.

בתכנות 236503 פרויקט

: אבולוציה עצי בשחזור השוואתי מחקר

תכנות מול קיימים אלגוריתמים

בשלמים2013אביב

מורן: שלמה מרצה : שילוח יוסי חיצוני מנחה

Website: http://webcourse.cs.technion.ac.il/236503/

2

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

Phylogenetic Reconstrutction

Problem

MPI, June 2012

4

AATCCTG

ATAGCTGAATGGGC

GAACGTA

AAACCGA

ACGGTCA

ACGGATA

ACGGGTA

ACCCGTG

ACCGTTG

TCTGGTA

TCTGGGA

TCCGGAA AGCCGTG

GGGGATT

AAAGTCA

AAAGGCG AAACACAAAAGCTG

Evolution is modeled by a Tree

(Species represented by their DNA sequences, consisting of {A,G,C,T})

MPI, June 2012

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AATCCTG

ATAGCTGAATGGGC

GAACGTA

AAACCGAACCGTTGTCTGGGA

TCCGGAA AGCCGTG

GGGGATT

Phylogenetic Reconstruction

MPI, June 2012

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B : AATCCTG

C : ATAGCTG

A : AATGGGC

D : GAACGTAE : AAACCGA

J : ACCGTTG

G : TCTGGGAH : TCCGGAA

I : AGCCGTG

F : GGGGATT

Goal: reconstruct the ‘true’ tree as accurately as possibleDistance Methods: use “evolutionary distances” between sequences

reconstruct

AB

C

FG

IH J

D

E

A

B

C F

G

I

H

J

D

E

(root)

Phylogenetic Reconstruction

MPI, June 2012

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Reconstructing weighted treeFrom exact interleaf distances

,

( , )u v S

D d u v

Exact (additive) distances

Between leaves

Reconstruction

(linear-tim

e)

Algorithm

MPI, June 2012

A

C

B

D

F

G

E

edge-weighted unknown tree

5

6

0.4

6

3 0.32 2

4

5

A

C

B

D

F

G

E

Reconstructed tree

5

6

0.4

6

3 0.32 2

4

5

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Formal statement of the problemfor exact distances

Input: an n×n distance matrix D=(d(i,j)): 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).

Output: If the distances can be realized by a weighted tree (i.e., the distances are additive) – return that tree.

Else – return nothing.

4 5

7 21

210 61

Distance based reconstruction methods:)since the 60’s(:

÷÷÷÷÷÷÷÷

ø

ö

çççççççç

è

æ

= ˆˆ ( , ) ( , )D u v d u vD̂

MPI, June 2012

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Solution for 3 objects

For n=3: Each distance metric can be realized by 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

v

i j k

i 0 a+b a+c

j 0 b+c

k 0

Distance metrics on 4 objects may not have a tree.

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The Four Points Condition

Definition: A distance metric on n objects satisfies the four points condition 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)

ik

lj

Theorem: A distance metric is additive iff it satisfies the four points condition

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Neighbor Joining Let i, j be neighboring leaves in a tree, let v be their parent, and let k

be any other leaf.

The formula

shows that we can compute the distances of v to all other leaves.

1

2( , ) [ ( , ) ( , ) ( , )]d k v d k i d k j d i j

d(k,v)

i

j

k

v

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Reconstructing trees byNeighbor Joining Algorithms

This suggest the following method to construct tree from a distance

matrix:

1. Find neighboring leaves i,j in the tree,

2. Replace i,j by their parent v and recursively construct a tree T

for the smaller set.

3. Add i,j as children of v in T.

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Neighbor Finding: Seitou&Nei method

Theorem (Saitou&Nei) Assume all internal edge weights are positive. If Q(i,j) is minimal (among all pairs of leaves), then i and j are neighboring leaves in the tree.

is a leaf

For a leaf , let

For leaves

2

( , ).

, :

( , ) ( ) ( , ) ( )

i

u

i j

i r d i u

i j

Q i j n d i j r r

Definitions

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S&N Neighbor Joining Algorithm If n =3, return tree of three vertices Compute Q(i,j) for all i,j Choose i,j such that Q(i,j) is minimal Create new vertex v, and set

ij

v

k

1 (for some

2 // or could be 0

1for each vertex ,

2

( , ) [ ( , ) ( , ) ( , )] )

( , ) ( , ) ( , ) ( , ) ( , )

( , ) [ ( , ) ( , ) ( , )]

d i v d i j d i r d j r r

d j v d i j d i v d i v d j v

k d v k d i k d j k d i j

remove i,j, and add v to the set of objectsRecursively construct a tree on the smaller set, then add i,j as children of v, at distances d(i,v) and d(j,v).

d(k,v)

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Initialization: θ(n2) to compute r(i) and Q(i,j) for all i,jL.

Each Iteration: O(n2) to find the maximal Q(i,j). O(n) to compute {D(v,k):k L} for the new node v,

and to update the matrix. O(n2) to update the values Q(i,j).

Total of O(n3).

Complexity of S&N Neighbor Joining Algorithm

ij

k

D(v,k)

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NEEDED:

Additive Distances

Between DNA Sequences

MPI, June 2012

Additive Evolutionary distance :The number of substitutions which occurred

during the sequence evolution

AC

CC

C G T A1 2 3

1

site 1

site 2

substitutions

Some substitutions are hidden, due to overwriting.Therefore, the exact number of subst. is usually larger than the number of observed changes.

site 30

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Edge weight = Expected number of substit’s per site

A A C A … G T C T T C G A G G C C Cu

v A G C A … G C C T A T G C G A C C T

MPI, June 2012

0 1 0 0 … 0 2 0 0 1 1 0 1 2 1 0 0 1

0.321 Number of substitutions per site

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When the exact number of substitutions between any two

sequences is known,

any algorithm which reconstructs trees from the exact

distances returns the correct evolutionary tree

Interleaf distances: sum of edge weights

vu0.5

0.42

0.3

d(u,v) = 1.12

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The expected number of substitutions

is estimated from the

observed number of substitutions

What we see is only the observed number of substitutions between pairs of leaf sequences.

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The estimation is based onSubstitution Model

The simplest model: Juke Cantor Model On each tree edge e, each letter is mutated to any other later by the same ratio re. The length of an edge is the expected number of mutations per site, i.e. t=3r

u

v

t

MPI, June 2012

T C G A

r r r - A

r r - r G

r - r r C

- r r r T

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The expected number of substitutions is estimated from

the observed changes by a correction formula

u A A C A … G T C T T C G A G G C C C

v A G C A … G C C T A T G C G A C C T

ˆ ˆ( , )d u v t

MPI, June 2012

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A

C

B

D

F

G

E

edge-weighted ‘true’ tree reconstructed tree

reconstruction

B

C

A

D

F

G

E

,

ˆˆ ( , )u v S

D d u v

5

6

0.4

6

3 0.32 2

4

5

Reconstruction from estimated distances:

Estimated distances

,

( , )u v S

D d u v

Exact (additive) distances

Between species

Distance estimationAssuming DNA

substitution model

MPI, June 2012

Challenge: minimizeReconstruction errors

A

C

B

D

F

G

E

edge-weighted ‘true’ tree T

5

6

0.4

6

3 0.32 2

4

5

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reconstructed tree T’

B

C

A

D

F

G

E

Correct and incorrect reconstruction of edges

MPI, June 2012

Each (internal) edge defines a split of the leaves:The edge {ABC | DEFG} is correctly reconstructedThe edge {ABCD | EFG} is false negativeThe edge {AC | BDEFG} is false positive.

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Robinson Foulds Distance

MPI, June 2012

False positives + false negativesTotal number of internal edges

Robison Foulds distance =

A

C

B

D

F

G

E

edge-weighted ‘true’ tree T

5

6

0.4

6

3 0.32 2

4

5

reconstructed tree T’

B

C

A

D

F

G

E

=

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Formal statement of the problemfor estimated distances

Input: an n×n distance matrix, which are estimations of tree (additive) distances.

Output: return a tree with small Robinson Foulds distance from the true tree.

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Project’s Goal Practice current algorithm (NJ) of phylogenetic reconstruction by

distance methods. Simulate evolutions of DNA sequences, and generate evolutionary

distances. Study a new method for tree reconstruction, based on mixed

integer programming with CPLEX. Compare the accuracy of this new method with that of Neighbor

Joining. You should use the PHYLIP phylogenetic package for most of the required tasks:

http://evolution.genetics.washington.edu/phylip.html

Time Line

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Mathematical programming

Algorithmicweek

construct distance matrix of a given tree )10-20 leaves(. Generate noisy distance matrices from exact ones.

1

general introduction to Mathematical programming

2

3

4

5

6789

1011121314

prepare final report and presentation.

Reconstruct trees from exact distances using the NJ algorithm. Repeat this for Noisy distances, and compute the RF )Robinson-Foulds( distance of the true tree from the reconstructed trees.

Study the reconstruction model, and use it to reconstruct trees from accurate distances

Construct evolutionary-distance matrix by simulating evolution using the Jukes Cantor Model, and then estimating the distances between the sequences using DNADIST

Reconstruct trees from noisy distances

Reconstruct trees from evolutionary distances using both reconstruction methods. Adjust the Integer Programming model to improve accuracy.

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Grading Scheme

10% - work plan 60% - final report + submitted code

Rough distribution of grade: 40% - meeting project requirements 10% - code organization and documentation 10% - innovation and creativeness

30% - final presentation