Sequence Alignment. COMPSCI 260 Spring 2016

Transcription

1 Sequence Alignment COMPSCI 260 Spring 2016

2 Why do we want to compare DNA or protein sequences? Find genes similar to known genes IdenGfy important (funcgonal) sequences by finding conserved regions As a step in genome assembly, and other sequence analysis tasks Understand evolugonary relagonships and distances (human is closer to chicken than to zebrafish) Partial CTCF protein Homologous sequences can be divided into two groups orthologous sequences: sequences that differ because they are found in different species paralogous sequences: sequences that differ because of a gene duplicagon event

3 Homology example: evolugon of globins Human α- globin and human β- globin are paralogs or orthologs? Paralogs Human α- globin and mouse α- globin are homologs or orthologs? Both

4 Homology example: sequence comparison can reveal structure 1dtk (a) 1dtk 5pti 5pti (b) 1dtk 1dtk XAKYCKLPLRIGPCKRKIPSFYYKWKAKQCLPFDYSGCGGNANRFKTIEECRRTCVG- 5pti RPDFCLEPPYTGPCKARIIRYFYNAKAGLCQTFVYGGCRAKRNNFKSAEDCMRTCGGA

5 What is the alignment problem? Input: Two protein or DNA sequences X = x 1 x 2 x m Y = y 1 y 2 y n where the x i and y i are chosen from a finite alphabet. For DNA sequences the alphabet is { A,C,G,T}. For protein sequences the alphabet is {A, C, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, Y}. Output: the opgmal alignment of the two sequences, in the form of a list of columns of the types x i y j or x i - or - y j X = CTATGCATCA Y = GTGCACCCA CTATGCAT- CA GT--GCACC CA

6 Sequence variagons Sequences may have diverged from a common ancestor through various types of mutagons: SubsGtuGon (single nucleogde) DeleGon (single nucleogde) InserGon (single nucleogde) Inversion TransposiGon (a piece is removed and then inserted somewhere else) DuplicaGon (a piece is put in twice, or perhaps a foreign body might be insergng its genegc material into various places...) What happens if a single nucleogde dele4on or inser4on occurs in the coding porgon of the genome? x i y j Match/ mismatch x i - DeleGon (in Y rel. to X) - y j InserGon (in Y rel. to X) What happens if a single nucleogde subs4tu4on occurs in the coding porgon of the genome?

7 What is the alignment problem? Input: Two protein or DNA sequences X = x 1 x 2 x m Y = y 1 y 2 y n where the x i and y i are chosen from a finite alphabet. For DNA sequences the alphabet is { A,C,G,T}. For protein sequences the alphabet is {A, C, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, Y}. Output: the op4mal alignment of the two sequences, in the form of a list of columns of the types x i y j or x i - or - y j We need a way to score any given alignment of X and Y

8 What is the alignment problem? Input: Two protein or DNA sequences X = x 1 x 2 x m Y = y 1 y 2 y n where the x i and y i are chosen from a finite alphabet. For DNA sequences the alphabet is { A,C,G,T}. For protein sequences the alphabet is {A, C, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, Y}. And a funcgon Score that assigns a score to any column of the types x i y j or x i - or - y j Output: the highest scoring alignment of the two sequences, where an alignment is defined as a list of columns of the types defined above, and Score(alignment) = Σ col Score(col).

9 Alignment scoring scheme Alignment scoring schemes reflect biological or stagsgcal observagons about the known sequences, and are frequently represented by scoring matrices Score(x i,y j ) = +1 if x i and y j is a match - > reward 1 if x i and y j is a mismatch - > penalty 2 if either x i or y j is a gap - > penalty In general, the gap penalty is denoted as g (or g) AAAC A-GC Score(A,A) + g + Score(A,G) + Score(C,C) = - 1 AAAC AGC- Score(A,A) + Score(A,G) + Score(A,C) + g = - 3?

10 Brute- force search for the opgmal alignment Given the two sequences X = x 1 x 2 x m and Y = y 1 y 2 y n we want to find the alignment that produces the best score according to the given scoring scheme. Brute- force solu4on: enumerate all the possible alignments, score each alignment, and select the alignment with the maximal score. AAAC AGC AAAC-- ---AGC AAA-C- ---AGC AAA--C ---AGC What is the total number of possible global alignments between X and Y? Idea: append n gaps to the sequence X to obtain X = x 1 x 2 x m Then we can pick n elements from X to align with the characters in Y. exponengal Gme

11 Brute- force search for the opgmal alignment Brute force: generate & score all possible alignments Time complexity: n Brute force Today s lecture , E E+58 10,000

12 OpGmal substructure property? The score is additive: for a given split (i, j), the best alignment can be computed as Best alignment of S1[1..i] and S2[1..j] + Best alignment of S1[ i+1..n] and S2[ j+1..m] Compute best alignment recursively

13 Divide and conquer? IdenGcal sub- problems! We should reuse our work!

14 SoluGon #1 MemoizaGon Create a big dicgonary (or table), indexed by aligned sequences When we encounter a new pair of sequences If it is in the dicgonary Look up the solugon If it is not in the dicgonary Compute the solugon Insert the solugon in the dicgonary Ensures that there is no duplicated work Only need to compute each sub- alignment once

15 SoluGon #2 Dynamic programming Strategy: reduce problem of best alignment of two sequences to best alignment of all prefixes of the sequences i- prefix of X is x 1 x 2 x i j- prefix of Y is y 1 y 2 y j Create a big table, indexed by (i,j) Fill it from the beginning all the way to the end The solugon is the element (n,m) Guaranteed to explore engre search space Ensures that there is no duplicated work Only need to compute each sub- alignment once! Very simply computagonally!

16 Needleman- Wunsch algorithm Consider last step in compugng the alignment of AAAC with AGC Three possible opgons; in each we will choose a different pairing (or column type) for end of alignment, and add this to best alignment of previous characters

17 Needleman- Wunsch algorithm Given an m- character sequence X, and an n- character sequence Y Construct an (m+1) (n+1) matrix F F ( i, j ) = score of the best alignment of x[1...i ] with y[1...j ]

18 Needleman- Wunsch algorithm Given an m- character sequence X, and an n- character sequence Y Construct an (m+1) (n+1) matrix F F ( i, j ) = score of the best alignment of x[1...i ] with y[1...j ] Y j X i

19 Needleman- Wunsch algorithm IniGalize first row and column of matrix Fill in rest of matrix from top to bo om, le} to right F(m, n) holds the opgmal alignment score For each F(i, j), save pointers to cells that resulted in best score Trace pointers back from F(m, n) to F(0, 0) to recover alignment Y j X i

20 IniGalizing the DP matrix

21 Global alignment example

22 Needleman- Wunsch algorithm Strategy: reduce problem of best alignment of two sequences to best alignment of all prefixes of the sequences i- prefix of X is x 1 x 2 x i j- prefix of Y is y 1 y 2 y j Matrix: F (i, j) = score of the best alignment of x 1 x 2 x i with y 1 y 2 y j

23 Needleman- Wunsch algorithm F (i, j) = score of the best alignment of x 1 x 2 x i with y 1 y 2 y j Update rule: Consider last step in compugng the alignment

24 Needleman- Wunsch algorithm IniGalize first row and column of matrix Fill in rest of matrix from top to bo om, le} to right F(m, n) holds the opgmal alignment score For each F(i, j), save pointers to cells that resulted in best score Trace pointers back from F(m, n) to F(0, 0) to recover alignment

25 Global alignment example

26 Global alignment example

27 More than one opgmal alignment? Graph search algorithms available on course website

28 Pairwise alignment via dynamic programming Works for either DNA or protein sequences, although the subsgtugon matrices are different Finds all opgmal alignments Time complexity: O(nm) Space complexity: O(nm) Can we do this faster? Yes, if we use the Four- Russians Speedup Arlazarov, Dinic, Kronrod, Faradzev (1970) Block alignment with blocks of size t = log(n) / 4 Running Gme goes from quadragc, O(n 2 ), to subquadragc: O(n 2 /logn)

29 Space complexity CompuGng the alignment requires quadragc space: O(nm) We need to keep all backtracking references in memory to reconstruct the path (backtracking)

30 Space complexity CompuGng just the score of the best alignment can be done in linear space We only need the previous column to calculate the current column, and we can then throw away that previous column once we are done using it Y X = x 1 x 2 x m Y = y 1 y 2 y n What is the space complexity? O(m) or O(n)? X How do we recover the path? We cannot. Can we compute the best alignment in linear space?

31 OpGmal global alignment in linear space The opgmal alignment must cross the middle line (i, m/2) We want to calculate the best alignment from (0,0) to (n,m) that passes through (i,m/2) where i = 0..n and represents the i- th row Define score(i) as the score of the opgmal alignment from (0,0) to (n,m) that passes through cell (i, m/2) Define (mid, m/2) as the cell where the opgmal alignment crosses the middle column score(mid) = score opgmal alignment = max 0 i n score(i)

32 OpGmal global alignment in linear space The opgmal alignment must cross the middle line (i, m/2) Prefix(i) is the score of the best alignment from (0,0) to (i,m/2) Compute Prefix(i) by DP in the le} half of the matrix Suffix(i) is the score of the best alignment from (i,m/2) to (n,m) = the score of the best alignment from (n,m) to (i,m/2) with all arrows reversed Compute Suffix(i) by DP in the right half of the reversed matrix score(i) = Prefix(i) + Suffix(i)

33 Finding the middle point

34 Finding the middle point again

35 And again

36 Space is linear. But how about Gme complexity? On first pass, the algorithm covers the engre area nm On second pass, the algorithm covers only half of the area: nm/2 CompuGng Prefix(i) CompuGng Suffix(i)

37 Space is linear. But how about Gme complexity? On third pass, only 1/4th is covered (nm/4) Geometric reducgon at each iteragon nm + nm/2 + nm/4 + < 2nm Running Gme: O(nm) Space: O(n+m) When referring to the longest common subsequence: Hirschberg s algorithm

38 Alignment Global alignment: find best match of both sequences in their entirety Semi-global alignment: find best match without penalizing gaps on the ends of the alignment Local alignment: find best subsequence match

39 Semi- global alignment We are aligning the following sequences CAGCACTTGGATTCTCGG CAGCGTGG One possible alignment CAGCACTTGGATTCTCGG CAGC-----G-T----GG We might prefer the alignment CAGCA-CTTGGATTCTCGG ---CAGCGTGG We need a new scoring scheme Score: 8(1)+0(- 1)+10(- 2) = - 12 Score: 6(1)+1(- 1)+12(- 2) = - 19 IntuiGvely, we do not penalize missing ends of the sequence We would like to model this intuigon

40 Semi- global alignment IniGalize first row and column according to gap penalty Fill in the matrix Global alignment Report F(m,n) as the opgmal score Semi- global alignment IniGalize first row and column with 0 Fill in the matrix, as above Report max over last row and last column as the opgmal score

41 Semi- global alignment IniGalize first row and column according to gap penalty Fill in the matrix Global alignment Report F(m,n) as the opgmal score Semi- global alignment IniGalize first row and column with 0 Fill in the matrix, as above Report max over last row and last column as the opgmal score

42 Semi- global alignment One possible alignment CAGCACTTGGATTCTCGG CAGC-----G-T----GG We might prefer the alignment CAGCA-CTTGGATTCTCGG Score: 8(1)+0(- 1)+10(- 2) = CAGCGTGG Score: 6(1)+1(- 1)+12(- 2) = - 19 With the new scoring scheme (semi- global alignment) CAGCA-CTTGGATTCTCGG ---CAGCGTGG Score: 6(1)+1(- 1)+1(- 2)+11(- 0) = 3

43 Semi- global alignments ApplicaGons: Finding a gene in a genome Aligning a read onto an assembly Finding the best alignment of a PCR primer query target These situagons have in common One sequence is much shorter than the other Alignment should span the engre length of the smaller sequence No need to align the engre length of the longer sequence Local sequence alignment: best alignment between subsequences of X and Y

44 Local alignment problem: definigon Input: Two protein or DNA sequences X = x 1 x 2 x m Y = y 1 y 2 y n where the x i and y i are chosen from a finite alphabet. For DNA sequences the alphabet is { A,C,G,T}. For protein sequences the alphabet is {A, C, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, Y}. And a funcgon Score that assigns a score to any column of the types x i y j or x i - or - y j Output: the highest scoring alignment between subsequences of X and Y, where an alignment is defined as a list of columns of the types defined above, and Score(alignment) = Σ col Score(col).

45 Local alignment Local alignment is much more common than global alignment Example: aligning two protein sequences that have a common domain but are otherwise different Compared to global alignment, the local alignment problem appears to be significantly more complex Naïve approach: given that we know how to compute the global alignment between two sequences in O(mn) Gme we can take all possible combinagons of substrings of X and substrings of Y, and run Needleman- Wunsch Smith-Waterman algorithm - Smith, T.F. & Waterman, M.S. (1981) Identification of common molecular subsequences. J. Mol. Biol. 147: Running Gme? O(m 3 n 3 ) We can improve this significantly by using a DP approach

46 Global vs. local alignment Global alignment Y F (i, j) = score of the best alignment of x[1...i ] with y[1...j ] X Local alignment If the best alignment up to some point has a negagve score, it is be er to start a new alignment (score 0)

47 Global vs. local alignment Global alignment Y F (i, j) = score of the best alignment of x[1...i ] with y[1...j ] X Local alignment F(i, j) = score of the best If alignment the best alignment of a suffix up of to x[1...i some ] point has a negagve score, it is be er with a suffix of y[1...j ] to start a new alignment (score 0)

48 Local alignment Smith- Waterman algorithm Update rule: F(i, j) = score of the best alignment of a suffix of x[1...i ] with a suffix of y[1...j ] Ini4aliza4on? Previously (global alignment) we inigalized with gap penalges Now (local alignment) we inigalize the first row and column with 0

49 Local alignment Smith- Waterman algorithm Update rule: F(i, j) = score of the best alignment of a suffix of x[1...i ] with a suffix of y[1...j ] Op4mal alignment score? Global: F(m,n) Semi- global: maximum over last row F(m,j) and column F(i,n) Local:???

50 Local alignment Smith- Waterman algorithm Update rule: F(i, j) = score of the best alignment of a suffix of x[1...i ] with a suffix of y[1...j ] Op4mal alignment score? Global: F(m,n) Semi- global: maximum over last row F(m,j) and column F(i,n) Local: maximum F(i,j) over all i=1..m and j=1..n

51 Local alignment Smith- Waterman algorithm Update rule: F(i, j) = score of the best alignment of a suffix of x[1...i ] with a suffix of y[1...j ] Ini4aliza4on: we inigalize the first row and column with 0 Op4mal alignment score: maximum F(i,j) over all i=1..m and j=1..n Traceback: Start from the cell containing the opgmal score Stop when we reach the value 0

52 Local alignment example

53 Local alignment example

54 Local alignment Smith- Waterman algorithm Time complexity: O(mn) Space complexity: O(mn), can be brought to O(m+n) Is the solugon opgmal? Have we searched the engre space of alignments between substrings of X and Y? Yes. Because a substring of a string is the suffix of a prefix. F(i, j) = score of the best alignment of a suffix of x[1...i ] with a suffix of y[1...j ] SoluGon: maximum F(i,j) over all i=1..m and j=1..n MulGple opgmal alignments

55 Local alignment Smith- Waterman algorithm Time complexity: O(mn) Space complexity: O(mn), can be brought to O(m+n) Is the solugon opgmal? Have we searched the engre space of alignments between substrings of X and Y? Yes. Because a substring of a string is the suffix of a prefix. F(i, j) = score of the best alignment of a suffix of x[1...i ] with a suffix of y[1...j ] SoluGon: maximum F(i,j) over all i=1..m and j=1..n MulGple opgmal alignments

56 Global and local alignment Global alignment F(i, j) = score of the best alignment of x[1...i ] with y[1...j ] Match/mismatch score Local alignment Gap penalty

57 Scoring matches/mismatches - revisited Alignment scoring schemes reflect biological or stagsgcal observagons about the known sequences, and are frequently represented by scoring matrices Purines Guanine (G) Adenine (A) Pyrimidines Cytosine (C) Thymine (T)

58 Scoring matches/mismatches - revisited DNA mutagons Transi4ons: subsgtugons that occur between the two- ring purines (A G and G A) or between the one- ring pyrimidines (C T and T C). Because these subsgtugons do not require a change in the number of rings, they occur more frequently than the other subsgtugons. Transversions: slower- rate subsgtugons that change a purine to a pyrimidine or vice versa (A C, A T, G C, and G T). Purines Guanine (G) Adenine (A) Pyrimidines Cytosine (C) Thymine (T)

59 Scoring matches/mismatches - revisited For protein sequence alignment, some amino acids have similar structures and can be more easily subsgtuted in nature Glutamic acid (D) Aspartic acid (D) Tryptophan (W)

60 Chemical structures of the 20 common amino acids

61 Protein subsgtugon matrices A biologist with good intuigon could come up with a decent scoring scheme (invent 210 scores) But we would like to have some theory behind these scores, that reflects the similarity between the residues SubsGtuGon scores can be derived from probabilisgc models of evolugon Durbin et al. Biological Sequence Analysis

62 SubsGtuGon matrices for protein sequences Two popular sets of matrices for protein sequences PAM matrices [Dayhoff et al., 1978] BLOSUM matrices [Henikoff & Henikoff, 1992] Both try to capture the relagve subsgtutability of amino acid pairs in the context of evolugon Point Accepted MutaGon 1 PAM unit = PAM 1 = one mutagon per 100 amino acids A}er 100 PAMs of evolugon, not every residue will have changed some residues may have mutated several Gmes some residues may have returned to their original state some residues may not have changed at all

63 SubsGtuGon matrices for protein sequences PAM250 is a widely used matrix:

64 SubsGtuGon matrices for protein sequences BLOSUM Blocks SubsGtuGon Matrix Scores derived from observaeons of the frequencies of subsgtugons in blocks of local alignments in related proteins Accounts for evolugonarily divergent sequences Matrix name indicates evolugonary distance BLOSUM62 was created using sequences sharing no more than 62% idengty

65 SubsGtuGon matrices for protein sequences BLOSUM50 Common aminoacids have low weights Positive for chemically similar amino-acids Rare amino-acids have high weights

66 Global and local alignment Global alignment F(i, j) = score of the best alignment of x[1...i ] with y[1...j ] Match/mismatch score Local alignment Gap penalty

67 Gap penalges vs. These have the same score But the second one is o}en more plausible A single insergon of GAAT into the first string could change it into the second Current scoring scheme assumes the gaps occurred independently

68 Gap penalges Current scoring scheme: the cost associated with a gap of length k is g*k This is a linear gap penalty We would like to have a gap penalty funcgon where long gaps are not penalized that heavily OpGon: penalty for opening a gap: h penalty for extending a gap: g < h The cost associated with a gap of length k: This is an affine gap penalty

69 Linear vs. affine gap penalty Linear gap: Affine gap: Y j F(i, j) = score of the best alignment of x[1...i ] with y[1...j ] X i

70 Linear vs. affine gap penalty Linear gap: Affine gap: Y j Y j X X i i

71 Linear vs. affine gap penalty Linear gap: Affine gap: Y j " $ $ $ $ F(i, j) = max # $ $ $ $ % F(i 1, j 1) + s(x i, y j ) F(i 1, j) + w(1) F(i 2, j) + w(2)... F(i, j 1) + w(1) F(i, j 2) + w(2)... X i

72 DP for the affine gap penalty case? Affine gap: Y j " $ $ $ $ F(i, j) = max # $ $ $ $ % F(i 1, j 1) + s(x i, y j ) F(i 1, j) + w(1) F(i 2, j) + w(2)... F(i, j 1) + w(1) F(i, j 2) + w(2)... X i We can use the same approach we used for the linear gap, but Running Gme increases from O(mn) to O(mn(m+n)) For m=n, the increase is from O(n 2 ) to O(n 3 )

73 DP for the affine gap penalty case We can reduce the Gme to O(mn), but we need 3 matrices instead of 1 M(i,j) = score of the best alignment of x[1...i] with y[1...j] given that x[i] is aligned to y[j] x i y j I x (i,j) = score of the best alignment of x[1...i] with y[1...j] given that x[i] is aligned to a gap I y (i,j) = score of the best alignment of x[1...i] with y[1...j] given that y[j] is aligned to a gap x i - - y j

74 DP for the affine gap penalty case x i y j x i - " $ M (i, j) = max # $ % $ " $ I x (i, j) = max # $ % $ M (i 1, j 1) + s(x i, y j ) I x (i 1, j 1) + s(x i, y j ) I y (i 1, j 1) + s(x i, y j ) M (i 1, j) + h + g I x (i 1, j) + g I y (i 1, j) + h + g - y j " $ I y (i, j) = max # $ % $ M (i, j 1) + h + g I x (i, j 1) + h + g I y (i, j 1) + g

75 DP for the affine gap penalty case - inigalizagon M(i,j) = score of the best alignment of x[1...i] with y[1...j] given that x[i] is aligned to y[j] I x (i,j) = score of the best alignment of x[1...i] with y[1...j] given that x[i] is aligned to a gap I y (i,j) = score of the best alignment of x[1...i] with y[1...j] given that y[j] is aligned to a gap x i y j x i - - y j M(0,0) = 0 M(0,j) = the score of best alignment between 0 characters of X and j characters of Y that ends in a match = - because no such alignment can exist = M(i,0) I x (i,0) = h + g*i I y (0,j) = h + g*j I x (0,j) = - I y (i,0) = -

76 DP for the affine gap penalty case - traceback M(i,j) = score of the best alignment of x[1...i] with y[1...j] given that x[i] is aligned to y[j] I x (i,j) = score of the best alignment of x[1...i] with y[1...j] given that x[i] is aligned to a gap I y (i,j) = score of the best alignment of x[1...i] with y[1...j] given that y[j] is aligned to a gap x i y j x i - - y j Start at largest of M(m,n), I x (m,n), I y (m,n) Stop at any of M(0,0), I x (0,0), I y (0,0) Note that pointers may traverse all three matrices

77 3- leveled Manha an grid

78 Global alignment example affine gap penalty

79 Global alignment example affine gap penalty

80 DP for the affine gap penalty case Global alignment x i y j x i - - y j " $ M (i, j) = max # $ % $ " $ I x (i, j) = max # $ % $ " $ I y (i, j) = max # $ % $ M (i 1, j 1) + s(x i, y j ) I x (i 1, j 1) + s(x i, y j ) I y (i 1, j 1) + s(x i, y j ) M (i 1, j) + h + g I x (i 1, j) + g I y (i 1, j) + h + g M (i, j 1) + h + g I x (i, j 1) + h + g I y (i, j 1) + g

81 DP for the affine gap penalty case x i y j Local alignment " M (i 1, j 1) + s(x i, y j ) $ $ I M (i, j) = max x (i 1, j 1) + s(x i, y j ) # $ I y (i 1, j 1) + s(x i, y j ) $ % 0 x i - - y j " $ I x (i, j) = max # $ % $ " $ I y (i, j) = max # $ % $ M (i 1, j) + h + g I x (i 1, j) + g I y (i 1, j) + h + g M (i, j 1) + h + g I x (i, j 1) + h + g I y (i, j 1) + g

82 DP for the affine gap penalty case IniGalizaGon Global alignment M(0,0) = 0 M(i,0) = M(0,j) = - I x (i,0) = h + g*i I y (0,j) = h + g*j Local alignment M(0,0) = 0 M(i,0) = M(0,j) = 0 I x (i,0) = 0 I y (0,j) = 0 Traceback Start at largest of: M(m,n), I x (m,n), I y (m,n) Stop at any of: M(0,0), I x (0,0), I y (0,0) Start at largest M(i,j) Stop at any M(i,j)=0

83 DP for the affine gap penalty case Complexity Fill in thee m x n matrices - > 3mn subproblems Each one takes constant Gme Total rungme O(mn) Space complexity O(mn)

84 Gap penalty funcgons Linear gap penalty: Time: O(n 2 ) Space: O(n 2 ) Affine gap penalty: Time: O(n 2 ) Space: O(n 2 ) h Concave gap penalty funcgon: Time: O(n 3 ) Space: O(n 2 )

85 Pairwise sequence alignment What type of alignment should we consider? (global, semi- global, local) Can be done in O(nm) Gme using dynamic programming, with either linear or affine gap penalges Can be done in O(nm) space, or even O(n+m) space We can find all alignments that have the best score Works for either DNA or protein sequences, with different subsgtugon matrices