an initial tree and swapping branches to
search for better ones
• The trees found are not guaranteed to be the
most parsimonious - they are best guesses
Finding optimal trees - heuristics
• Stepwise addition
Asis - the order in the data matrix
Closest -starts with shortest 3-taxon tree adds taxa in
order that produces the least increase in tree length
(greedy heuristic)
Simple - the first taxon in the matrix is a taken as a
reference - taxa are added to it in the order of their
decreasing similarity to the reference
Random - taxa are added in a random sequence, many
different sequences can be used
• Recommend random with as many (e.g. 10-100) addition
sequences as practical
Finding most parsimonious trees heuristics
• Branch Swapping:
Nearest neighbor interchange (NNI)
Subtree pruning and regrafting (SPR)
Tree bisection and reconnection (TBR)
Ratchet
Tree fusing
Tree drifting
Sectorial searches
Finding optimal trees - heuristics
• Nearest neighbor interchange (NNI)
C
A
D
E
F
B
G
A
D
C
C
E
A
D
E
F
B
G
F
B
G
Finding optimal trees heuristics
• Subtree pruning and regrafting
(SPR)
A
C
D
E
F
B
G
C
D
E
C
F
G
E
F
G
B
D
A
Finding optimal trees - heuristics
• Tree bisection and reconnection
(TBR) C D
E
A
F
B
G
E
A
C
A
B
G
F
D
F
B
G
D
C
E
Tree fusing
•
•
Needs to have some trees in memory, typically
from RAS+TBR searches
Resembles genetic algorithms
1. Pick two trees
2. Exchange one compatible branch between the
trees, and make SPR-search
3. Repeat 1. several times
4. Calculate the lenght of all trees, and pick the
shortest one
Tree drifting
• Also known as simulated annealing
• While rearranging the tree, even
suboptimal rearrangements (such that
make the tree longer) can be accepted,
although with a small probability.
Sectorial searches
• Select a smaller data set from the tree,
typically 35-55 taxa.
• Make a few RAS+TBR search for this
subset, and put the subtree back to its
correct place.
• Rearrange the whole tree using TBR
• Repeat the whole cycle a few dozon times.
Practical considerations
• Small dataset (<50 taxa)
– Make 100 RAS
– Rearrange with TBR
• Big dataset (>50-100 taxa)
– Make at least a 100 RAS
– Rearrange using TBR
• Save one tree per RAS replicate
– Use Tree drifting and sectorial searches to
further optimize these 100 trees
Missing data
• Missing data is ignored in tree building but can lead to alternative equally
parsimonious optimizations in the absence of homoplasy
1
A
?
B
single
origin
0 => 1
on any
one of 3
branches
*
*
*
?
C
0
D
0
E
Abundant missing data can
lead to multiple equally
parsimonious trees.
This can be a serious
problem with morphological
data but is unlikely to arise
with molecular data unless
analyses are of incomplete
data
Consensus trees
Multiple optimal trees
• Many methods can yield multiple
equally optimal trees
• We can further select among these
trees with additional criteria, but
• Typically, relationships common to all
the optimal trees are summarised with
consensus trees
Consensus methods
• A consensus tree is a summary of the
agreement among a set of fundamental
trees
• There are many consensus methods that
differ in:
1. the kind of agreement
2. the level of agreement
• Consensus methods can be used with
multiple trees from a single analysis or from
multiple analyses
Strict consensus methods
TWO FUNDAMENTAL TREES
A
B
C
D
E
A
F
B
C
B
A
G
D
E
C
F
E
D
F
G
G
STRICT COMPONENT CONSENSUS TREE
Majority rule consensus
THREE FUNDAMENTAL TREES
A
B
C
D
E
F
G
B
A
A
E
C
B
C
D
F
E
D
F
B
C
E
D
G
66
100
66
Numbers indicate frequency of
clades in the fundamental trees
A
G
66
66
MAJORITY-RULE COMPONENT CONSENSUS TREE
F
G
Consensus methods
strict (component)
Three fundamental trees
Ochromonas
Symbiodinium
Prorocentrum
Loxodes
Tetrahymena
Tracheloraphis
Spirostomum
Euplotes
Gruberia
Ochromonas
Symbiodinium
Prorocentrum
Loxodes
Tetrahymena
Spirostomumum
Tracheloraphis
Euplotes
Gruberia
Ochromonas
Symbiodinium
Prorocentrum
Loxodes
Tetrahymena
Spirostomumum
Euplotes
Tracheloraphis
Gruberia
Ochromonas
Symbiodinium
Prorocentrum
Loxodes
Tetrahymena
Euplotes
Spirostomumum
Tracheloraphis
Gruberia
majority-rule
100
100
66
66
10
0
100
Ochromonas
Symbiodinium
Prorocentrum
Loxodes
Tetrahymena
Spirostomum
Euplotes
Tracheloraphis
Gruberia
Checking ”reliability”
Bootstrapping
• 123456
• ACGTAC
• TGCATG
122456
ACCTAC
TGGATG
123446
ACGTTC
TGCAAG
• Creation of n, say 100, new random
sequence alignments from the original
one.
• Sampling with replacement
– The same column can appear multiple time
in the same random sequence alignment.
Bootstrapping -analysis
• Every randomized dataset is analyzed
exactly as the original one was.
• Results consists of, say 100, trees.
• These trees are combined into a single
tree using majority rule consensus
method.
• Tree contains numbers indicating how
many times (out of 100 tree) a specified
group is found from the trees as such
Bootstrapping result
Jackknifing
• Similar to bootstrapping
• Often used with parsimony method
• Generates a number of randomized data
sets that are sampled without
replacements -> each data set is smaller
than the original.