Alignment ABC. Most slides are modified from Serafim s lectures

Transcription

1 Alignment ABC Most slides are modified from Serafim s lectures

2 Complete genomes

3 Evolution

4 Evolution at the DNA level C ACGGTGCAGTCACCA ACGTTGCAGTCCACCA SEQUENCE EDITS REARRANGEMENTS

5 Sequence conservation implies function Interleukin region in human and mouse

6 Sequence Alignment AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGGTCGATTTGCCCGAC -AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC Definition Given two strings x = x 1 x 2...x M, y = y 1 y 2 y N, an alignment is an assignment of gaps to positions 0,, N in x, and 0,, N in y, so as to line up each letter in one sequence with either a letter, or a gap in the other sequence

7 What is a good alignment? Alignment: The best way to match the letters of one sequence with those of the other How do we define best? Alignment: A hypothesis that the two sequences come from a common ancestor through sequence edits Parsimonious explanation: Find the minimum number of edits that transform one sequence into the other

8 Scoring Function Sequence edits: Mutations Insertions Deletions Scoring Function: Match: +m Mismatch: -s Gap: -d AGGCCTC AGGACTC AGGGCCTC AGG.CTC Score F = (# matches) m - (# mismatches) s (#gaps) d

9 Scoring the gaps more accurately Current model: (n) Gap of length incurs penalty n n d However, gaps usually occur in bunches Convex gap penalty function: (n) (n): for all n, (n + 1) (n) (n) (n 1)

10 How do we compute the best alignment? AGTGCCCTGGAACCCTGACGGTGGGTCACAAAACTTCTGGA AGTGACCTGGGAAGACCCTGACCCTGGGTCACAAAACTC Too many possible alignments: O( 2 M+N )

11 Dynamic Programming We will now describe a dynamic programming algorithm Suppose we wish to align x 1 x M y 1 y N Let F(i,j) = optimal score of aligning x 1 x i y 1 y j

12 Dynamic Programming (cont d) Notice three possible cases: 1. x i aligns to y j x 1 x i-1 y 1 y j-1 x i y j F(i,j) = F(i-1, j-1) + m, if x i = y j -s, if not 2. x i aligns to a gap x 1 x i-1 y 1 y j - x i 3. y j aligns to a gap x 1 x i - y 1 y j-1 y j F(i,j) = F(i-1, j) - d F(i,j) = F(i, j-1) - d

13 Dynamic Programming (cont d) How do we know which case is correct? Inductive assumption: F(i, j-1), F(i-1, j), F(i-1, j-1) are optimal Then, F(i, j) = max F(i-1, j-1) + s(x i, y j ) F(i-1, j) d F( i, j-1) d Where s(x i, y j ) = m, if x i = y j ; -s, if not

14 Example x = AGTA m = 1 y = ATA s = -1 F(i,j) i = d = -1 j = A G T A A T A Optimal Alignment: F(4,3) = 2 AGTA A - TA

15 The Needleman Wunsch Algorithm 1. Initialization. a. F(0, 0) = 0 b. F(0, j) = - j d c. F(i, 0) = - i d 2. Main Iteration. Filling-in partial alignments a. For each i = 1 M For each j = 1 N F(i-1,j) d [case 1] F(i, j) = max F(i, j-1) d [case 2] F(i-1, j-1) + s(x i, y j ) [case 3] UP, if [case 1] Ptr(i,j) = LEFT if [case 2] DIAG if [case 3] 3. Termination. F(M, N) is the optimal score, and from Ptr(M, N) can trace back optimal alignment

16 Performance Time: O(NM) Space: O(NM) There are more efficient methods

17 A variant of the basic algorithm: Maybe it is OK to have an unlimited # of gaps in the beginning and end: CTATCACCTGACCTCCAGGCCGATGCCCCTTCCGGC GCGAGTTCATCTATCAC--GACCGC--GGTCG Then, we don t want to penalize gaps in the ends

18 Different types of overlaps

19 The Overlap Detection variant x 1 x M Changes: y 1 y N 1. Initialization For all i, j, F(i, 0) = 0 F(0, j) = 0 2. Termination max i F(i, N) F OPT = max max j F(M, j)

20 Bounded Dynamic Programming Initialization: F(i,0), F(0,j) undefined for i, j > k y 1 y Nx1 xm k(n) Iteration: For i = 1 M For j = max(1, i k) min(n, i+k) F(i, j) = max Termination: F(i 1, j 1)+ s(x i, y j ) F(i, j 1) d, if j > i k(n) F(i 1, j) d, if j < i + k(n) same Easy to extend to the affine gap case

21 The local alignment problem Given two strings x = x 1 x M, y = y 1 y N Find substrings x, y whose similarity (optimal global alignment value) is maximum e.g. x = aaaacccccgggg y = cccgggaaccaacc

22 Why local alignment Genes are shuffled between genomes Portions of proteins (domains) are often conserved

23 The Smith Waterman algorithm Idea: Ignore badly aligning regions Modifications to Needleman Wunsch: Initialization: F(0, j) = F(i, 0) = 0 0 Iteration: F(i, j) = max F(i 1, j) d F(i, j 1) d F(i 1, j 1) + s(x i, y j )

24 The Smith Waterman algorithm Termination: 1. If we want the best local alignment F OPT = max i,j F(i, j) 2. If we want all local alignments scoring > t For all i, j find F(i, j) > t, and trace back

25 Multiple Sequence Alignments

26 Definition Given N sequences x 1, x 2,, x N : Insert gaps ( ) in each sequence x i, such that All sequences have the same length L Score of the global map is maximum The sum of pairs score of an alignment is the sum of the scores of all induced pairwise alignments s(m k, m l ): S(m) = k<l s(m k, m l ) score of induced alignment (k,l)

27 Multidimensional Dynamic Example: in 3D (three sequences): Programming 7 neighbors/cell F(i,j,k) = max{ F(i-1,j-1,k-1)+S(x i, x j, x k ), F(i-1,j-1,k )+S(x i, x j, - ), F(i-1,j,k-1)+S(x i, -, x k ), F(i-1,j,k )+S(x i, -, - ), F(i,j-1,k-1)+S( -, x j, x k ), F(i,j-1,k )+S( -, x j, x k ), F(i,j,k-1)+S( -, -, x k ) }

28 Progressive Alignment Multiple Alignment is NP complete Most used heuristic: Progressive Alignment Algorithm: 1. Align two of the sequences x i, x j 2. Fix that alignment 3. Align a third sequence x k to the alignment x i,x j 4. Repeat until all sequences are aligned Running Time: O( N L 2 )

29 Progressive Alignment When evolutionary tree is known: Align closest first, in the order of the tree x y z w Example: Order of alignments: 1. (x,y) 2. (z,w) 3. (xy, zw)

30 Progressive Alignment: CLUSTALW CLUSTALW: most popular multiple protein alignment Algorithm: 1. Find all d ij : alignment dist (x i, x j ) 2. Construct a tree (Neighbor joining hierarchical clustering) 3. Align nodes in order of decreasing similarity + a large number of heuristics

31 CLUSTALW

32 MLAGAN: Multiple Alignment 1. Multi Anchoring 2. Progressive Alignment 3. Iterative Refinement

33 1. Multi anchoring To anchor the (X/Y), and (Z) alignments: X Z Y Z X/Y Z

34 2. Progressive Alignment Human Baboon Mouse Rat Given N sequences, phylogenetic tree Align pairwise, in order of the tree (LAGAN)

35 3. Iterative Refinement For each sequence y 1. Remove y 2. Anchor good spots 3. Realign y using LAGAN y z x x,z fixed projection

36 Cystic Fibrosis (CFTR), 12 species The zoo project Cow Chicken Chimp Pig Human Baboon Cat Dog Rat Fugu Mouse Human sequence length: 1.8 Mb Total genomic sequence: 13 Mb Zebrafish

37 Performance in the CFTR region Exons Perfect Exons > 90% TIME (sec) MAX MEMORY (Mb) MUMmer Mammals 25% 40% Chicken & Fishes 0% 0% 7 40 AVID Mammals 95% 98% Chicken & Fishes 23% 27% LAGAN Mammals 98% 99.7% Chicken & Fishes 80% 84% MLAGAN Mammals 98% 99.8% Chicken & Fishes 82% 91%

38 MLAGAN