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

2. Multiple Sequence Alignment (cont)
Input: Sequences S1 , S2 ,…, Sk over the same alphabet
Output: Gapped sequences S’1 , S’2 ,…, S’k of equal length
|S’1|= |S’2|=…= |S’k|
Removal of spaces from S’i obtains Si
Sum-of-pairs (SP) score for a multiple global alignment is the sum of scores of all pairwise alignments induced by it.

3. Multiple Sequence Alignment Example
Consider the following alignment:
AC-CDB-
-C-ADBD
A-BCDAD
Scoring scheme: match - 0
mismatch/indel - -1
SP score: -3 -5 -4
=-12

4. Multiple Sequence Alignment Complexity
Given k strings of length n, there is a generalization of the DP algorithm that finds an optimal SP alignment:
Instead of a 2-dimensional table we have a k-dimensional table
Each dimension is of length ‘n’+1
Each entry depends on 2k-1 adjacent entries
Complexity: O(2knk)
This problem is known to be NP-hard (no polynomial-time algorithm)

5. Multiple Sequence Alignment Approximation Algorithm
We use cost instead of score
 Find alignment of minimal cost
Assumption: the cost function δ is a distance function
δ(x,x) = 0
δ(x,y) = δ(y,x) ≥ 0
δ(x,y) + δ(y,z) ≥ δ(x,z) (triangle inequality)
(e.g. cost of MM ≤ cost of two indels)
D(S,T) - cost of minimum global alignment between S and T

6. Multiple Sequence Alignment Approximation Algorithm
The ‘star’ algorithm:
Input: Γ - set of k strings S1, …,Sk.
Find the string S’ (center) that minimizes
Denote S1=S’ and the rest of the strings as S2, …,Sk
Iteratively add S2, …,Sk to the alignment as follows:
Suppose S1, …,Si-1 are already aligned as S’1, …,S’i-1
Align Si to S’1 to produce S’i and S’’1 aligned
Adjust S’2, …,S’i-1 by adding spaces where spaces were added to S’’1
Replace S’1 by S’’1

7. Multiple Sequence Alignment Approximation Algorithm
Time analysis:
Choosing S1 – execute DP for all sequence-pairs - O(k2n2)
Adding Si to the alignment - execute DP for Si , S’1 - O(i·n2).
(In the ith stage the length of S’1 can be up-to i· n)
total complexity

8. Multiple Sequence Alignment Approximation Algorithm
Approximation ratio:
M* - optimal alignment
M - The alignment produced by this algorithm
d(i,j) - the distance M induces on the pair Si,Sj
For all i: d(1,i)=D(S1,Si)
(we perform optimal alignment between S’1 and Si and δ(-,-) = 0 )

9. Multiple Sequence Alignment Approximation Algorithm
Triangle inequality
Approximation ratio:
Definition of S1:

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

11. Multiple Sequence Alignment Reminder
Input: Sequences S1 , S2 ,…, Sk over the same alphabet
Output: Gapped sequences S’1 , S’2 ,…, S’k of equal length
|S’1|= |S’2|=…= |S’k|
Removal of spaces from S’i obtains Si
Sum-of-pairs (SP) score for a multiple global alignment is the sum of scores of all pairwise alignments induced by it.

12. Multiple Sequence Alignment Reminder
The ‘star’ algorithm:
Input: Γ - set of k strings S1, …,Sk.
Find the string S1 (center) that minimizes
Iteratively add S2, …,Sk to the alignment
Finds MA costing at most twice the optimal cost!
Problem: Conventional MA does not model correctly evolutionary relationships

13. Tree Alignment Input: X - set of sequences
T – phylogenetic tree on X (leaves labeled by X)
Output: labels on internal vertices of T, s.t. sum of costs of all edges of T is minimal.
How do we label internal vertices?
Sequences
Profiles (multiple alignments)

14. : 3 Profile Alignment A T G C -
A profile of a MA of length n over alphabet Σ is a (| Σ |+1)*n table.
Column i holds the distribution of Σ (and gap) in that position A - T G C A 1 T 2 G 3 C -
: 3

15. (same goes for aligning two profiles)
Profile Alignment
Aligning a sequence to a profile:
Matching letter to position: weighted average of scores
Indels: introducing new columns gets special consideration
(same goes for aligning two profiles) A 1 T 2 G 3 C -
: 3

16. Clustal Algorithm Iteratively constructs MA for intermediate nodes
At each point holds profiles for all leaves
Chooses closest pair of neighbors
neighbors – have common father in T
distance - cost of optimal (pairwise) alignment
Aligns the two profiles to get the ‘father-profile’
Replaces the two leaves with their father
Analysis:
Initialization – O(k2) alignments
k-1 iterations
Iteration i involves k-i-1 new pairwise alignments
ClustalW – more advanced version.
Sequences/profiles are weighted

17. Lifted Tree Alignments
each internal node is labeled by one of the labels of its daughters
Internal nodes are sequences and not profiles
Example:
We’ll show:
DP algorithm for optimal lifted tree alignment
Optimal lifted alignment is 2-approximation of optimal tree alignment S1 S2 S3 S4 S6 S5

18. Lifted Tree Alignments Algorithm
Input: X - set of sequences
T – phylogenetic tree on X (leaves labeled by X)
Output: lifted labels on internal vertices of T, s.t. sum of costs of all edges of T is minimal.
Basic principle: calculate for every node v in T, and sequence S in X:
d(v,S) - the optimal cost of v’s subtree when it is labeled by S
The cost of optimal tree is

19. Lifted Tree Alignments Algorithm
d(v,S) - the optimal cost of v’s subtree when it is labeled by S
Initialization: for leaf v labeled Sv -
Recurrence: for internal node v with daughters u1,…ul -
Correctness: check for suboptimal solution property
Complexity: O(k2) pairwise alignments - O(n2k2) .
k-1 iterations
For internal node v - O(kv2) work
Total: O(k2(n2+depth(T)))
O(k2depth(T))=O(k3)

20. Lifted Tree Alignments Approximation analysis
Claim: Optimal LTA 2-approximates general tree alignments
We’ll show construction of LTA which costs at most twice the optimal TA with sequence-labeled nodes
(? can be generalized for profile-labeled nodes ?)
Notations:
T* - optimal TA labels
Sv* - label of node v in T*
TL – our constructed LTA
SvL - label of node v in TL

21. Lifted Tree Alignments Approximation analysis
Construction:
We label the nodes bottom-up.
For node v with daughters u1,…ul –
we choose the label (from Su1L ,…,SulL) closest to Sv*
We need to show: D(TL) ≤ 2D(T*)

22. Lifted Tree Alignments Approximation analysis
Some edges in TL have cost 0
Observe edges (v,u) of cost > 0:
Si- label of father(v)
Sj- label of daughter (u)
P(v,u) – the path in T* from v to the leaf labeled by Sj
D(Si,Sj) ≤ D(Si,Sv*) + D(Sj,Sv*) ≤ 2D(Sj,Sv*) ≤ 2D(P(v,u))
triangle inequality
choice of i
triangle inequality

23. Lifted Tree Alignments Approximation analysis
D(Si,Sj) ≤ 2D(P(v,u))
If (u,v) and (u’,v’) are two different edges with cost > 0 in TL, then P(u,v) and P(u’,v’) are mutually disjoint in edges
Q.E.D.
Final Remarks:
Lifted tree alignment TL is only conceptual (we don’t have T*)
Optimal LTA cannot cost more than TL
In case of profile-labeled nodes:
construction and analysis OK when cost is still distance function