Bioinformatics for Computer Scientists (Part 2 Sequence Alignment) Sepp Hochreiter

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1 Bioinformatics for Computer Scientists (Part 2 Sequence Alignment) Institute of Bioinformatics Johannes Kepler University, Linz, Austria

2 Sequence Alignment 2. Sequence Alignment

3 Sequence Alignment 2.1 Motivation Kind of mutations for a DNA example point mutations: CCGTCAGTTACGCCGTATCGTCTAGCT CCGCTCAGTTATCGCCGTAGTCGTCTACGCT deletion: CCGTCAGTTACGCCGTATCGTCTAGCT CCGTCAGTT CCGTCAGTTACGCCGTATCGTCTAGCT CCGTATCGTCTAGCT insertion: CCGTCAGTTACG CCGTATCGTCTAGCT CCGTCAGTT CCGTCAGTTCCGTATCGTCTAGCTCCGTATCGTCTAGCT Deletion and insertion are indistinguishable: Indel Goal: optimal position of the blanks (max. score)

4 Sequence Alignment 2.2 Sequence Similarities and Scoring BIOINFORMATICS BOILING FOR MANICS BIOI--N-FORMATICS B-OILINGFORMANICS 12 identical letters out of the 14 letters of BIOINFORMATICS Mutations: (1) delete I BOINFORMATICS (2) insert LI BOILINFORMATICS (3) insert G BOILINGFORMATICS (4) change T into N BOILINGFORMANICS Is I deleted form the first string or inserted in the second? Indels are denoted by -

5 Sequence Alignment Identity Matrix Dot matrix: one sequence on the top and the other vertically letters of the sequences are paired (all pairs) each matching pair of letters receives a dot

6 Sequence Alignment Identity Matrix B I O I N F O R M A T I C S B O I L I N G F O R M A N I C S

7 Sequence Alignment Identity Matrix Which pairs correspond to the optimal alignment? Each path through the matrix is an alignment and vice versa Goal: search path with most dots A simple game, where you can move horizontally (a - in the vertical sequence) vertically (a - in the horizontal sequence) diagonal only if you at the position of a dot (matches) Task: hit as many dots as possible if you run from the upper left corner to the lower right corner.

8 Sequence Alignment Identity Matrix B I O I N F O R M A T I C S B & O & I & L I N & G F & O & R & M & A & N I & C & S &

9 Sequence Alignment Identity Matrix dots on diagonals correspond to matching regions Example: triosephosphate isomerase (TIM) human Yeast E. coli (bacteria) archaeon

10 Sequence Alignment Identity Matrix

11 Sequence Alignment PAM Matrices A R N D C Q E G H I L K M F P S T W Y V A 2 R -2 6 N PAM250 log-odd-scores D C Q E G H I L K M F P S T W Y V

12 Sequence Alignment BLOSUM Matrices PAM matrices: very similar sequences and generalized to less similar by matrix multiplication BLOSUM (BLOck SUbstitution Matrix, Henikoff and Henikoff, 1992) is based on the Blocks database BLOSUM directly determines the similarity BLOSUM p: p% identity of the blocks BLOSUM 62 (62% identity) is most popular

13 Sequence Alignment BLOSUM Matrices Calculation of the BLOSUM matrices: 1. Sequences with at least p% identity are clustered. Each cluster provides a sequence of frequencies. In the following we only consider the case without frequencies. 2. Frequency sequences are compared and pairs (i,j) counted by c i,j according to column k: n k i amino acids i and nk j amino acids j c k i,j = µ n k i 2 = 1 2 n k i 2 for i = j n k i nk j for i>j n k i n k i n k i n k i where the factor 1/2 accounts for symmetry and subtracts the original sequence (no mutation)

14 Sequence Alignment BLOSUM Matrices 3. For N sequences of length L compute c i,j = X c k i,j, Z = X c i,j = k i,j<i q i,j LN (N 1) 2 = c i,j Z, q j,i = q i,j i>j q i,j =2p(i, j), p i,j = p(j i) 4. probability to observe amino acid i is q i = q i,i + X j6=i q i,j is divided by 2: mutations from i to j and j to i in step 2. q i,j 2 5. Likelihood ratios and the log-odd ratios q i,i q 2 i, q i,j /2 q i q j, BLOSUM i,j = 2log 2 q i,i q 2 i 2log 2 for i = j q i,j 2 q i q j for i 6= j BLOSUM values are rounded to integers

15 Sequence Alignment BLOSUM Matrices example for computing BLOSUM100 matrix 1 NFHV 2 DFNV 3 DFKV 4 NFHV 5 KFHR R N D H K F V R N D H K F V Z = = 40 = X i j c i,j 1x2+1x1

16 Sequence Alignment BLOSUM Matrices R N D H K F V R N D H K F V R 0.05 N 0.15 D 0.1 H 0.15 K 0.1 F 0.25 V 0.2 N: ( ) = 0.15

17 Sequence Alignment BLOSUM Matrices R N D H K F V R N D H K F V R N D H K F V R N D H K F V likelihood ratio log-odd ratios

18 Sequence Alignment BLOSUM Matrices Now we consider clusters and frequencies fi,l k :frequency(aminoacidi, kth column, lth cluster) c k i,j = X X l f k i,l l,m:l6=m X m:m6=l f k i,l f k j,m = f k j,m = n k i nk j X l f k i,l f k j,l, nk i = X l f k i,l Ã! c k i,i = 1 n k 2 X i f k 2 2 i,l l Other computations remain the same

19 Sequence Alignment BLOSUM Matrices A R N D C Q E G H I L K M F P S T W Y V A R N D C Q E G H I L K M F P S T W Y V BLOSUM62 Scoring matrix

20 Sequence Alignment BLOSUM Matrices BLOSUM and PAM compared: PAM100 BLOSUM90 PAM120 BLOSUM80 PAM160 BLOSUM60 PAM200 BLOSUM52 PAM250 BLOSUM45 PAM: context dependent, dependency between substitutions low probability mutations are not as well observed subsequences of very similar sequences (bias to mutation) BLOSUM: not model based evolutionary relationships not considered

21 Sequence Alignment Gap Penalties sequence similarities pointwise more complex scores? simple scores lead to efficient algorithms

22 Sequence Alignment Gap Penalties BIOINFORMATICS BOILING FOR MANICS Gap: maximal substring of - BIOI--N-FORMATICS B-OILINGFORMANICS gaps contribute negatively to the score but how? linear gap penalty: - l d (l is gap length, d is cost)

23 Sequence Alignment Gap Penalties However: neighboring indels may result from a single mutation and are statistically not independent Sequence with introns and exons may be compared to a measured sequence (x-ray, NMR) --> missing introns Affine gap penalty: d (l 1) e d: gap open penalty e: gap extension penalty

24 Sequence Alignment Gap Penalties Parameter: gap penalty gap opening penalty d and gap extension penalty e d=20 and e=1: RKFFVGGNWKMNGDKKSLNGAKLSADTEVVCGAPSIYLDF. :..:..:.... :. : RTFFVGGNFK LNTASIPENVEVVICPPATYLDY d=1 and e=1: RKFFVGGNWKMNGDKKSL--NGAKLSADTEVV-CGAPSIYLDF. : :.. : :. :. : RTFFVGGNFKLN--TASIPEN---V----EVVIC-PPATYLDY d=4 and e=4: RKFFVGGNWKMNGDKKSLNGAKLSADTEVVCGAPSIYLDF. : :.. :.: :.:. :. :. : RTFFVGGNFKLN--TASI--PE-NVEV-VIC-PPATYLDY few gaps: e<d

25 Sequence Alignment 2.3 Global - Needleman-Wunsch 2.3 Alignment Algorithms: Global Alignment using the Needleman-Wunsch Algorithm How to compute where to insert gaps to obtain the best score?

26 Sequence Alignment 2.3 Global - Needleman-Wunsch T C A G A C A T G A T x y x y y x T T C A G G A A C A T (i,j) (0,0) (1,1) (2,1) (3,1) (4,2) (5,3) (6,3) (7,4)

27 Sequence Alignment 2.3 Global - Needleman-Wunsch T C A G A C A T G A T S(i,j) TCAG T i j

28 Sequence Alignment 2.3 Global - Needleman-Wunsch 1970 Needleman and Wunsch: Dynamic Programming alignment of two sequences of length n and m can be reduced to the alignment of two shorter sequences match gap gap x?x x- y y-?y x = T C A G A C A = T G A T y T C A G A C A T G A T T C A G A C A T G A T T C A G A C A T G A T Either the ends match or the end of one sequence is more to the right then the end of the other sequence.

29 Sequence Alignment 2.3 Global - Needleman-Wunsch optimal score S(n, m) x y : 1.sequence with : 2. sequence with d : gap penalty (linear gap) s : scoring function x i,1 i n y i,1 i m Recursion for S: S(i, j) = max S(i 1,j 1) + s(x i,y j ) S(i 1,j) d S(i, j 1) d S(0, 0) = 0 and S( 1, j) = S(i, 1) = S(0,j)= jd and S(i, 0) = id

30 Sequence Alignment 2.3 Global - Needleman-Wunsch 0 x 1... x i 1 x i... x n 0 S(0, 0) S(1, 0) S(n, 0) y 1 S(0, 1) S(1, 1) S(n, 1) y y y j 1 S(i 1,j 1) S(i, j 1) & y j S(i 1,j) S(i, j) y m S(0,m) S(1,m) S(n, m)

31 Sequence Alignment 2.3 Global - Needleman-Wunsch Recursion to compute optimal score plus path Problem: multiple computations exponential complexity

32 Sequence Alignment 2.3 Global - Needleman-Wunsch T C A G A C A T = 0-2= = Enter optimal score G 4-2-2=-4-2-2= A T = =-4 +2 for x l = y l : s(x l,y l ) = -1 for x l y l : -2 for x l, y l = : and 1-2=-1 u = u shortend +s(x l,y l ) maximum

33 Sequence Alignment 2.3 Global - Needleman-Wunsch During filling the matrix the path must be memorized. The S(i 1,j 1),S(i 1,j),S(i, j 1) from which S(i, j) was computed must be stored in a variable B: B(i, j) =(i 1,j 1) or (i 1,j) or (i, j 1) This variable allows to generate the alignment through backtracking starting from (n,m): x (i 1,j 1) then print i y j if B(i, j) = (i 1,j) then print (i, j 1) then print x i y j

34 Sequence Alignment 2.3 Global - Needleman-Wunsch T C A G A C A backtracking T x y G A x T y T T C A G G A A C A T T T C A G G A A C T A

35 Sequence Alignment 2.3 Global - Needleman-Wunsch Algorithm 1 Needleman-Wunsch with linear gap Input: two sequences x and y with length n and m, respectively; scoring matrix s, gappenaltyd Output: optimal global alignment and its score BEGIN INITIALIZATION S(0, 0) = 0, S(0,j)= jd,1 j m, ands(i, 0) = id,1 i n END INITIALIZATION BEGIN PROCEDURE for 1 i n do for 1 j m do a(i 1,j 1) = S(i 1,j 1) + s(x i,y j ),a(i 1,j) = S(i 1,j) d, a(i, j 1) = S(i, j 1) d S(i, j) = max{a(i 1,j 1),a(i 1,j),a(i, j 1)} B(i, j) = argmax{a(i 1,j 1),a(i 1,j),a(i, j 1)} end for end for print Score: S(n, m) (i, j) = (n, m) while (i, j) 6= (0, 0) do if B(i, j) = (i, j) = B(i, j) end while END PROCEDURE (i 1,j 1) then print (i 1,j) then print (i, j 1) then print x i y j x i y j

36 Sequence Alignment 2.3 Global - Needleman-Wunsch G A K L S A D T E V V C G A P S I Y L D F R T F F V G G N F K L N T A S I P E N V E V V I C P P A T Y L D Y

37 Sequence Alignment 2.3 Global - Needleman-Wunsch R K F F V G G N W K M N G D K K S L N G R T F F V G G N F K L N T A S I P E N V E V V I C P P A T Y L D Y

38 Sequence Alignment 2.3 Global - Needleman-Wunsch Affine Gap Penalty Problem: long term dependencies Introducing a gap implies a gap opening event earlier in the sequence and all earlier events must be considered: S(i 1,j 1) + s(x i,y j ) S(i, j) = max S(i k, j) d (k 1) e, 1 k i S(i, j k) d (k 1) e, 1 k j two sequences of length n: complexity O n 3 because all S(i, j) must considered ( O n 2 ) and checking all previous gap openings O (n) for

39 Sequence Alignment 2.3 Global - Needleman-Wunsch Idea: propagate 3 matrices best score up to position (i, j) : S(i, j) best score up to position (i, j) with an opened gap in x at position i: G x (i, j) best score up to position (i, j) with an opened gap in y at position j: G y (i, j) G x G y For and it must be checked whether extending an existing gap or to introduce a new gap gives a better score

40 Sequence Alignment 2.3 Global - Needleman-Wunsch recursion equations: G x (i, j) = max ½ S(i, j 1) d G x (i, j 1) e, G y (i, j) = max initialization: ½ S(i 1,j) d G y (i 1,j) e S(0, 0) = 0,G y (0, 0) = and G x (0, 0) =, S(i, 0) = G y (i, 0) = d (i 1) e, G x (i, 0) =, S(0,j)=G x (0,j)= d (j 1) e, G y (0,j)= and S(i, j) = max{s(i 1,j 1),G y (i 1,j 1), G x (i 1,j 1)} + s(x i,y j )

41 Sequence Alignment 2.3 Global - Needleman-Wunsch Algorithm 1 Needleman-Wunsch with affine gap Input: x,y length n, m; scorings, gap opening extend penalty d and e Output: optimal global alignment and its score BEGIN INITIALIZATION S(0, 0) = 0 G x (0,j)=S(0,j)= d (j 1) e, G y (0,j)= d (n + m) e, 1 j m G y (i, 0) = S(i, 0) = d (i 1) e, G x (i, 0) = d (m + n) e, 1 i n, END INITIALIZATION BEGIN PROCEDURE define s(x n+1,y m+1 ) = 0 for 1 i n +1do for 1 j m +1do G x (i, j) = max{s(i, j 1) d, G x (i, j 1) e} if G x (i, j) = S(i, j 1) d then B x (i, j) = d else B x (i, j) = x G y (i, j) = max{s(i 1,j) d, G y (i 1,j) e} if G y (i, j) = S(i, j 1) d then B y (i, j) = d else B y (i, j) = y S(i, j) = max{s(i 1,j 1),G y (i 1,j 1),G x (i 1,j 1)} + s(x i,y j ) if S(i, j) = S(i 1,j 1) + s(x i,y j ) then B(i, j) = d if S(i, j) = G y (i 1,j 1) + s(x i,y j ) then B(i, j) = y if S(i, j) = G x (i 1,j 1) + s(x i,y j ) then B(i, j) = x end for end for print Score: S(n +1,m+1) Time and memory complexity of O(n m) (i, j) = (n +1,m+1), t = B(i, j) while (i, j) 6= (0, 0) do d then print if t = y then print x then print end while Bioinformatics END PROCEDURE for Computer Scientists x i y j ; i = i 1, j = j 1, t = B(i, j) x i ; i = i 1, t = B y(i, j) y j ; j = j 1, t = B x (i, j)

42 Sequence Alignment 2.3 Global - Needleman-Wunsch G A K L S A D T E V V C G A P S I Y L D F R T F F V G G N F K L N T A S I P E N V E V V I C P P A T Y L D Y

43 Sequence Alignment 2.3 Global - Needleman-Wunsch R K F F V G G N W K M N G D K K S L N G R T F F V G G N F K L N T A S I P E N V E V V I C P P A T Y L D Y

44 Sequence Alignment 2.4 Multiple Alignment and Phylogeny Compare more than two sequences Different from pairwise alignment because best solution may lead to suboptimal pairwise alignments

45 Sequence Alignment 2.4 Multiple Alignment and Phylogeny Compare more than two sequences: arranged sequences so that the amino acids for every the columns match as good as possible

46 Sequence Alignment 2.4 Multiple Alignment and Phylogeny

47 Sequence Alignment 2.4 Multiple Alignment and Phylogeny

48 Sequence Alignment 2.4 Multiple Alignment and Phylogeny

49 Sequence Alignment 2.4 Multiple Alignment and Phylogeny Phylogenetics: based on sequence alignment

50 Sequence Alignment 2.4 Multiple Alignment and Phylogeny

51 Sequence Alignment 2.4 Multiple Alignment and Phylogeny The beginning of life