Phylogenetic Trees Lecture 12 - Technion
Transcript of Phylogenetic Trees Lecture 12 - Technion
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Phylogenetic TreesLecture 12
Based on pages 160-176 in Durbin et al (the black text book).This class has been edited from Nir Friedman’s lecture which was available at www.cs.huji.ac.il/~nir. Pictures from Tal Pupko slides. Changes by Dan Geiger and Shlomo Moran. 2
EvolutionEvolution of new organisms
is driven byDiversity
Different individuals carry different variants of the same basic blue print
MutationsThe 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
Sour
ce: A
lber
ts e
t al
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Tree of life- a better picture
D’après Ernst Haeckel, 1891
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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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Morphological vs. MolecularClassical phylogenetic analysis: morphological features:number of legs, lengths of legs, etc.
Modern biological methods allow to use molecular featuresGene sequencesProtein sequences
Analysis based on homologous sequences (e.g., globins) in different species
Important for many aspects of biologyClassificationUnderstanding biological mechanisms
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Morphological topology(Based on Mc Kenna and Bell, 1997)
BonoboChimpanzee
ManGorillaSumatran orangutanBornean orangutanCommon gibbon
Barbary 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 hedgehog
HedgehogGymnureMoleArmadilloBandicootWallarooOpossumPlatypus
Archonta
Glires
CarnivoraInsectivoraXenarthra
Ungulata
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From sequences to a phylogenetic tree
Rat QEPGGLVVPPTDA
Rabbit QEPGGMVVPPTDA
Gorilla QEPGGLVVPPTDA
Cat REPGGLVVPPTEG
There are many possible types of sequences to use (e.g. Mitochondrial vs Nuclear proteins).
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Mitochondrial topology(Based on Pupko et al.,)
DonkeyHorseIndian rhinoWhite rhinoGrey sealHarbor sealDogCatBlue whaleFin whaleSperm whaleHippopotamusSheepCowAlpacaPig
Little 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
CarnivoraCetartiodactyla
Rodentia 1
HedgehogsRodentia 2
Primates
ChiropteraMoles+ShrewsAfrotheria
XenarthraLagomorpha+ Scandentia
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Nuclear topology(Based on Pupko et al. slide)(tree by Madsenl)
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
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Cetartiodactyla
Afrotheria
ChiropteraEulipotyphla
Glires
Xenarthra
CarnivoraPerissodactyla
Scandentia+Dermoptera
Pholidota
Primate
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Theory of EvolutionBasic 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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Phylogenenetic trees
Aardvark Bison Chimp Dog Elephant
Leafs - current day speciesNodes - hypothetical most recent common ancestorsEdges length - “time” from one speciation to the next
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Dangers in Molecular PhylogeniesGene and protein sequences can be homologous for various reasons:
Orthologs -- sequences diverged after a speciationevent. Indicative of a new specie.Paralogs -- sequences diverged after a duplicationevent.Xenologs -- sequences diverged after a horizontaltransfer (e.g., by virus).
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Gene PhylogeniesPhylogenies can be constructed to describe evolution genes.
Species Phylogeny
Speciation events
Gene Duplication
1A 2A 3A 3B 2B 1B
Three species termed 1,2,3.Two paralog genes A and B.
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Dangers of ParalogsIf we happen to consider only species 1A, 2B, and 3A, 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.
Speciation events
Gene Duplication
2B 1B3A 3B2A1A
In the sequel we assume all given sequences are orthologs.16
Types of TreesA natural model to consider is that of rooted trees
CommonAncestor
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Types of treesUnrooted tree represents phylogeny without the root node
Depending on the model, data from current day species does not distinguish between different placements of the root.
In this example there are seven possible ways to place a root.18
Rooted versus unrooted treesTree bTree a
ab
c
Represents the three rooted trees
Tree c
Slide by Tal Pupko
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Positioning Roots in Unrooted TreesWe 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
Falcon
Proposed root
Aardvark Bison Chimp Dog Elephant
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Type of Data Distance-based
Input is a matrix of distances between speciesCan be fraction of residue they disagree on, or alignment score between them, or …
Character-basedExamine 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.http://evolution.genetics.washington.edu/phylip.html
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Distance-Based (1st type Method)Input: distance matrix between speciesOutline:
Cluster species togetherInitially clusters are singletonsAt each iteration combine two “closest” clusters to get a new one
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UPGMA ClusteringLet 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 daughter nodes at depth d(Ci,Cj)/2
ääÍ Í
=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.Claim (exercise): A node is never placed below its children.
2 3
98
0.5d(7,8)0.5d(2,3)
4 5 1
6 7
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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 ClockUPGMA constructs trees that satisfy a molecular clock, even if the true tree does not satisfy a molecular clock.
2 3 4 1
1
23
4
UPGMA
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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. 28
AdditivityMolecular clock defines additive distances, namely,
distances between objects can be realized by a tree:
ab
c
i
j
k
cbkjdcakidbajid
+=
+=
+=
),(),(),(
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Basic property of AdditivitySuppose input distances are additiveFor any three leaves
Thus
m
cbmjdcamidbajid
+=+=+=
),(),(),(
a
cb j
k
i
)),(),(),((21),( jidmjdmidmkd -+=
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Constructing additive trees:The neighbor finding problem
Can we use this fact to construct trees assuming only additivity (but not a molecular clock)?Yes. The formula shows that if we knew that i and j are neighboring leaves, then we can construct their parent node k and compute the distances of k to all other leaves m.
We remove nodes i,j and add k.
)),(),(),((21),( jidmjdmidmkd -+=
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Neighbor FindingHow can we find from distances alone that a pair of nodes i,j
are neighboring leaves? Closest nodes aren’t necessarily neighbors.
A B
CD
Next we show one way to find neighbors from additive distances.32
Neighbor Finding
is a leafFor a leaf , let ( , ).i
mi r d i m= ä
Definition: Let , be leaves Then( , ) ( 2) ( , ) ( )
where is the number of leaves ini j
i jD i j L d i j r r
L T= - - +
Theorem (Saitou&Nei) Assume all edge weights are positive. If D(i,j) is minimal (among all pairs of leaves), then i and j areneighboring leaves in the tree.
ij
kl
m
T1T2
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Neighbor Joining AlgorithmSet L to contain all leaves
Iteration:Choose i,j such that D(i,j) is minimalCreate new node k, and set
remove i,j from L, and add kTerminate:
when |L| =2, connect two remaining nodes
1( , ) ( ( , ) ( , ) ( , )) (for some )2
( , ) ( , ) ( , )1for each node , ( , ) ( ( , ) ( , ) ( , ))2
d i k d i j d i m d j m m
d j k d i j d i k
m d k m d i m d j m d i j
= + -
= -
= + -
ij
k
m
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Neighbor FindingNotations used in the proof
p(i,j) = the path from vertex i to vertex j;P(D,C) = (e1,e2,e3) = (D,E,F,C)
For a vertex i, and an edge e=(i’,j’):Ni(e) = |{k : e is on p(i,k)}|.ND(e1) = 3, ND(e2) = 2, ND(e3) = 1NC(e1) = 1
A B
C D
e1e3
e2EF
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Neighbor FindingNotation: For e=(i,m), we denote d(i,m) by d(e).
ij
kl
Rest of T is a leaf
Observe that ( , ) ( ) ( ),i im e E
r d i m d e N eÍ
= =ä ä
[ ]=-=- äÍ ),(
)()()(jipe
jiji eNeNedrr
Lemma: For leaves i,j connected by a path (i,l,…,k,j),
[ ] [ ] [ ])1(1),(1)1(),()()()(),(
--+--+-= äÍ
LjkdLlideNeNedklpe
ji
[ ] [ ]äÍ
-+--=-),(
)()()(),(),()2(klpe
jiji eNeNedjkdlidLrr36
Neighbor FindingProof of Theorem: Assume by contradiction that D(i,j) is minimal for i,j which are not neighboring leaves.Let (i,l,...,k,j) be the path from i to j. Let T1 and T2 be the subtrees rooted at l and k.
Let |T| denote the number of leaves in T.
ij
kl
T1T2
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Neighbor FindingCase 1: i or j has a neighboring leaf. WLOG j and m are such leaves.A. D(i,j) - D(m,j)=(L-2)(d(i,j) - d(j,m) ) – (ri+rj) + rm+ rj {Definition}
=(L-2)(d(i,k)-d(k,m) )+rm-ri {Figure}
i j
kl
mT2
B. rm-ri (L-2)(d(k,m)-d(i,l)) + (4-L)d(k,l) {Lemma+Figure}
(since for each edge eÍP(k,l), Nm(e) 2 and Ni(e) ¢ L-2,
so Nm(e)- Ni(e ) 4-L )Substituting B in A:
D(i,j) - D(m,j) (L-2)(d(i,k)-d(i,l))+ (4-L)d(k,l) = 2d(k,l) > 0,
contradicting the minimality assumption.
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Neighbor FindingCase 2: Not case 1. Then both T1 and T2 contain 2 neighboring leaves.We show that if D(i,j) is minimal, then we must have both |T1| > |T2| and |T2| > |T1| - which is a contradiction, hence D(i,j) is not minimal.
i j
kl
mn
p
T1
T2We prove that |T1| > |T2| by assuming that |T1| |T2| and reaching a contradiction.The proof that |T2| > |T1| is similar.Let n,m be neighboring leaves in T1.
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Neighbor FindingA. 0 D(m,n) - D(i,j)= (L-2)(d(m,n) - d(i,j) ) + (ri+rj) – (rm+rn)
i j
kl
mn
p
T1
T2
C. ri-rn < (L-2)(d(i,k) – d(n,p)) + (|T1|-|T2|)d(l,p)
Adding B and C, noting that d(l,p)>d(k,p) and using the assumption |T1| - |T2| 0:D. (ri+rj) – (rm+rn) < (L-2)(d(i,j)-d(n,m)) +
2(|T1|-|T2|)d(k,p)
Substituting D in the right hand side of A:0 D(m,n) - D(i,j)< 2(|T1|-|T2|)d(k,p),hence |T1|-|T2| > 0, a contradiction.
B. rj-rm< (L-2)(d(j,k) – d(m,p)) + (|T1|-|T2|)d(k,p)(Because Nj(e)- Nm(e ) < |T1|-|T2|).