Multiple Sequence Alignment

Transcription

1 . Multiple Sequence lignment utorial #4 Ilan ronau Multiple Sequence lignment Reminder S = S = S = Possible alignment Possible alignment

2 Multiple Sequence lignment Reminder Input: Sequences S, S,, S over the same alphabet Output: apped sequences S, S,, S of equal length. S = S = = S. Removal of spaces from S i obtains S i Sumofpairs (SP) score for a multiple global alignment is the sum of scores of all pairwise alignments induced by it. Multiple Sequence lignment pproximation lgorithm he star algorithm: Input: Γ set of strings S,,S.. For each i<j calculate D(S i,s j ).. Find the string S (center) that minimizes S Γ\ { S '} ( ', S). Denote S =S and the rest of the strings as S,,S. Iteratively add S,,S to the alignment as follows: a. Suppose S,,S i are already aligned as S,,S i b. lign S i to S to produce S i and S aligned c. djust S,,S i by adding spaces where spaces were added to S d. Replace S by S D S 4

3 Multiple Sequence lignment pproximation lgorithm ime analysis: hoosing S execute DP for all sequencepairs O( n ) dding S i to the alignment execute DP for S i, S O(i n ). (In the i th stage the length of S can be upto i n) ( n ) = O( n ) O i total complexity 5 Multiple Sequence lignment pproximation lgorithm pproximation ratio: M* optimal alignment M he alignment produced by this algorithm d(i,j) the distance M induced on the pair S i,s j v ( M ) = d ( i, j ) = d ( i, j ) j= i< j For all i: d(,i) D(S,S i ) d(,i) is minimal cost of alignment between S and S i an alignment between S and S i implies an alignment between S and S i 6

4 Multiple Sequence lignment pproximation lgorithm pproximation ratio: v ( M) = d( i, j) d(, i) + d(, j) = ( ) v j= l= d (, l) l= ( ) ( ) D S, S = ( ) j= * * ( M ) = d ( i j) j= j= ( ) D S, S j riangle inequality j=, D( S i, S j ) j= ( ) D S, S j l i : j= v( M ) ( ) v( M ) Definition of S : * (, S j) D( Si, S j) D S j= 7 Multiple Sequence lignment Reminder Problem: onventional M does not model correctly evolutionary relationships 8 4

5 5 9 Input: X set of sequences phylogenetic tree on X (leaves labeled by X) Output: labels on internal vertices of, s.t. sum of costs of all edges of is minimal. How do we label internal vertices? Sequences Profiles (multiple alignments) ree lignment profile of a M of length n over alphabet Σ is a ( Σ +)*n table. olumn i holds the distribution of Σ (and gap) in that position Profile lignment :

6 Profile lignment ligning 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) Solve using standard DP algorithms for pairwise alignment : lustal lgorithm Progressive M using a phylogenetic tree: t each point hold profiles for all leaves hoose neighboring leaves neighbors have common father in lign the two profiles to get the fatherprofile New profile replaces the two old ones in set of leafprofiles How do we obtain the phylogenetic tree? From pairwise distances between sequences lgorithms such as UPM, NeighborJoining, etc We discuss such algorithms later in the course lustalw more advanced version. Sequences/profiles are weighted 6

7 Lifted ree lignments Lifted tree alignment each internal node is labeled by one of the labels of its daughters Internal nodes are sequences and not profiles Example: S We ll show:. DP algorithm for optimal lifted tree alignment. Optimal lifted alignment is approximation of optimal tree alignment S S S S 6 Lifted ree lignments lgorithm S Input: X set of sequences S S S S 6 phylogenetic tree on X (leaves labeled by X) Output: lifted labels on internal vertices of, s.t. sum of costs of all edges of is minimal. Basic principle: calculate for every node v in, and sequence S in X: d(v,s) the optimal cost of v s subtree when it is labeled by S he cost of optimal tree is min{ d( root, S) } S X 4 7

8 Lifted ree lignments lgorithm S S S S S 6 d(v,s) the optimal cost of v s subtree when it is labeled by S Initialization: for leaf v labeled S v S = S d( v, S) = S S v v Recurrence: for internal node v with daughters u, u l d( v, S) = l min S ' X { D( S, S') + d( u, S') } orrectness: chec for suboptimal solution property omplexity: O( ) pairwise alignments O(n ). iterations O( depth())=o( ) For internal node v O( v ) wor v number of leaves in subtree of v i otal: O( (n +depth())) 5 Lifted ree lignments pproximation analysis S laim: Optimal L approximates general tree alignments We ll show construction of L which costs at most twice the optimal with sequencelabeled nodes (? can be generalized for profilelabeled nodes?) S S S S 6 Notations: * optimal labels S v * label of node v in * L our constructed L S vl label of node v in L 6 8

9 Lifted ree lignments pproximation analysis S S S S S 6 onstruction: We label the nodes bottomup. For node v with daughters u, u l we choose the label (from S L u,,s ull ) closest to S v * We need to show: D( L ) D(*) 7 Lifted ree lignments pproximation analysis S S S S S 6 nalysis: Some edges in L have cost Observe edges (v,u) of cost > : (v parent of u) P(v,u) the path in * from v to the leaf labeled by S u D(S v,s u ) D(S v,s v *) + D(S u,s v *) D(S u,s v *) D(P(v,u)) triangle inequality choice of S v triangle inequality D(S v,s u ) D(P(v,u)) If (u,v) and (u,v ) are two different edges with cost > in L, then P(u,v) and P(u,v ) are mutually disjoint in edges Q.E.D. 8 9

10 Lifted ree lignments pproximation analysis S Final Remars: S S S S 6 Lifted tree alignment L is only conceptual (we don t have *) Optimal L cannot cost more than L In case of profilelabeled nodes: construction and analysis OK when cost is still distance function 9