Multiple Sequence Alignment
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- Patience Howard
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1 Multiple Sequence Alignment Reference: Gusfield, Algorithms on Strings, Trees & Sequences Some slides from: Jones, Pevzner, USC Intro to Bioinformatics Algorithms S. Batzoglu, Stanford Geiger, Wexler, Technion Ruzzo, Tompa U. Washington CSE 590bi Poch, Strasbourg A. Drummond, Auckland, NZ 1
2 Multiple Alignment vs. Pairwise Alignment Up until now we have only tried to align two sequences. What about more than two? And what for? A faint similarity between two sequences becomes significant if present in many Multiple alignments can reveal subtle similarities that pairwise alignments do not reveal 2
3 Multiple Alignment vs. Pairwise Alignment Pairwise alignment whispers multiple alignment shouts out loud Hubbard, Lesk, Tramontano, Nature Structural Biology
4 Multiple Alignment Definition Input: Sequences S 1, S 2,, S k over the same alphabet Output: Gapped sequences S 1, S 2,, S k of equal length 1. S 1 = S 2 = = S k 2. Removal of spaces from S i gives S i for all i 4
5 5 Example S 1 =AGGTC S 2 =GTTCG S 3 =TGAAC Possible alignment A - T G G G G - - T T A - T A C C C - G - Possible alignment A G - G T T G T G T - A - - A C C A - G C
6 6
7 Example Multiple sequence alignment of 7 neuroglobins using clustalx 7
8 Aggregation of deamidated human βb2-crystallin and incomplete rescue by α- crystallin chaperone. Michiel et al. Experimental Eye Research
9 Protein Phylogenies Example Kinase domain 9
10 Motivation again Common structure, function or origin may be only weakly reflected in sequence multiple comparisons may highlight weak signals Major uses: Identify and represent protein families Identify and represent conserved sequence or structure elements (e.g. domains) Deduce evolutionary history 10
11 MSA : central role in biology Comparative genomics Phylogenetic studies Hierarchical function annotation: homologs, domains, motifs Gene identification, validation MSA Structure comparison, modelling RNA sequence, structure, function Interaction networks Human genetics, SNPs DBD Therapeutics, drug design insertion domain Therapeutics, drug discovery binding sites / mutations LBD
12 Scoring alignments Given input seqs. S 1, S 2,, S k find a multiple alignment of optimal score Scores preview: Sum of pairs Consensus Tree 12
13 Sum of Pairs score Def: Induced pairwise alignment A pairwise alignment induced by the multiple alignment Example: Induces: x: AC-GCGG-C y: AC-GC-GAG z: GCCGC-GAG x: ACGCGG-C; x: AC-GCGG-C; y: AC-GCGAG y: ACGC-GAC; z: GCCGC-GAG; z: GCCGCGAG S(M) = Σ k<l σ(s k, S l ) 13
14 SOP Score Example Consider the following alignment: AC-CDB- -C-ADBD A-BCDAD Scoring scheme: match - 0 mismatch/indel - -1 SP score: =-12 14
15 Alignments = Paths Align 3 sequences: ATGC, AATC,ATGC A -- T G C A A T -- C -- A T G C 15
16 Alignment Paths A -- T G C x coordinate A A T -- C -- A T G C 16
17 Alignment Paths Align 3 sequences: ATGC, AATC,ATGC A -- T G C A A T -- C x coordinate y coordinate -- A T G C 17
18 Alignment Paths A -- T G C A A T -- C A T G C x coordinate y coordinate z coordinate Resulting path in (x,y,z) space: (0,0,0) (1,1,0) (1,2,1) (2,3,2) (3,3,3) (4,4,4) 18
19 Aligning Three Sequences Same strategy as aligning two sequences Use a 3-D Manhattan Cube, with each axis representing a sequence to align For global alignments, go from source to sink source sink 19
20 2-D vs 3-D Alignment Grid V W 2-D edit graph 3-D edit graph 20
21 2-D cell versus 2-D Alignment Cell In 2-D, 3 edges in each unit square In 3-D, 7 edges in each unit cube 21
22 Architecture of 3-D Alignment Cell (i-1,j-1,k-1) (i-1,j,k-1) (i-1,j-1,k) (i-1,j,k) (i,j-1,k-1) (i,j,k-1) (i,j-1,k) (i,j,k) 22
23 s i,j,k = max Multiple Alignment: Dynamic Programming s i-1,j-1,k-1 + δ(v i, w j, u k ) s i-1,j-1,k + δ (v i, w j, _ ) s i-1,j,k-1 + δ (v i, _, u k ) s i,j-1,k-1 + δ (_, w j, u k ) s i-1,j,k + δ (v i, _, _) s i,j-1,k + δ (_, w j, _) s i,j,k-1 + δ (_, _, u k ) cube diagonal: no indels face diagonal: one indel edge diagonal: two indels δ(x, y, z) is an entry in the 3-D scoring matrix 23
24 Running Time For 3 sequences of length n, the run time is O(n 3 ) For k sequences, build a k-dimensional cube, with run time O(2 k n k ) [another 2 k factor for affine gaps] Impractical for most realistic cases NP-hard (Elias 03 for general matrices) 24
25 Minimum cost SOP We use min cost instead of max score Find alignment of minimal cost Assumption (for the approximation algs to follow): the cost function δ is a distance function δ(x,x) = 0 (also for gaps) δ(x,y) = δ(y,x) 0 δ(x,y) + δ(y,z) δ(x,z) (triangle inequality) 26
26 Forward Dynamic Programming An alternative approach to DP. Useful for pairwise (and multiple) alignment: D(v) opt value of path source v p(w) best-yet solution of path source w When D(v) is computed, send its value forward on the arcs exiting from v: For v w: p(w)=min{p(w),d(v)+cost(v,w)} Once p(w) has been updated by all incoming edges that value is optimal; set as D(w) 27
27 Forward Dynamic Programming (2) Maintain a queue of nodes whose D is not set yet For the node w at the head of the queue: Set D(w) p(w) and remove out-neighbor x of w update p; if x is not in the queue add it at the end Breaking ties lexicographically Only x-s with some forward transmission are added to the queue Same complexity as the regular (backwards) DP 28
28 Faster DP Algorithm for MultiAlign Carillo-Lipman 88, demonstrated on three sequences Key: after computing D(v), with a little extra computation, we may already know that v will not on any optimal solution. k,l, k<l compute f kl (i,j) = opt pairwise alignment score of suffixes S k (i+1,..n 1 ), S l (j+1,..n 2 ). Use forward DP. If a known soln of cost z, and if D(i,j,k)+ f 12 (i,j) + f 13 (i,k) +f 23 (j,k) > z Do not send D(i,j,k) forward Guarantees opt soln no improved time bound, but often saves a lot in practice. 29
29 An approximation algorithm Gusfield 93 Compute a center string, minimizing the sum of pairwise distances to the other strings Use it as a pivot for the alignment D(S,T) - cost of minimum pairwise global alignment between S and T 30
30 The Center Star algorithm Input: Γ - set of k strings S 1,,S k. 1. Find the string S* Γ (center) that minimizes 2. Denote S 1 =S* and the rest of the strings as S 2,,S k 3. Iteratively add S 2,,S k to the alignment as follows: a. Suppose S 1,,S i-1 are already aligned as S 1,,S i-1 b. Optimally align S i to S 1 to produce S i and S 1 aligned ( *, S ) c. Adjust S 2,,S i-1 by adding spaces where spaces were added to S 1 { S } S Γ\ * D S d. Replace S 1 by S 1 31
31 Inheriting gaps x: AGAC y: ATGA center z: ATGGA 1: y: ATGAx: A-GAC 2: y: ATG-A Z: ATGGA y: ATG-Ax: A-G-AC Z: ATGGA- 32
32 Running time Choosing S 1 execute DP for all sequence-pairs - O(k 2 n 2 ) Adding S i to the alignment - execute DP for S i, S 1 - O(i n 2 ). (In the i th stage the length of S 1 can be at most i n) k 1 i= 1 O i ( 2 ) = ( 2 2 n O k n ) total complexity 33
33 Approximation ratio M* - An optimal alignment M For all i: d(1,i)=d(s 1,S i ) (we perform optimal alignment between S 1 and S i and δ(-,-) = 0 ) - The alignment produced by this algorithm d(i,j) - The distance M induces on the pair S i,s j k k ( M ) = d( i, j) = 2 d( i j) v, i= 1 j= 1 j i recall D(S,T) min cost of alignment between S and T i< j 34
34 Approximation ratio (2) v = v k k ( M ) = d( i, j) d( 1, i) + d( 1, j) 2( k 1) i= 1 j= 1 j i k l= 2 k d k ( 1, l) = = 2( k 1) k k ( ) i= 1 j= 1 j i k j= 2 k k l= 2 ( ) D S1, ( * ) * M = d ( i, j) (, ) k k i= 1 j= 2 i= 1 j= 1 j i ( ) D S1, S j Triangle inequality + symmetry k k i= 1 j= 1 j i ( ) D S1, S j S l D S i S j i : k j= 2 v( M ) v( M ) Definition of S 1 : 2( k 1) k * k ( 1, S j ) D( Si, S j ) D S j= 1 j i 35 2
35 Theorem (Gusfield 93) We have proved: The center star algorithm is a polynomial algorithm that guarantees a solution at most twice the optimum. a 2-approximation an approximation ratio of 2 36
36 Steiner/consensus string Input: Γ - set of k strings S 1,,S k. D(X,Y) score of aligning X, Y. S arbitrary sequence (unrelated to Γ) The consensus error of S relative to Γ: E(S) = Σ i k D(S, S i ) S* is an optimal Steiner string for Γ if it minimizes E(S) Different objective function linear no of terms No direct relation to multialign! (for now) 37
37 Thm: Assume D satisfies triangle ineq. Then S Γ that gives a 2-approximation. E ( S) = S D( S, Si ) = Optimal Steiner String: Approximation Pf: Pick S Γ ( k S i 2) D( S, S*) + ( D ( S, S *) D( S*, S )) + S D( S, S i S*) + S = ( k 2) DSS (, *) + ES ( *) i S D( S*, S i i ) E( S) ( k 2) Pick S Γ closest to S* (not constructively) + 1< * E( S ) k S Γ E ( S*) = D( S*, S ) k D( S, S*) i i 38 2
38 Optimal Steiner String: Approximation Pick S Γ that minimizes E(S). S gives a 2-approximation. 39
39 Consensus string from MA The consensus string of a MA is obtained by taking the most frequent character in each position S*: AC-GC-GAG x: AC-GCGG-C y: AC-GC-GAG z: GCCGA-GAG u: AC-T-GGCA v: -CAGT-GAG w: AC-GC-GAG Alignment error: S(M) = Σ k σ(s k, S*) The opt consensus MA of the set of input 40 sequences Γ: MA with least alignment error
40 Pf: ex. Consensus multiple alignment Thm: opt soln of consensus MA = Steiner string (up to spaces) 41
41 Tree Multiple Alignment Input: Tree T, a string for each leaf Phylogenetic (also Tree) alignment for T: Assignment of a string to each internal node Score (weighted) sum of scores along edges Goal: find tree alignment of optimal score Consensus =? GTTG CTGG GTTG tree alignment where T is a star CTGG CCGG CTTG GTTC 43
42 Tree MA complexity NP-hard Poly time approximations: 2-approximation Better approximation with more time (PTAS) 44
43 Lifted alignment The seq. label at every internal node is lifted from one of its children Lifted: CTGG Not lifted: GTTG CTGG GTTG GTTG CCGG CTTG GTTC CTTG GTTG CTGG CCGG CTTG GTTC 45
44 A 2-approximation to Tree MSA [Jiang, Wang, Lawler 1996] Assumes triangle inequality (non constructively) Transform an optimal tree T* to a lifted alignment T L : At each internal node v, assign seq. of a child that is closest to the optimal label of v S* v S S 1 S 2 S 3 S 4 S 1 S 2 S 3 S 4 Claim: T L has at most twice the distance 46 of T*
45 Pf sketch: cost(t L ) 2 cost(t*) In T L, take e=(v,w), v=pa(w) with labels S j for v, S i for w, S i S j D(S j,s i ) D(S j,s * v) + D(S * v,s i ) 2D(S i,s * v) (why?) Path P e from leaf labeled S i up to v has cost: D(S j,s i ) in T L At least D(S * v,s i ) in T* Paths {P e } are edge disjoint and cover all nonzero edges in T L 47
46 Dynamic Programming alg for optimal lifted alignment d(v,s) distance of the best lifted alignment of T v s.t. string S is assigned to node v d(v,s) = Σ w min T [D(S,T)+d(w,T)] here w child of v, T string at a leaf of T w Complexity: k = no of leaves, tot length N Compute all pairwise leaf distances in O(N 2 ) Computation per internal node: O(k 2 ) O(N 2 +k 3 ) (can do O(N 2 +k 2 )) 48
47 Wrapping up lifted alignment a lifted alignment that is 2 OPT We can find the min cost lifted alignment in poly time That alignment is also 2 OPT Thm: lifted alignment alg gives a poly-time 2-approximation to Tree Alignment 49
48 Profile Representation of MA - 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 Alternatively, use log odds: p i (a) = fraction of a s in col i p(a) = fraction of a s overall log p i (a)/p(a) 50
49 51
50 Aligning a sequence to a profile Key in pairwise alignment is scoring two positions x,y: σ(x,y) For a letter x and a column C in a profile, σ(x,c)=value of x in col. C Invent a score for σ(x,-) Run the DP alg for pairwise alignment 52
51 Aligning alignments Given two alignments, how can we align them? Hint: use DP on the corresponding profiles. x GGGCACTGCAT y GGTTACGTC-- Alignment 1 z GGGAACTGCAG w GGACGTACC-- Alignment 2 v GGACCT----- x GGGCACTGCAT y GGTTACGTC-- z GGGAACTGCAG w GGACGTACC-- v GGACCT
52 Profile-profile scoring Fix a position in the alignment p i prob (i in 1 st profile); q i in 2 nd profile Expected score: Σ ij p i q j s ij Other scores in use: Euclidean distance Pearson correlation KL-divergence (relative entropy) 54
53 Multiple Alignment: Greedy Heuristic Choose most similar pair of sequences and combine into a profile, thereby reducing alignment of k sequences to an alignment of k-1 sequences/profiles. Repeat u 1 = ACGTACGTACGT u 1 = ACg/tTACg/tTACg/cT k u 2 = TTAATTAATTAA u 3 = ACTACTACTACT u 2 = TTAATTAATTAA u k = CCGGCCGGCCGG k-1 u k = CCGGCCGGCCGG 56
54 Progressive Alignment A variation of greedy algorithm with a somewhat more intelligent strategy for choosing the order of alignments. 57
55 Progressive alignment Align sequences (pairwise) in some (greedy) order Decisions (1) Order of alignments (2) Alignment of group to group (3) Method of alignment, and scoring function 58
56 Guide tree A this? B C D E A or this? B C D E F 59
57 60 Multiple sequence alignment (MSA) A B C D E Pairwise distance table Guide tree A B C D E progressive MSA E. Privman Phylogeny workshop TAU 09
58 ClustalW Thompson, Higgins, Gibson 94 Popular multiple alignment tool today Three-step process 1.) Construct pairwise alignments 2.) Build guide tree 3.) Progressive alignment guided by the tree 74
59 Step 1: Pairwise Alignment Aligns each pair of sequences, giving a similarity matrix Similarity = exact matches / sequence length (percent identity) v 1 v 2 v 3 v 4 v 1 - v v v (.17 means 17 % identical) 75
60 Step 2: Guide Tree Use the similarity method to create a guide tree by applying some clustering method* Guide tree roughly reflects evolutionary relations *ClustalW uses the neighbor-joining method (to be described later in the course) 76
61 Step 2: Guide Tree (cont d) v 1 v 2 v 3 v 4 v 1 - v v v v 1 v 3 v 4 v 2 Calculate: v 1,3 = alignment (v 1, v 3 ) v 1,3,4 = alignment((v 1,3 ),v 4 ) v 1,2,3,4 = alignment((v 1,3,4 ),v 2 ) 77
62 Step 3: Progressive Alignment Start by aligning the two most similar sequences Using the guide tree, add in the most similar pair (seq-seq, seq-prof or prof-prof) Insert gaps as necessary Many ad-hoc rules: weighting, different matrices, special gap scores. FOS_RAT FOS_MOUSE FOS_CHICK FOSB_MOUSE FOSB_HUMAN PEEMSVTS-LDLTGGLPEATTPESEEAFTLPLLNDPEPK-PSLEPVKNISNMELKAEPFD PEEMSVAS-LDLTGGLPEASTPESEEAFTLPLLNDPEPK-PSLEPVKSISNVELKAEPFD SEELAAATALDLG----APSPAAAEEAFALPLMTEAPPAVPPKEPSG--SGLELKAEPFD PGPGPLAEVRDLPG-----STSAKEDGFGWLLPPPPPPP LPFQ PGPGPLAEVRDLPG-----SAPAKEDGFSWLLPPPPPPP LPFQ.. : **. :.. *:.* *. * **: Dots and stars show how well-conserved a column is. 78
63 79 Iterative Multiple sequence alignment A B C D E Pairwise distance table Take sequences out; realign to the rest Guide tree A B C D E MSA E. Privman Phylogeny workshop TAU 09
64 Multiple Alignment: History Sankoff Formulated multiple alignment problem and gave dynamic programming solution Carrillo- Lipman Branch and Bound approach for MSA Feng- Doolittle Progressive alignment Thompson- Higgins- Gibson- ClustalW > 40K citations! Most popular multiple alignment program Morgenstern et al. - DIALIGN Segment-based multiple alignment 2000 Notredame- Higgins- Heringa- T- coffee Using the library of pairwise alignments 2002 MAFFT 2004 MUSCLE 2005 ProbCons Clustal Omega Still a lot to be done! 80
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