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SIS Phylogenetic Reconstruction:

Distance Matrix Methods

Anders Gorm PedersenAnders Gorm Pedersen

Molecular Evolution GroupMolecular Evolution Group

Center for Biological Sequence AnalysisCenter for Biological Sequence Analysis

Technical University of DenmarkTechnical University of Denmark

gorm@cbs.dtu.dkgorm@cbs.dtu.dk

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Trees: terminology

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Trees: terminology

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Trees: representations

Three different representations of the same tree

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Trees: rooted vs. unrooted

• A rooted tree has a single node (the root) that represents a point in time that is earlier than any other node in the tree.

• A rooted tree has directionality (nodes can be ordered in terms of “earlier” or “later”).

• In the rooted tree, distance between two nodes is represented along the time-axis only (the second axis just helps spread out the leafs)

Early Late

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Trees: rooted vs. unrooted

• A rooted tree has a single node (the root) that represents a point in time that is earlier than any other node in the tree.

• A rooted tree has directionality (nodes can be ordered in terms of “earlier” or “later”).

• In the rooted tree, distance between two nodes is represented along the time-axis only (the second axis just helps spread out the leafs)

Early Late

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Trees: rooted vs. unrooted

• A rooted tree has a single node (the root) that represents a point in time that is earlier than any other node in the tree.

• A rooted tree has directionality (nodes can be ordered in terms of “earlier” or “later”).

• In the rooted tree, distance between two nodes is represented along the time-axis only (the second axis just helps spread out the leafs)

Early Late

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Trees: rooted vs. unrooted

• In unrooted trees there is no directionality: we do not know In unrooted trees there is no directionality: we do not know if a node is earlier or later than another nodeif a node is earlier or later than another node

• Distance along branches directly represents node distanceDistance along branches directly represents node distance

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Trees: rooted vs. unrooted

• In unrooted trees there is no directionality: we do not know In unrooted trees there is no directionality: we do not know if a node is earlier or later than another nodeif a node is earlier or later than another node

• Distance along branches directly represents node distanceDistance along branches directly represents node distance

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Reconstructing a tree using non-contemporaneous data

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Reconstructing a tree using present-day data

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Data: molecular phylogeny

• DNA sequencesDNA sequences– genomic DNAgenomic DNA– mitochondrial DNAmitochondrial DNA– chloroplast DNAchloroplast DNA

• Protein sequencesProtein sequences

• Restriction site polymorphismsRestriction site polymorphisms

• DNA/DNA hybridizationDNA/DNA hybridization

• Immunological cross-reactionImmunological cross-reaction

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Morphology vs. molecular data

African white-backed vulture(old world vulture)

Andean condor (new world vulture)

New and old world vultures seem to be closely related based on morphology.

Molecular data indicates that old world vultures are related to birds of prey (falcons, hawks, etc.) while new world vultures are more closely related to storks

Similar features presumably the result of convergent evolution

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Molecular data: single-celled organisms

Molecular data useful for analyzing single-celled organisms (which have only few prominent morphological features).

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Distance Matrix Methods

1.1. Construct multiple Construct multiple alignment of sequencesalignment of sequences

2.2. Construct table listing all Construct table listing all pairwise differences pairwise differences (distance matrix)(distance matrix)

3.3. Construct tree from Construct tree from pairwise distancespairwise distances

Gorilla : ACGTCGTAHuman : ACGTTCCTChimpanzee: ACGTTTCG

GoGo HuHu ChCh

GoGo -- 44 44

HuHu -- 22

ChCh --

Go

Hu

Ch

2

11

1

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Finding Optimal Branch Lengths

SS11 SS22 SS33 SS44

SS11 -- DD1212 DD1313 DD1414

SS22 -- DD2323 DD2424

SS33 -- DD3434

SS44 --Observed distance

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S2

S4

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d e

Distance along tree

D12 d12 = a + b + cD13 d13 = a + dD14 d14 = a + b + eD23 d23 = d + b + cD24 d24 = c + eD34 d34 = d + b + e

Goal:

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Optimal Branch Lengths: Least Squares

• Fit between given tree Fit between given tree and observed distances and observed distances can be expressed as “sum can be expressed as “sum of squared differences”:of squared differences”:

Q = Q = (D(Dijij - d - dijij))22

• Find branch lengths that Find branch lengths that minimize Q - this is the minimize Q - this is the optimal set of branch optimal set of branch lengths for this tree.lengths for this tree.

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S3

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S4

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Distance along tree

D12 d12 = a + b + cD13 d13 = a + dD14 d14 = a + b + eD23 d23 = d + b + cD24 d24 = c + eD34 d34 = d + b + e

Goal:

j>i

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Least Squares Optimality Criterion

• Search through all (or many) tree topologiesSearch through all (or many) tree topologies

• For each investigated tree, find best branch For each investigated tree, find best branch lengths using least squares criterionlengths using least squares criterion

• Among all investigated trees, the best tree is the Among all investigated trees, the best tree is the one with the smallest sum of squared errors. one with the smallest sum of squared errors.

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Exhaustive search impossible for large data sets

No. No. taxataxa

No. treesNo. trees

33 11

44 33

55 1515

66 105105

77 945945

88 10,39510,395

99 135,135135,135

1010 2,027,0252,027,025

1111 34,459,42534,459,425

1212 654,729,075654,729,075

1313 13,749,310,57513,749,310,575

1414 316,234,143,225316,234,143,225

1515 7,905,853,580,6257,905,853,580,625

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Heuristic search

1.1. Construct initial tree; determine sum of squaresConstruct initial tree; determine sum of squares

2.2. Construct set of “neighboring trees” by making small Construct set of “neighboring trees” by making small rearrangements of initial tree; determine sum of squares for rearrangements of initial tree; determine sum of squares for each neighboreach neighbor

3.3. If any of the neighboring trees are better than the initial tree, If any of the neighboring trees are better than the initial tree, then select it/them and use as starting point for new round of then select it/them and use as starting point for new round of rearrangements. (Possibly several neighbors are equally good)rearrangements. (Possibly several neighbors are equally good)

4.4. Repeat steps 2+3 until you have found a tree that is better Repeat steps 2+3 until you have found a tree that is better than all of its neighbors.than all of its neighbors.

5.5. This tree is a “local optimum” (not necessarily a global This tree is a “local optimum” (not necessarily a global optimum!) optimum!)

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Clustering Algorithms

• Starting point: Distance matrixStarting point: Distance matrix

• Cluster least different pair of sequences:Cluster least different pair of sequences:– Tree: pair connected to common ancestral node, compute branch Tree: pair connected to common ancestral node, compute branch

lengths from ancestral node to both descendantslengths from ancestral node to both descendants

– Distance matrix: combine two entries into one. Compute new Distance matrix: combine two entries into one. Compute new

distance matrix, by finding distance from new node to all other distance matrix, by finding distance from new node to all other nodesnodes

• Repeat until all nodes are linkedRepeat until all nodes are linked

• Results in only one tree, there is no measure of tree-Results in only one tree, there is no measure of tree-goodness.goodness.

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Neighbor Joining Algorithm

• For each tip compute For each tip compute uuii = = jj DDijij/(n-2)/(n-2)

(this is essentially the average distance to all other tips, except the (this is essentially the average distance to all other tips, except the denominator is n-2 instead of n)denominator is n-2 instead of n)

• Find the pair of tips, i and j, where Find the pair of tips, i and j, where DDijij-u-uii-u-ujj is smallest is smallest

• Connect the tips i and j, forming a new ancestral node. The branch Connect the tips i and j, forming a new ancestral node. The branch lengths from the ancestral node to i and j are:lengths from the ancestral node to i and j are:

vvii = 0.5 D = 0.5 Dijij + 0.5 (u + 0.5 (uii-u-ujj))

vvjj = 0.5 D = 0.5 Dijij + 0.5 (u + 0.5 (ujj-u-uii))

• Update the distance matrix: Compute distance between new node and Update the distance matrix: Compute distance between new node and each remaining tip as follows:each remaining tip as follows:

DDij,kij,k = (D = (Dikik+D+Djkjk-D-Dijij)/2)/2

• Replace tips i and j by the new node which is now treated as a tipReplace tips i and j by the new node which is now treated as a tip

• Repeat until only two nodes remain.Repeat until only two nodes remain.

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Superimposed Substitutions

• Actual number of Actual number of

evolutionary events:evolutionary events: 55

• Observed number of Observed number of

differences:differences: 22

• Distance is (almost) always Distance is (almost) always underestimatedunderestimated

ACGGTGC

C T

GCGGTGA

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Model-based correction for superimposed substitutions

• Goal: try to infer the real number of evolutionary Goal: try to infer the real number of evolutionary events (the real distance) based onevents (the real distance) based on

1.1. Observed data (sequence alignment)Observed data (sequence alignment)

2.2. A model of how evolution occursA model of how evolution occurs

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Jukes and Cantor Model

• Four nucleotides assumed to Four nucleotides assumed to be equally frequent (f=0.25)be equally frequent (f=0.25)

• All 12 substitution rates All 12 substitution rates assumed to be equalassumed to be equal

• Under this model the Under this model the corrected distance is:corrected distance is:

DDJCJC = -0.75 x ln(1-1.33 x = -0.75 x ln(1-1.33 x DDOBSOBS))

• For instance:For instance:

DDOBSOBS=0.43 => =0.43 => DDJCJC=0.64=0.64

A C G T

A -3

C -3

G -3

T -3

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Other models of evolution