.

Multiple Sequence Multiple Sequence AlignmentsAlignments

2

The Global Alignment problem

AGTGCCCTGGAACCCTGACGGTGGGTCACAAAACTTCTGGA

AGTGACCTGGGAAGACCCTGACCCTGGGTCACAAAACTC

x

y

z

3

Multiple Sequence Alignment

S1=AGGTC

S2=GTTCG

S3=TGAACPossible alignment

A-T

GGG

G--

TTA

-TA

CCC

-G-

Possible alignment

AG-

GTT

GTG

T-A

--A

CCA

-GC

4

Motivations• Protein databases categorized by protein families

Collection of proteins with similar structure, function, or evolutionary history

• Comparing a new protein with a family requires to construct a representation of the family and then compare the new protein with the family representation

• How to score a multiple alignment ?• Consensus Distance• Evolutionary Tree Distance• Sum-of-Pairs Distance

5

Definition

Given N sequences x1, x2,…, xN: Insert gaps (-) in each sequence xi, such that

All sequences have the same length LScore of the global map is maximum

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

Multiple alignments can help improve the pairwise alignments

6

Multiple Sequence AlignmentDefinition: Given stings S1, S2, …,Sk a multiple (global) alignment

map them to strings S’1, S’2, …,S’k that may contain spaces, where:

1. |S’1|= |S’2|=…= |S’k|

2. The removal of spaces from S’i leaves Si

Definition: The sum-of-pairs (SP) value for a multiple global

alignment A of k strings is the sum of the values of all pairwise alignments induced by A

2k

7

Scoring Function: Sum Of Pairs Definition: Induced pairwise alignment

A pairwise alignment induced by the multiple alignment

Example:

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

8

Multiple Sequence AlignmentGiven k strings of length n, there is a generalization of the dynamic

programming algorithm that finds an optimal SP alignment.

NP completeness:

• Instead of a 2-dimensional table we now have a k-dimensional table to fill. O(nk) cells to fill

• Each dimension’s size is n+1. Each entry depends on 2k - 1adjacent entries.

Time Complexity: O(k2knk)

9

Multidimensional Dynamic Programming

Generalization of Needleman-Wunsh:

S(m) = i S(mi)

(sum of column scores)

F(i1,i2,…,iN): Optimal alignment up to (i1, …, iN)

F(i1,i2,…,iN) = max(all neighbors of cube)(F(nbr)+S(nbr))

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Example: in 3D (three sequences):

7 neighbors/cells

F(i,j,k) = max{ F(i-1,j-1,k-1)+S(xi, xj, xk),F(i-1,j-1,k )+S(xi, xj, - ),F(i-1,j ,k-1)+S(xi, -, xk),F(i-1,j ,k )+S(xi, -, - ),F(i ,j-1,k-1)+S( -, xj, xk),F(i ,j-1,k )+S( -, xj, xk),F(i ,j ,k-1)+S( -, -, xk) }

Multidimensional Dynamic Programming

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Multiple alignmentsWe use a matrix to represent the alignment of k sequences,

K=(x1,...,xk). We assume no columns consists solely of blanks.

M Q _ I L L L

M L R - L L -

M K _ I L L L

M P P V L I L

The common scoring functions give a score to each column, and set: score(K)= ∑i score(column(i))

For k=10, a scoring function has 2k -1 > 1000 entries to specify. The scoring function is symmetric - the order of arguments need not matter: score(I,_,I,V) = score(_,I,I,V).

x1

x2

x3

x4

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SUM OF PAIRS

M Q _ I L L L

M L R - L L -

M K _ I L L L

M P P V L I L

A common scoring function is SP – sum of scores of the projected pairwise alignments: SPscore(K)=∑i<j score(xi,xj).

In order for this score to be written as ∑i score(column(i)),we set score(-,-) = 0. Why ?

Because these entries appear in the sum of columns but not in the sum of projected pairwise alignments (rows).

Note that we need to specify the score(-,-) because a column may have several blanks (as long as not all entries are blanks).

13

SUM OF PAIRS

M Q _ I L L L

M L R - L L -

M K _ I L L L

M P P V L I L

Definition: The sum-of-pairs (SP) value for a multiple global

alignment A of k strings is the sum of the values of all projected

pairwise alignments induced by A where the pairwise alignment

function score(xi,xj) is additive.

2k

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Example

Consider the following alignment:

a c - c d b -- c - a d b da - b c d a d

Using the edit distance and for ,

this alignment has a SP value of

0, xx 1, yx yx

3 + 4 + 5 = 12

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Running Time:

1. Size of matrix: LN;

Where L = length of each sequence N = number of sequences

2. Neighbors/cell: 2N – 1

Therefore………………………… O(2N LN)

Multidimensional Dynamic Programming How do affine gaps generalize?

VERY badly! Require 2N states, one per combination

of gapped/ungapped sequences Running time: O(2N 2N LN) = O(4N

LN)

XY XYZ Z

Y YZ

X XZ

16

Multiple Sequence AlignmentGiven k strings of length n, there is a natural generalization of the

dynamic programming algorithm that finds an alignment that maximizes

SP-score(K) = ∑i<j score(xi,xj).

Instead of a 2-dimensional table, we now have a k-dimensional table to fill.

For each vector i =(i1,..,ik), compute an optimal multiple alignment for the k prefix sequences x1(1,..,i1),...,xk(1,..,ik).

The adjacent entries are those that differ in their index by one or zero. Each entry depends on 2k-1 adjacent entries.

17

The idea via K=2

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

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

1jtj1iV1is1jiV

1jt1isjiV1j1iV

])..[],..[(],[ j1ti1sdjiV

V[i,j] V[i+1,j]

V[i,j+1] V[i+1,j+1] Note that the new cell index (i+1,j+1) differs from previous indices by one of 2k-1 non-zero binary vectors (1,1), (1,0), (0,1).

Recall the notation:

and the following recurrence for V:

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The idea for arbitrary k

Order the vectors i=(i1,..,ik) by increasing order of the sum ∑ij. Set s(0,..,0)=0, and for i > (0,...,0):

( ) max{ ( ) ( ( , ))}s s score columnb

i i b i b

•The vector b ranges over all non-zero binary vectors. •The vector i-b is the non-negative difference of i and b. •The jth entry of column(i,b) equals cj= xj(ij) if bi=1, and cj= ‘-’ otherwise.(Reflecting that b is 1 at location j if that location changed in the “current comparison”).

Where

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Complexity of the DP approach

Number of cells nk.Number of adjacent cells O(2k).Computation of SP score for each column(i,b) is O(k2)

Total run time is O(k22knk) which is utterly unacceptable !

Not much hope for a polynomial algorithm because the problem has been shown to be NP complete.

Need heuristic to reduce time.

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Time saving heuristics: Relevance testsHeuristic: Avoid computing score(i) for irrelevant vectors.

M Q _ I L L L

M L R - L L -

M K _ I L L L

M P P V L I L

x1

x2

x3

x4

Let L be a lower bound on the optimal SP score of a multiple alignment of the k sequences. A lower bound L can be obtained from an arbitrary multiple alignment, computed in any way.

Main idea: Compute upper bounds H(u,v) for the optimal score for every two sequences s=xu and t=xv, 1 u < v k. When processing vector i=(..iu,..iv…), the relevant cells are such that in every projection on xu and xv, the optimal pairwise score is above a value based on H(u,v) and L.

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Recall the Linear Space algorithm V[i,j] = d(s[1..i],t[1..j]) B[i,j] = d(s[i+1..n],t[j+1..m]) F[i,j] + B[i,j] = score of best alignment through

(i,j)t

s

These computations done in linear space.

),( vuxxiia vuBuild such a table

for every two sequences s=xu and t=xv, 1 u, v k. This entry encodes the optimum through (iu,iv).

22

Let S(u,v) the score of the alignment of xu and xv in the multiple alignment.

Then, we have: L ≤ S(u,v) – H(u,v) +

And then: S(u,v) ≥ L + H(u,v) –

Now for each pair u,v we want to consider only the cells Iu and Iv for which the best pairwise alignment score that can be obtained through them (that is ) is greater than the above value:

Time saving heuristics:Relevance test

''

)','(vu

vuH

''

)','(vu

vuH

),( vuxxiia vu

),( vuxxiia vu

≥ L + H(u,v) - ''

)','(vu

vuH

23

A Profile Representation of Multiple Alignment

Given a multiple alignment M = m1…mn

Replace each column mi with profile entry pi Frequency of each letter in # gaps Optional: # gap openings, extensions, closings

- 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 .2O .2 .8 .4 .4E .4C .2 .8 .4 .2

24

Multiple Alignments With Profile Consider the MSA

a b c - aa b a b aa c c b -c b - b c

• Its corresponding profile P is C1 C2 C3 C4 C5a 75% 25% 50%b 75% 75%c 25% 25% 50% 25%− 25% 25% 25%

Aligning a string S to a profile P will tell us how well S fits P. Given the column positions C of P, the alignment consists of inserting spaces into S and C=(1,2,3,4,5) as in pure string alignment. For instance, an alignment of aabbc to P is:

a a b - b c1 - 2 3 4 5

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String-to-Profile Alignment Scoring a column j is equivalent to aligning S j to each character

at column j. σ(j) = sum{over all i}σ(Sj, ij)pij

pij is frequency of i-th character in column j,

Score of an alignment = sum of all column scores σ(j).

Use Dynamic Programming as before (NW, SW, …) to do a string-to-profile alignment

Except that you should use this scoring function defined above.

Profile-to-Profile Alignments?

26

Progressive Alignment

When evolutionary tree is known:

Align closest first, in the order of the tree In each step, align two sequences x, y, or profiles px, py, to generate a new

alignment with associated profile presult

Weighted version: Tree edges have weights, proportional to the divergence in that edge New profile is a weighted average of two old profiles

x

w

y

z

27

Example

Profile: (A, C, G, T, -)px = (0.8, 0.2, 0, 0, 0)py = (0.6, 0, 0, 0, 0.4)

s(px, py) = 0.8*0.6*s(A, A) + 0.2*0.6*s(C, A) + 0.8*0.4*s(A, -) + 0.2*0.4*s(C, -)

Result: pxy = (0.7, 0.1, 0, 0, 0.2)

s(px, -) = 0.8*1.0*s(A, -) + 0.2*1.0*s(C, -)

Result: px- = (0.4, 0.1, 0, 0, 0.5)

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Progressive Alignment

When evolutionary tree is unknown:

Perform all pairwise alignments Define distance matrix D, where D(x, y) is a measure of evolutionary

distance, based on pairwise alignment Construct a tree (we will describe more in detail later in the course) Align on the tree

x

w

y

z?

29

CLUSTALW (1). Perform pairwise alignments of all sequences (2). Use alignment scores to produce a phylogenetic tree (3). Align the sequences sequentially by the tree that is

based on genetic distances.

-- The most closely related sequences are aligned first, then additional sequences or groups are added according to initial alignments

-- Genetic distance: no. of mismatched positions divided by the total no. of matched positions (positions opposite a gap are not scored)

-- Sequence contributions to MSA are weighted according to their relationships on the tree-- weighting scheme: the more distant, the higher the weight-- Context (neighbor amino acid) is taken into account for the gap penality-- Gap score is adapted to force gaps to open at the same position.

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S1 S3 S2 S5 S6

DP alignment

S1 sequenceS3 sequence

S2 sequenceS5 sequenceS6 sequence

First align S1 and S3, S5 and S6, then align (S5,S6) and S2. Finally

Pairwise alignment

31

Tree AlignmentsAssume that there is a tree T=(V,E) whose leaves are the sequences. • Associate a sequence in each internal node.• Tree-score(K) = ∑(i,j)Escore(xi,xj).

Finding the optimal assignment of sequences to the internal nodes is NP Hard.

We will meet again this problem in the study ofPhylogenetic trees

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Star Alignments

Rather then summing up all pairwise alignments, select a fixed sequence x0 as a center, and set

Star-score(K) = ∑j>0score(x0,xj).

The algorithm to find optimal alignment: at each step, add another sequence aligned with x0, keeping old gaps and possibly adding new ones.

33

Multiple Sequence Alignment – Approximation Algorithm

Polynomial time algorithm:

assumption: the cost function δ is a distance function:

•

•

• (triangle inequality)

Let D(S,T) be the value of the minimum global alignment between S and T.

0),( xx

),(),(),( yxzyzx

0),(),( xyyx

34

Multiple Sequence Alignment – Approximation Algorithm (cont.)

Polynomial time algorithm:The input is a set Γ of k strings Si.1. Find the string S1 that minimizes

1\

1,SS

SSD

2. Call the remaining strings S2, …,Sk.

3. Add a string to the multiple alignment that initially contains only S1 as follows:

• Suppose S1, …,Si-1 are already aligned as S’1, …,S’i-1. Add Si by running dynamic programming algorithm on S’1 and Si to produce S’’1 and S’i.

• Adjust S’2, …,S’i-1 by adding spaces to those columns where spaces were added to get S’’1 from S’1.

• Replace S’1 by S’’1.

35

Multiple Sequence Alignment – Approximation Algorithm (cont.)

Time analysis:• Choosing S1 – running dynamic programming algorithm times – O(k2n2)• When Si is added to the multiple alignment, the length of S1 is at most in, so the time to add all k strings is

2k

1

1

22k

i

nkOninO

36

Multiple Sequence Alignment – Approximation Algorithm (cont.)

Error analysis:

• M - The alignment produced by this algorithm.

For all i, d(1,i)=D(S1,Si)

(we performed optimal alignment between S’1 and Si and )0),(

k

i

k

ijj

jidMv1 1

,•

• d(i,j) - the distance M induces on the pair Si,Sj.

• M* - optimal alignment.

37

Multiple Sequence Alignment – Approximation Algorithm (cont.)

Error analysis:

k

llSSDk

21,)1(2

k

jjSSDk

21,

2)1(2)()(

*

k

kMvMv

k

i

k

ijj

jidMv1 1

, jdidk

i

k

ijj

,1,11 1

k

l

ldk2

,1)1(2

Triangle inequality

k

i

k

ijj

jidMv1 1

** ,

k

i

k

ijj

ji SSD1 1

,

k

i

k

ijj

jSSD1 1

1,

Definition of S1

38

Iterative Refinement

Algorithm (Barton-Stenberg):

1. Align most similar xi, xj

2. Align xk most similar to (xixj)3. Repeat 2 until (x1…xN) are aligned

4. For j = 1 to N,Remove xj, and realign to x1…xj-1xj+1…xN

5. Repeat 4 until convergence

Note: Guaranteed to converge

39

40

Some ResourcesGenome Resources

Annotation and alignment genome browser at UCSChttp://genome.ucsc.edu/cgi-bin/hgGateway

Specialized VISTA alignment browser at LBNLhttp://pipeline.lbl.gov/cgi-bin/gateway2

ABC—Nice Stanford tool for browsing alignmentshttp://encode.stanford.edu/~asimenos/ABC/

Protein Multiple Aligners

http://www.ebi.ac.uk/clustalw/ CLUSTALW – most widely used

http://phylogenomics.berkeley.edu/cgi-bin/muscle/input_muscle.py MUSCLE – most scalable

http://probcons.stanford.edu/ PROBCONS – most accurate