Combinatorial Pattern Matching

Transcription

1 Combinatorial Pattern Matching

2 Outline Hash Tables Repeat Finding Exact Pattern Matching Keyword Trees Suffix Trees Heuristic Similarity Search Algorithms Approximate String Matching Filtration Comparing a Sequence Against a Database Algorithm behind BLAST Statistics behind BLAST PatternHunter and BLAT

3 Genomic Repeats Example of repeats: ATGGTCTAGGTCCTAGTGGTC Motivation to find them: Genomic rearrangements are often associated with repeats Trace evolutionary secrets Many tumors are characterized by an explosion of repeats

4 Genomic Repeats The problem is often more difficult: ATGGTCTAGGACCTAGTGTTC Motivation to find them: Genomic rearrangements are often associated with repeats Trace evolutionary secrets Many tumors are characterized by an explosion of repeats

5 l-mer Repeats Long repeats are difficult to find Short repeats are easy to find (e.g., hashing) Simple approach to finding long repeats: Find exact repeats of short l-mers (l is usually 10 to 13) Use l-mer repeats to potentially extend into longer, maximal repeats

6 l-mer Repeats (cont d) There are typically many locations where an l-mer is repeated: GCTTACAGATTCAGTCTTACAGATGGT The 4-mer TTAC starts at locations 3 and 17

7 Extending l-mer Repeats GCTTACAGATTCAGTCTTACAGATGGT Extend these 4-mer matches: GCTTACAGATTCAGTCTTACAGATGGT Maximal repeat: TTACAGAT

8 Maximal Repeats To find maximal repeats in this way, we need ALL start locations of all l-mers in the genome Hashing lets us find repeats quickly in this manner

9 Hashing: What is it? What does hashing do? For different data, generate a unique integer Store data in an array at the unique integer index generated from the data Hashing is a very efficient way to store and retrieve data

10 Hashing: Definitions Hash table: array used in hashing Records: data stored in a hash table Keys: identifies sets of records Hash function: uses a key to generate an index to insert at in hash table Collision: when more than one record is mapped to the same index in the hash table

11 Hashing: Example Where do the animals eat? Records: each animal Keys: where each animal eats

12 Hashing DNA sequences Each l-mer can be translated into a binary string (A, T, C, G can be represented as 00, 01, 10, 11) After assigning a unique integer per l-mer it is easy to get all start locations of each l- mer in a genome

13 Hashing: Maximal Repeats To find repeats in a genome: For all l-mers in the genome, note the start position and the sequence Generate a hash table index for each unique l-mer sequence In each index of the hash table, store all genome start locations of the l-mer which generated that index Extend l-mer repeats to maximal repeats

14 Hashing: Collisions Dealing with collisions: Chain all start locations of l-mers (linked list)

15 Hashing: Summary When finding genomic repeats from l-mers: Generate a hash table index for each l-mer sequence In each index, store all genome start locations of the l-mer which generated that index Extend l-mer repeats to maximal repeats

16 Pattern Matching What if, instead of finding repeats in a genome, we want to find all sequences in a database that contain a given pattern? This leads us to a different problem, the Pattern Matching Problem

17 Pattern Matching Problem Goal: Find all occurrences of a pattern in a text Input: Pattern p = p 1 p n and text t = t 1 t m Output: All positions 1< i < (m n + 1) such that the n-letter substring of t starting at i matches p Motivation: Searching database for a known pattern

18 Exact Pattern Matching: A Brute-Force Algorithm PatternMatching(p,t) 1 n ß length of pattern p 2 m ß length of text t 3 for i ß 1 to (m n + 1) 4 if t i t i+n-1 = p 5 output i

19 Exact Pattern Matching: An Example PatternMatching algorithm for: Pattern GCAT Text CGCATC GCAT CGCATC GCAT CGCATC GCAT CGCATC GCAT CGCATC GCAT CGCATC

20 Exact Pattern Matching: Running Time PatternMatching runtime: O(nm) Probability-wise, it s more like O(m) Rarely will there be close to n comparisons in line 4 Better solution: suffix trees Can solve problem in O(m) time Conceptually related to keyword trees

21 Keyword Trees: Example Keyword tree: Apple

22 Keyword Trees: Example (cont d) Keyword tree: Apple Apropos

23 Keyword Trees: Example (cont d) Keyword tree: Apple Apropos Banana

24 Keyword Trees: Example (cont d) Keyword tree: Apple Apropos Banana Bandana

25 Keyword Trees: Example (cont d) Keyword tree: Apple Apropos Banana Bandana Orange

26 Keyword Trees: Properties Stores a set of keywords in a rooted labeled tree Each edge labeled with a letter from an alphabet Any two edges coming out of the same vertex have distinct labels Every keyword stored can be spelled on a path from root to some leaf

27 Keyword Trees: Threading (cont d) Thread appeal appeal

28 Keyword Trees: Threading (cont d) Thread appeal appeal

29 Keyword Trees: Threading (cont d) Thread appeal appeal

30 Keyword Trees: Threading (cont d) Thread appeal appeal

31 Keyword Trees: Threading (cont d) Thread apple apple

32 Keyword Trees: Threading (cont d) Thread apple apple

33 Keyword Trees: Threading (cont d) Thread apple apple

34 Keyword Trees: Threading (cont d) Thread apple apple

35 Keyword Trees: Threading (cont d) Thread apple apple

36 Multiple Pattern Matching Problem Goal: Given a set of patterns and a text, find all occurrences of any of patterns in text Input: k patterns p 1,,p k, and text t = t 1 t m Output: Positions 1 < i < m where substring of t starting at i matches p j for 1 < j < k Motivation: Searching database for known multiple patterns

37 Multiple Pattern Matching: Straightforward Approach Can solve as k Pattern Matching Problems Runtime: O(kmn) using the PatternMatching algorithm k times m - length of the text n - average length of the pattern

38 Multiple Pattern Matching: Keyword Tree Approach Or, we could use keyword trees: Build keyword tree in O(N) time; N is total length of all patterns With naive threading: O(N + nm) Aho-Corasick algorithm: O(N + m)

39 Keyword Trees: Threading To match patterns in a text using a keyword tree: Build keyword tree of patterns Thread the text through the keyword tree

40 Keyword Trees: Threading (cont d) Threading is complete when we reach a leaf in the keyword tree When threading is complete, we ve found a pattern in the text

41 Suffix Trees=Collapsed Keyword Trees Similar to keyword trees, except edges that form paths are collapsed Each edge is labeled with a substring of a text All internal edges have at least two outgoing edges Leaves labeled by the index of the pattern.

42 Suffix Tree of a Text Suffix trees of a text is constructed for all its suffixes ATCATG TCATG CATG ATG TG G Keywor d Tree Suffix Tree

43 Suffix Tree of a Text Suffix trees of a text is constructed for all its suffixes ATCATG TCATG CATG ATG TG G Keywor d Tree Suffix Tree How much time does it take?

44 Suffix Tree of a Text Suffix trees of a text is constructed for all its suffixes ATCATG TCATG CATG ATG TG G quadratic Keywor d Tree Suffix Tree Time is linear in the total size of all suffixes, i.e., it is quadratic in the length of the text

45 Suffix Trees: Advantages Suffix trees of a text is constructed for all its suffixes Suffix trees build faster than keyword trees ATCATG TCATG CATG ATG TG G quadratic Keywor d Tree linear (Weiner suffix tree algorithm) Suffix Tree

46 Use of Suffix Trees Suffix trees hold all suffixes of a text i.e., ATCGC: ATCGC, TCGC, CGC, GC, C Builds in O(m) time for text of length m To find any pattern of length n in a text: Build suffix tree for text Thread the pattern through the suffix tree Can find pattern in text in O(n) time! O(n + m) time for Pattern Matching Problem Build suffix tree and lookup pattern

47 Pattern Matching with Suffix Trees SuffixTreePatternMatching(p,t) 2 Build suffix tree for text t 3 Thread pattern p through suffix tree 4 if threading is complete 5 output positions of all p-matching leaves in the tree 6 else 7 output Pattern does not appear in text

48 Suffix Trees: Example

49 Multiple Pattern Matching: Summary Keyword and suffix trees are used to find patterns in a text Keyword trees: Build keyword tree of patterns, and thread text through it Suffix trees: Build suffix tree of text, and thread patterns through it

50 Approximate vs. Exact Pattern Matching So far all we ve seen exact pattern matching algorithms Usually, because of mutations, it makes much more biological sense to find approximate pattern matches Biologists often use fast heuristic approaches (rather than local alignment) to find approximate matches

51 Heuristic Similarity Searches Genomes are huge: Smith-Waterman quadratic alignment algorithms are too slow Alignment of two sequences usually has short identical or highly similar fragments Many heuristic methods (i.e., FASTA) are based on the same idea of filtration Find short exact matches, and use them as seeds for potential match extension Filter out positions with no extendable matches

52 Dot Matrices Dot matrices show similarities between two sequences FASTA makes an implicit dot matrix from short exact matches, and tries to find long diagonals (allowing for some mismatches)

53 Dot Matrices (cont d) Identify diagonals above a threshold length Diagonals in the dot matrix indicate exact substring matching

54 Diagonals in Dot Matrices Extend diagonals and try to link them together, allowing for minimal mismatches/indels Linking diagonals reveals approximate matches over longer substrings

55 Approximate Pattern Matching Problem Goal: Find all approximate occurrences of a pattern in a text Input: A pattern p = p 1 p n, text t = t 1 t m, and k, the maximum number of mismatches Output: All positions 1 < i < (m n + 1) such that t i t i+n-1 and p 1 p n have at most k mismatches (i.e., Hamming distance between t i t i+n-1 and p < k)

56 Approximate Pattern Matching: A Brute- Force Algorithm ApproximatePatternMatching(p, t, k) 2 n ß length of pattern p 3 m ß length of text t 4 for i ß 1 to m n dist ß 0 6 for j ß 1 to n 7 if t i+j-1!= p j 8 dist ß dist if dist < k 10 output i

57 Approximate Pattern Matching: Running Time That algorithm runs in O(nm). Landau-Vishkin algorithm: O(kn) We can generalize the Approximate Pattern Matching Problem into a Query Matching Problem : We want to match substrings in a query to substrings in a text with at most k mismatches Motivation: we want to see similarities to some gene, but we may not know which parts of the gene to look for

58 Query Matching Problem Goal: Find all substrings of the query that approximately match the text Input: Query q = q 1 q w, text t = t 1 t m, n (length of matching substrings), k (maximum number of mismatches) Output: All pairs of positions (i, j) such that the n-letter substring of q starting at i approximately matches the n-letter substring of t starting at j, with at most k mismatches

59 Approximate Pattern Matching vs Query Matching

60 Query Matching: Main Idea Approximately matching strings share some perfectly matching substrings. Instead of searching for approximately matching strings (difficult) search for perfectly matching substrings (easy).

61 Filtration in Query Matching We want all n-matches between a query and a text with up to k mismatches Filter out positions we know do not match between text and query Potential match detection: find all matches of l-tuples in query and text for some small l Potential match verification: Verify each potential match by extending it to the left and right, until (k + 1) mismatches are found

62 Filtration: Match Detection If x 1 x n and y 1 y n match with at most k mismatches, they must share an l-tuple that is perfectly matched, with l = ën/(k + 1)û Break string of length n into k+1 parts, each each of length ën/(k + 1)û k mismatches can affect at most k of these k+1 parts At least one of these k+1 parts is perfectly matched

63 Filtration: Match Detection (cont d) Suppose k = 3. We would then have l=n/(k+1)=n/4: 1 l l +1 2l 2l +1 3l 3l +1 n 1 2 k k + 1 There are at most k mismatches in n, so at the very least there must be one out of the k+1 l tuples without a mismatch

64 Filtration: Match Verification For each l -match we find, try to extend the match further to see if it is substantial Extend perfect match of length l query until we find an approximate match of length n with k text mismatches

65 Filtration: Example l -tuple length k = 0 k = 1 k = 2 k = 3 k = 4 k = 5 n n/2 n/3 n/4 n/5 n/6 Shorter perfect matches required Performance decreases

66 Local alignment is to slow Quadratic local alignment is too slow while looking for similarities between long strings (e.g. the entire GenBank database)

67 Local alignment is to slow Quadratic local alignment is too slow while looking for similarities between long strings (e.g. the entire GenBank database)

68 Local alignment is to slow Quadratic local alignment is too slow while looking for similarities between long strings (e.g. the entire GenBank database)

69 Local alignment is to slow Quadratic local alignment is too slow while looking for similarities between long strings (e.g. the entire GenBank database) Guaranteed to find the optimal local alignment Sets the standard for sensitivity

70 Local alignment is to slow Quadratic local alignment is too slow while looking for similarities between long strings (e.g. the entire GenBank database) Basic Local Alignment Search Tool Altschul, S., Gish, W., Miller, W., Myers, E. & Lipman, D.J. Journal of Mol. Biol., 1990 Search sequence databases for local alignments to a query

71 BLAST Great improvement in speed, with a modest decrease in sensitivity Minimizes search space instead of exploring entire search space between two sequences Finds short exact matches ( seeds ), only explores locally around these hits

72 What Similarity Reveals BLASTing a new gene Evolutionary relationship Similarity between protein function BLASTing a genome Potential genes

73 BLAST algorithm Keyword search of all words of length w from the query of length n in database of length m with score above threshold w = 11 for DNA queries, w =3 for proteins Local alignment extension for each found keyword Extend result until longest match above threshold is achieved Running time O(nm)

74 BLAST algorithm (cont d) keyword Query: KRHRKVLRDNIQGITKPAIRRLARRGGVKRISGLIYEETRGVLKIFLENVIRD neighborhood score threshold (T = 13) extension GVK 18 GAK 16 GIK 16 GGK 14 GLK 13 GNK 12 GRK 11 GEK 11 GDK 11 Neighborhood words Query: 22 VLRDNIQGITKPAIRRLARRGGVKRISGLIYEETRGVLK DN +G + IR L G+K I+ L+ E+ RG++K Sbjct: 226 IIKDNGRGFSGKQIRNLNYGIGLKVIADLV-EKHRGIIK 263 High-scoring Pair (HSP)

75 Original BLAST Dictionary All words of length w Alignment Ungapped extensions until score falls below some statistical threshold Output All local alignments with score > threshold

76 Original BLAST: Example w = 4 Exact keyword match of GGTC Extend diagonals with mismatches until score is under 50% Output result GTAAGGTCC GTTAGGTCC From lectures by Serafim Batzoglou (Stanford) C T G A T C C T G G A T T G C G A A C G A A G T A A G G T C C A G T

77 Gapped BLAST : Example Original BLAST exact keyword search, THEN: Extend with gaps around ends of exact match until score < threshold Output result GTAAGGTCCA GT GTTAGGTC- AGT From lectures by Serafim Batzoglou (Stanford) C T G A T C C T G G A T T G C G A A C G A A G T A A G G T C C A G T

78 Incarnations of BLAST blastn: Nucleotide-nucleotide blastp: Protein-protein blastx: Translated query vs. protein database tblastn: Protein query vs. translated database tblastx: Translated query vs. translated database (6 frames each)

79 Incarnations of BLAST (cont d) PSI-BLAST Find members of a protein family or build a custom position-specific score matrix Megablast: Search longer sequences with fewer differences WU-BLAST: (Wash U BLAST) Optimized, added features

80 Assessing sequence similarity Need to know how strong an alignment can be expected from chance alone Chance relates to comparison of sequences that are generated randomly based upon a certain sequence model Sequence models may take into account: G+C content Poly-A tails Junk DNA Codon bias Etc.

81 BLAST: Segment Score BLAST uses scoring matrices (d) to improve on efficiency of match detection Some proteins may have very different amino acid sequences, but are still similar For any two l-mers x 1 x l and y 1 y l : Segment pair: pair of l-mers, one from each sequence Segment score: S l i=1 d(x i, y i )

82 BLAST: Locally Maximal Segment Pairs A segment pair is maximal if it has the best score over all segment pairs A segment pair is locally maximal if its score can t be improved by extending or shortening Statistically significant locally maximal segment pairs are of biological interest BLAST finds all locally maximal segment pairs with scores above some threshold A significantly high threshold will filter out some statistically insignificant matches

83 BLAST: Statistics Threshold: Altschul-Dembo-Karlin statistics Identifies smallest segment score that is unlikely to happen by chance # matches above q has mean E(q) = Kmne - lq ; K is a constant, m and n are the lengths of the two compared sequences Parameter l is positive root of: S x,y in A (p x p y ed(x,y) ) = 1, where p x and p y are frequenceies of amino acids x and y, and A is the twenty

84 P-values The probability of finding b HSPs with a score S is given by: (e -E E b )/b! For b = 0, that chance is: e -E Thus the probability of finding at least one HSP with a score S is: P = 1 e -E

85 Query: 121 KE-FTPPVQAAYQKVVAGVANALAHKYH F VQ A+QK +A V +AL +YH Sbjct: 121 QAGFNADVQEAWQKFLAVVVSALCRQYH 148 Sample BLAST output Blast of human beta globin protein against zebra fish Sequences producing significant alignments: Score E (bits) Value gi ref NP_ ba1 globin [Danio rerio] >gi e-44 gi ref NP_ ba2 globin; SI:dZ118J2.3 [Danio rer e-44 gi emb CAE SI:bY187G17.6 (novel beta globin) [D e-44 gi gb AAH Ba1 protein [Danio rerio] 168 3e-43 ALIGNMENTS >gi ref NP_ ba1 globin [Danio rerio] Length = 148 Score = 171 bits (434), Expect = 3e-44 Identities = 76/148 (51%), Positives = 106/148 (71%), Gaps = 1/148 (0%) Query: 1 MVHLTPEEKSAVTALWGKVNVDEVGGEALGRLLVVYPWTQRFFESFGDLSTPDAVMGNPK 60 MV T E++A+ LWGK+N+DE+G +AL R L+VYPWTQR+F +FG+LS+P A+MGNPK Sbjct: 1 MVEWTDAERTAILGLWGKLNIDEIGPQALSRCLIVYPWTQRYFATFGNLSSPAAIMGNPK 60 Query: 61 VKAHGKKVLGAFSDGLAHLDNLKGTFATLSELHCDKLHVDPENFRLLGNVLVCVLAHHFG 120 V AHG+ V+G + ++DN+K T+A LS +H +KLHVDP+NFRLL + + A FG Sbjct: 61 VAAHGRTVMGGLERAIKNMDNVKNTYAALSVMHSEKLHVDPDNFRLLADCITVCAAMKFG 120

86 Sample BLAST output (cont d) Blast of human beta globin DNA against human DNA Sequences producing significant alignments: Score E (bits) Value gi gb AF Homo sapiens gamma A hemoglobin (HBG e-75 gi gb M HUMHBG3E Human gamma-globin mrna, 3' end 289 1e-75 gi gb AY Homo sapiens A-gamma globin (HBG1) ge e-72 gi emb V HSGGL1 Human messenger RNA for gamma-globin 260 1e-66 gi ref NR_ Homo sapiens hemoglobin, beta pseud e-34 gi gb AF Homo sapiens haplotype PB26 beta-glob e-33 ALIGNMENTS >gi ref NG_ Homo sapiens beta globin region (HBB@) on chromosome 11 Length = Score = 149 bits (75), Expect = 3e-33 Identities = 183/219 (83%) Strand = Plus / Plus Query: 267 ttgggagatgccacaaagcacctggatgatctcaagggcacctttgcccagctgagtgaa 326 Sbjct: ttcggaaaagctgttatgctcacggatgacctcaaaggcacctttgctacactgagtgac Query: 327 ctgcactgtgacaagctgcatgtggatcctgagaacttc 365 Sbjct: ctgcactgtaacaagctgcacgtggaccctgagaacttc 54507

87 Timeline 1970: Needleman-Wunsch global alignment algorithm 1981: Smith-Waterman local alignment algorithm 1985: FASTA 1990: BLAST (basic local alignment search tool) 2000s: BLAST has become too slow in genome vs. genome comparisons - new faster algorithms evolve! Pattern Hunter BLAT

88 PatternHunter: faster and even more sensitive BLAST: matches short consecutive sequences (consecutive seed) Length = k Example (k = 11): Each 1 represents a match PatternHunter: matches short non-consecutive sequences (spaced seed) Increases sensitivity by locating homologies that would otherwise be missed Example (a spaced seed of length 18 w/ 11 matches ): Each 0 represents a don t care, so there can be a match or a mismatch

89 Spaced seeds Example of a hit using a spaced seed: How does this result in better sensitivity?

90 Why is PH better? BLAST: redundant hits PatternHunter This results in > 1 hit and creates clusters of redundant hits This results in very few redundant hits

91 Why is PH better? BLAST may also miss a hit GAGTACTCAACACCAACATTAGTGGGCAATGGAAAAT GAATACTCAACAGCAACATCAATGGGCAGCAGAAAAT 9 matches In this example, despite a clear homology, there is no sequence of continuous matches longer than length 9. BLAST uses a length 11 and because of this, BLAST does not recognize this as a hit! Resolving this would require reducing the seed length to 9, which would have a damaging effect on speed

92 Advantage of Gapped Seeds 11 positions 11 positions 10 positions

93 Why is PH better? Higher hit probability Lower expected number of random hits

94 Use of Multiple Seeds Basic Searching Algorithm 2. Select a group of spaced seed models 3. For each hit of each model, conduct extension to find a homology.

95 Another method: BLAT BLAT (BLAST-Like Alignment Tool) Same idea as BLAST - locate short sequence hits and extend

96 BLAT vs. BLAST: Differences BLAT builds an index of the database and scans linearly through the query sequence, whereas BLAST builds an index of the query sequence and then scans linearly through the database Index is stored in RAM which is memory intensive, but results in faster searches

97 BLAT: Fast cdna Alignments Steps: 1. Break cdna into 500 base chunks. 2. Use an index to find regions in genome similar to each chunk of cdna. 3. Do a detailed alignment between genomic regions and cdna chunk. 4. Use dynamic programming to stitch together detailed alignments of chunks into detailed alignment of whole.

98 BLAT: Indexing An index is built that contains the positions of each k-mer in the genome Each k-mer in the query sequence is compared to each k-mer in the index A list of hits is generated - positions in cdna and in genome that match for k bases

99 Indexing: An Example Here is an example with k = 3: Genome: cacaattatcacgaccgc 3-mers (non-overlapping): cac aat tat cac gac cgc Index: aat 3 gac 12 cac 0,9 tat 6 cgc 15 cdna (query sequence): aattctcac 3-mers (overlapping): aat att ttc tct ctc tca cac Position of 3-mer in query, genome Multiple instances map to single index Hits: aat 0,3 cac 6,0 cac 6,9 clump: cacaattatcacgaccgc

100 However BLAT was designed to find sequences of 95% and greater similarity of length >40; may miss more divergent or shorter sequence alignments

101 PatternHunter and BLAT vs. BLAST PatternHunter is times faster than Blastn, depending on data size, at the same sensitivity BLAT is several times faster than BLAST, but best results are limited to closely related sequences

102 Resources tandem.bu.edu/classes/ 2004/papers/pathunter_grp_prsnt.ppt