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512236פרויקט בתכנות מתקדם –

פונקציות מרחק אופטימליות לשיחזור עצי

אבולוציה

2010סמסטר אביב http://webcourse.cs.technion.ac.il/236512/

דואר אלקטרוני

חדר טלפון  

moran@cs 4363 639 טאוב   שלמה מורן 

ddoerr@cs 4319 534טאוב דניאל דור

.

ההשפעה של פונקציות מרחק על שיחזור עצי אבולוציה

לאחר שלב ההודעות, נעביר היום קורס בזק מקוצר על:DNAעצי אבולוציה: הגדרות ומודלים מבוססי 1.שיטות מבוססות מרחקים לבניית עצי אבולוציה2.פונקציות מרחק למודלים אבולוציוניים3.הערכת מרחקים בין זנים על סמך השוני בין סדרות 4.

DNAה

לאחר ה"קורס המזורז" עדיין תזדקקו להשלמות מסוימות בהמשך הסמסטר.

במהלך "קורס הבזק" יוצגו הפרויקטים.

נושא הפרויקט

.

, הסתברות1אלגוריתמים : דרישות קדם אלגוריתמים בביולוגיה רצוי )אך לא הכרחי(:

חישובית

.ככלל, הפרויקטים יעשו בזוגותתוך שבוע הודיעונו על החלוקה לזוגות )בדוא"ל(

.בחירת פרוייקט: יהיו שני כיוונים עיקריים .השלב הראשוני דומה בשני הכיוונים

התמקדות בכיוון מסוים תעשה בהמשך )תוך .כחודש(

)מכאן והלאה שקפים באנגלית(

אדמיניסטרציה

4

Crash course on evolutionary distances

5

The

Phylogenetic Reconstrutction

Problem

6

AATCCTG

ATAGCTGAATGGGC

GAACGTA

AAACCGA

ACGGTCA

ACGGATA

ACGGGTA

ACCCGTG

ACCGTTG

TCTGGTA

TCTGGGA

TCCGGAA AGCCGTG

GGGGATT

AAAGTCA

AAAGGCG AAACACAAAAGCTG

Evolution is modeled by DNA sequences which evolve along an Evolution Tree (Phylogeny)

)All our sequences are DNA sequences, consisting of {A,G,C,T}(

7

AATCCTG

ATAGCTGAATGGGC

GAACGTA

AAACCGAACCGTTGTCTGGGA

TCCGGAA AGCCGTG

GGGGATT

Phylogenetic Reconstruction

8

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 possible

reconstruct

AB

C

FG

IH J

D

E

A

B

C F

G

I

H

J

D

E

(root)

Phylogenetic Reconstruction

9

Three Methods of Tree Construction

Parsimony – A tree with minimum number of mutations.

Maximum likelihood - Finding the “most probable” tree.

Distance- A weighted tree that realizes the distances between the species.

10

A

C

B

D

F

G

E

edge-weighted ‘true’ tree

,

) , (T T i j SD d i j

reconstructed tree

reconstruction

B

C

A

D

F

G

E

,

ˆˆ ) , (i j S

D d i j

noise α

ˆ) , ( ) , ( ) , (Td i j d i j i j

5

6

0.4

6

3 0.32 2

4

5

Major problem: sensitivity to noise

reconstruction

in O(n2)

Distance Based Reconstruction:Exact vs. approximate distances

Exactdistances

11

A

C

B

D

F

G

E

edge-weighted ‘true’ tree

,

) , (T T i j SD d i j

1

5

6

0.4

6

3 0.32 2

4

5

reconstruction

in O(n2)

The Algorithmic Aspect

Exactdistances

Many algorithms can reconstruct a weighted tree from the exact distances.In this project we will use the “Saitou&Nei Neighbor Joining algorithm”, or simply the “NJ algorithm”.

12

Evolutionary Distances:- How are they defined?- How are they extracted from the DNA sequences?

We’ll show this on a specific model the Kimura 2 Parameters (K2P) model

The Distance Estimation Aspect

,

) , (T T i j SD d i j

,

ˆˆ ) , (i j S

D d i j

noise α

ˆ) , ( ) , ( ) , (Td i j d i j i j

13

The Kimura 2 Parameter )K2P( model [Kimura80]:each edge corresponds to a “Rate Matrix”

{ }A G

{ }C T

Transitions

Transversions

Transitions

Transitions/Transversions ratio = / 2 1R

-αββT

α-ββC

ββ-αG

ββα-A

TCGA

-αββT

α-ββC

ββ-αG

ββα-A

TCGA

K2P generic rate matrixu

v

14

K2P standard distance: Δtotal = Total substitution rate

u v w

The total substitution rate of a K2P rate matrix R is

This is the expected number of mutations per site. It is an additive distance.

+

) ( 2 : ~ total expected number of mutationstotal uvR

α + 2β α’ + 2β’

)α+α’( + 2)β+ β’(

15

The distance Δtotal(Ruv) = dK2P(u,v) is estimated from the aligned sequences

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

2ˆ ˆˆ) , ( 2K Pd u v

K2P total rate“distance correction”

procedure

since mutations may overwrite each other,

this is a “noisy” process

A basic question:How good is a reconstruction method which uses K2P distances?

A C

B D

wsep

The performance of tree reconstructions method is often tested on quartets, which are trees with 4 taxa.A quartet contains a single internal edge, which defines the quartet-split.

17

A correct reconstruction of the quartet requires finding of the true quartet-split

A C

B D

A B

C D

A C

D B

Distance methods reconstruct the true split by the

4-point condition:

There are 3 possible splits:

wsep

The 4-point condition for noisy distances is:

2 2 2 2 2 2) , ( ) , ( min ) , ( ) , ( , ) , ( ) , (K P K P K P K P K P K Pd d d d d d A B C D A C B D A D B C

2 2 2 2 2 2) , ( ) , ( ) , ( ) , ( ) , ( ) , (2K P K P K P K P K Pse K Ppd d dwd d d A B C D A C B D A D B C

18

We evaluate the accuracy of the K2P distance estimation

by Split Resolution Test:

root

D

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

t

10t

CA

B

10t 10t10t

t-αββC

α-ββT

ββ-αG

ββα-A

CTGA

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

-αββC

α-ββT

ββ-αG

ββα-A

CTGA

t is “evolutionary time”

The diameter of the quartet is 22t

19

Phase A: simulate evolution

DC

AB

CCCGGAGCTTCTG…ACAA CCCGGAGCTTCTG…ACAA

CCCGGAGCTTCTG…ACAA CCCGGAGCTTCTG…ACAA

CCCGGAGCTTCTG…ACAA CCCGGAGCTTCTG…ACAA

CCCGGAGCTTCTG…ACAA CCCGGAGCTTCTG…ACAA

CCCGGAGCTTCTG…ACAA CCCGGAGCTTCTG…ACAA

CCCGGAGCTTCTG…ACAA CCCGGAGCTTCTG…ACAACCCGGAGCTTCTG…ACAA CCCGGAGCTTCTG…ACAA

20

Phase B: reconstruct the split by the 4p condition

DCCCGGAGCTTCTG…ACAA CCCGGAGCTTCTG…ACAA

CCCCGGAGCTTCTG…ACAA CCCGGAGCTTCTG…ACAA

BCCCGGAGCTTCTG…ACAA CCCGGAGCTTCTG…ACAA

ACCCGGAGCTTCTG…ACAA CCCGGAGCTTCTG…ACAA

÷÷÷÷÷÷÷÷

øçççççççç

è

2ˆˆ ) , ( ) , (K P i jD i j d s s

Apply the 4p condition.Was the correct split found?

compute distances between sequences,

Repeat this process 10,000 times,

count number of failures

21

root

D

t

10t

CA

B

10t 10t 10t

t

root

D

t

10t

CA

B

10t 10t 10t

t

the split resolution test was applied on the model quartet with various diameters

For each diameter, mark the fraction )percentage( of the simulations

in which the 4p condition failed )next slide(

root

D

t

10t

CA

B

10t 10t 10t

t

root

D

t

10t

CA

B

10t 10t 10t

t

root

D

t

10t

CA

B

10t 10t 10t

t

root

D

t

10t

CA

B

10t 10t 10t

t

root

D

t

10t

C

AB

10t 10t 10t

t

root

D

t

10t

C

AB

10t 10t 10t

t … …

22

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

quartet diameter )total rate between furthest leaves(

Fra

ctio

n of

failu

res

out o

f 100

00 e

xper

imen

tsperformance of K2P standard distance method in resolving quartets, R=10

Performance of K2P distances in resolving quartets, small diameters: 0.01-0.2

root

D

t

10t

CA

B

10t 10t 10t

t

root

D

t

10t

CA

B

10t 10t 10t

t

Templatequartet

23

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

quartet diameter (=mutations rate between furthest leaves)

Fract

ion

of fa

ilure

s out

of 10

000 si

mul

atio

nsperformance of K2P standard distance method in resolving quartets,

For quartet ratio 0.1, R=10

Performance for larger diameters

“site saturation”

24

{ }A G

{ }C T

Transitions

Transversions

Transitions

When β < α, we can postpone the “site saturation” effect. For this, use another distance function for the same model, Δtv , which counts only transversions:

{0}

{1}

This is the CFN model

[Cavendar78, Farris73, Neymann71]

α

α

β

25

Apply the same split resolution test on the transversions only distance:

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) , (trd u v

Transversions onlyDistance correction

procedure

26

transversions only performs better on large, worse on small rates

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

quartet diameter

Fract

ion

of Fa

ilure

s out o

f 10

000

exper

imen

ts

performance of distance methods in resolving quartets, R=10

Transversions only

total K2P rate

.

4 5

7 21

210 61

Conclusion: Distance based reconstruction methods should be

adaptive:

Find a distance function d which is good for the input ÷

÷÷÷÷÷÷÷

ø

ö

çççççççç

è

æ

= ˆˆ ) , ( ) , (D u v d u vD

Projects goal: Evaluate the performance of distance

functions in reconstructing phylogenies

28

1st step in finding good distance functions )for the K2P model(:

Characterize the available distance functions.

Ideally, we would like to use the K2P distance associated

with the rate matrix of each edge, but...

29

Rate matrices are hard to observe, hence we use Substitution matrices

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

-αββT

α-ββC

ββ-αG

ββα-A

TCGA

-αββT

α-ββC

ββ-αG

ββα-A

TCGA

C

T

G

A

CTGA

C

T

G

A

CTGA

p

p

p

p

p

p p

p

p

p

p p1 2 p p

1 2 p p

1 2 p p

1 2 p p

uvP uvR

Evolution of a finite sequence by unknown model parameters α, β

A stochastic substitution matrix Puv

30

Subtitution matrices are extended to paths:

uvP

vwP

u

v

w

uw uv vwP P P

31

Substitution matrices are converted to distances

by a Substitution Rate function

uvP

vwP

u

v

w

SR function need to satisfy the following for all

substitution matrices P,Q in K2P:

1. Δ)PQ( = Δ)P(+ Δ)Q( )additivity(

2. Δ)P(>0 )positivity(

32

To define SR functions which are additive:

Δ)PQ( = Δ)P(+ Δ)Q(

We use some linear algebra

33

Lemma: There is a matrix U which diagonalizes each K2P Substitution Matrix P:

1 0 0 0

0 λP 0 0

0 0 μP 0

0 0 0 μP

Where:

U-1 PU =

C

T

G

A

CTGA

C

T

G

A

CTGA

p

p

p

p

p

p p

p

p

p

p p1 2 p p

1 2 p p

1 2 p p

1 2 p p

P =

0 < λP <10 < μP < 1

4

2 2

1 4

1 2 2

P

P

p e

p p e

34

μP000

0μP00

00λP0

0001

U-1 P U =

μQ000

0μQ00

00λQ0

0001

U-1 Q U =

U-1 PQ U =

Let P,Q be two matrices in K2P. Then:

μP μQ

000

0μP μQ00

00λP λQ0

0001

U-1 PQ U =

35

Proof: Dλ (PQ) = -ln)λPλQ( = -ln)λP( -ln)λQ( = Dλ (P)+ Dλ (Q)

And the same for Dμ (P (= -ln)μP(

Hence, the functions:Dλ)P(= -ln)λP( , Dμ (P)=-ln)μP(are additive distance functions

For the K2P model

36

Moreover, Each positive linear combination of Dλ and Dμ

is an additive distance function

uvP

vwP

u

v

w

1 2

1 2

1 2

) , ( ln) ( ln) (

) , ( ln) ( ln) (

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

) , ( ) , (

uv uv

vw vw

uv vw uv vw

P P

P P

P P P P

D u v c c

D v w c c

D u w c c

D u v D v w

Our goal: given set of input sequences, select D which guarantees best reconstruction of the true tree.

37

ACGGTCA

ACGGATA

GGGGATT

The approximate distance function is defined by the observable noisy version of the substitution matrices

w

v

u uvP

vwP

We would like to use functions which minimize the influence of the “noise” on the reconstruction.Such a function can be defined&computed analytically for a single distance . Computing it for even small trees looks hard.

uvP

vwP

38

Summary • We have infinitely many additive distance functions

for the K2P model.• Which one should we use for the given input DNA

sequences?• If we have the exact substitution matrices for all

pairs of taxa, then all functions are equally good.• But we have only finite sequences, whose

alignments provide only estimations of the true substitution matrices

39

3 phases of the project

• Phase 1: Distance functions on simulated quartets :1 month• Phase 2: Distance functions on larger simulated trees: )1+( month

• Phase 3: Extensions to real data and/or different models: 1 month

Phase 2 and 3 are flexible

40

Phase I: Quartets (~one month)

• Study the relevant info in “Towards Optimal....”http://webcourse.cs.technion.ac.il/236512/Spring2010/ho/WCFiles/optimal_distance_functions.pdf.

• Write a program )in MATLAB or C..( which compute optimal

distance functions as in the above paper

• Repeat the “quartet resolution test” given in this presentation, and

extend it to include optimal distance functions.

• Feel free modify the simulation by your judgment.

41

Phase II: Reconstructing Larger Trees using the Neighbor Joining Algorithm

1. Study the Neighbor Joining algorithm

2. Newick trees representations, and Robinson Fould measure.

3. Make similar tests, but this time on larger trees.

4. Implementation of NJ, and “Tree Templates” can be

downloaded from the www.

More information will be given later, either via the course site

or in a meeting.

42

Phase III: Trees from Real Data

1. Get Homologeous DNA sequences from existing databases.

2. Align the sequences using public domain software.

3. Select appropriate distance functions, and estimate

distances between the aligned sequences, using appropriate

distance functions

4. Use the various distance functions to reconstruct the trees,

and compare their perfomance.