Phylogenetic Analysis

2

Introduction

• Intension– Using powerful algorithms to reconstruct the

evolutionary history of all know organisms.• Phylogenetic tree

– It can help understand the evolutionary relationships among species of organisms.

– But we have to infer the evolutionary history of current organisms.

Campanulaceae (bluebell) family

Herpesviruses

4

Ancestral Node or ROOT of

the TreeInternal Nodes orDivergence Points

(represent hypothetical ancestors of the taxa)

Branches or Lineages

Terminal Nodes

A

B

C

D

E

Represent theTAXA (genes,populations,species, etc.)used to inferthe phylogeny

Common Phylogenetic Tree Terminology

5

Taxon A

Taxon B

Taxon C

Taxon D

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1

6

3

5

genetic change

Taxon A

Taxon B

Taxon C

Taxon D

time

Taxon A

Taxon B

Taxon C

Taxon D

no meaning

Three types of trees

Cladogram Phylogram Ultrametric tree

All show the same evolutionary relationships, or branching orders, between the taxa.

Phylogenetic trees diagram the evolutionary relationships between the taxa

((A,(B,C)),(D,E)) = The above phylogeny as nested parentheses

Taxon A

Taxon B

Taxon C

Taxon E

Taxon D

No meaning to thespacing between thetaxa, or to the order inwhich they appear fromtop to bottom.

This dimension either can have no scale (for ‘cladograms’),can be proportional to genetic distance or amount of change(for ‘phylograms’ or ‘additive trees’), or can be proportionalto time (for ‘ultrametric trees’ or true evolutionary trees).

These say that B and C are more closely related to each other than either is to A,and that A, B, and C form a clade that is a sister group to the clade composed ofD and E. If the tree has a time scale, then D and E are the most closely related.

7

Completely unresolvedor "star" phylogeny

Partially resolvedphylogeny

Fully resolved,bifurcating phylogeny

A A A

B

B B

C

C

C

E

E

E

D

D D

Polytomy or multifurcation A bifurcation

The goal of phylogeny inference is to resolve the branching orders of lineages in evolutionary trees:

C-B Stewart, NHGRI lecture, 12/5/00

There are three possible unrooted trees for four taxa (A, B, C, D)

A C

B D

Tree 1A B

C D

Tree 2A B

D C

Tree 3

Phylogenetic tree building (or inference) methods are aimed at discovering which of the possible unrooted trees is "correct".We would like this to be the “true” biological tree — that is, one that accurately represents the evolutionary history of the taxa.However, we must settle for discovering the computationally correct or optimal tree for the phylogenetic method of choice.

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The number of unrooted trees increases in a greater than exponential manner with number of taxa

(2N - 5)!! = # unrooted trees for N taxa(2N- 3)!! = # rooted trees for N taxa

CA

B D

A B

C

A D

B E

C

A D

B E

C

F

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Introduction• NP-Hard optimization problem

– Unrooted trees # of n organisms = TU(n)– Edges # of unrooted trees of n organisms = E(n)

= 2n-3 , n>=2– TU(n) = TU(n-1)*E(n-1) = ΠE(i) = Π(2i-5)– Ex.

– Rooted trees # of n organisms = TR(n)= TU(n)*E(n) = TU(n+1)

x y

z

x y

zt

x y

z

t

x y

zt

n-1 n

i=2 i=3

add t

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Inferring evolutionary relationships between the taxa requires rooting the tree:

To root a tree mentally, imagine that the tree is made of string. Grab the string at the root and tug on it until the ends of the string (the taxa) fall opposite the root: A

BC

Root D

A B C D

RootNote that in this rooted tree, taxon A is no more closely related to taxon B than it is to C or D.

Rooted tree

Unrooted tree

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Now, try it again with the root at another position:

A

BC

Root

D

Unrooted tree

Note that in this rooted tree, taxon A is most closely related to taxon B, and together they are equally distantly related to taxa C and D.

C D

Root

Rooted tree

A

B

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An unrooted, four-taxon tree theoretically can be rooted in five different places to produce five different rooted trees

The unrooted tree 1:

A C

B D

Rooted tree 1d

C

D

A

B

4

Rooted tree 1c

A

B

C

D

3

Rooted tree 1e

D

C

A

B

5

Rooted tree 1b

A

B

C

D

2

Rooted tree 1a

B

A

C

D

1

These trees show five different evolutionary relationships among the taxa!

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All of these rearrangements show the same evolutionary relationships between the taxa

B

A

C

D

A

B

D

C

B

C

AD

B

D

AC

B

ACD

Rooted tree 1aB

A

C

D

A

B

C

D

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Molecular phylogenetic tree building methods:Are mathematical and/or statistical methods for inferring the divergence order of taxa, as well as the lengths of the branches that connect them. There are many phylogenetic methods available today, each having strengths and weaknesses. Most can be classified as follows:

COMPUTATIONAL METHODClustering algorithmOptimality criterion

DA

TA T

YPE

Cha

ract

ers

Dis

tanc

es

PARSIMONY

MAXIMUM LIKELIHOOD

UPGMA

NEIGHBOR-JOINING

MINIMUM EVOLUTION

LEAST SQUARES

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parsimony

• model complexity vs. sample size• minimize Hamming distance summed over

all edges of the tree• justification: minimum possible number of

evolutionary events• subject of serious dispute by systematic

biologists

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Method– Maximum parsimony (MP)

• Seek the tree that minimizes the total number of evolutionary events on the edges of tree

• Ex.

• Require two algorithms– Search over tree topology– The computation of a cost for a given tree

1 1

1AAA

AAG

AAA

GGA

AGA

AGA

AAA

AAA

AAG

AAA

AAA

AAA

GGAAGA

1 1 2

AAA

AAG

AAA

AAA

AAA

AGAGGA

1 12

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maximum likelihood

• estimate probability that a specific evolutionary model will produce a particular phylogeny yielding the observed sequences

• many evolutionary models

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Method– Maximum likelihood (ML)

• Seek the tree that maximizes likelihood P(data|tree)• Ex.

– Compute likelihoodP(x1,x2,x3|T,t1,t2,t3,t4)

– x•: a set of sequences– T: a tree– t•: edge lengths of tree

• Require two algorithms– Search over tree topology– Search over all possible lengths of edges t• to compute likeliho

od

X1X2

X5

X4

X3

root

t1t2 t3

t4

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Distance Matrix Methods

• produce a tree such that the path distance between leaves i and j (sum of edge weights in the path between i and j) equals Dij

• this the additive property for a distance matrix -- of course real distance matrices may not be additive

• most methods use agglomerative clustering -- successively choosing pairs of nodes to combine

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Ultrametric trees

• path distance from the root to each leaf is the same

• strong molecular clock assumption - distance is proportional to evolutionary time

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Example Tree and Additive Matrix

a

e

c

b

d 2

3 3

5

3

2 1

1

A B C D EA 0 10 12 10 7B 0 4 4 13C 0 6 15D 0 13E 0

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Distance Matrix Methods

• UPGMA• Neighbor Joining• Fitch Margoliash• Quartet Puzzling• Witness-Anitwitness• Double Pivot

many are “not yet in use by the systematic biology community”

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Distance Measures

• DNA hybridization amounts• immunological distances• genetic distances • sequence distances

(DNA, RNA, protein)

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…what distance?

• need distance measure that reflects the actual number of point mutations on the path between the leaves

• particular problem with sequence data - Hamming distance and assumption of no reversals

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UPGMA

• Unweighted Pair-Group Method with Arithmetic mean

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UPGMA Step 1combine B and C

a

e c b

d

A B C D EA 0 10 12 10 7B 0 4 4 13C 0 6 15D 0 13E 0

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UPGMA step 2combine BC and D

a

e c b

d

2 2

A BC D EA 0 11 10 7BC 0 5 14D 0 13E 0

(10+12)/2

(4+6)/2

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UPGMA step 3combine A and E

A BCD EA 0 10.5 7BCD 0 13.5E 0

a

e c b

d

2

2.5 0.5

2

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UPGMA step 4combine AE and BCD

a

e

c b

d 3.5

3.5

2

2.5 .5

2

AE BCDAE 0 12BCD 0

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UPGMA Result

a

e

c b

d 3.5

3.5

2

2.5 .5

2

2.5

1.5 A B C D E

A 0 10 12 10 7B 0 4 4 13C 0 6 15D 0 13E 0

3.5

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UPGMA Result

a

e

c b

d 3.5

3.5

2

2.5 .5

2

2.5

1.5

a

e

c

b

d 2

3 3

5

3

2 1

1

3.5

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Method

• Phylogenetic reconstruction techniques– NJ (neighbor-joining method)

• A star tree is successively inserted branches between a pair of closest neighbors and the remaining terminals in the tree

• Character– The fastest reconstruction method– Poor accuracy when the distance matrix contains

large value

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Method• Ex.

– The cost save by pairing S1 and S2 = New connection cost (NC) – Old connection cost (OC) = 2.34 NC = ½(average(S1)+average(S2)+d(S1,S2))=6.33 OC = average(S1) +average(S2) = 8.67

– The largest cost save by pairing S3 and S4 = 2.67Thus we pair S3 and S4

S1 S2 S3 S4S1 0 4 4 3S2 0 6 5S3 0 2S4 0Distance matrix Star tree

S2

S1 S3

S45

3.67

3.33

4

X

S2

S1 S3

S4X

XS2

S1

Pair S1 and S2

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Neighbor-Joining Result

a

e

c b

d

2 5

3

1

6

1.5

2

a

e

c

b

d 2

3 3

5

3

2 1

1

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Genome Rearragement– Generalized Nadean-Tayor (GNT) evolution model

• P(transpostion) = α• P(inverted trans.) = β• P(inversion) = 1-(α+β)• events # on edge :

according to Poissondistributionf(x) = ; x=1,2,..

Genome rearrangement

λx•e-3 x!

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Improving reconstruction algorithms

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Improving reconstruction algorithms– Estimators of true evolutionary distance

• Exact-IEBP (inverting the expected breakpoint distance)ML estimate of the breakpoint distance after K rearrangements

• Approx-IEBPapproximate Exact-IEBP

• EDE (empirically derived estimator)empirical estimate of the inversion distance after K rearrangements

produced a nonlinear regression formula that computes the expected distance given that K random rearrangements

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Conclusion

• New generation of phylogenetic software needs– More sophisticated models of evolution– Faster optimization algorithms– High performance algorithm engineering– Powerful modes of user interaction