1. Multiple Sequence Alignments.

2. The Global Alignment problem
AGTGCCCTGGAACCCTGACGGTGGGTCACAAAACTTCTGGA x y z
AGTGACCTGGGAAGACCCTGACCCTGGGTCACAAAACTC

3. Multiple Sequence Alignment
S1=AGGTC
Possible alignment A - T G C
S2=GTTCG
S3=TGAAC
Possible alignment A G - T C

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 L
Score 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 Alignment
Definition: Given stings S1, S2, …,Sk a multiple (global) alignment map them to strings S’1, S’2, …,S’k that may contain spaces, where:
|S’1|= |S’2|=…= |S’k|
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

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-GCGAG
y: ACGC-GAC; z: GCCGC-GAG; z: GCCGCGAG

8. Multiple Sequence Alignment
Given 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 - 1 adjacent 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))

10. Multidimensional Dynamic Programming
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) }

11. Multiple alignments
We use a matrix to represent the alignment of k sequences, K=(x1,...,xk). We assume no columns consists solely of blanks.
The common scoring functions give a score to each column, and set: score(K)= ∑i score(column(i)) x1 x2 x3 x4 M Q _ I L R - K P V
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).

12. SUM OF PAIRS
A common scoring function is SP – sum of scores of the projected pairwise alignments: SPscore(K)=∑i<j score(xi,xj). M Q _ I L R - K P V
Note that we need to specify the score(-,-) because a column may have several blanks (as long as not all entries are blanks).
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).

13. SUM OF PAIRS
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. M Q _ I L R - K P V

14. Example Consider the following alignment: a c - c d b - - c - a d b d
a - b c d a d
Using the edit distance and for ,
this alignment has a SP value of
= 12

15. 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)
Running Time:
Size of matrix: LN;
Where L = length of each sequence
N = number of sequences
Neighbors/cell: 2N – 1
Therefore………………………… O(2N LN) Y YZ XY
XYZ Z X XZ

16. Multiple Sequence Alignment
Given 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 Recall the notation:
and the following recurrence for V:
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).

18. 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):
Where
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”).

19. 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.

20. Time saving heuristics: Relevance tests
Heuristic: Avoid computing score(i) for irrelevant vectors. x1 x2 x3 x4 M Q _ I L R - K P V
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.

21. 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.
Build 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. Time saving heuristics: Relevance test
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:
≥ L + H(u,v) -

23. A 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 C G T O E C
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

24. Multiple Alignments With Profile
Consider the MSA
a b c - a
a b a b a
a c c b -
c b - b c
Its corresponding profile P is
C1 C C3 C C5
a 75% % %
b % %
c 25% 25% 50% %
− % 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 c

25. String-to-Profile Alignment
Scoring a column j is equivalent to aligning Sj 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 Alignmentx y z w
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

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)

28. Progressive Alignmentx y ? z w
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

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.

30. DP alignment Pairwise alignment S1 sequence S3 sequence S2 sequence
S1 S3 S2 S5 S6
First align S1 and S3, S5 and S6, then align (S5,S6) and S2. Finally

31. Tree Alignments
Assume 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 of
Phylogenetic trees

32. Star-score(K) = ∑j>0score(x0,xj).
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.

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
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

36. Multiple Sequence Alignment – Approximation Algorithm (cont.)
Error analysis:
M - The alignment produced by this algorithm.
d(i,j) - the distance M induces on the pair Si,Sj.
M* - optimal alignment.
For all i, d(1,i)=D(S1,Si)
(we performed optimal alignment between S’1 and Si and )

37. Multiple Sequence Alignment – Approximation Algorithm (cont.)
Triangle inequality
Error analysis:
Definition of S1

38. Iterative Refinement Remove xj, and realign to x1…xj-1xj+1…xN
Algorithm (Barton-Stenberg):
Align most similar xi, xj
Align xk most similar to (xixj)
Repeat 2 until (x1…xN) are aligned
For j = 1 to N,
Remove xj, and realign to x1…xj-1xj+1…xN
Repeat 4 until convergence
Note: Guaranteed to converge

40. Some Resources Genome Resources Protein Multiple Aligners
Annotation and alignment genome browser at UCSC
Specialized VISTA alignment browser at LBNL
ABC—Nice Stanford tool for browsing alignments
Protein Multiple Aligners
CLUSTALW – most widely used
MUSCLE – most scalable
PROBCONS – most accurate