From [4] D. Gusfield. Algorithms on Strings, Trees and Sequences. Cambridge University Press, New York, 1997.

Transcription

1 Multiple Alignment From [4] D. Gusfield. Algorithms on Strings, Trees and Sequences. Cambridge University Press, New York, Multiple Sequence Alignment Introduction Very active research area Very important: extracting and representing biologically important similarities from a set of strings Algorithms for three basic variants of the problem Profile representation Consensus Sequence representation Signature representation 2 1

2 Multiple Sequence Alignment Shows multiple similarities: Common structure of protein product Common function Common evolutionary process Family of proteins => more information than single sequences Characterization: signatures of protein families 3 Multiple Sequence Alignment Characterization: signatures of protein families Identification of potential members of these families Similarities may be very faint and dispersed become more obvious in a comparison of related sequences Can be used to do better searches: The inverse problem of 2-string alignment: Goal is to deduce unknown conserved sub patterns from a set of strings that is known to be biologically related. Compare to 2-string alignment Common sub patterns Matching strings that may not have been known to be biologically related 4 2

3 Biological Background: Transcription Gene Expression from DNA to Protein Translation Transcription 1 st Fact of Biological Sequence Analysis [4]: Sequence similarity usually implies significant structural or functional similarity - 2-sequence comparison - Biological database search - BUT: the reverse is not true! 5 Sequence similarity 2 nd Fact of Biological Sequence Analysis [4]: Evolutionary and functionally related molecular strings can differ significantly throughout much of the string and yet preserve: 1. the same three dimensional structures 2. the same two-dimensional substructures (motifs, domains) 3. the same active sites 4. the same or related dispersed residues (DNA or amino acid) 6 3

4 Evolutionary Conservation Evolutionary preserved features: 3-dimensional structures (well preserved) 2-dimensional substructures (well preserved) active sites: functions (less common) Amino acid sequence (least common) Three biological uses: Representation of protein families Identification of conserved sequence features correlating with structure and function Deduction of evolutionary history 7 Example: Hemoglobin Hemoglobin: An almost universal protein found in insects, mammals, etc. 4 chains of ~140 amino acids Functions the same in both insects as well as mammals: binds and transports oxygen Insects and mammals diverged ~600 million years ago => On average 100 amino acids mutations per chain Pair wise alignment: human - chimpanzee equal Mammal - mammal suggests functional similarity Insect-mammal very little similarity! Secondary and 3-dimensional structure well preserved 8 4

5 human beta globin horse beta globin human alpha globin horse alpha globin cyano haemoglobin whale myoglobin leghaemoglobin Multiple Alignment α-helices Alignments of globins produced by Clustal. 9 From: Protein Data Bank 10 5

6 aligning several sequences acidic ribosomal protein P0 homolog (L10E) encoded by the Rplp0 gene 11 Multiple Alignment Examples Related sequences have so few conserved or dispersed matching amino acids: statistically indistinguishable from the best alignment of 2 random strings. For example this is true for: Hemoglobin, immunoglobulin (antibody proteins) E-Cadherin (adhesion molecule) Compare to: 1 DNA nucleotide change: Sickle cell anemia 12 6

7 Definition (Global) Multiple Alignment A multiple alignment of a set of strings {S 1,S 2,,S k } is a series of strings S`1,S`2,,S`k such that 1. S`1 = S`2 = = S`k (all S`i have the same length) 2. For every i: S`i is an extension of S i obtained by insertion of spaces. A multiple alignment of strings ACBCBD, CADDB, and ACABCD: Note: similarly, local multiple alignment (fro substrings) can be defined. 13 Multiple Alignment Definition Induced Pair-Wise Alignment: Given a multiple alignment M, the induced pair-wise alignment of two string S i and S is obtained from M by removing all rows except rows i an. Note: two opposing spaces can be removed Definition Score of Induced Pair-Wise Alignment: The score of an induced pair-wise alignment is determined using any chosen scoring scheme for 2-string alignment in the standard manner. 14 7

8 Multiple Alignment: Sum-of-Pairs Definition: Sum of Pairs Score The sum-of-pairs (SP) score of a multiple alignment M is the sum of the scores of pair-wise global alignments induced by M. Problem: The SP Alignment Problem Compute a global multiple alignment M with minimum sum-of-pairs score. Note: Intuitively this is a reasonable score, but: no theoretical ustification! 15 An Exact Solution of the SP Alignment Problem A Dynamic Programming Approach The best multiple alignment of r sequences is calculated using an r-dimensional hypercube D, where: Each dimension is formed by one of the r sequences. The nodes ( 1, 2,, r ) of the hypercube hold the best score D( 1, 2,, r ) for aligning the prefixes of the sequences x 1, x 2,, x r of lengths 1, 2,, r, respectively. 16 8

9 Multiple Alignment: Dynamic Programming The best score D can be calculated using dynamic programming on the following recursion: D( 0,0,...,0) 0 Where D(, 2,..., r ) min { D( 1 1, 2 2,..., ) ( 1,,..., )} r r r x 1 2x 2 rx 1 r {0,1}, 0 with ρ the cost function (for example Sum of Pairs (SP), i.e. pairwise summation of the induced alignment at that position), and r ( 1, 2,..., r ) {0,1 } Note: There is no generally accepted score for a multiple alignment. We will discuss some candidates. Biological relevance of the multiple alignment is of maor importance. 17 Multiple Alignment: Dynamic Programming (4,3,2) ε =(1,0,1) (3,3,1) (3,2,0) ε =(0,1,1) (1,0,0) (2,1,0) ε =(1,1,0) ε = (1,1,0) Space complexity O(n r ).(O(r-dimensional cost function ρ calculation)); Time complexity O(2 r.n r ) (consider 2 r previously calculated values) 18 9

10 Exact Multiple Alignment Complexity The Exact Multiple Alignment problem solved by Dynamic Programming has: Space complexity O(n r ).(O(ρ calculation)) Time complexity O(2 r.n r ) This exact solution using dynamic programming is only useful for a small number of strings, i.e., a small r. In general: The exact multiple alignment problem using sum-of-pairs or evolutionary-tree scoring is NP-complete [9]. 19 Scoring Metrics Possible ρ functions: Distance from Consensus Let C be the consensus sequence, then the total distance is defined as: i {1,..., r} Evolutionary Tree Alignment Define D(S v,s w ) as the number of changes between S v and S w, then the weight of an evolutionary tree is defined as: Then the weight of the lightest evolutionary tree that can be constructed from the sequences is taken. Sum of Pairs Defined as the sum of pairwise distances between all pairs of sequences: D( S i, C) {1,..., r}, i D( S i, S ) ( v, w) T D( S v, S w ) 20 10

11 Faster Dynamic Programming A heuristic by Carrillo and Lipman, 1988 [1] Implemented in MSA, Lipman et al., 1989 [6] Idea: For similar strings the alignment path is close to the diagonal (of the hypercube). Upper bound on best alignment-cost enables pruning of the alignments that are a priori too costly 21 Implementation of the MSA Heuristic Let A an alignment of sequences x 1,x 2,,x r. A the pair of rows x i and x from A. c(a ) the induced cost of the pair wise alignment of sequences x i and x. (alignment of space against space is ignored, cost function D), then the total cost of A, c(a) can be defined as: c( A) c( Assume known that for an optimal alignment A * : c(a * ) c, then: * * * * c' c( A ) c( A ) c( A ) c( A ) Hence: c( A c( A ) {1,..., r}, i A i, ) {1,..., r}, i D( x, x * u, v i ) {1,..., r}, i,( ) ( u, v) * u, v ) c' D( x x ) {1,..., r}, i,( ) ( u, v) u, v {1,..., r}, i,( ) ( u, v) optimal pair-wise alignment 22 11

12 Implementation of the MSA Heuristic Assume known that for an optimal alignment A * : c(a * ) c, then: * c( A ) c' D( x, x u, v i ) {1,..., r}, i,( ) ( u, v) Now define: B( u, v) c' D( x i, x ) {1,..., r}, i,( ) ( u, v) Note that given c` we can calculate B(u,v) by calculating optimal pair-wise alignment cost D(x i, x ) for each i and. 23 Implementation of the MSA Heuristic Now consider a node from the hyper-cube: q = (i 1,i 2,,i u =s,,i v =t,,i r ) then clearly (s,t) is the proection of q onto the (u,v)-plane If the best alignment A * goes through q then the proection of the A*-path to the (u,v)-plane, A* u,v, goes through (s,t). Now let best(u,v) s,t an upper bound on the optimal score for an alignment through (s,t) in the (u,v)-plane, then: best(u,v) s,t c(a* u,v ) B(u,v) B( u, v) c' D( x i, x ) Let d(k 1,k 2 ) the cost of matching characters k 1 and k 2 then: best(u,v) s,t = D(x u,1,,x u,i-1, x v,1,,x v,-1 ) + d(x u,i, x v, ) + D(x u,i+1,,x u,n(u), x v,+1,,x v,n(v) ) Can be computed by forward dynamic programming, keeping a queue of nodes with uncompleted D-values. {1,..., r}, i,( ) ( u, v) 24 12

13 Multiple Sequence Alignment Approximations Sketch computation by forward dynamic programming while keeping a queue of nodes with uncompleted D-values: The algorithm will set the D value of the cell w at the head of the queue and remove that cell. It also updates the shortest distance from cell (0,..., 0) to all neighbors of w, whose D value can be influenced by w. If any of the neighbors is not yet in the queue, it is added to the end of the queue. If at some point best(u,v) s,t > B(u, v), then it is clear that the best alignment A* cannot pass through node (i 1, i 2,..., i u = s,..., i v = t,..., i r ) for any i 1, i 2,..., i u 1, i u+1,..., i v 1, i v+1,..., i r => These cells can be discarded from the computation Note: the bound c is found by using heuristics giving promising solutions first. In practice, MSA [6] can align 6 sequences of length The Center Star Method for (SP) Alignment. An approximation algorithm for the optimal alignment under the Sum of Pairs metric [4, pp ] achieving an approximation ratio of 2. Sum of Pairs metric Defined as the sum of induced pair-wise distances between all pairs of sequences: D( S i, S ) Where D(S,T) the best score of aligning S with T, using a scoring function d(x,y) defined by: d(-,-)=0 d(x,y) = d(y,x) d(x,y) d(x,z) + d(z,y) (triangle inequality) {1,..., r}, i 26 13

14 star alignment - We put a string in center (which one?) - Compute pair-wise alignments - Extend pairs into multiple alignment - Where once a gap, always a gap Complexity: O(k 2 n 2 ) k sequences, n length Theory proves: within factor 2 of optimal alignment 27 Star Alignment Problem: The Sum of Pairs alignment problem Input: A set of sequences {S 1,S 2,,S k } Question: Compute a global multiple alignment M with minimum sum-of-pairs score. Definition: Given a set of k strings S, define a center string S c (element of S) as a string that minimizes: Definition: Define the center star to be a star tree of k nodes, with the center node S c and with each of the remaining k-1 nodes labeled by a distinct string from S\{S c }. S S D ( S, S ) c Definition: Define the multiple alignment M c of the set of strings S to be the multiple alignment consistent with the center star

15 The Center Star Method for (SP) Alignment. S 4 S 5 S 2 S 3 S 6 S 1 A generic center start for 6 strings with center string S C = S 3. [4, p349]. 29 The Center Star Algorithm Star Alignment 1. Find S t from S minimizing and let M={S t }. 2. Add the sequences in S\{S t } to M one by one such that the alignment of every newly added sequence with S t is optimal. Add spaces when needed, to all pre-aligned sequences. it D ( S i, S t ) 30 15

16 star alignment 1 ATTGCCATT 2 ATGGCCATT 3 ATCCAATTTT 4 ATCTTCTT 5 ACTGACC 1 ATTGCCATT 2 ATGGCCATT 1 ATTGCCATT-- 3 ATC-CAATTTT 1 ATTGCCATT 4 ATCTTC-TT 1 ATTGCCATT 5 ACTGACC-- <= center 1 ATTGCCATT-- 2 ATGGCCATT-- 3 ATC-CAATTTT 1 ATTGCCATT-- 2 ATGGCCATT-- 3 ATC-CAATTTT 4 ATCTTC-TT-- 1 ATTGCCATT-- 2 ATGGCCATT-- 3 ATC-CAATTTT 4 ATCTTC-TT-- 5 ACTGACC The Center Star Algorithm as D(x,y) is the optimal distance between x and y Multiple alignment of the center start produces a multiple alignment with a value at most 2(k-1)/k times the optimal value

17 The Center Star Algorithm (S 1 minimizes this sum) any other S i does worse * Triangle inequality: D(S i,s ) D(S i,s 1 ) + D(S 1,S ) 33 Multiple Alignment with Consensus Consensus string S* does not need to be An element of the set of strings S

18 Multiple Alignment with Consensus Steiner string capture common characteristics of the set S. S* not necessarily element of S. Approximation algorithm with worst case approximation ratio Multiple Alignment with Consensus Multiple alignment with consensus string A B C D E D 36 18

19 Multiple Alignment: Consensus Strings S * the Steiner String that minimizes the consensus error. Lemma 4.2 shows that actually you can use a string from the set S as a consensus string as an approximation. S c of the previous algorithm will do. 37 Multiple Alignment: Consensus Strings Triangle inequality Furthermore: 38 19

20 Consensus Strings from Multiple Alignment Column (character) wise consensus. ( See Theorem 4.3 ) 39 Common Alignment Methods Aligning a String to a Profile Iterative Pair-wise Alignment Progressive Alignment Feng-Doolittle (1987)[2] CLUSTALW 40 20

21 Example: Signature Profiles Helicases protein to unwound DNA for further read for duplication, transcription, recombination or repair. Werner s syndrome an aging disease is believed to be due to a gene that codes for a helicase protein. A Signature Profile for Helicases Conserved sequence signatures or motifs Some of these motifs are unique identifiers for helicases Maybe functional units 41 Multiple Alignment Profile Multiple Alignment Profile: Character frequencies given per column p i (a) is the fraction of a s in column i p(a) is the fraction of a s overall Log likelihood ratio log( p i (a)/p(a) ) can be used

22 Aligning a String to a Profile consensus Profile for Classical Chromo Domains [10]. 43 Iterative Pair-wise Alignment Algorithm 1. Align some pair 2. While (not done) (a) Pick an unaligned string which is near some aligned one(s). (b) Align with the profile of the previously aligned group. Resulting new spaces are inserted into all strings in the group. This approach uses pair-wise alignment scores to iteratively add one additional string to a growing multiple alignment. 1. We start by aligning the two strings whose edit distance is the minimum over all pairs of strings. 2. Then we iteratively consider the string with the smallest distance to any of the strings already in the multiple alignment

23 Progressive Alignment Feng-Doolittle (1987) CLUSTALW 45 Feng-Doolittle (1987)[2] Feng-Doolittle Algorithm (sketch): 1. Calculate the pairwise alignment scores, and convert them to distances. 2. Use an incremental clustering algorithm [3] to construct a tree from the distances. 3. Traverse the nodes in their order of addition to the tree, progressively aligning the sequences. This way, the most similar pair is aligned first, followed by the addition of the next most similar sequence or set of sequences. Features of the Feng-Doolittle Heuristic: The order of aligning of sequences, or sets of sequences, is determined by their highest scoring pairwise alignment. Once a gap, always a gap 46 23

24 CLUSTALW ClustalW is a software package for multiple alignment Implemention of an algorithm of Thompson, Higgins, Gibson 1994 [8]. The basic idea is the same as in the Feng-Doolittle algorithm: k 1. Calculate the pairwise alignment scores, and convert them to 2 distances. 2. Use a neighbor-oining algorithm to build a tree from the distances. 3. Align sequence - sequence, sequence - profile, profile - profile in decreasing similarity order. This algorithm makes use of many ad-hoc rules such as weighting, different matrix scores and special gap scores. 47 Alignment Tree by CLUSTALW 48 24

25 Multiple Sequence Alignment: HMM known Multiple Sequence Alignment Problem: Given sequence S 1,, S n, how can they be optimally aligned? Assume a profile HMM P is known, then: - Align each sequence S i to the profile separately - Accumulate the obtained alignments to a multiple alignment - Hereby insert runs are not aligned. (Just leftustify insert regions.) 49 Multiple Sequence Alignment: HMM unknown Multiple Sequence Alignment Problem: Given sequence S 1,, S n, how can they be optimally aligned? Assume a profile HMM P is not known, then obtain an HMM profile P from S 1,, S n as follows: - Choose a length L for the profile HMM and intialize the transition and emission probabilities. - Train the HMM using Baum- Welch on all sequences S 1,, S n. Now obtain the multiple alignment using this HMM P as in the previous case: - Align each sequence S i to the profile separately - Accumulate the obtained alignments to a multiple alignment - Hereby insert runs are not aligned. (Just left-ustify insert regions.) 50 25

26 Bibliography [1] H. Carrillo and D. Lipmann. The multiple sequence alignment problem in biology. SIAM J. Appl. Math, 48: , [2] D. Feng and R. F. Doolittle. Progressive sequence alignment as a prerequisite to correct phylogenetic trees. J. Mol. Evol., 25: , [3] W. M. Fitch and E. Margoliash. Construction of phylogenetic trees. science, 15: , [4] D. Gusfield. Algorithms on Strings, Trees and Sequences. Cambridge University Press, New York, [5] T. Jiang, L. Wang, and E. L. Lawler. Approximation algorithms for tree alignment with a given phylogeny. Algorithmica, 16: , [6] D. J. Lipman, S. Altshul, and J. Kececiogly. A tool for multiple sequence alignment. Proc. Natl. Academy Science, 86: , [7] M. Murata, J.S. Richardson, and J.L. Sussman. Three protein alignment. Medical Information Sciences, 231:9, [8] J. D. Thompson, D. G. Higgins, and T. J. Gibson. Clustal w: improving the sensitivity of progressive multiple sequence alignment through sequence weighting, position-specific gap penalties and weight matrix choice. Nucleic Acids Res, 22: , [9] L. Wang and T. Jiang. On the complexity of multiple sequence alignment. Computational Biology, 1: , [10] [11] [12] L. Sterck, ProCoGen Training Workshop, Umea, Hidden Markov Models Profiles 52 26

27 model structure A G C T many states & fully connected training seldom works because of local maxima use knowledge about the problem e.g. linear model for profile alignment 53 silent states skipping nodes [square states: emitting] silent states (no emission) linear size vs. quadratic size but less modeling possibilities high/low 54 27

28 Forward Algorithm: silent states: Forward Algorithm q as before transition / emission q for silent states no silent loops (!): update in topological order 55 profile alignment (1) begin M 2 end Case 1: ungapped transition prob s 1 trivial alignment of the HMM to sequence VGAHAGEY VTGNVDEV VEADVAGH VKSNDVAD VYSTVETS FNANIPKH IAGNGAGV profile HMM P has a dedicated topology e i (b) = probability of observing symbol b at pos i 56 28

29 profile alignment (2) Case 2: with unmatched inserts I insert state M M +1 match states VGA--HAGEY VNA--NVDEV VEA--DVAGH VKG--NYDED VYS--TYETS FNA--NIPKH IAGADNGAGV emission: background probabilities or based on alignment affine gap model open gap extension 57 profile alignment (3) Case 3: with unmatched inserts and deletions VGA--HAGEY V----NVDEV VEA--DVAGH VKG------D VYS--TYETS FNA--NIPKH IAGADNGAGV I M M +1 D -1 D insert state match states delete state (silent) M -1 M M +1 adapt Viterbi => 58 29

30 HMM for profiles / multiple alignment Deletion (D) Insertion (I) D I Viterbi Match (M) begin M end M Y v ( i) em ( x i ). max v 1( i 1). ty Y M, I, D 1 I Y v ( i) p( xi ). max v ( i 1). ty I Y M, I, D 1 D Y v ( i) max v 1( i). ty D Y M, I, D 1 M same level same position 59 given multiple alignment Insertion / Deletion states profile alignment M i : VGA--HAGEY V----NVDEV VEA--DVAGH VKG------D VYS--TYETS FNA--NIPKH IAGADNGAGV counting transitions M 1 M / 10 M 1 I / 10 M 1 D / 10 emissions F 1+1 2/ 27 I 1+1 2/ 27 V 5+1 6/ 27 other 17x 0+1 1/ 27 Laplace correction, i.e., adding 1 for each frequency to avoid dividing by