Multiple Sequence Alignment

Transcription

1 Multiple Sequence Alignment Reference: Gusfield, Algorithms on Strings, Trees & Sequences, chapter 14 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 Revised Nov

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 1996.

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 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 - S 1 S 2 S 3 Possible alignment A G - G T T G T G T - A - - A C C A - G C S 1 S 2 S 3 S 1 = S 2 = S 3 5

6 Example 6

7 Example 1 Multiple sequence alignment of 7 neuroglobins using clustalx Identify and represent protein families. 7

8 Example 2 Aggregation of deamidated human βb2-crystallin and incomplete rescue by α-crystallin chaperone. Michiel et al. Experimental Eye Research 2010 Identify and represent conserved motifs (conserved common biological function). Shamir nro GC 8

9 Protein Phylogenies Example 3 Kinase domain Deduce evolutionary history 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 seq. elements (e.g. domains) Deduce evolutionary history

11 MSA : central role in biology Comparative genomics Phylogenetic studies Hierarchical function annotation: homologs, domains, motifs Gene identification, validation MACS 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 AC-GCGAG GCCGCGAG x: ACGCGG-C; x: AC-GCGG-C; y: y: ACGC-GAC; z: GCCGC-GAG; z: S(M) = Σ k<l σ(s k, S l ) k<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 2 sequences: ATGC, AATC A -- T G C A A T -- C 15

16 Alignment Paths Align 2 sequences: ATGC, AATC A -- T G C x coordinate A A T -- C 16

17 Alignment Paths Align 2 sequences: ATGC, AATC A -- T G C A A T -- C x coordinate y coordinate 17

18 Alignment Paths A -- T G C A A T -- C x coordinate y coordinate Resulting path in (x,y) space: (0,0) (1,1) (1,2) (2,3) (3,3) (4,4) 18

19 Alignments = Paths Align 3 sequences: ATGC, AATC,ATGC A -- T G C A A T -- C -- A T G C 19

20 Alignment Paths Align 3 sequences: ATGC, AATC,ATGC A -- T G C x coordinate A A T -- C -- A T G C 20

21 Alignment Paths Align 3 sequences: ATGC, AATC,ATGC A -- T G C A A T -- C x coordinate y coordinate -- A T G C 21

22 Alignment Paths Align 3 sequences: ATGC, AATC,ATGC 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) 22

23 2-D vs 3-D Alignment Grid V W 2-D edit graph 3-D edit graph 23

24 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 24

25 3-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 25

26 Architecture of 3-D Alignment (i-1,j-1,k-1) Cell (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) 26

27 Architecture of 3-D Alignment (i-1,j-1,k-1) Cell (i-1,j,k-1) (i-1,j-1,k) (i-1,j,k) Cube diagonal: no indels (i,j-1,k-1) (i,j,k-1) Edge: 2 indels Face diagonal: 1 indels (i,j-1,k) (i,j,k) 27

28 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: two indels δ(x, y, z) is an entry in the 3-D scoring matrix 28

29 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) 29

30 Minimum cost SOP We use min cost instead of max score Find alignment of minimal cost Observe: opt multialign score sum of optimal pairwise scores Shamir nro GC 30

31 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) 31

32 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 32

33 Faster DP Algorithm for MultiAlign Carillo-Lipman 88 Idea: 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. 33

34 Approximation Algorithms - assumption We use min cost instead of max 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 34

35 The Center Star algorithm Gusfield 1993 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 c. Adjust S 2,,S i-1 by adding spaces where spaces were added to S 1 d. Replace S 1 by S 1 { S } S Γ\ * ( *, S) D S 35

36 The Center Star algorithm (demonstration) X : A-G-AC AC x: AGAC y: ATGA <- center z: ATGGA w: AGTGA w Y : A-TG TG-A- X : A-- --G-AC Z : A-TGGA TGGA- W : AGTG-Ay Y : A-TG TG-A- W : AGTG-A- Inheriting gaps x z Y : ATG-A- Z : ATGGA-

37 The Center Star algorithm 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 up-to i n) k 1 i= 1 O i ( 2 ) = ( 2 2 n O k n ) total complexity 37

38 The Center Star algorithm Approximation ratio M* M - An optimal alignment - The alignment produced by this algorithm d(i,j) - The distance M induces on the pair S i,s j ( M ) = d( i, j) = 2 d( i j) recall D(S,T) min cost of alignment between S and T For all i: d(1,i)=d(s 1,S i ) (we perform optimal alignment between S 1 and S i and δ(-,-) = 0 ) k k v, i= 1 j= 1 j i i< j 38

39 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, The Center Star algorithm Approximation ratio (2) 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 39 2

40 The Center Star algorithm 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 40

41 Steiner String and Consensus MA 41

42 Consensus error & Steiner string -definitions Input: set of k strings Γ ={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 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) 42

43 Thm: Assume D satisfies triangle ineq. Then S Γ that guarantees an approximation ratio 2. = S ( ( ( ) ( )) D S, S * + D S*, S E S) D( S, S ) = Optimal Steiner String: Approximation Pf: Pick S Γ ( k S i 2) D( S, i S*) + S D( S, S i S*) + S i S D( S*, S i i ) = ( k 2) D( S, S*) + E( S*) 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 43 2

44 Optimal Steiner String: Approximation Resulting algorithm: Pick S c Γ that minimizes E(S c ) (S c is the center string). Approximation: S c gives a 2-approximation. The center string has a consensus error at most 2 times the error of the optimal Steiner string. Shamir nro GC 44

45 Consensus String Given multiple alignment, the consensus character i is the character that minimize the summed distance to it from all the characters in column i of the MA. For example, given scoring scheme with match = 0, mismatch or indel=1, the consensus character is the most frequent character. The consensus string derived from alignment is the concentration of consensus characters.

46 Consensus string versus Steiner string definitions d(s,t) - The distance that a given multiple alignment M induces on the pair S,T D(S,T) min cost of alignment between S and T Steiner string S*: minimizes Σ i D(S*, Si) Consensus string S*: minimizes Σ i d(s*, S i)

47 Consensus multiple alignment 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 d(s k k, S*) The opt consensus MA: one with least alignment error 47

48 Pf: ex. Consensus multiple alignment Thm: opt soln of consensus MA = Steiner string (up to spaces) Shamir nro GC 48

49 A summary S c Center string S* S con Optimal Steiner string Optimal consensus string Center star algorithm Optimal consensus alignment Alignment error <= Σ i d(s con, S i) <= Σ i d(sc, S i) = Σ i D(Sc, Si) Consensus error Σ i D(Sc, Si) 2- appr Optimal consensus error == Pf: ex. Optimal (consensus) alignment error Σ i D(S*, Si) Σ i d(s con, S i)

50 A summary Center string Optimal Steiner string Optimal consensus string Optimal SOP alignment Center star algorithm Optimal consensus alignment Optimal SOP score 2- appr SOP score Alignment error <= Consensus error 2- appr Optimal consensus error == Pf: ex. Optimal (consensus) alignment error 2-approximation

51 A summary Center string Optimal Steiner string Optimal consensus string Optimal SOP alignment Center star algorithm Optimal consensus alignment Optimal SOP score 2- appr SOP score Alignment error <= Consensus error 2- appr Optimal consensus error == Pf: ex. Optimal (consensus) alignment error 2-approximation

52 Theorem Assuming the triangle inequality, the multiple alignment created by the center star method has: - A sum-of-pairs (SOP) score that is never more than 2 times the SOP score of the optimal SOP alignment - An alignment error and that is never more than 2 times the alignment error of the optimal consensus alignment.

53 Tree MA Input: Tree T, a string for each leaf Phylogenetic alignment for T: Assignment of a string to each internal node GTTG CTGG GTTG CTGG CCGG CTTG GTTC Score (weighted) sum of scores along edges Goal: find tree alignment of optimal score Consensus = tree Alignment where T is a star 54

54 NP-hard Tree MA complexity Poly time approximations: 2-approximation Better approximation with more time (PTAS) 55

55 Lifted alignment The seq. label at every internal node is lifted from one of its children Lifted: Not lifted: Shamir nro GC GTTC CTGG GTTC GTTG CTGG CCGG CTTG GTTC CTTG GTTG CTGG CCGG CTTG GTTC 56

56 A 2-approximation to Tree MSA [Jiang, Wang, Lawler 1996] Assumes triangle inequality Suppose we knew an optimal tree T*. We transform it into a lifted alignment T L in a postorder traversal: 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 twice the distance of T* 57 Shamir nro GC

57 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

58 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 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 ))

59 Wrapping up lifted alignment a lifted alignment L T that is 2 OPT We can find a min cost lifted L T* alignment in poly time Cost(L T* ) cost(l T ) 2 OPT Thm: lifted alignment alg gives a polytime 2-approximation to Tree Alignment

60 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) Shamir nro GC 61

61 Shamir nro GC 62

62 Aligning a sequence to a profile Key in pairwise alignment is scoring two letters 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

63 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----- Shamir nro GC 64

64 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 σ(i,j) Other scores in use: Euclidean distance Pearson correlation KL-divergence (relative entropy)

65 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

66 Progressive Alignment A variation of greedy algorithm with a somewhat more intelligent strategy for choosing the order of alignments.

67 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 Shamir nro GC 69

68 Guide tree A this? B C D E A or this? B C D E F Shamir nro GC 70

69 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 Shamir nro GC 85

70 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) Shamir nro GC 86

71 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) Shamir nro GC 87

72 Step 2: Guide Tree (cont d) v 1 v 2 v 3 v 4 v 1 - v v v v 1,3 v 1,3,4 v 1,2,3,4 v 1 v 3 v 4 v 2 Calculate: 1,3 = alignment (v 1, v 3 ) 1,3,4 = alignment((v 1,3 ),v 4 ) 1,2,3,4 = alignment(( ((v 1,3,4 ),v 2 ) 1,3,4 Shamir nro GC 88

73 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 PEEMSVTS-LDLTGGLPEATTPESEEAFTLPLLNDPEPK-PSLEPVKNISNMELKAEPFD FOS_MOUSE PEEMSVAS-LDLTGGLPEASTPESEEAFTLPLLNDPEPK-PSLEPVKSISNVELKAEPFD FOS_CHICK SEELAAATALDLG----APSPAAAEEAFALPLMTEAPPAVPPKEPSG--SGLELKAEPFD FOSB_MOUSE PGPGPLAEVRDLPG-----STSAKEDGFGWLLPPPPPPP LPFQ FOSB_HUMAN PGPGPLAEVRDLPG-----SAPAKEDGFSWLLPPPPPPP LPFQ.. : **. :.. *:.* *. * **: Shamir nro GC Dots and stars show how well-conserved a column is. 89

74 Multiple Alignment: History 1975 Sankoff Formulated multiple alignment problem and gave dynamic programming solution 1988 Carrillo-Lipman Branch and Bound approach for MSA 1990 Feng-Doolittle Progressive alignment 1994 Thompson-Higgins-Gibson-ClustalW >40K citations! Most popular multiple alignment program 1998 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 2011 Clustal Omega Shamir nro GC Still a lot to be done! 91