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Phylogenetic Trees
Lecture 1
Credits: N. Friedman, D. Geiger , S. Moran,
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Evolution
Evolution of new organisms is driven by
Diversity Different individuals
carry different variants of the same basic blue print
Mutations The DNA sequence
can be changed due to single base changes, deletion/insertion of DNA segments, etc.
Selection bias
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The Tree of Life
Sou
rce:
Alb
erts
et
al
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D’après Ernst Haeckel, 1891
Tree of life- a better picture
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Primate evolution
A phylogeny is a tree that describes the sequence of speciation events that lead to the forming of a set of current day species; also called a phylogenetic tree.
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Historical Note Until mid 1950’s phylogenies were constructed by
experts based on their opinion (subjective criteria)
Since then, focus on objective criteria for constructing phylogenetic trees
Thousands of articles in the last decades
Important for many aspects of biology Classification Understanding biological mechanisms
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Morphological vs. Molecular
Classical phylogenetic analysis: morphological features: number of legs, lengths of legs, etc.
Modern biological methods allow to use molecular features
Gene sequences Protein sequences
Analysis based on homologous sequences (e.g., globins) in different species
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Morphological topology
BonoboChimpanzeeManGorillaSumatran orangutanBornean orangutanCommon gibbonBarbary apeBaboonWhite-fronted capuchinSlow lorisTree shrewJapanese pipistrelleLong-tailed batJamaican fruit-eating batHorseshoe bat
Little red flying foxRyukyu flying foxMouseRatVoleCane-ratGuinea pigSquirrelDormouseRabbitPikaPigHippopotamusSheepCowAlpacaBlue whaleFin whaleSperm whaleDonkeyHorseIndian rhinoWhite rhinoElephantAardvarkGrey sealHarbor sealDogCatAsiatic shrewLong-clawed shrewSmall Madagascar hedgehogHedgehogGymnureMoleArmadilloBandicootWallarooOpossumPlatypus
Archonta
Glires
Ungulata
Carnivora
Insectivora
Xenarthra
(Based on Mc Kenna and Bell, 1997)
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Rat QEPGGLVVPPTDA
Rabbit QEPGGMVVPPTDA
Gorilla QEPGGLVVPPTDA
Cat REPGGLVVPPTEG
From sequences to a phylogenetic tree
There are many possible types of sequences to use (e.g. Mitochondrial vs Nuclear proteins).
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DonkeyHorseIndian rhinoWhite rhinoGrey sealHarbor sealDogCatBlue whaleFin whaleSperm whaleHippopotamusSheepCowAlpacaPigLittle red flying foxRyukyu flying foxHorseshoe batJapanese pipistrelleLong-tailed batJamaican fruit-eating bat
Asiatic shrewLong-clawed shrew
MoleSmall Madagascar hedgehogAardvarkElephantArmadilloRabbitPikaTree shrewBonoboChimpanzeeManGorillaSumatran orangutanBornean orangutanCommon gibbonBarbary apeBaboon
White-fronted capuchinSlow lorisSquirrelDormouseCane-ratGuinea pigMouseRatVoleHedgehogGymnureBandicootWallarooOpossumPlatypus
Perissodactyla
Carnivora
Cetartiodactyla
Rodentia 1
HedgehogsRodentia 2
Primates
ChiropteraMoles+ShrewsAfrotheria
XenarthraLagomorpha
+ Scandentia
Mitochondrial topology(Based on Pupko et al.,)
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Nuclear topology
Round Eared Bat
Flying Fox
Hedgehog
Mole
Pangolin
Whale
Hippo
Cow
Pig
Cat
Dog
Horse
Rhino
Rat
Capybara
Rabbit
Flying Lemur
Tree Shrew
Human
Galago
Sloth
Hyrax
Dugong
Elephant
Aardvark
Elephant Shrew
Opossum
Kangaroo
1
2
3
4
Cetartiodactyla
Afrotheria
Chiroptera
Eulipotyphla
Glires
Xenarthra
CarnivoraPerissodactyla
Scandentia+Dermoptera
Pholidota
Primate
(tree by Madsenl)
(Based on Pupko et al. slide)
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Theory of Evolution
Basic idea speciation events lead to creation of different
species. Speciation caused by physical separation into
groups where different genetic variants become dominant
Any two species share a (possibly distant) common ancestor
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Basic Assumptions Closer related organisms have more similar
genomes.
Highly similar genes are homologous (have the same ancestor).
A universal ancestor exists for all life forms.
Molecular difference in homologous genes (or protein sequences) are positively correlated with evolution time.
Phylogenetic relation can be expressed by a dendrogram (a “tree”) .
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Phylogenenetic trees
Leafs - current day species Nodes - hypothetical most recent common ancestors Edges length - “time” from one speciation to the next
Aardvark Bison Chimp Dog Elephant
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Dangers in Molecular Phylogenies
We have to emphasize that gene/protein sequence can be homologous for several different reasons:
Orthologs -- sequences diverged after a speciation event
Paralogs -- sequences diverged after a duplication event
Xenologs -- sequences diverged after a horizontal transfer (e.g., by virus)
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Species Phylogeny
Gene Phylogenies
Speciation events
Gene Duplication
1A 2A 3A 3B 2B 1B
Phylogenies can be constructed to describe evolution genes.
Three species termed 1,2,3.Two paralog genes A and B.
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Dangers of Paralogs
Speciation events
Gene Duplication
1A 2A 3A 3B 2B 1B
If we happen to consider genes 1A, 2B, and 3A of species 1,2,3, we get a wrong tree that does not represent the phylogeny of the host species of the given sequences because duplication does not create new species.
In the sequel we assume all given sequences are orthologs.
S
SS
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Types of Trees
A natural model to consider is that of rooted trees
CommonAncestor
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Types of treesUnrooted tree represents the same phylogeny without
the root node
Depending on the model, data from current day species does not distinguish between different placements of the root.
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Rooted versus unrooted treesTree A
ab
Tree B
c
Tree C
Represents the three rooted trees
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Positioning Roots in Unrooted Trees
We can estimate the position of the root by introducing an outgroup:
a set of species that are definitely distant from all the species of interest
Aardvark Bison Chimp Dog Elephant
Falcon
Proposed root
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Type of Data
Distance-based Input is a matrix of distances between species Can be fraction of residue they disagree on, or
alignment score between them, or …
Character-based Examine each character (e.g., residue)
separately
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Three Methods of Tree Construction
Distance- A tree that recursively combines two nodes of the smallest distance.
Parsimony – A tree with a total minimum number of character changes between nodes.
Maximum likelihood - Finding the best Bayesian network of a tree shape. The method of choice nowadays. Most known and useful software called phylip uses this method.
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Distance-Based Method
Input: distance matrix between species
Outline: Cluster species together Initially clusters are singletons At each iteration combine two “closest”
clusters to get a new one
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Unweighted Pair Group Method using Arithmetic Averages (UPGMA)
UPGMA is a type of Distance-Based algorithm.
Despite its formidable acronym, the method is simple and intuitively appealing.
It works by clustering the sequences, at each stage amalgamating two clusters and, at the same time, creating a new node on the tree.
Thus, the tree can be imagined as being assembled upwards, each node being added above the others, and the edge lengths being determined by the difference in the heights of the nodes at the top and bottom of an edge.
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An example showing how UPGMA producesa rooted phylogenetic tree
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An example showing how UPGMA producesa rooted phylogenetic tree
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An example showing how UPGMA producesa rooted phylogenetic tree
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An example showing how UPGMA producesa rooted phylogenetic tree
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An example showing how UPGMA producesa rooted phylogenetic tree
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UPGMA Clustering Let Ci and Cj be clusters, define distance between them to
be
When we combine two cluster, Ci and Cj, to form a new cluster Ck, then
Define a node K and place its children nodes at depth
d(Ci, Cj)/2
i jCp Cqji
ji qpdCC
1CCd ),(
||||),(
||||
),(||),(||),(
ji
ljjliilk CC
CCdCCCdCCCd
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Example
UPGMA construction on five objects.
The length of an edge = its (vertical) height.
2 3 4 1
6
8
9
d(7,8) / 2
5
7d(2,3) / 2
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Molecular clock
This phylogenetic tree has all leaves in the same level. When this property holds, the phylogenetic tree is said to satisfy a molecular clock. Namely, the time from a speciation event to the formation of current species is identical for all paths (wrong assumption in reality).
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Molecular Clock
1
2
3
4
UPGMA
2 3 4 1
UPGMA constructs trees that satisfy a molecular clock, even if the true tree does not satisfy a molecular clock.
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Restrictive Correctness of UPGMAProposition: If the distance function is derived by adding edge distances in a tree T with a molecular clock, then UPGMA will reconstruct T.
Proof idea: Move a horizontal line from the bottom of the T to the top. Whenever an internal node is formed, the algorithm will create it.
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Additivity
Molecular clock defines additive distances, namely,
distances between objects can be realized by a tree:
ab
c
i
j
k
cbkjd
cakid
bajid
),(
),(
),(
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What is a Distance Matrix?
Given a set M of L objects with an L× L
distance matrix:
d(i, i) = 0, and for i ≠ j, d(i, j) > 0d(i, j) = d(j, i). For all i, j, k, it holds that d(i, k) ≤ d(i, j)+d(j, k).
Can we construct a weighted tree which realizes these distances?
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Additive Distances
We say that the set M with L objects is additive if there is a tree T, L of its nodes correspond to the L objects, with positive weights on the edges, such that for all i, j, d(i, j) = dT(i, j), the length of the path from i to j in T.
Note: Sometimes the tree is required to be binary, and then the edge weights are required to be non-negative.
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Three objects sets are additive:
For L=3: There is always a (unique) tree with one internal node.
( , )( , )( , )
d i j a bd i k a cd j k b c
ab
c
i
j
k
m
Thus0
2
1 )],(),(),([),( jidkjdkidmkdc
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How about four objects?
L=4: Not all sets with 4 objects are additive:
e.g., there is no tree which realizes the below distances.
i j k l
i 0 2 2 2
j 0 2 2
k 0 3
l 0
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The Four Points Condition
Theorem: A set M of L objects is additive iff any subset of four objects can be labeled i,j,k,l so that:
d(i, k) + d(j, l) = d(i, l) +d(k, j) ≥ d(i, j) + d(k, l)
We call {{i,j}, {k,l}} the “split” of {i, j, k, l}.
ik
lj
Proof:Additivity 4P Condition: By the figure...
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4P Condition Additivity:
Induction on the number of objects, L.For L ≤ 3 the condition is empty and tree exists.
Consider L=4. B = d(i, k) +d(j, l) = d(i, l) +d(j, k) ≥ d(i, j) + d(k, l) = A
Let y = (B – A)/2 ≥ 0. Then the tree should look as follows:We have to find the distances a,b, c and f.
a b
i j
k
m
c
y
l
n
f
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Tree construction for L = 4
a
b
i
j
k
m
c
y
l
n
f
Construct the tree by the given distances as follows:
1. Construct a tree for {i, j, k}, with internal vertex m
2. Add vertex n ,d(m,n) = y
3. Add edge (n, l), c+f = d(k, l)
n
f
n
f
n
fRemains to prove:
d(i,l) = dT(i,l)d(j,l) = dT(j,l)
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Proof for “L = 4”
a
b
i
j
k
m
c
y
l
n
f
By the 4 points condition and the definition of y :d(i,l) = d(i,j) + d(k,l) +2y - d(k,j) = a + y + f = dT(i,l) (the middle equality holds since d(i,j), d(k,l) and d(k,j) are realized by the tree)d(j, l) = dT(j, l) is proved similarly.
B = d(i, k) +d(j, l) = d(i, l) +d(j, k) ≥ d(i, j) + d(k, l) = A,y = (B – A)/2 ≥ 0.
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Induction step for “L > 4” : Remove Object L from the set By induction, there is a tree, T’, for {1, 2, … , L-1}. For each pair of labeled nodes (i, j) in T’, let aij, bij, cij
be defined by the following figure:
aij
bij
cij
i
j
L
mij
1[ ( , ) ( , ) ( , )]
2ijc d i L d j L d i j
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Induction step: Pick i and j that minimize cij.
T is constructed by adding L (and possibly mij) to T’, as in the figure. Then d(i,L) = dT(i,L) and d(j,L) = dT(j,L)
Remains to prove: For each k ≠ i, j : d(k,L) = dT(k,L).
aij
bij
cij
i
j
L
mij
T’
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Induction step (cont.)
Let k ≠ i, j be an arbitrary node in T’, and let n be the branching point of k in the path from i to j.
By the minimality of cij , {{i,j},{k,L}} is NOT a “split” of {i,j,k,L}. So assume WLOG that {{i,L},{j,k}} is a
“split” of {i,j, k,L}.
aij
bij
cij
i
j
L
mij
T’
k
n
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Induction step (end)Since {{i,L},{j,k}} is a split, by the 4 points condition
d(L,k) = d(i,k) + d(L,j) - d(i,j)
d(i,k) = dT(i,k) and d(i,j) = dT(i,j) by induction hypothesis, and
d(L,j) = dT(L,j) by the construction.
Hence d(L,k) = dT(L,k). QED
aij
bij
cij
i
j
L
mij
T’
k
n