1. Multiple alignment
One of the most essential tools in molecular biology
Finding highly conserved subregions or embedded patterns of a set of biological sequences
Conserved regions usually are key functional regions, prime targets for drug developments
Estimation of evolutionary distance between sequences
Prediction of protein secondary/tertiary structure
Practically useful methods only since 1987 (D. Sankoff)
Before 1987 they were constructed by hand
Dynamic programming is expensive
MSA gives stronger signal for sequence similarity than pairwise alignment – conserved pattern may be so dissimilar or dispersed that it cannot be detected when just two strings are aligned
Build protein family – collection of proteins with similar structure, function and evolutionary history
Consensus sequence – representative sequence constructed from MSA of members of protein family
Used in databases
Estimate evolutionary distance by assuming consensus sequence is ancestor
Other uses: Sequence assembly, as in ARACHNE

2. . Alignment between globins (human beta globin, horse beta globin,
An Alignment between globins produces by an alignment program Clustal.
The proteins that appear in the alignment are human beta globin,
horse beta globin, human alpha globin, horse alpha globin,
cyanohaemoglobin, whale myoglobin and leghaemoglobin in that order.
The boxes mark the seven alpha helices composing each globin.
Alignment between globins (human beta globin, horse beta globin,
human alpha globin, horse alpha globin, cyanohaemoglobin,
whale myoglobin, leghaemoglobin) produced by Clustal.
Boxes mark the seven alpha helices composing each globin. .

4. Definition
Given strings x1, x2 … xk a multiple (global) alignment maps them to strings x’1, x’2 … x’k that may contain spaces where
|x’1| = |x’2| = … = |xk’|
The removal of all spaces from x’i leaves xi, for 1 i  k

5. Definitions Multiple Alignment Motif
A rectangular arrangement, where each row consists of one protein sequence padded by gaps, such that the columns highlight similarity/conservation between positions
Motif
A conserved element of a protein sequence alignment that usually correlates with a particular function
Motifs are generated from a local multiple protein sequence alignment corresponding to a region whose function or structure is known

6. Example of motif
NAYCDEECK
NAYCDKLC-
-GYCN-ECT
NDYC-RECR
Motifs are conserved and hence predictive of any subsequent occurrence of such a structural/functional region in any other novel protein sequence

7. Scoring multiple alignments
Ideally, a scoring scheme should
Penalize variations in conserved positions higher
Relate sequences by a phylogenetic tree
Tree alignment
Usually assume
Independence of columns
Quality computation
Entropy-based scoring
Compute the Shannon entropy of each column
Minimize the total entropy
Steiner string
Sum-of-pairs (SP) score

8. Tree alignment Ideally:
Find alignment that maximizes probability that sequences evolved from common ancestor x y ? z w v

9. Tree alignment
Model the k sequences with a tree having k leaves (1 to 1 correspondence)
Compute a weight for each edge, which is the similarity score
Sum of all the weights is the score of the tree
Assign sequences to internal nodes so that score is maximized

10. Tree alignment example
Match +1, gap -1, mismatch 0
If x=CT and y=CG, score of 6
CAT
CTG x y
GT/CT = +1, CAT/CT=+1, CG/CG=+2, CTG/CG=+1,CT/CG=+1 CG GT

11. Analysis The tree alignment problem is NP-complete
Holds even for the special case of star alignment
“lifting alignment” gives a 2-approximate algorithm
The generalized tree alignment problem (find the best tree) is also NP-complete
Special cases for different kinds of scoring metrics
Size of alphabet
Triangle inequality

12. Consensus representations
Relative frequencies of symbols in each column
Adds up to 1 in each column
Steiner string
Minimize the consensus error
May not belong to the set of input strings
Consensus string for a given multiple alignment
Choose optimal character in every column
Consensus string is the concatenation of these characters
Alignment error of a column is the distance-sum to the optimal character of all symbols in the column
Alignment error of a consensus string is the sum of all column errors
Optimal consensus string: optimize over all multiple alignments
Signature representation
Regular expression
Helicase protein: [&H][&A]D[DE]xn[TSN]x4[QK]Gx7[&A]
& is any amino acid in {I,L,V,M,F,Y,W}

13. Steiner string and consensus error metric
Minimize Σ D(s,xi), over all possible strings s
String smin is called the Steiner string
May not belong to the set of inputs
NP-complete
Consensus error metric based on similarity to the steiner string
center string provides an approximation factor of 2 i

14. Relating alignment error and consensus error
Let s be the steiner string for a string set X = {xi} and c be the optimal consensus string
For any multiple alignment M of X,
Let xM be the consensus string
Alignment error of xM = consensus error using xM ≥ consensus error using s
Consider the star multiple alignment N using s
Alignment error of N using s = consensus error using s
Alignment error of N using s ≤ Alignment error of any multiple alignment
N is the optimal multiple alignment and s (after removing gaps) is the consensus string
Steiner string provides the optimal consensus string

15. Aligning to family representations
Profile
Apply dynamic programming
Score depends on the profile
Consensus string
Signature representations
Align to regular expressions / CFG/ …

16. 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

17. Sum of Pairs (cont’d)
The sum-of-pairs (SP) score of a multiple alignment A is the sum of the scores of all induced pairwise alignments
S(A) = i<j S(Aij)
Aij is the induced alignment of xi, xj
Drawback: no evolutionary characterization
Every sequence derived from all others

18. Optimal solution for SP scores
Multidimensional Dynamic Programming
Generalization of pair-wise alignment
For simplicity, assume k sequences of length n
The dynamic programming array is k-dimensional hyperlattice of length n+1 (including initial gaps)
The entry F(i1, …, ik) represents score of optimal alignment for s1[1..i1], … sk[1..ik]
Initialize values on the faces of the hyperlattice

19. k= k –1=7 A S V

20. Complexity Space complexity: O(nk) for k sequences each n long.
Computing at a cell: O(2k). cost of computing δ.
Time complexity: O(2knk). cost of computing δ.
Finding the optimal solution is exponential in k
Proven to be NP-complete for a number of cost functions

21. Algorithms Faster Dynamic Programming Star alignment
Carrillo and Lipman 88 (CL)
Pruning of hyperlattice in DP
Practical for about 6 sequences of length about 200.
Star alignment
Progressive methods
CLUSTALW
PILEUP
Iterative algorithms
Hidden Markov Model (HMM) based methods

22. CL algorithm Find pairwise alignment
Trial multiple alignment produced by a tree, cost = d
This provides a limit to the volume within which optimal alignments are found
Specifics
Sequences x1,..,xr.
Alignment A, score = s(A)
Optimal alignment A*
Aij = induced alignment on xi,..,xj on account of A
D(xi,xj) = score of optimal pairwise alignment of xi,xj ≥ s(Aij )

23. CL algorithm d ≤ s(A*) = s(A*uv) + Σ Σ s(A*ij) ≤
s(A*uv) + Σ Σ D(xi,xj)
s(A*uv) ≥ d - Σ Σ D(xi,xj) = B(u,v)
Compute B(u,v) for each (u,v) pair
Consider any cell f with projection (s,t) on u,v plane.
If A* passes through f then A*uv passes through (s,t)
beststuv = best pairwise alignment of xu,xv that passes through (s,t).
beststuv = score of the prefixes up to (s,t) + cost(xsi,xsj) + score of suffixes after (s,t) i
i < j
(i,j) ≠ (u,v) i
i < j
(i,j) ≠ (u,v)

24. CL algorithm If beststuv < B(u,v), then
A* cannot pass through cell f
Discard such cells from computation of DP
Can prune for all (u,v) pairs

25. Star alignment Heuristic method for multiple sequence alignments
Select a sequence c as the center of the star
For each sequence x1, …, xk such that index i  c, perform a Needleman-Wunsch global alignment
Aggregate alignments with the principle “once a gap, always a gap.”
Consider the case of distance (not scores)
Find multiple alignment with minimum distance

26. Star alignment example
MPE
| |
MKE
MSKE
| ||
M-KE
S1: MPE
S2: MKE
S3: MSKE
S4: SKE s1 s3 s2
SKE ||
MKE
M-PE
M-KE
MSKE
S-KE
M-PE
M-KE
MSKE
MPE
MKE s4

27. Choosing a center
Try them all and pick the one with the least distance
Let D(xi,xj) be the optimal distance between sequences xi and xj.
Given a multiple alignment A, let c(Aij) be the distance between xi and xj that is induced on account of A.
Calculate all O(k2) alignments, and pick the sequence xi that minimizes the following as xc
Σ D(xi,xj)
The resulting multiple alignment A has the property that c(Aci) = D(xc,xi).
j ≠ i

28. Analysis Assuming all sequences have length n
O(k2n2) to calculate center
Step i of iterative pairwise alignment takes O((i.n).n) time
two strings of length n and i.n
O(k2n2) overall cost
Produces multiple sequence alignments whose SP values are at most twice that of the optimal solutions, provided triangle inequality holds.

29. Bound analysis Let M = Σ c(A1i) = Σ D(x1,xi), assume x1 is the center
2 c(A) = Σ Σ c(Aij) ≤ Σ Σ [c(A1i) + c(A1j)] =
2(k-1) Σ c(A1i) = 2(k-1) M
2 c(A*) = Σ Σ c(A*ij) ≥ Σ Σ D(xi,xj) ≥ k Σ c(A1i) = k M
c(A)/c(A*) <= 2(k-1)/k <= 2
i = 2
i = 2 i
j ≠ i i
j ≠ i
i = 2 i
j ≠ i i
j ≠ i
i = 2

30. Consensus error
Center string c also provides an approximation factor of 2 under consensus error (score) metric
Assume triangle inequality
Let E(x) denote the consensus error wrt string x.
Let z be the Steiner string
E(z) = Σ D(z,xi) i

31. Consensus error For any string y in the input set,
E(y) = Σ D(y,xi) ≤ Σ [D(y,z) + D(z,xi)] =
(k-2) D(y,z) + D(y,z) + Σ D(z,xi) = (k-2) D(y,z) + E(z)
Pick y* from input set that is closest to z.
E(z) = Σ D(z,xi) ≥ k D(y*,z)
E(y*)/E(z) ≤ [(k-2) D(y*,z) +E(z)]/E(z)
≤ (k-2) D(y*,z) / [k D(y*,z)] + 1 ≤ 2-2/k <= 2
E(c) ≤ E(y*) i
y ≠ xi
y ≠ xi i

32. ClustalW Progressive alignment 3 steps:
All pairs of sequences are aligned to produce a distance matrix (or a similarity matrix)
A rooted guide tree is calculated from this matrix by the neighbor-joining (NJ) method
Neighbor Joining – Saitou, 1987
The sequences are aligned progressively according to the branching order in the guide tree

33. ClustalW example
S1 ALSK
S2 TNSD
S3 NASK
S4 NTSD

34. ClustalW example S1 ALSK S2 TNSD S3 NASK S4 NTSD S1 S2 S3 S4 9 4 7 8 3
All pairwise
alignments S1 S2 S3 S4 9 4 7 8 3
Distance Matrix

35. ClustalW example S1 ALSK S2 TNSD S3 NASK S4 NTSD S1 S2 S3 S4 9 4 7 8 3
All pairwise
alignments S1 S2 S3 S4 9 4 7 8 3 S3 S1 S2 S4
Rooted Tree
Neighbor
Joining
Distance Matrix

36. Multiple Alignment Steps
ClustalW example
S1 ALSK
S2 TNSD
S3 NASK
S4 NTSD
Multiple Alignment Steps
Align S1 with S3
Align S2 with S4
Align (S1, S3) with (S2, S4)
All pairwise
alignments S1 S2 S3 S4 9 4 7 8 3 S3 S1 S2 S4
Rooted Tree
Neighbor
Joining
Distance Matrix

37. Multiple Alignment Steps
ClustalW example
Multiple Alignment Steps
-ALSK
NA-SK
S1 ALSK
S2 TNSD
S3 NASK
S4 NTSD
-ALSK
-TNSD
NA-SK
NT-SD
Align S1 with S3
Align S2 with S4
Align (S1, S3) with (S2, S4)
-TNSD
NT-SD
All pairwise
alignments
Multiple
Alignment S1 S2 S3 S4 9 4 7 8 3
Neighbor
Joining
Rooted Tree
Distance Matrix

38. Other progressive approaches
PILEUP
Similar to CLUSTALW
Uses UPGMA to produce tree

39. Problems with progressive alignments
Depend on pairwise alignments
If sequences are very distantly related, much higher likelihood of errors
Care must be made in choosing scoring matrices and penalties

40. Iterative refinement in progressive alignment
One problem of progressive alignment:
Initial alignments are “frozen” even when new evidence comes
Example:
x: GAAGTT
y: GAC-TT
z: GAACTG
w: GTACTG
Frozen!
Now clear that correct y = GA-CTT

41. Multiple alignment tools
Clustal W (Thompson, 1994)
Most popular
PRRP (Gotoh, 1993)
HMMT (Eddy, 1995)
DIALIGN (Morgenstern, 1998)
T-Coffee (Notredame, 2000)
MUSCLE (Edgar, 2004)
Align-m (Walle, 2004)
PROBCONS (Do, 2004)

42. Evaluating multiple alignments
Balibase benchmark (Thompson, 1999)
De-facto standard for assessing the quality of a multiple alignment tool
Manually refined multiple sequence alignments
Quality measured by how good it matches the core blocks
Clustal W performs the best
Problems of Clustal W
Once a gap, always a gap
Order dependent

43. Computationally challenging problems
Scalable multiple alignment
Dynamic programming is exponential in number of sequences
Practical for about 6 sequences of length about 200.

44. Quick Primer on NP completeness
Polynomial-time Reductions
If we could solve X in polynomial time, then we could also solve Y in polynomial time
YP X
Class NP
Set of all problems for which there exists an efficient certifier
P = NP?
General transformation of checking a solution to finding a solution
Can arbitrary instances of Y be solved using a polynomial number of standard computational steps plus a polynomial number of calls to a black box that solves X
Efficient checking vs efficient solving

45. NP-completeness X is NP-complete if
XNP
For all YNP, YPX
If X is NP-complete, X is solvable in polynomial time iff P=NP
Satisfiability is NP-complete
If Y is NP-complete and X is in NP with the property that YPX, then X is NP complete