Pairwise Sequence alignment Basic Algorithms

Transcription

1 Pairwise Sequence alignment Basic Algorithms

2 Agenda - Previous Lesson: Minhala - + Biological Story on Biomolecular Sequences - + General Overview of Problems in Computational Biology - Reminder: Dynamic Programming Today: -Algorithms for Global and Local Sequence Alignment + variants -Bioinformatic Motivation for Sequence Alignment

3 Literature list Alberts, B et al. Essential Cell Biology: An introducton to the Molecular Biology of the Cell. Mount, D.W. Bioinformatics: Sequence and Genome Analysis. Jones N.C & Pevzner P.A. An introduction to Bioinformatics algorithms. R. Durbin, S. Eddy, A. Krogh, and G. Mitchison. Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Dan Gusfield. Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology. 3

4 Move to Slides On Dynamic Programming

5 Sequence Comparison (cont) We seek the following similarities between sequences : Find similar proteins Allows to predict function & structure Locate similar subsequences in DNA Allows to identify (e.g) regulatory elements Locate DNA sequences that might overlap Helps in sequence assembly g 1 g 2 /64

6 Sequence Modifications Three types of changes Substitution (point mutation) Insertion Indel (replication slippage) Deletion TCCGT TCAGT TCGAGT TCAGT TCGT 6

7 Choosing Alignments There are many possible alignments For example, compare: -GCGC-ATGGATTGAGCGA TGCGCCATTGAT-GACC-A to GCGCATGGATTGAGCGA TGCGCC----ATTGATGACCA-- Which one is better? 7/64

8 Another example Given two sequences: X: TGCATAT Y: ATCCGAT Question: How can X be transformed into Y? Or, How did Y evolve from X? 8/64

9 One possible transformation TGCATAT TGCATA TGCAT ATGCAT ATCCAT ATCCGAT delete T delete A insert A G C insert G 9/64 operations Alignment: -TGC-ATAT ATCCGAT--

10 Another possible transformation TGCATAT ATGCATAT ATGCAAT ATGCGAT ATCCGAT insert A delete T A G G C 4 operations Alignment: -TGCATAT ATCCG-AT Which one is better? 10/64

11 In order to align two sequences we need a quantitive model to evaluate similarity between sequences. How do we quantitate sequence similarity? 11

12 Scoring Similarity Assume independent mutation model Each site considered separately Score at each site Positive if the same Negative if different Sum to make final score GTAGTC CTAGCG Can be positive or negative Significance depends on sequence length 12

13 Pairwise Alignment - Identity Human Hemoglobin (HH) vs Sperm Whale Myoglobin (SWM): (HH) VLSPADKTNVKAAWGKVGAHAGYEG (SWM) VLSEGEWQLVLHVWAKVEADVAGHG Percent Identity: ( only)

14 D and E are similar: Pairwise 1. structure is similar. Alignment - Similarity 2. both are acidic and hydrophilic 3. one mutation can separate them from one to the other. (HH) VLSPADKTNVKAAWGKVGAHAGYEG. (SWM) VLSEGEWQLVLHVWAKVEADVAGHG Percent Similarity: ( and.) Percent Identity: ( only)

15 Pairwise Alignment Gap insertion (HH) VLSPADKTNVKAAWGKVGAH-AGYEG. (SWM) VLSEGEWQLVLHVWAKVEADVAGH-G Gaps: 2 Percent Similarity: Percent Identity: (12/26)

16 Pairwise Alignment - Scoring The final score of the alignment is the sum of the positive scores and penalty scores: + Number of Identities + Number of Similarities - Number of gap insertions Alignment score

17 Pairwise Alignment - Scoring (HH) VLSPADKTNVKAAWGKVGAH-AGYEG. (SWM) VLSEGEWQLVLHVWAKVEADVAGH-G Final score: (V,V) + (L,L) + (S,S) + (D,E) + - (penalty for gap insertion)*(number of gaps) - (penalty for gap extension)*(extension length) We are interested in both the score and the alignment trace.

18 Optimum Alignment The score of an alignment is a measure of its quality Optimum alignment problem: Given a pair of sequences X and Y, find an alignment (global or local) with maximum score The similarity between X and Y, denoted sim(x,y), is the maximum score of an alignment of X and Y 18

19 Computing Optimal Score How can we compute the optimal score? If s = n and t = m, the number A(m,n) of possible legal alignments is large! A( m, n) A( n, n) m of the order of n Exercise 2n n 2 2 n Stirling s formula: n x 12 x x! 2 x e we perform dynamic programming to compute the optimal score efficiently. 19/64

20 Manhattan Tourist Problem (MTP) Imagine seeking a path (from source to sink) to travel (only eastward and southward) with the most number of attractions (*) in the Manhattan grid Source * * * * * * * * * * * * Sink

21 Manhattan Tourist Problem (MTP) Imagine seeking a path (from source to sink) to travel (only eastward and southward) with the most number of attractions (*) in the Manhattan grid Source * * * * * * * * * * * * Sink

22 Manhattan Tourist Problem: Formulation Goal: Find the longest (highest scoring) path in a weighted grid. Input: A weighted grid G with two distinct vertices, one labeled source and the other labeled sink Output: A longest path in G from source to sink

23 i coordinate source 0 MTP: An Example j coordinate sink

24 MTP: Greedy Algorithm Is Not Optimal source promising start, but leads to bad choices! sink

25 MTP: Dynamic Programming source j 0 1 i S 0,1 = 1 1 S 1,0 = Calculate optimal path score for each vertex in the graph Each vertex s score is the maximum of the prior vertices score plus the weight of the respective edge in between

26 MTP: Dynamic Programming (cont d) source j i S 0,2 = S 1,1 = S 2,0 = 8

27 MTP: Dynamic Programming (cont d) source j i S 3,0 = S 1,2 = S 2,1 = S 3,0 = 8

28 MTP: Dynamic Programming (cont d) source j i S 1,3 = S 2,2 = greedy alg. fails! 9 S 3,1 = 9

29 MTP: Dynamic Programming (cont d) source j i S 2,3 = S 3,2 = 9

30 MTP: Dynamic Programming (cont d) source j i Done! (showing all back-traces) S 3,3 = 16

31 MTP: Recurrence Computing the score for a point (i,j) by the recurrence relation: s i, j = max s i-1, j + weight of the edge between (i-1, j) and (i, j) s i, j-1 + weight of the edge between (i, j-1) and (i, j) The running time is n x m for a n by m grid (n = # of rows, m = # of columns)

32 Adding Diagonal Edges to the Grid A 2 A 3 A 1 B What about diagonals? The score at point B is given by: s B = max of s A1 + weight of the edge (A 1, B) s A2 + weight of the edge (A 2, B) s A3 + weight of the edge (A 3, B)

33 Adding Diagonal Edges to the Grid More generally, computing the score for point x is given by the recurrence relation: s x = max of s y + weight of vertex (y, x) where y є Predecessors(x) Predecessors (x) set of vertices that have edges leading to x

34 Traveling in the Grid The only hitch is that one must decide on the order in which visit the vertices By the time the vertex x is analyzed, the values s y for all its predecessors y should be computed otherwise we are in trouble. We need to traverse the vertices in some order Try to find such order for a directed acyclic grid graph???

35 Traversing the Manhattan Grid 3 different strategies: a) Column by column b) Row by row c) Along diagonals a) b) c)

36 Comparison methods Global alignment Finds the best alignment across the whole two sequences. Local alignment Finds regions of similarity in parts of the sequences. Global Local

37 Global Alignment Algorithm of Needleman and Wunsch (1970) Finds the alignment of two complete sequences: ADLGAVFALCDRYFQ ADLGRTQN-CDRYYQ Some global alignment programs trim ends

38 Local Alignment Algorithm of Smith and Waterman (1981). Makes an optimal alignment of the best segment of similarity between two sequences. ADLG CDRYFQ ADLG CDRYYQ Can return a number of highly aligned segments.

39 Global Alignment: Algorithm S T 1.. i 1.. j Prefix oflength i of S Prefix oflength j of T C ( i, j ) Cost of optimum alignment of S and T 1..i 1..j w ( a, b) if if a b a b 39

40 Theorem. C(i,j) satisfies the following relationships: Initial conditions: C(i,0) i C(0, j) j Recurrence relation: For 1 i n, 1 j m: C(i, j) C(i 1, j 1) max C(i 1, j) C(i, j 1) w(s,t ) i j 40

41 Example Case 1: Line up S i with T j i - 1 S: C A T T C A C T: C - T T C A G j -1 j Case 2: Line up S i with space i - 1 i S: C A T T C A - C T: C - T T C A G - Case 3: Line up T j with space i S: C A T T C A C - T: C - T T C A - G j j -1 i j 41

42 Justification: Optimal Substructure Property Followed S 1 S 2... S i-1 S i T 1 T 2... T j-1 T j S 1 S 2... S i-1 S i T 1 T 2... T j C(i-1,j-1) + w(s i,t j ) C(i-1,j) S 1 S 2... S i T 1 T 2... T j-1 T j C(i,j-1) 42

43 Computation Procedure C(0,0) C(i-1,j-1) C(i-1,j) C(i,j-1) C(i,j) C(n,m) C(i, j) max C(i 1, j 1) w(s,t ), i j C(i 1, j), C(i, j 1) 43

44 λ C T C G C A G C λ C A T T C A C for match, -2 for mismatch, - for space 44

45 λ C T C G C A G C λ C A T T C A C * * Traceback can yield both optimum alignments 4