20 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, Global and local alignment of two sequences using dynamic programming
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1 20 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, Pairwise alignment We will discuss: 1. Strings 2. Dot matrix method for comparing sequences 3. Edit distance 4. Global and local alignment of two sequences using dynamic programming 4.1 Strings Definition An alphabet Σ is a finite, non-empty set. The elements of an alphabet are called symbols or letters. A string S over an alphabet Σ is obtained by writing a finite sequence S = s 1 s 2... s n of symbols from Σ. The length of a string S is the number of symbols in S, denoted by S. The alphabet for DNA is Σ = {A, G, C, T } and e.g. is a string over Σ of length 15. S = AT GGAAT GCT AAT AG Note: the words String and sequence are used as synonyms in bioinformatics. Usually we will use the word sequence in the biological context (such as DNA and protein sequences) and string in more abstract settings. Definition Let S and T be two strings over an alphabet Σ. concatenation of S and T. We will use ST to denote the Note: Generally ST T S for strings. Definition Let S and T be two strings over an alphabet Σ. We call S a substring of T if there exist strings U, V over Σ such that T = USV. Definition Let Σ be an alphabet and let S = s 1... s n with s i Σ. For all i, j {1,..., n}, i < j, we denote the substring s i... s j by S[i, j]. Example: T = AT GGAAT GCT AAT AG then S = AAT GCT is a substring of T, Definition Let S = s 1... s n be a sequence of length n over the DNA alphabet Σ = {A, G, C, T }. The sequence S C = s c n... s c 1 with A if s i = T s c G if s i = i = C T if s i = A C if s i = G is called the reverse complement of S.
2 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, Sequence formats The most widely used format for DNA and protein sequences is FASTA (pronounced fast-ay). Single FASTA format: Each record has a header line that starts with the symbol > and contains an identifier for the sequence. The following lines contain the actual sequence. Empty lines are skipped. Example: >gi sp P09184 VSR_ECOLI PATCH REPAIR PROTEIN MADVHDKATRSKNMRAIATDTAIEKRLASLLTGQGLAFRVQDASLPGRPDFVV DEYRCVIFTHGCFWHHHHCYLFKVPATRTEFWLEKIGKNVERDRRDISRLQEL GWRVLIVWECALRGREKLTDEALTERLEEWICGEGASAQIDTQGIHLLA Multiple-FASTA-Format: A file in multiple-fasta format contains a series of FASTA records. Example: >SEQUENCE 1 ACTCFTGCGCGCGC... >SEQUENCE 2 ACGCGCGCTCFTGCGCGCGC... >SEQUENCE 3 GTTGGGACTCFTGCGCGCGC... >SEQUENCE 4 TGCGCACTCFTGCGCGCGC Dot matrix sequence comparison A dot matrix analysis is primarily a method for comparing two sequences. An (n m) matrix relating two sequences of length n and m repectively is produced: by placing a dot at each cell for which the corresponding symbols match. Here is an example for the two sequences IMISSMISSISSIPPI and MYMISSISAHIPPIE: Definition Let S = s 1 s 2... s n and T = t 1... t m be two strings of length n and m respectively. A (simple) dot matrix is an n m matrix M defined as: for all i, j with 1 i n, 1 j m. M[i, j] = { 1 for si = t j 0 else,
3 22 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, 2008 Note: The longest common substring within the two strings S and T is then the longest matrix subdiagonal containing only 1s. However, rather than drawing the digit 1 we draw a dot. Some of the properties of a dot plot are the visualization is easy to understand it is easy to find common substrings, they appear as contiguous dots along a diagonal it is easy to find reversed substrings (see assignment) it is easy to discover displacements it is easy to find repeats Example: DNA sequences which encode the Bacteriophage lambda and Bacteriophage P22 repressor proteins: Dot plots of biological sequences will usually contain lots of dots, many of which are considered as noise. To reduce the noise, a window size w and a stringency s are used and a dot is only drawn at point (x, y) if in the next w positions at least s characters are equal. w = 1, s = 1 w = 11, s = 7 w = 23, s = 15 Dot matrix analysis of human LDL receptor protein 1 against itself: 1 The low density lipoprotein (LDL) receptor is a cell surface protein that plays a central role in the metabolism of cholesterol in humans and animals. Mutations affecting its structure and function give rise to one of the most prevalent human genetic diseases, familial hypercholesterolemia.
4 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, w = 1, s = 1 w = 23, s = 7 The dot plot on the right reveals many repeats in the first 300 positions. 4.4 Limitations of dot plots Only a visualization tool... No score to quantify identical or similar strings Runtime is quadratic; more efficient algorithms to identify identical substrings exist (e.g. based on suffix trees) 4.5 Comparing sequences The comparison of biological sequences is one of the most important operations in computational biology Biology s paradigm: High sequence similarity implies similar structure and/or function In an alignment of two sequences: place one sequences above the other one such that similar or identical characters are in the same column and non-identical/non-similar characters are either placed in the same column as a mismatch or opposite a gap in the other sequence. Example: Two strings: Alignment: IMISSMISSISSIPPI I-MISSMISSISIPPI- MYMISSISAHIPPIE MYMISS-ISAH-IPPIE Definition Given two sequences X and Y over an alphabet Σ. An alignment A of X and Y is obtained by inserting dashes ( - ) so that both resulting strings X and Y are of equal length and thus can be written one above the other in such a way that each character in the one string is opposite a unique character in the other string. Note: Usually, we require that no two dashes are aligned in this way. Example: is one alignment, another one is X= Y E - S T E R D A Y Y= - E A S T E R S - -
5 24 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, 2008 Which one is the best alignment? X= Y E - S T E R D A Y Y= - E A S T E R - S - In order to evaluate an alignment we need a scoring system. The score of two aligned characters in biology is mostly a similarity score, however we will first start with the mathematical concepts of distances of sequences and/or characters in sequences Hamming distance Definition Let X = (x 1,..., x n ) and Y = (y 1,..., y n ) be two strings of length n over an alphabet Σ. Then d H (X, Y ) = {i : x i y i } is called the Hamming distance. Example: X =ATGGAACTA Y =ATGCAAGAT d H (X, Y ) = Edit or Levenshtein distance One way of defining the distance between two strings follows from the possibilities and number of editing operations needed to transform one string into the other. Definition The edit distance (also known as Levenshtein distance) d edit between two sequences X and Y is is the minimum number of edit operations of type Replacement, Insertion, or Deletion, that one needs to transform sequence X into sequence Y : d edit (X, Y ) = min{r(x, Y ) + I(X, Y ) + D(X, Y )} Using M for match, an edit transcript is a string over the alphabet I, D, R, M that describes a transformation of X to Y. Example: Given two strings X = Y = YESTERDAY EASTERS Here is a minimum edit transcript for the above example: edit transcript= D M I M M M M R D D. X= Y E S T E R D A Y Y= E A S T E R S The edit distance d edit (X, Y ) of X, Y is 5. Edit transcripts and alignments are mathematically equivalent ways of describing a relationship between two strings.
6 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, However, an edit transcript implies a set of putative mutational events, whereas an alignment presents a static picture of the relationship. To obtain an algorithm for computing the edit distance, it is best to aim at computing an optimal alignment Dynamic programming calculation of the edit distance Given two sequences X = x 1... x n and Y = y 1... y m. We want to compute the edit distance D edit (X, Y ) between X and Y. Let D(i, j) denote the edit distance of the two prefixes x 1... x i and y 1... y j. We want to obtain D edit (X, Y ) = D(n, m) by computing D(i, j) for all pairs of prefixes x 1... x i and y 1... y j. We will use the standard dynamic programming approach that has three essential components: the recurrence relation, the tabular computation, and the traceback. The recursion is computed in tabular form: 0 x 1 x 2 x 3... x i 1 x i... x n 0 D(0, 0) y 1 y 2 y 3... y j 1 D(i 1, j 1) D(i, j 1) y j D(i 1, j) D(i, j)... y m The recurrence relation determines how we can obtain the value for D(i, j) from values for smaller indices. When there are no smaller indices, then we must explicitly state base conditions. Base conditions: We set D(i, 0) = i for all 0 i n. This corresponds to an alignment in which the first i characters of X are aligned to the left of the first character of Y. x 1 x 2... x i x i+1... y 1 We set D(0, j) = j for all 1 j m. This corresponds to an alignment in which the first j characters of Y occur to the left of the first character of X.... x 1 y 1 y 2... y j y j+1
7 26 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, The recurrence relation The recurrence relation for D(i, j) when both i and j are positive, is given by D(i, j) = min {D(i, j 1) + 1, D(i 1, j) + 1, D(i 1, j 1) + t(i, j)}, { 0 if xi = y where t(i, j) = j, 1 else. Example: We can compute the edit score between sequences GATTAG and ATTAC by filling the following matrix: Distance: D 0 G A T T A G 0 A T T A C Proof of the recurrence relation Theorem The above recurrence relation is correct. For the proof we will show: Lemma 1 The value of D(i, j) must be either D(i, j 1) + 1, D(i 1, j) + 1 or D(i 1, j 1) + t(i, j). Lemma 2 D(i, j) min {D(i, j 1) + 1, D(i 1, j) + 1, D(i 1, j 1) + t(i, j)}. Proof of Lemma 1 Given a minimum edit transcript t 1... t r of x 1... x i to y 1... y j. There are four possibilities for the last symbol t r : I The last operation was to insert y j at the end of the first string. Hence, t 1... t r 1 is a minimum edit transcript of x 1... x i to y 1... y j 1. By definition, the latter contains D(i, j 1) edit operations, thus D(i, j) = D(i, j 1) + 1. D The last operation was to delete x i. Hence, t 1... t r 1 is a minimum edit transcript of x 1... x i 1 to y 1... y j. By definition, the latter contains D(i 1, j) edit operations, thus D(i, j) = D(i 1, j) + 1. R, M The last operation was to replace (or match) x i by y j. Thus t 1... t r 1 is a minimum edit transcript of x 1... x i 1 to y 1... y j 1 that contains D(i 1, j 1) + t(i, j) edit operations. Proof of Lemma 2 We show that all three scores can be achieved, thus so can their minimum:
8 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, There exists a transcript with score D(i, j 1) + 1: transform x 1... x i to y 1... y j 1 using D(i, j 1) edit operations, then use one more to insert y j. There exists a transcript with score D(i 1, j) + 1: transform x 1... x i 1 to y 1... y j using D(i 1, j) edit operations, then use one more to delete x i. There exists a transcript with score D(i 1, j 1) + t(x i, y j ): transform x 1... x i 1 to y 1... y j 1 using D(i 1, j 1) edit operations, then replace by, or match, x i and y j, using one or zero more edit operations, respectively Traceback How to obtain an actual edit transcript that corresponds to the minimal edit distance? Starting at the last cell in the table, trace-back through the table via the predecessor cells that gave rise to the values of the cells Traceback D 0 G A T T A G A T T A C Edit transcript: I M M M M R Alignment: G A T T A G - A T T A C Weighted edit distance We can generalize the edit distance by weighting each of the edit operations I, D and R by a number. The operation weighted edit distance between two sequences X and Y is the minimum sum of weights of any edit transcript from X to Y. A second important generalization is obtained by making the score of an edit operation depend on the characters involved. This gives rise to the alphabet weighted edit distance. Definition Let Σ be a finite alphabet and d a metric on Σ. Let ɛ Σ denote the gap symbol. Then for two strings X and Y of length n and m, respectively, D(i, j):=min {D(i, j 1)+d(x i, ɛ), D(i 1, j)+d(y j, ɛ), D(i 1, j 1)+d(x i, y j )}, and D(n, m) is the alphabet-weighted edit distance between X and Y.
9 28 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, Sequence similarity We have seen how to express string relatedness using the edit distance. In biology, we are usually interested in similarity rather than distance. Given two sequences X = x 1... x n and Y = y 1... y m over an alphabet Σ. A similarity score matrix s : Σ {ɛ} Σ {ɛ} R assigns a similarity score to each pair of characters in Σ {ɛ}. Let A be an alignment of X and Y : X = x 1... x l and Y = y 1... y l denote the two strings obtained after inserting dashes (for the gap symbol ɛ), both of length l. The score S(A) of A is defined as S(A) = l s(x i, y i). i=1 4.7 Similarity alignment Example: for Σ = {A, B, L, } we use the similarity score matrix: s A B L A B L Example of an alignment and the calculation of its score: X = B L A - B L A Y = A L A B B L = Pairwise alignment: Example Alignment between very similar human alpha- and beta globins: HBA_HUMAN HBB_HUMAN GSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKL G+ +VK+HGKKV A+++++AH+D LS+LH KL GNPKVKAHGKKVLGAFSDGLAHLDNLKGTFATLSELHCDKL Here there are many positions at which the two corresponding residues are identical. Many others are functionally conserved, e.g. the D-E pair, both negatively charged amino acids. Plausible alignment to leghaemoglobin from yellow lupin: HBA_HUMAN GSAQVKGHGKKVADALTNAVAHV---D--DMPNALSALSDLHAHKL H+ KV + +A ++ +L+ L+++H+ K LGB2_LUPLU NNPELQAHAGKVFKLVYEAAIQLQVTGVVVTDATLKNLGSVHVSKG
10 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, These two proteins are known to be evolutionarily related, have the same 3D structure and both have the same function, thus this is most probably a biologically meaningful alignment. However, there are only a few identities and many gaps. A spurious high-scoring alignment of human alpha globin to a nematode glutathione S- transferase homologue: HBA_HUMAN F11G11.2 GSAQVKGHGKKVADALTNAVAHVDDMPNALSALSD----LHAHKL GS+ + G + +D L ++ H+ D+ A +AL D ++AH+ GSGYLVGDSLTFVDLLVAQHTADLL--AANAALLDEFPQFKAHQE This alignment has a similar number of identities and conservative changes as in the previous example. However, it is an alignment between two proteins that have completely different structure and function, and is thus considered spurious. 4.9 Substitution matrices To be able to score an alignment, we need to determine score terms for each aligned residue pair. Definition A substitution matrix S over an alphabet Σ = {a 1,..., a κ } has κ κ entries, where each entry (i, j) assigns a score for a substitution of the letter a i by the letter a j in an alignment. Substitution matrices are generated as follows: Consider a database of high quality non-gapped alignments. Compute the frequency f(a i ) of each symbol a i and the frequency f(a i, a j ) of each substitution a i a j in the database. Based on this counts, compare a null/random model against the match model Consider non-gapped alignments of the form: X = x 1 x 2... x n Y = y 1 y 2... y n The null hypothesis is: the two sequences are unrelated (not homologous). So, the alignment is then random with a probability described by a random model R: each letter a occurs independently with some probability p a, and the probability of the two sequences is the product: P (X, Y R) = p xi p yj. i j The alternative hypothesis is the match model M: the two sequences are related (homologous). Aligned pairs of residues occur with a joint probability p ab, the probability that a and b have each evolved from some (unknown) common ancestor residue c. In this case, the probability for the whole alignment is: P (X, Y M) = i p xi y i.
11 30 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, 2008 The ratio of the two gives a measure of the relative likelihood that the sequences are related (model M) as opposed to being unrelated (model R). This ratio is called the odds ratio: P (X, Y M) P (X, Y R) = i p x i y i i p x i i p y i = i p xi y i p xi p yi To obtain an additive scoring scheme, we take the logarithm (base 2 is usually chosen) to get the log-odds ratio: P (X, Y M) S = log( P (X, Y R) ) = log( p xi y i ) = s(x i, y i ), p i xi p yi i with ( ) pab s(a, b) := log. p a p b In this approach, a matrix S = s(a, b) determines the score for each aligned residue pair, known as a score or substitution matrix. For amino-acid alignments, commonly used matrices are the PAM and BLOSUM matrices BLOSUM matrices A popular family of substitution matrices are the BLOSUM (=BLOcks SUbstitution Matrix) matrices. These were empirically derived as described above from the BLOCKS database (Henikoff and Henikoff, 1992). Blocks are multiply aligned ungapped segments corresponding to the most highly conserved regions of proteins. Different members of the BLOSUM family of matrices were created by considing alignments of sequences with different levels of identity. For example, the most commonly used matrix is BLOSUM62. To create this matrix, sequences where clustered so that any two sequences of 62% sequence identity are placed in the same cluster and all counts are based on comparisons between pair of sequences contained in different clusters BLOSUM62 BLOSUM matrices are scaled so that their values are in half-bits, that is, the log-odds were multiplied by 2/ log 2 and then rounded to the nearest integer value. 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
12 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, Classification of amino acids Notice that the values in the BLOSUM62 matrix reflect some of the similarities between different amino-acids shown here: 4.13 Gap penalties Gaps are undesirable and thus penalized. The standard cost associated with a gap of length g is given either by a linear score γ(g) = gd or an affine score γ(g) = d (g 1)e, where d is the gap open penalty and e is the gap extension penalty. Usually, e < d, with the result that less isolated gaps are produced, as shown in the following comparison: Linear gap penalty: GSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKL GSAQVKGHGKK VA--D----A-SALSDLHAHKL Affine gap penalty: GSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKL GSAQVKGHGKKVADA SALSDLHAHKL 4.14 The number of possible alignments How many different gapped alignments are possible between two sequences x = (x 1, x 2,..., x n ) and y = (y 1, y 2,..., y m )? A sequences of pairs (i 1, j 1 ), (i 2, j 2 ),..., (i r, j r ) is called a subsequence of indices of x and y, if 1 i 1 i 2 i r n and if 1 j 1 j 2 j r m. We use such a subsequence to specify the set of all positions that are paired by a given alignment of x and y. Example: Alignment Subsequence a - c g t g t a - c a c g t g t a c \ / / a g c - t - g a c c a g c t g a c c
13 32 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, 2008 Here the subsequence is: (1, 1), (2, 3), (4, 4), (6, 5), (7, 6), (8, 8). Lemma The number of possible alignments between x and y equals the number of possible subsequences of indices, min(n,m) ( )( ) n m N(n, m) =. r r r=0 Proof: For each r {0, 1,..., min(n, m)}: the number of ordered selections of i 1, i 2,..., i r in 1, 2,..., n is ( ) n r and the number of ordered selections of j1, j 2,..., j r in 1, 2,..., m is ( ) m r. All these possibilities can be combined. The number N(n, n) = n ( n n ( r=0 r)( r) = 2n ) n can be approximated by 2 2n πn, using Stirling s formula. Hence, N(1000, 1000) Alignment algorithms Given a scoring scheme, we need to have an algorithm that computes the highest-scoring alignment of two sequences. As for the edit distance-based alignments we will discuss alignment algorithms based on dynamic programming. They are guaranteed to find the optimal scoring alignment. However, for large sequences they can be too slow and heuristics (such as BLAST, FASTA, MUMMER etc) are then used that usually perform very well, but will miss the best alignment for some sequence pairs. In particular we will study an algorithm for the computation of a global alignment and one for the computation of a local alignment Global alignment: Needleman-Wunsch algorithm The Needleman-Wunsch algorithm 2 is a dynamic program that solves the problem of obtaining the best global alignment of two sequences. Idea: Build up an optimal alignment using previous solutions for optimal alignments of smaller substrings. Given two sequences X = (x 1, x 2,..., x n ) and Y = (y 1, y 2,..., y m ). We will compute a matrix F : {1, 2,..., n} {1, 2,..., m} R in which F (i, j) equals the best score of the alignment of the two prefixes (x 1, x 2,..., x i ) and (y 1, y 2,..., y j ) Needleman-Wunsch algorithm This will be done recursively by setting F (0, 0) = 0 and then computing F (i, j) from F (i 1, j 1), F (i 1, j) and F (i, j 1): 2 Saul Needleman and Christian Wunsch (1970), improved by Peter Sellers (1974).
14 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, x 1 x 2... x i 1 x i... x n 0 F (0, 0) y 1 y 2... y j 1 F (i 1, j 1) F (i, j 1) y j F (i 1, j) F (i, j)... y m The recursion There are three ways in which an alignment can be extended up to (i, j): x i aligns to y j : x i aligns to a gap: y j aligns to a gap: A G A x i A G G y j A A G A x i A G G y j - T G A x i - T G G C y j We obtain F (i, j) as the largest score arising from these three options: F (i 1, j 1) + s(x i, y j ) F (i, j) := max F (i 1, j) d F (i, j 1) d. This is applied repeatedly until the whole matrix F (i, j) is filled with values The recursion To complete the description of the recursion, we need to set the values of F (i, 0) and F (0, j) for i 0 and j 0: We set F (i, 0) = for i = 0, 1,..., n and we set F (0, j) = for j = 0, 1,..., m. The final value F (n, m) contains the score of the best global alignment between X and Y. To obtain an alignment corresponding to this score, we must find the path of choices that the recursion made to obtain the score using traceback Example of a global alignment matrix Needleman-Wunsch matrix of the sequences GATTAG and ATTAC, scoring values s(a, a) = 1, s(a, b) = 1 and a linear gap cost of d = 2:
15 34 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, 2008 F 0 G A T T A G 0 0 A T T A C Score: ; Alignment: Pseudo code of Needleman-Wunsch algorithm Input: two sequences X and Y Output: optimal alignment and score α Initialization: Set F (i, 0) := i d for all i = 0, 1, 2,..., n Set F (0, j) := j d for all j = 0, 1, 2,..., m For i = 1, 2,..., n do: For j = 1, 2,..., m do: 8 < Set F (i, j) := max : F (i 1, j 1) + s(x i, y j) F (i 1, j) d F (i, j 1) d Set backtrace T (i, j) to the maximizing pair (i, j ) The best score is α := F (n, m) Set (i, j) := (n, m) repeat if T (i, j) = (i 1, j 1) print `x i 1 y j 1 else if T (i, j) = (i 1, j) print `x i 1 Set (i, j) := T (i, j) until (i, j) = (0, 0). else print ` y j Complexity Complexity of the Needleman-Wunsch algorithm: We need to store (n + 1) (m + 1) numbers. Each number takes a constant number of calculations to compute: three additions and a max. Hence, the algorithm requires O(nm) time and memory. Something to think about: if we are only interested in the best score, but not the actual alignment, then it is easy to reduce the space requirement to linear Local alignment Global alignment is applicable when we have two similar sequences that we want to align from endto-end, e.g. two homologous genes from related species. Often, however, we have two sequences X and Y and we would like to find the best match between substrings of both. For example, we may want to find the position of a fragment of DNA in a genomic sequence:
16 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, AATGATCGTAGCGCTGAGAGGGGGAGGAGTTTAGGCGATGA GCTCAGAGAGGGAC Here the score of an alignment between two substrings would be larger than the score of an alignment between the full lengths strings Smith-Waterman algorithm The Smith-Waterman 3 local alignment algorithm is obtained by making two simple modifications to the global alignment algorithm. (1) In the main recursion, we set the value of F (i, j) to zero, if all attainable values at position (i, j) are negative: 0, F (i 1, j 1) + s(x F (i, j) = max i, y j ), F (i 1, j) d, F (i, j 1) d. The value F (i, j) = 0 indicates that we should start a new alignment at (i, j). This implies for the base conditions: set F (i, 0) = and F (0, j) = for all i = 0, 1, 2,..., n and j = 0, 1, 2,..., m Smith-Waterman algorithm (2) Instead of starting the traceback at (n, m), we start it at the cell with the highest score: arg max F (i, j). The traceback ends upon arrival at a cell with score 0, with corresponds to the start of the alignment. For this algorithm to work, we require that the expected score for a random match is negative, i.e. that p a p b s(a, b) < 0, a,b Σ where p a and p b are the probabilities for the seeing the symbol a or b respectively, at any given position. Otherwise, matrix entries will tend to be positive, producing long matches between random sequences Example Smith-Waterman matrix of the sequences GATTAG and ATTAC with s(a, a) = 1, s(a, b) = 1 and s(a, ) = s(, a) = 2: 3 Smith, T. and M. Waterman, Identification of common molecular subsequences. J. Mol. Biol. 147: , 1981
17 36 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, 2008 F 0 G A T T A G 0 A T T A C Score: ; Alignment = Pseudo code of Smith-Waterman algorithm Input: two sequences X and Y Output: optimal local alignment and score α Initialization: Set F (i, 0) := 0 for all i = 0, 1, 2,..., n Set F (0, j) := 0 for all j = 1, 2,..., m For i = 1, 2,..., n do: For j = 1, 2,..., m do: 8 >< Set F (i, j) := max >: 0 F (i 1, j 1) + s(x i, y j) F (i 1, j) d F (i, j 1) d Set backtrace T (i, j) to the maximizing pair (i, j ) Set (i, j) := arg max{f (i, j) i = 1, 2,..., n, j = 1, 2,..., m} The best score is α := F (i, j) repeat if T (i, j) = (i 1, j 1) print `x i 1 y j 1 else if T (i, j) = (i 1, j) print `x i 1 Set (i, j) := T (i, j) until F (i, j) = 0. else print ` y j Summary We have discussed: the dot matrix for visual comparison of two sequences, the edit distance and the dynamic programming algorithm for its computation, global alignments and the Needleman-Wunsch algorithm, and local alignments and the Smith-Waterman algorithm. An implementation of any of the dynamic programming algorithms uses O(nm) time and O(nm) space.
18 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, Affine gap scores The standard alternative to using the above recursion is to use an affine gap score γ(g) = d (g 1)e, with d the gap-open score and e the gap-extension score. We will discuss how to modify the Needleman-Wunsch algorithm for global alignment so as to incorporate affine gap costs. Approach is due to Osamu Gotoh (1982) 4 Instead of using one matrix F (i, j) to represent the best score attainable up to x i and y j, we will now use three matrices M, I x and I y : Case 1 Case 2 Case 3 x i aligns to y j : x i aligns to a gap: y j aligns to a gap: I G A x i L G V y j A I G A x i A G V y j - S G A x i - S L G V y j 1. M(i, j) is the best score up to (i, j), given that x i is aligned to y j, 2. I x (i, j) is the best score up to (i, j), given that x i is aligned to a gap, and 3. I y (i, j) is the best score up to (i, j), given that y j is aligned to a gap Recursion for global alignment with affine gap costs Initialization: M(0, 0) = 0, I x (0, 0) = I y (0, 0) = I x (i, 0) = d (i 1)e, M(i, 0) = I y (i, 0) =, for i = 1,..., n, and I y (0, j) = d (j 1)e, M(0, j) = I x (0, j) =, for j = 1,..., m. Recursion: { M(i 1, j) d, I x (i, j) = max I x (i 1, j) e; { M(i, j 1) d, I y (i, j) = max I y (i, j 1) e; M(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 ). Here we assume that a gap in one string is not immediately followed by a gap in the other, which is ok, if d e is < the lowest mismatch score Example of a global alignment with affine gap costs Given two sequences X = TTAGAT and Y = TTG. We use s(a, a) = 1, s(a, b) = 1, γ(g) = d (g 1)e with d = 4 and e = 1 for the match, mismatch, gap-open and gap-extension scores, respectively: 4 Gotoh, O. An improved algorithm for matching biological sequences. J. Mol. Biol. 162: , 1982.
19 38 Grundlagen der Bioinformatik, SS 08, D. Huson, May 27, T T G M I x I y M I x I y M I x I y M I x I y 0 T T A G T 4.23 Automaton description We can interpret the three matrices M, I x and I y as three different states in a finite state automaton. This automaton emits two aligned characters on transition across an edge: gap open gap open start substitution substitution gap open gap extension I x substitution M gap open I y gap extension substitution end We do not want a gap in x to be immediately followed by a gap in y. Therefore, there is no transition between I x and I y.
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