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www.bioalgorithms.info An Introduction to Bioinformatics Algorithms

Combinatorial Pattern Matching

An Introduction to Bioinformatics Algorithms www.bioalgorithms.info

Outline

1. Repeat Finding and Hash Tables 2. Exact Pattern Matching 3. Keyword Trees 4. Suffix Trees 5. Heuristic Similarity Search Algorithms 6. Approximate String Matching 7. Query Matching and Filtration 8. Comparing a Sequence Against a Database 9. BLAST Algorithm/Statistics 10. PatternHunter and BLAT


Section 1: Repeat Finding and

Hash Tables


Genomic Repeats

• Repeat: A sequence of DNA that occurs more than once in a genome.

• Example: ATGGTCTAGGTCCTAGTGGTC


Genomic Repeats

• Repeat: A sequence of DNA that occurs more than once in a genome.

• Example: ATGGTCTAGGTCCTAGTGGTC

• Why do we want to find repeats? 1. Understand more about evolution. 2. Many tumors are characterized by an explosion of repeats. 3. Genomic rearrangements are often associated with repeats.


Genomic Repeats

• Note: Often, a repeat will refer to a segment of DNA that occurs very often with slight modifications.

• As we might imagine, this is a more difficult computational problem.


Genomic Repeats

• Note: Often, a repeat will refer to a segment of DNA that occurs very often with slight modifications.

• As we might imagine, this is a more difficult computational problem.

• Example: Say our target segment is GGTC and we allow one substitution: • ATGGTCTAGGACCTAGTGTTC


Extending l-mer Repeats

• Long repeats are difficult to find, but short repeats are easy.

• Simple approach to finding long repeats: 1. Find exact repeats of short l-mers (l is usually 10 to 13). 2. Use l-mer repeats to potentially extend into longer,

maximal repeats.



• Example: We start with a repeat of length 4.

• GCTTACAGATTCAGTCTTACAGATGGT





• Extend both these 4-mers:






• Extend both these 4-mers:


• Maximal repeat: CTTACAGAT


Maximal Repeats: Issue

• In order 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.


Hashing

• Hashing is a very efficient way to store and retrieve data.

• What does hashing do?

• Say we are given a collection of data.

• For different entry, generate a unique integer and store the entry in an array at this integer location.


Hashing: Definitions

• Records: data stored in a hash table.

• Keys: Integers identifying set of records.

• Hash Table: The array of keys used in hashing.

• Hash Function: Assigns each record to a key.

• Collision: Occurs when more than one record is mapped to the same index in the hash table.


Hashing: Example

• Records: Animals

• Keys: Where each animal eats.


Hashing DNA sequences

• Each l-mer can be translated into a binary string: • A can be represented as 00 • T can be represented as 01 • G can be represented as 10 • C can be represented as 11 • Example: ATGCTA = 000110110100

• After assigning a unique integer to each l-mer, it is easy to obtain all start locations of each l-mer in a genome.


Hashing: Maximal Repeats

• To find repeats in a genome: 1. For all l-mers in the genome, note the start position and the

sequence. 2. Generate a hash table index for each unique l-mer

sequence. 3. At each index of the hash table, store all genome start

locations of the l-mer that generated the index. 4. Anywhere there are collisions in the table, extend

corresponding l-mers to maximal repeats.

• How do we deal with collisions?


Hashing: Collisions

• In order to deal with collisions:

• “Chain” all start locations of l-mers.

• This can be done via a data structure called a linked list.


Section 2: Exact Pattern Matching


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.


Pattern Matching Problem

• Goal: Find all occurrences of a pattern in a text.

• Input: • Pattern p = p1…pn of length n • Text t = t1…tm of length m

• Output: All positions 1< i < (m – n + 1) such that the n-letter substring of text t starting at i matches the pattern p.

• Motivation: Searching database for a known pattern.


Exact Pattern Matching: A Brute Force Algorithm

PatternMatching(p,t) 1 n length of pattern p 2 m length of text t 3 for i 1 to (m – n + 1) 4 if ti…ti+n-1 = p 5 output i


• PatternMatching algorithm for: • Pattern GCAT • Text AGCCGCATCT

Exact Pattern Matching: An Example


• PatternMatching algorithm for: • Pattern GCAT • Text AGCCGCATCT

• AGCCGCATCT • GCAT

Exact Pattern Matching: An Example


• PatternMatching runtime: O(nm) • In the worst case, we have to check n characters at each of

the m letters of the text. • Example: Text = AAAAAAAAAAAAAAAA, Pattern = AAAC

• This is rare; on average, the run time is more like O(m). • Rarely will there be close to n comparisons at each step.

• Better solution: suffix trees…solve in O(m) time.

• First we need keyword trees.

Exact Pattern Matching: Running Time


Section 3: Keyword Trees


Keyword Trees: Example

• Keyword tree: • Apple



• Keyword tree: • Apple • Apropos



• Keyword tree: • Apple • Apropos • Banana



• Keyword tree: • Apple • Apropos • Banana • Bandana



• Keyword tree: • Apple • Apropos • Banana • Bandana • Orange


Keyword Trees: Properties

• Stores a set of keywords in a rooted labeled tree.

• Each edge is 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.

• Furthermore, every path from root to leaf gives a keyword.


Keyword Trees: Threading

• Thread “appeal” • appeal



• Thread “apple” • apple


Multiple Pattern Matching (MPM) Problem

• Goal: Given a set of patterns and a text, find all occurrences of any of patterns in text.

• Input: k patterns p1,…,pk, and text of m characters t = t1…tm

• Output: Positions 1 < i < m where the substring of text t starting at i matches one of the patterns pj for 1 < j < k.

• Motivation: Searching database for known multiple patterns.


MPM: Naïve Approach

• We could always solve MPM as k “Pattern Matching Problems.”

• Runtime: Using the PatternMatching algorithm k times gives O(kmn): • m = length of the text. • n = average pattern length.


MPM: Keyword Tree Approach

• Alternatively we could use a keyword approach.

• Let N = total length of all patterns.

• We can build the keyword tree in O(N) time.

• Total Runtime: • With naive threading: O(N + nm) • Aho-Corasick algorithm: Improvement to O(N + m)



• To match patterns in a text using a keyword tree: • Build keyword tree of patterns • “Thread” the text through the

keyword tree.



• To match patterns in a text using a keyword tree: • Build keyword tree of patterns • “Thread” the text through the

keyword tree.

• Threading is “complete” when we reach a leaf in the keyword tree. • When threading is “complete,”

we’ve found a pattern in the text.


Section 4: Suffix Trees


Suffix Trees = Collapsed Keyword Trees

• Suffix Tree: Similar to keyword tree, except edges that form paths are collapsed. • Each edge is labeled with

a substring of the text. • All internal edges have at

least two outgoing edges. • Leaves are labeled by the

index of the pattern.

Keyword Tree Suffix Tree


Suffix Tree of a Text

Keyword Tree

Suffix Tree

• The suffix trees of a text is constructed for all its suffixes.

• Example: Suffix tree of ATCATG:

ATCATG TCATG CATG ATG TG G


Suffix Trees: Example

• Example: Suffix tree of TCATACATGGCATACATGG


Suffix Trees: Runtime

ATCATG TCATG CATG ATG TG G

Keyword Tree

Suffix Tree

• Say the length of the text is m. • Construction of the keyword tree takes O(m2) time. • Constructing the suffix tree takes O(m) time. • So our method takes O(m2 + m) = O(m2) time. • However, there exists a method of calculating the suffix tree in

O(m) time without needing the keyword tree for the suffixes.


Pattern Matching with Suffix Trees

• To find any pattern in a text: 1. Build suffix tree for text of length m—O(m) time 2. Thread the pattern of length n through the suffix tree of the

text—O(n) time.

• Therefore the runtime of the Pattern Matching Problem is only O(m + n)!


Pattern Matching with Suffix Trees

SuffixTreePatternMatching(p,t) 1. Build suffix tree for text t 2. Thread pattern p through suffix tree 3. if threading is complete 4. output positions of all p-matching leaves in the tree 5. else 6. output “Pattern does not appear in text”


Threading ATG through a Suffix Tree


Multiple Pattern Matching: Summary

• Keyword and suffix trees are used to find patterns in a text.

• Keyword trees: • Build the keyword tree of patterns, and thread the text

through it.

• Suffix trees: • Build the suffix tree of the text, and thread the patterns

through it.


Section 5: Heuristic Similarity Search

Algorithms


Approximate vs. Exact Pattern Matching

• So far all we’ve seen exact pattern matching algorithms.

• Usually, because of mutations, it makes sense to find approximate pattern matches.

• Biologists often use fast heuristic approaches (rather than local alignment) to find approximate matches.


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 (e.g. BLAST) 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.


Dot Matrices

• Dot matrices show similarities between two sequences.

• FASTA makes an implicit dot matrix from short exact matches, and tries to find long approximate diagonals (allowing for some mismatches).


Dot Matrices: Example

• Identify diagonals above a threshold length.

• Diagonals in the dot matrix indicate exact substring matching.


Diagonals in Dot Matrices

• Extend diagonals and try to link them together, allowing for minimal mismatches/indels.

• Linking exact diagonals reveals approximate matches over longer substrings.


Section 6: Approximate String

Matching


Approximate Pattern Matching Problem

• Goal: Find all approximate occurrences of a pattern in a text.

• Input: A pattern p = p1…pn, text t = t1…tm, and k, the maximum allowable number of mismatches.

• Output: All positions 1 < i < (m – n + 1) such that ti…ti+n-1 and p1…pn have at most k mismatches. i.e., HammingDistance(ti…ti+n-1, p) ≤ k.


Approximate Pattern Matching: Brute Force

ApproximatePatternMatching(p, t, k) 1. n length of pattern p 2. m length of text t 3. for i 1 to m – n + 1 4. dist 0 5. for j 1 to n 6. if ti+j-1 ≠ pj 7. dist dist + 1 8. if dist < k 9. output i


Approximate Pattern Matching: Running Time

• The brute force algorithm runs in O(mn) time.

• Landau-Vishkin algorithm: Gives improvement to O(km).

• We will next generalize the “Approximate Pattern Matching Problem” into a “Query Matching Problem.” • Rather than patterns, 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.


Section 7: Query Matching and

Filtration


Query Matching Problem

• Goal: Find all substrings of a query that approximately match the text.

• Input: Query q = q1…qw Text t = t1…tm, n = length of matching substrings k = maximum number of mismatches

• Output: All pairs of positions (i, j) such that the n-letter substring of query q starting at i approximately matches the n-letter substring of text t starting at j, with at most k mismatches.


Approximate Pattern Matching vs Query Matching


Filtration in 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).


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.

• Filtration involves two processes: 1. Potential match detection: Find all exact matches of l-

tuples in query and text for some small l. 2. Potential match verification: Verify each potential match

by extending it to the left and right, until (k + 1) mismatches are found or we have n – k matches.


Step 1: Match Detection

• If x1…xn and y1…yn match with at most k mismatches, they must share an l-tuple that is perfectly matched, where we have l = ⎣n/(k + 1)⎦. • Here ⎣ j ⎦ denotes the largest integer q with q ≤ j

• Where does this formula come from? • Break string of length n into k+1 parts, each each of length

⎣n/(k + 1)⎦ • k mismatches can affect at most k of these k+1 parts. • At least one of these k+1 parts is perfectly matched.


Step 1: Match Detection

1…l l +1…2l 2l +1…3l 3l +1…n 1 2 k k + 1

• Suppose k = 3. We would then have l = ⎣n / (k+1)⎦ =⎣n/4⎦:

• 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.


Step 2: Match Verification

Query

Extend perfect match of length l until we find an approximate match of length n with k mismatches

Text

• For each l-match we find, try to extend the match further to see if it is substantial.


Filtration: Changing l Changes Performance

l -tuple length

Shorter perfect matches required

Performance decreases

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Section 8: Comparing a Sequence

Against a Database


Local alignment is too 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.


Local alignment is too slow…

• BLAST: Basic Local Alignment Search Tool

• Created in 1990 by Altschul, Gish, Miller, Myers, & Lipman.

• Idea: Search sequence databases for local alignments to a query.


Features of 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”), and only explores locally around these “hits.”


Section 9: BLAST Algorithm/

Statistics


BLAST Algorithm

• Keyword search of all words of length w from a query of length n in database of length m with score above threshold. • w = 11 for DNA queries • w =3 for proteins

• Perform local alignment extension for each keyword.

• Extend results until longest match above a given threshold is achieved.

• Runtime: O(nm)


BLAST Algorithm

Query: 22 VLRDNIQGITKPAIRRLARRGGVKRISGLIYEETRGVLK 60 +++DN +G + IR L G+K I+ L+ E+ RG++K Sbjct: 226 IIKDNGRGFSGKQIRNLNYGIGLKVIADLV-EKHRGIIK 263

Query: KRHRKVLRDNIQGITKPAIRRLARRGGVKRISGLIYEETRGVLKIFLENVIRD

keyword

GVK 18 GAK 16 GIK 16 GGK 14 GLK 13 GNK 12 GRK 11 GEK 11 GDK 11

neighborhood score threshold

(T = 13)

Neighborhood words

High-scoring Pair (HSP)

extension


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


Original BLAST: Example

From lectures by Serafim Batzoglou (Stanford)

• w = 4 • Exact keyword

match of GGTC

A C G A A G T A A G G T C C A G T

G

A T

C

C T

G

G

A

T

T G

C

G

A




match of GGTC


G

A T

C

C T

G

G

A

T

T G

C

G

A





match of GGTC • Extend diagonals

with mismatches until score is under 50%.


G

A T

C

C T

G

G

A

T

T G

C

G

A





G

A T

C

C T

G

G

A

T

T G

C

G

A


match of GGTC • Extend diagonals

with mismatches until score is under 50%.

• Output result: • GTAAGGTCC • GTTAGGTCC



Gapped BLAST: Example

• Original BLAST exact keyword search, THEN:

• Extend with gaps around ends of exact match until score < threshold

• Output result GTAAGGTCCAGT GTTAGGTC-AGT


G

A T

C

C T

G

G

A

T

T G

C

G

A



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)


Incarnations of BLAST

• 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.


Assessing Sequence Similarity

• We need to know how strong an alignment can be expected to be 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.


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 x1…xl and y1…yl : • Segment pair: Pair of l-mers, one from each sequence. • Segment score:

�

d xi,yi( )i=1

l∑


BLAST: Locally Maximal 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.


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(pxpyed(x,y)) = 1, where px and py are frequencies of amino acids x and y, and A is the twenty letter amino acid alphabet


P-values

• The probability of finding b HSPs with a score ≥S is given by:

• For b = 0, that chance is:

• Thus the probability of finding at least one HSP with a score ≥ S is:

�

(e−EE b )b!

�

e−E

�

P =1− e−E


Sample BLAST output

Score E Sequences producing significant alignments: (bits) Value

gi|18858329|ref|NP_571095.1| ba1 globin [Danio rerio] >gi|147757... 171 3e-44 gi|18858331|ref|NP_571096.1| ba2 globin; SI:dZ118J2.3 [Danio rer... 170 7e-44 gi|37606100|emb|CAE48992.1| SI:bY187G17.6 (novel beta globin) [D... 170 7e-44 gi|31419195|gb|AAH53176.1| Ba1 protein [Danio rerio] 168 3e-43

ALIGNMENTS >gi|18858329|ref|NP_571095.1| 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

• BLAST of human betaglobin protein against zebra fish:


Sample BLAST Output

Score E Sequences producing significant alignments: (bits) Value

gi|19849266|gb|AF487523.1| Homo sapiens gamma A hemoglobin (HBG1... 289 1e-75 gi|183868|gb|M11427.1|HUMHBG3E Human gamma-globin mRNA, 3' end 289 1e-75 gi|44887617|gb|AY534688.1| Homo sapiens A-gamma globin (HBG1) ge... 280 1e-72 gi|31726|emb|V00512.1|HSGGL1 Human messenger RNA for gamma-globin 260 1e-66 gi|38683401|ref|NR_001589.1| Homo sapiens hemoglobin, beta pseud... 151 7e-34 gi|18462073|gb|AF339400.1| Homo sapiens haplotype PB26 beta-glob... 149 3e-33

ALIGNMENTS >gi|28380636|ref|NG_000007.3| Homo sapiens beta globin region (HBB@) on chromosome 11 Length = 81706 Score = 149 bits (75), Expect = 3e-33 Identities = 183/219 (83%) Strand = Plus / Plus

Query: 267 ttgggagatgccacaaagcacctggatgatctcaagggcacctttgcccagctgagtgaa 326 || ||| | || | || | |||||| ||||| ||||||||||| |||||||| Sbjct: 54409 ttcggaaaagctgttatgctcacggatgacctcaaaggcacctttgctacactgagtgac 54468

• BLAST of human betaglobin protein against zebra fish:


Section 10: PatternHunter and BLAT


Timeline

• 1970: Needleman-Wunsch global alignment algorithm

• 1981: Smith-Waterman local alignment algorithm

• 1985: FASTA

• 1990: BLAST (basic local alignment search tool)

• 2000s: Faster algorithms evolve for genome/genome: • Pattern Hunter • BLAT


BLAT vs. BLAST

• BLAT (BLAST-Like Alignment Tool): Same idea as BLAST—locate short sequence hits and extend.

• Differences between BLAT and BLAST: • BLAST builds an index of the query sequence and then

scans linearly through the database. • BLAT builds an index of the database and scans linearly

through the query sequence.

• Index is stored in RAM, resulting in faster searches.


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 – matching k-mer positions in cDNA and in genome.


• Steps: 1. Break cDNA into 500 base chunks. 2. Use an index to find regions in genome similar to each

chunk of cDNA. 3. Construct alignment between genomic regions and cDNA

chunks. 4. Use dynamic programming to stitch together alignments

of chunks into the alignment of the whole.

BLAT: Fast cDNA Alignments


Indexing: An Example

Multiple instances map to single index"

Position of 3-mer in query, genome"

• 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 0 1 2 3 4 5 6

Hits: aat 0,3 cac 6,0 cac 6,9 clump: cacAATtatCACgaccgc


However…

• BLAT was designed to find sequences of 95% and greater similarity of length > 40.

• Therefore we may miss more divergent or shorter sequence alignments.


PatternHunter: faster and even more sensitive

• BLAST: matches short consecutive sequences (consecutive seed)

• Length = k • Example (k = 11):

11111111111 • 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 with 11 matches):

111010010100110111 • Each 0 represents a “don’t

care”, so there can be a match or a mismatch.


Spaced Seeds

• Example of a hit using a spaced seed:

• How does this result in better sensitivity?


Why is PH Better?

• BLAST: redundant hits

• This results in > 1 hit and creates clusters of redundant hits.

• Example:

• PatternHunter: very few redundant hits.

• Example:


Why is PH Better?

GAGTACTCAACACCAACATTAGTGGGCAATGGAAAAT

|||||||||||||||||||||||||||||||||||||

GAATACTCAACAGCAACATCAATGGGCAGCAGAAAAT

• In this example, despite a clear similarity, 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. • Example:

9 matches


Advantage of Gapped Seeds

11 positions

11 positions

10 positions


Why is PH Better?

1. Higher hit probability. 2. Lower expected number of random hits.


Use of Multiple Seeds

• Basic Search Algorithm: 1. Select a group of spaced seed models. 2. For each hit of each model, conduct extension to find a

homology.


PatternHunter and BLAT vs. BLAST

• PatternHunter is 5-100 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.


Resources

• tandem.bu.edu/classes/ 2004/papers/pathunter_grp_prsnt.ppt!

• http://www.jax.org/courses/archives/2004/gsa04_king_presentation.pdf!

• http://www.genomeblat.com/genomeblat/blatRapShow.pps!

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