Trees, Stars, and Multiple Biological Sequence Alignment

Jesse Wolfgang

CSE 497

February 19, 2004

02/19/20042

Importance?

RNA folding (Trifonov, Bolshoi) Gene regulation (Galas et al.) Protein structure-function relationships(Wu, Kabat)

Molecular evolution (Dayhoff)

02/19/20043

Introduction

Original sequence unknown– Must consider all possible transformations– Including insertions, deletions, and replacements

Choose the most likely set of transformations– With a given model of protein evolution

02/19/20044

Sequences and Alignments

An alignment of the sequences

is written asnSS ,...,1

nSS ,...,1

K-sequence: sequence of k characters ),...,( 1 knnS =

Each is obtained from– Blanks are inserted in positions where some of the other

sequences have a nonblank character– At least one must be nonblank for each

is the length of the aligned sequences

iS iS

jS lj ,...,1=l

02/19/20045

Alignments

D Q L FD N V QQ G L

1S

2S

3S

D - - Q – L FD N V Q - - -- - - Q G L -

1S

2S

3S

Ex: sequences DQLF, DNVQ, QGL

02/19/20046

Lattices and Paths

– Cartesian product of strings of squaresn

A path between the sequences is a set of connected line segments (connected broken line)

),...,( 1 nSSγnSS ,...,1

A lattice of sequences with lengths),...,( 1 nSSL nSS ,...,1

nkk ,...,1

n

– Consists of -dimensional hypercubesn

– Forms an -dimensional parallelepipedn

02/19/20047

Paths

2 dimensions 3 dimensions

3 possible paths

7 possible paths

= 2n-1 = O(2n)

02/19/20048

Paths

DQ

G

L

N V Q

DQ

LF

3-dimensional parallelepiped

sublattice

Sequences DQLF, DNVQ, QGL

DD-

-N-

QQQ

--G

L-L

F--

-V-

02/19/20049

Sequences: ABCD, ABD, BCD

Paths and Sequence Length

Note:– Where is the length of

nn kklkk ++≤≤ ...},...,{max 11

ik iS

4}3,3,4max{ ==l

A B C DA B – D- B C D

A B C D

AB

D

B

C

D

02/19/200410

Sequences: ABCD, EFGH, IJK

Paths and Sequence Length

Note:– Where is the length of

nn kklkk ++≤≤ ...},...,{max 11

ik iS

EI

J

K

F G H

AB

CD 11344 =++=l

A B C D – - - - - - -- - - - E F G H - - -- - - - - - - - I J K

02/19/200411

Sequences DQLF, DNVQ, QGL

Projections

DQ

G

L

N V Q

DQ

LF

denotes an alignment of and)),...,(( 1 nij SSp γiS jS

D Q – L F- Q G L -

D Q L F

Q

G

L

02/19/200412

Optimal Paths

is a measure assigned to)(γM γ– Measure of the similarity among based upon a particular metric

nSS ,...,1

For each measure there is at least one path with attaining a minimum value at , the optimal path

M),...,( 1*

nSSγ)(γM

*γ

02/19/200413

))(),...,(( 11 nn iSiSL

DQ

G

L

N V Q

DQ

LF

Each vertex in L is an end corner of the sublattice

Calculating Optimal Paths

First: compute score of each of the possible paths for the cube that has a vertex at the original corner Next: using this information, compute minimum score to reach the vertices of the adjacent cubes to the original corner

02/19/200414

Problems with This Algorithm

Calculates a weighted sum of its projected pairwise alignments– Called “Sum-of-the-Pairs” (SP)

Other methods fit biological intuition more closely

02/19/200415

Tree-Alignment

Treat sequences as leaves of an evolutionary tree

Reconstruct ancestral sequences which minimize the cost of the tree– Must assign sequences to internal nodes

Align the given and reconstructed sequences Star-alignment: only one internal node

02/19/200416

Tree-Alignment

Many different methods for calculating tree alignments

Discuss version used by ClustalX

02/19/200417

Tree-Alignment in ClustalX

Three main parts

1. Perform pairwise alignment on all sequences to calculate a distance matrix

2. Use distance matrix to calculate a guide tree

3. Sequences are progressively aligned using the branching order in the guide tree

http://bimas.dcrt.nih.gov/clustalw/clustalw.html

02/19/200418

Calculating Distance Matrix

Use standard dynamic programming to find the best alignment

– Gap penalties for opening a gap and continuing a gap (possibly different)

Divide number of matches by total number of residues compared (excluding gaps)

Convert to distances by dividing by 100 and subtracting from 1

Gives one entry in the n by n matrix

02/19/200419

Calculating Distance Matrix

Ex: sequences ATCG, ATCC, AGGC, AGCC

A T C GA T C C

= 3/4 = .75/100 = 1-.0075 = .9925

A T C GA G G C

= 1/4 = .25/100 = 1-.0025 = .9975

02/19/200420

Calculating Distance Matrix

ATCG ATCT AGGC GCAA

ATCG -- -- -- --

ATCT .9925 -- -- --

AGGC .9975 .9975 -- --

GCAA 1 1 1 --

02/19/200421

Calculating a Guide Tree

Using Nearest-Neighbor method to group sequences– Results in an unrooted tree– Branch lengths proportional to estimated

divergence “Mid-point” method used to determine root

– Means of the branch lengths to each side of the root are equal (or approximately equal)

02/19/200422

Calculating a Guide Tree

ATCG ATCT

ATCG AGGC

AGCC GCAA

AGAA

.9925.9925

.9975/2 .9975

1/3 1

ATCG = 1.8245ATCT = 1.8245

AGGC = 1.33081.6599

GCAA = 1

02/19/200423

Calculating a Guide Tree

ATCG = 1.4911ATCT = 1.4911

1.4911

AGGC = 1.4986GCAA = 1.4986

1.4986

ATCG ATCT

ATCG

AGGC

AGCC

GCAA

AGAA.9925.9925 1 1

.9975/2.9975/2

02/19/200424

Progressive Alignment

Perform a series of pairwise alignments– Slowly align larger and larger groups of

sequences

Follow the branching order of the tree– From leaves to root

02/19/200425

Progressive Alignment

ATCG ATCT

ATCG

AGCC

AGGC GCAA

AGAA

02/19/200426

Alignment Costs

A C

A

A

C

A, A, A, C, C

--

6

A

A

A

A

AC

C

C

A, A, A, C, C

A, A, C

1

C

C

A

A

A

A

A, A, A, C, C

A

2

Traditional

Input seq

Reconstructedseq

Missmatches

Traditional (SP) Tree-Alignment Star-Alignment

02/19/200427

Alignment Inconsistencies

Different definitions of multiple alignments can yield different optimal alignments

Optimal tree-alignments minimize number of mutations from theorized common ancestors

SP-alignments maximize number of positions where aligned sequences agree– Sometimes makes more biological sense since

certain regions of proteins more likely to mutate

02/19/200428

Alignment Inconsistencies

Ex: cost of 1 for aligning two different letters, cost of 2 for aligning a letter with a null

Sequences: ACC, ACC, TCT, ATCT

Input sequences

Reconstructedsequences

- A C C- A C C- T C TA T C T

--

Traditional (SP)

A C C -A C C -T C T -A T C T

A C C -

Star-Alignment

02/19/200429

ClustalX Demo

Multiple sequence alignment program For more information on ClustalX

– http://www.at.embnet.org/embnet/progs/clustal/clustalx.htm