Pairwise Sequence Alignments

Comparison methods

Global alignment

Local alignment

Topics to be Covered

Introduction to Alignment

Analyze the similarities and differences at the individual base level or amino acid level

Aim is to infer structural, functional and evolutionary relationships among sequences

982 TGTTTGCTAAAGCTTCAGCTATCCACAACCCAATTGACCTCTAC 1022 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 961 TCTTTGCTAAGACCGCCTCCATCTACAACCCAATCA - - - TCTAC 1001

Two sequences written out , one on top of the otherIdentical or similar characters placed in same columnNonidentical characters either placed in same column as mismatch or opposite gap in the other sequenceOverall quality of the alignment is then evaluated based on a formula that counts the number of identical (or similar) pairs minus the number of mismatches and gaps

Sequence Alignment

Pairwise Sequence AlignmentsWhy to compare

Similarity search is necessary for:

• Family assignment

• Sequence annotation

• Construction of phylogenetic trees

• Learn about evolutionary relationships

• Classify sequences

• Identify functions

• Homology Modeling

Essential Elements of an Alignment Algorithm

• Defining the problem (Global, local alignment)

• Scoring scheme (Gap penalties)

• Distance Matrix (PAM, BLOSUM series)

Global and Local AlignmentsGlobal – attempt is made to align the entire sequence using as many characters as possible, up to both ends of the sequencesLocal – stretches of sequence with the highest density of matches are aligned

L G P S S K Q T G K G S – S R I W D N| | | | | | | Global AlignmentL N – I T K S A G K G A I M R L G D A

- - - - - - - T G K G - - - - - - | | | Local Alignment- - - - - - - A G K G - - - - - -

Local vs. Global Alignment (cont’d)

• Global Alignment

• Local Alignment—better alignment to find conserved segment

--T—-CC-C-AGT—-TATGT-CAGGGGACACG—A-GCATGCAGA-GAC | || | || | | | ||| || | | | | |||| | AATTGCCGCC-GTCGT-T-TTCAG----CA-GTTATG—T-CAGAT--C

TCCCAGTTATGTCAGGGGACACGAGCATGCAGAGAC ||||||||||||

AATTGCCGCCGTCGTTTTCAGCAGTTATGTCAGATC

• Global - When two sequences are of approximately equal length. Here, the goal is to obtain maximum score by completely aligning them

• Local - When one sequence is a sub-string of the other or the goal is to get maximum local score

• Protein motif searches in a database

Global and Local Alignments

Dynamic programming algorithm

• Dynamic programming =

Build up optimal alignment using previous solutions for optimal alignments of subsequences

Aligning Sequences without Insertions and Deletions: Hamming Distance

Given two DNA sequences v and w :v :

• The Hamming distance: dH(v, w) = 8 is large but the sequences are very similar

A T A T A T A TAT A T A T A Tw :

Aligning Sequences with Insertions and Deletions

v : A T A T A T A TAT A T A T A Tw : ----

By shifting one sequence over one position:

• The edit distance: dH(v, w) = 2.

• Hamming distance neglects insertions and deletions in DNA

Edit Distance

Levenshtein (1966) introduced edit distance between two strings as the minimum number of elementary operations (insertions, deletions, and substitutions) to transform one string into the other

d(v,w) = MIN number of elementary operations

to transform v w

Edit Distance vs Hamming Distance

V = ATATATAT

W = TATATATA

Hamming distance always compares i-th letter of v with i-th letter of w

Hamming distance: d(v, w)=8Computing Hamming distance is a trivial task.

Edit Distance vs Hamming Distance

V = ATATATAT

W = TATATATA

Hamming distance: Edit distance: d(v, w)=8 d(v, w)=2

Computing Hamming distance Computing edit distance

is a trivial task is a non-trivial task

W = TATATATA

Just one shift

Make it all line up

V = - ATATATAT

Hamming distance always compares i-th letter of v with i-th letter of w

Edit distance may compare i-th letter of v with j-th letter of w

Edit Distance vs Hamming Distance

V = ATATATAT

W = TATATATA

Hamming distance: Edit distance: d(v, w)=8 d(v, w)=2

(one insertion and one deletion)

How to find what j goes with what i ???

W = TATATATA

V = - ATATATAT

Hamming distance always compares i-th letter of v with i-th letter of w

Edit distance may compare i-th letter of v with j-th letter of w

Edit Distance: Example

TGCATAT ATCCGAT in 5 steps

TGCATAT (delete last T) TGCATA (delete last A) TGCAT (insert A at front)ATGCAT (substitute C for 3rd G)ATCCAT (insert G before last A) ATCCGAT (Done)

Edit Distance: ExampleTGCATAT ATCCGAT in 5 steps

TGCATAT (delete last T)TGCATA (delete last A)TGCAT (insert A at front)ATGCAT (substitute C for 3rd G)ATCCAT (insert G before last A) ATCCGAT (Done)What is the edit distance? 5?

Edit Distance: Example (cont’d)

TGCATAT ATCCGAT in 4 steps

TGCATAT (insert A at front)

ATGCATAT (delete 6th T)

ATGCATA (substitute G for 5th A)

ATGCGTA (substitute C for 3rd G)

ATCCGTA (Done)

Edit Distance: Example (cont’d)TGCATAT ATCCGAT in 4 steps

TGCATAT (insert A at front)

ATGCATAT (delete 6th T)

ATGCAAT (substitute G for 5th A)

ATGCGAT (substitute C for 3rd G)

ATCCGAT (Done)

Can it be done in 3 steps???

The Alignment Grid

– Every alignment path is from source to sink

Alignment as a Path in the Edit Graph

0 1 2 2 3 4 5 6 7 70 1 2 2 3 4 5 6 7 7 A T _ G T T A T _A T _ G T T A T _ A T C G T _ A _ CA T C G T _ A _ C0 1 2 3 4 5 5 6 6 7 0 1 2 3 4 5 5 6 6 7

(0,0) , (1,1) , (2,2), (2,3), (3,4), (4,5), (0,0) , (1,1) , (2,2), (2,3), (3,4), (4,5), (5,5), (6,6), (7,6), (7,7)(5,5), (6,6), (7,6), (7,7)

1

0

2

3

4

5

6

7

10 2 3 4 5 6 7

GGA T C AAT Cw

AA

T

T

G

T

AA

T

v

- Corresponding path -

Alignment as a Path in the Edit Graph

1

0

2

3

4

5

6

7

10 2 3 4 5 6 7

GGA T C AAT Cw

AA

T

T

G

T

AA

T

v

Every path in the edit graph corresponds to an alignment:

Alignment as a Path in the Edit Graph

1

0

2

3

4

5

6

7

10 2 3 4 5 6 7

GGA T C AAT Cw

AA

T

T

G

T

AA

T

vOld AlignmentOld Alignment 01223012234545677677v= AT_Gv= AT_GTTTTAT_AT_w=ATCGw=ATCGT_T_A_CA_C 01234012345555667667

New AlignmentNew Alignment 01223012234545677677v= AT_Gv= AT_GTTTTAT_AT_w=ATCGw=ATCG_T_TA_CA_C 01234012344545667667

From LCS to Alignment: Change up the Scoring

• The Longest Common Subsequence (LCS) problem—the simplest form of sequence alignment – allows only insertions and deletions (no mismatches).

• In the LCS Problem, we scored 1 for matches and 0 for indels

• Consider penalizing indels and mismatches with negative scores

• Simplest scoring schema: +1 : match premium -μ : mismatch penalty -σ : indel penalty

Simple Scoring

• When mismatches are penalized by –μ, indels are penalized by –σ, and matches are rewarded with +1,

the resulting score is:

#matches – μ(#mismatches) – σ (#indels)

Dynamic programming algorithm

• define a matrix Fij:

Fij is the optimal alignment of

subsequence A1...i and B1...j

• iterative build up: F(0,0) = 0

• define each element i,j from

(i-1,j): gap in sequence A

(i, j-1): gap in sequence B

(i-1, j-1): alignment of Ai to Bj

Dynamic programming

Sequence Comparison Scoring Matrices

• The choice of a scoring matrix can strongly influence the outcome of sequence analysis

• Scoring matrices implicitly represent a particular theory of evolution

• Elements of the matrices specify the similarity or the

distance of replacing one residue (base) by another

• Distance and similarity matrices are inter-convertible by some mathematical transformation.

Protein Scoring Matrices

• The two most popular matrices are the PAM and the BLOSUM matrix

T A T G T G G A A T G A

Scoring Insertions and Deletions

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

A T G T A A T G C A

T A T G T G G A A T G A

The creation of a gap is penalized with a negative score value.

insertion / deletion

• The optimal alignment of two similar sequences is usually that which

• maximizes the number of matches and• minimizes the number of gaps.

• Permitting the insertion of arbitrarily many gaps can lead to high scoring alignments of non-homologous sequences.

• Penalizing gaps forces alignments to have relatively few gaps.

Why Gap Penalties?

1 GTGATAGACACAGACCGGTGGCATTGTGG 29 ||| | | ||| | || || |1 GTGTCGGGAAGAGATAACTCCGATGGTTG 29

Why Gap Penalties?

Gaps allowed but not penalized Score: 88

Gaps not permitted Score: 0

1 GTG.ATAG.ACACAGA..CCGGT..GGCATTGTGG 29 ||| || | | | ||| || | | || || |1 GTGTAT.GGA.AGAGATACC..TCCG..ATGGTTG 29

Match = 5Mismatch = -4

Gap Penalties Linear gap penalty score:

γ(g) = - gd

Affine gap penalty score:

γ(g) = -d - (g -1)e

γ(g) = gap penalty score of a gap of length g d = gap opening penalty

e = gap extension penalty

g = gap length

Scoring Indels: Naive Approach

• A fixed penalty σ is given to every indel:– -σ for 1 indel, – -2σ for 2 consecutive indels– -3σ for 3 consecutive indels, etc.

Can be too severe penalty for a series of 100 consecutive indels

Affine Gap Penalties

• In nature, a series of k indels often come as a single event rather than a series of k single nucleotide events:

ATA__GCATATTGC

ATAG_GCAT_GTGC

Normal scoring would give the same score for both alignments

This is more likely.

This is less likely.

Accounting for Gaps

• Gaps- contiguous sequence of spaces in one of the rows

• Score for a gap of length x is: -(ρ + σx) where ρ >0 is the penalty for introducing a gap: gap opening penalty ρ will be large relative to σ: gap extension penalty because you do not want to add too much of a

penalty for extending the gap.

Affine Gap Penalty Recurrencessi,j = s i-1,j - σ

max s i-1,j –(ρ+σ)

si,j = s i,j-1 - σ

max s i,j-1 –(ρ+σ)

si,j = si-1,j-1 + δ (vi, wj)

max s i,j

s i,j

Continue Gap in w (deletion)

Start Gap in w (deletion): from middle

Continue Gap in v (insertion)

Start Gap in v (insertion):from middle

Match or Mismatch

End deletion: from top

End insertion: from bottom

Scoring Insertions and Deletions

A T G T T A T A C

T A T G T G C G T A T A

Total Score: 4

Gap parameters:

d = 3 (gap opening)

e = 0.1 (gap extension)

g = 3 (gap lenght)

γ(g) = -3 - (3 -1) 0.1 = -3.2

T A T G T G C G T A T A

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

insertion / deletion

match = 1mismatch = 0

Total Score: 8 - 3.2 = 4.8

Modification of Gap Penalties

1 V...LSPADKFLTNV 12 | |||| | | |1 VFTELSPA.K..T.V 11

gap opening penalty = 0gap extension penalty = 0.1score = 11.3

Score Matrix: BLOSUM62

1 ...VLSPADKFLTNV 12 |||| 1 VFTELSPAKTV.... 11

gap opening penalty = 3gap extension penalty = 0.1score = 6.3

Pairwise Sequence AlignmentLocal Alignment

Semi-Global Alignment

Local Alignment

A local Alignment between sequence s and

sequence t is an alignment with maximum

similarity between a substring of s and a

substring of t.

T. F. Smith & M. S. Waterman, “Identification of Common Molecular Subsequences”, J. Mol. Biol., 147:195-197

43

Why choose a local alignment algorithm?

• More meaningful – point out conserved regions between two sequences

• Aligns two sequences of different lengths to be matched

• Aligns two partially overlapping sequences

• Aligns two sequences where one is a subsequence of another

43

44

Dynamic ProgrammingLocal Alignment

• Si,j = MAXIMUM[ Si-1, j-1 + s(ai,bj) (match/mismatch in the diagonal),

Si,j-1 + w (gap in sequence #1),

Si-1,j + w (gap in sequence #2),

0]

44

45

Initialization Step

46

Matrix Fill Step

47

Traceback Step

48

Traceback Step

49

Traceback Step

An Introduction To Multiple Sequence Alignment (MSA)

Topics To Be Discussed

• Motivation for MSA

• What is MSA

• Extension of Dynamic Programming

• The STAR Method

• Progressive Alignment

• Scoring Multiple Alignments

Multiple Alignment versus Pairwise Alignment

• Up until now we have only tried to align two sequences.

• Up until now we have only tried to align two sequences.

• What about more than two? And what for?

Multiple Alignment versus Pairwise Alignment

• Up until now we have only tried to align two sequences.

• What about more than two? And what for?

• A faint similarity between two sequences becomes significant if present in many

• Multiple alignments can reveal subtle similarities that pairwise alignments do not reveal

Multiple Alignment versus Pairwise Alignment

Motivation For MSA

• A natural extension of Pairwise Sequence Alignment

• MSA gives Biologist the ability to extract biologically important

but perhaps widely dispersed sequence similarities that can give

biologist hints about the evolutionary history of certain sequences.

• In pairwise alignment, when two sequences align, it is concluded

that there is probably a functional relationship between the two

sequences. Whereas for MSA, if it is known that there is a

functional similarity amongst a number of sequences, we can use

MSA to find out where the similarity comes from.

What is MSA• MSA is the alignment of N sequences (Protein/Nucleotide)

simultaneously, where N > 2 .

• Let Si denote a sequence than the Global Multiple Sequence

Alignment of N > 2 sequences S = { S1 , …, SN } is obtained

by inserting gaps denoted by “ - “ at any possibly the beginning

or end, position.

• The new set of N sequences denoted by

S’ = { S1’ , …, SN

’ } will all have length L

Ovar STCVLSAYWKD-LNNYHBota STCVLSAYWKD-LNNYHSusc STCVLSAYWRNELNNFHHosa STCMLGTY-QD-FNKFHRano STCMLGTY-QD-LNKFHSasa STCVLGKLSQE-LHKLQ

Interpretation of positions

• Generally there are two interpretations of a position in a multiple sequence alignment:

• Evolutionary/historical

• Functional/structural

In many cases these are the same, but they may not be.

Multiple sequence alignment algorithm

• Ideal approach to multiple sequence alignment is to extend dynamic programming.

• Instead of aligning two sequences (two dimensional grid) we align k sequences (k dimensional grid)

• Extension is relatively straightforward

Dynamic programming for sequence alignment

• Recurrence relation

• Tabular computation

• Traceback

• Pairwise recurrence relation

S(i,j) = max[S(i-1, j-1) + m(i,j), S(i-1, j) + g, S(i, j-1) + g]

m(i,j) = similarity matrix eg BLOSUM

g = gap penalty

Aligning Three Sequences• Same strategy as

aligning two sequences

• Use a 3-D “Manhattan Cube”, with each axis representing a sequence to align

• For global alignments, go from source to sink

source

sink

2-D cell versus 2-D Alignment Cell

In 3-D, 7 edges in each unit cube

In 2-D, 3 edges in each unit square

Architecture of 3-D Alignment Cell(i-1,j-1,k-1)

(i,j-1,k-1)

(i,j-1,k)

(i-1,j-1,k) (i-1,j,k)

(i,j,k)

(i-1,j,k-1)

(i,j,k-1)

Multiple Alignment: Dynamic Programming

• si,j,k = max

• δ(x, y, z) is an entry in the 3-D scoring matrix

si-1,j-1,k-1 + δ(vi, wj, uk)

si-1,j-1,k + δ (vi, wj, _ )

si-1,j,k-1 + δ (vi, _, uk)

si,j-1,k-1 + δ (_, wj, uk)

si-1,j,k + δ (vi, _ , _)

si,j-1,k + δ (_, wj, _)

si,j,k-1 + δ (_, _, uk)

cube diagonal: no indels

face diagonal: one indel

edge diagonal: two indels

Extending dynamic programming

• Based on the extrapolation from two to three sequences, we can define the recurrence relation for any number of sequences in the same way

• The other steps - tabular computation and traceback - are done in the same way as for pairwise alignment

There are seven cases when aligning three sequences

1 2 3 4 5 6 7 I I I - I -

- J J - J - J

- K - K K - -

K

23 -1 to choose the maximum similarity

Three sequence recurrence relation

S(i,j,k) = max[S(i-1, j-1, k-1) + m(i,j) + m(i,k) + m(j,k), S(i-1, j-1, k) + m(i,j) + g, S(i-1, j, k-1) + m(i,k) + g, S(i, j-1, k-1) + m(j,k) + g, S(i-1, j, k)+ g + g, S(i, j-1, k) + g + g, S(i, j, k-1) + g + g]m(i,j) = similarity matrix eg BLOSUMg = gap penalty

Dynamic programming time increases exponentially

• Time taken for alignment by dynamic programming is O(n * m) for two sequences n, m characters long.

• Time taken for alignment by dynamic programming is O(n * m * p) for three sequences n, m, p characters long.

Dynamic programming time increases exponentially

• Clearly, for N sequences, each sequence Li

characters long, the time required will be

N O( Π Li )

i=1This is exponential - O( LN )We need to fill out each ‘box’ in the grid

Pairwise Dynamic ProgrammingComparing Similar Sequences

• Faster algorithm for aligning similar sequences.

• If two sequences are similar, the best alignments have their paths near the main diagonal of the dynamic programming matrix.

• To compute the optimal score and alignment, it is not necessary to fill in the entire matrix.

• A narrow band around the main diagonal should suffice

Global Alignment: Comparing Similar Sequences

Match = 5, Mismatch = -4, Gap w= -7, K=2

Global Alignment: Comparing Similar Sequences

Match = 5, Mismatch = -4, Gap w= -7, K=2

Heuristic multiple sequence alignment

• Currently, most practical methods are hierarchial methods

• For example, pairwise alignments, defining hierarchy followed by progressive addition of sequences to alignment

Multiple Alignment Induces Pairwise Alignments

Every multiple alignment induces pairwise alignments

x: AC-GCGG-C y: AC-GC-GAG z: GCCGC-GAG

Induces:

x: ACGCGG-C; x: AC-GCGG-C; y: AC-GCGAGy: ACGC-GAC; z: GCCGC-GAG; z: GCCGCGAG

Reverse Problem: Constructing Multiple Alignment from Pairwise Alignments

Given 3 arbitrary pairwise alignments:

x: ACGCTGG-C; x: AC-GCTGG-C; y: AC-GC-GAGy: ACGC--GAC; z: GCCGCA-GAG; z: GCCGCAGAG

can we construct a multiple alignment that inducesthem?

Reverse Problem: Constructing Multiple Alignment from Pairwise Alignments

Given 3 arbitrary pairwise alignments:

x: ACGCTGG-C; x: AC-GCTGG-C; y: AC-GC-GAGy: ACGC--GAC; z: GCCGCA-GAG; z: GCCGCAGAG

can we construct a multiple alignment that inducesthem? NOT ALWAYS

Pairwise alignments may be inconsistent

Inferring Multiple Alignment from Pairwise Alignments

• From an optimal multiple alignment, we can infer pairwise alignments between all pairs of sequences, but they are not necessarily optimal

• It is difficult to infer a ``good” multiple alignment from optimal pairwise alignments between all sequences

Combining Optimal Pairwise Alignments into Multiple Alignment

Can combine pairwise alignments into multiple alignment

Can not combine pairwise alignments into multiple alignment

The STAR Alignment Method

• Using a pairwise alignment method (DP,etc) find the sequence that

is most similar to all the other sequences.

• Using this “best” sequence as the center (of a star, hence the name)

align the other sequences following the once a gap always a gap

rule .

• For example consider the following set of sequences

S1 A T T G C C A T T

S2 A T G G C C A T T

S3 A T C C A A T T T T

S4 A T C T T C T T

S5 A C T G A C C

STAR Alignment - 2

• Now Consider the following similarity matrix for the pairwise

comparing of the sequences.

S1 S2 S3 S4 S5 SUM sim(Si, Sj)

I≠J

S1 - 7 -2 0 -3 2

S2 7 - -2 0 -4 1

S3 -2 -2 - 0 -7 -11

S4 0 0 0 - -3 -3

S5 -3 -4 -7 -3 - -17

For this example S1 is the center of the STAR

STAR Alignment - 3

• Next we get the best alignment between S1 and the other sequences as follows:

S1 | A T T G C C A T T S1 | A T T G C C A T T

S2 | A T G G C C A T T S5 | A C T G A C C - -

S1 | A T T G C C A T T - -

S3 | A T C - C A A T T T T

S1 | A T T G C C A T T

S4 | A T C T T C - T T

STAR Alignment 4

• Next to build the MSA we start with S1 & S2 as

A T T G C C A T T

A T G G C C A T T adding S3 using once a gap always a gap

A T T G C C A T T - -

A T G G C C A T T - -

A T C - C A A T T T T continuing in this fashion we obtain

for our MSA of all the sequences

Star Alignment 5

A T T G C C A T T - -

A T G G C C A T T - -

A T C - C A A T T T T

A T C T T C - T T - -

A C T G A C C - - - -

• Clearly, using the STAR method the time complexity is

dominated by computing the pairwise alignment which again for

N sequences we have O(N2) pairs. We consider each pairwise

alignment to take L2 time where again L is the length of each

sequence.

STAR Alignment - 6

• Thus the time complexity for computing all pairwise alignments

will be O[(NL)2]

• We still have to consider the time it takes to merge the sequences into a MSA . If Lmax is the upper bound of the alignment length

then it will take N2(Lmax) time to merge the sequences into a MSA.

• Thus the time complexity for STAR is O( N2L2 + N2Lmax )

• Clearly for large N, L this is less than the time complexity for

SP which is O[ (2L)N (N2)]

• Recall SP is optimal whereas STAR is not, thus there is a trade-off between optimization and practicality .

Profile Representation of Multiple Alignment

- A G G C T A T C A C C T G T A G – C T A C C A - - - G C A G – C T A C C A - - - G C A G – C T A T C A C – G G C A G – C T A T C G C – G G

A 1 1 .8 C .6 1 .4 1 .6 .2G 1 .2 .2 .4 1T .2 1 .6 .2- .2 .8 .4 .8 .4

Profile Representation of Multiple Alignment

In the past we were aligning a sequence against a sequence

Can we align a sequence against a profile?

Can we align a profile against a profile?

- A G G C T A T C A C C T G T A G – C T A C C A - - - G C A G – C T A C C A - - - G C A G – C T A T C A C – G G C A G – C T A T C G C – G G

A 1 1 .8 C .6 1 .4 1 .6 .2G 1 .2 .2 .4 1T .2 1 .6 .2- .2 .8 .4 .8 .4

Aligning alignments

• Given two alignments, can we align them?

x GGGCACTGCATy GGTTACGTC-- Alignment 1 z GGGAACTGCAG

w GGACGTACC-- Alignment 2v GGACCT-----

Aligning alignments

• Given two alignments, can we align them?

• Hint: use alignment of corresponding profiles

x GGGCACTGCATy GGTTACGTC-- Combined Alignment z GGGAACTGCAG

w GGACGTACC-- v GGACCT-----

Multiple Alignment: Greedy Approach

• Choose most similar pair of strings and combine into a profile , thereby reducing alignment of k sequences to an alignment of of k-1 sequences/profiles. Repeat

• This is a heuristic greedy methodu1= ACGTACGTACGT…

u2 = TTAATTAATTAA…

u3 = ACTACTACTACT…

…

uk = CCGGCCGGCCGG

u1= ACg/tTACg/tTACg/cT…

u2 = TTAATTAATTAA…

…

uk = CCGGCCGGCCGG…

k

k-1

Greedy Approach: Example

• Consider these 4 sequencess1 GATTCAs2 GTCTGAs3 GATATTs4 GTCAGC

Greedy Approach: Example (cont’d)

• There are = 6 possible alignments

s2 GTCTGAs4 GTCAGC (score = 2)

s1 GAT-TCAs2 G-TCTGA (score = 1)

s1 GAT-TCAs3 GATAT-T (score = 1)

s1 GATTCA--s4 G—T-CAGC(score = 0)

s2 G-TCTGAs3 GATAT-T (score = -1)

s3 GAT-ATTs4 G-TCAGC (score = -1)

Greedy Approach: Example (cont’d)

s2 and s4 are closest; combine:

s2 GTCTGAs4 GTCAGC

s2,4 GTCt/aGa/cA (profile)

s1 GATTCAs3 GATATTs2,4 GTCt/aGa/c

new set of 3 sequences:

Progressive Alignment

• Progressive alignment is a variation of greedy algorithm with a somewhat more intelligent strategy for choosing the order of alignments.

• Progressive alignment works well for close sequences, but deteriorates for distant sequences• Gaps in consensus string are permanent• Use profiles to compare sequences

ClustalW

• Popular multiple alignment tool today

• ‘W’ stands for ‘weighted’ (different parts of alignment are weighted differently).

• Three-step process

1.) Construct pairwise alignments

2.) Build Guide Tree

3.) Progressive Alignment guided by the tree

The CLUSTALW Algorithm

• Step 1 : Determine all pairwise alignment between sequences and determine degrees of similarity between each pair.

• Step 2 : Construct a similarity tree * .

• Step 3 : Combine the alignments starting from the most closely related groups to the most distantly related groups, as in STAR we use the once a gap always a gap rule .

• * The PILEUP program is similar to CLUSTALW but uses a different method for producing the similarity tree .

Heuristic Multiple Alignment Methods

Clustal W progressive multiple alignment

• Align two sequences to each other

• Align a sequence to an existing alignment

• Align two alignments to each other

Multiple Alignments: Scoring

• As in the pairwise case, not all MSA’s are equally good.

• We need a method of scoring for determining when one MSA is better than another one.

• Number of matches (multiple longest common subsequence score)

• Entropy score

• Sum of pairs (SP-Score)

Multiple LCS Score• A column is a “match” if all the letters in the

column are the same

• Only good for very similar sequences

AAAAAAAATATC

Entropy• Define frequencies for the occurrence of each

letter in each column of multiple alignment• pA = 1, pT=pG=pC=0 (1st column)

• pA = 0.75, pT = 0.25, pG=pC=0 (2nd column)

• pA = 0.50, pT = 0.25, pC=0.25 pG=0 (3rd column)

• Compute entropy of each column AAAAAAAATATC

Entropy: Example

Best case

Worst case

Multiple Alignment: Entropy Score Entropy for a multiple alignment is the sum of entropies of its columns:

Σ over all columns Σ X=A,T,G,C pX logpX

Entropy of an Alignment: Example

column entropy: -( pAlogpA + pClogpC + pGlogpG + pTlogpT)

•Column 1 = -[1*log(1) + 0*log0 + 0*log0 +0*log0] = 0

•Column 2 = -[(1/4)*log(1/4) + (3/4)*log(3/4) + 0*log0 + 0*log0] = -[ (1/4)*(-2) + (3/4)*(-.415) ] = +0.811

•Column 3 = -[(1/4)*log(1/4)+(1/4)*log(1/4)+(1/4)*log(1/4) +(1/4)*log(1/4)] = 4* -[(1/4)*(-2)] = +2.0

•Alignment Entropy = 0 + 0.811 + 2.0 = +2.811

A A A

A C C

A C G

A C T

Sum of Pairs Score(SP-Score)

• Consider pairwise alignment of sequences

ai and aj

imposed by a multiple alignment of k sequences

• Denote the score of this suboptimal (not necessarily optimal) pairwise alignment as

s*(ai, aj)

• Sum up the pairwise scores for a multiple alignment:

s(a1,…,ak) = Σi,j s*(ai, aj)

Computing SP-Score

Aligning 4 sequences: 6 pairwise alignments

Given a1,a2,a3,a4:

s(a1…a4) = Σs*(ai,aj) = s*(a1,a2) + s*(a1,a3) + s*(a1,a4) + s*(a2,a3) + s*(a2,a4) + s*(a3,a4)

SP-Score: Examplea1

.ak

ATG-C-AATA-G-CATATATCCCATTT

Pairs of Sequences

A

A A

11

1

G

C G

1−μ

−μ

Score=3 Score = 1 – μ

Column 1 Column 3

To calculate each column:

• Consider aligning the following 4 portein sequences

S1 = AQPILLLV

S2 = ALRLL

S3 = AKILLL

S4 = CPPVLILV

• Next consider the following MSA matrix M

A Q P I L L L V

A L R - L L - -

A K - I L L L -

C P P V L I L V

SP-Score: Example

• Assume s(match) = 1 , s(mismatch) = -1 , and s(gap) = -2 ,

also assume s(-, -) = 0 to prevent the double counting of gaps.

• Then the SP score for the 4th column of M would be

SP(m4) = SP(I, -, I, V)

= s(I,-) + s(I,I) + s(I,V) + s(-,I) + s(-, V) + s(I,V)

= -2 + 1 + (-1) + (-2) + (-2) +(-1)

= -7

• To find SP(M) we would find the score of each mi and then SUM

all the SP(mi) scores to get the score M .

• To find the optimal score using this method we need to consider

all possible MSA matrices. We say more about this later.

SP-Score: Example

Some Problems with the SP Score

• Consider column 1 of our example ie A,A,A,C for this column

we get SP(m4) = SP(A,A,A,C)

= 1 + 1 + (-1) + 1 + (-1) + (-1)

= 0

whereas if we had A,A,A,A we get a score of

SP(A,A,A,A) = 1+1+1+1+1+1 = 6 , thus we get a difference of

6 for what could be explained by a single mutation.

• The SP method tends to overweight the influence of mutations

• The major problem with the SP method is that finding the optimal MSA is very time consuming.