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Linear-Space Alignment

Linear-Space Alignment. Subsequences and Substrings. Definition A string x’ is a substring of a string x, if x = ux’v for some prefix string u and suffix string v (similarly, x’ = x i …x j , for some 1  i  j  |x|) A string x’ is a subsequence of a string x

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Linear-Space Alignment

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  1. Linear-Space Alignment

  2. Subsequences and Substrings Definition A string x’ is a substring of a string x, if x = ux’v for some prefix string u and suffix string v (similarly, x’ = xi…xj, for some 1  i  j  |x|) A string x’ is a subsequence of a string x if x’ can be obtained from x by deleting 0 or more letters (x’ = xi1…xik, for some 1  i1  …  ik  |x|) Note: a substring is always a subsequence Example:x = abracadabra y = cadabr; substring z = brcdbr; subseqence, not substring

  3. Hirschberg’s algortihm Given a set of strings x, y,…, a common subsequence is a string u that is a subsequence of all strings x, y, … • Longest common subsequence • Given strings x = x1 x2 … xM, y = y1 y2 … yN, • Find longest common subsequence u = u1 … uk • Algorithm: F(i – 1, j) • F(i, j) = max F(i, j – 1) F(i – 1, j – 1) + [1, if xi = yj; 0 otherwise] • Ptr(i, j) = (same as in N-W) • Termination: trace back from Ptr(M, N), and prepend a letter to u whenever • Ptr(i, j) = DIAG and F(i – 1, j – 1) < F(i, j) • Hirschberg’s original algorithm solves this in linear space

  4. F(i,j) Introduction: Compute optimal score It is easy to compute F(M, N) in linear space Allocate ( column[1] ) Allocate ( column[2] ) For i = 1….M If i > 1, then: Free( column[ i – 2 ] ) Allocate( column[ i ] ) For j = 1…N F(i, j) = …

  5. Linear-space alignment To compute both the optimal score and the optimal alignment: Divide & Conquer approach: Notation: xr, yr: reverse of x, y E.g. x = accgg; xr = ggcca Fr(i, j): optimal score of aligning xr1…xri & yr1…yrj same as aligning xM-i+1…xM & yN-j+1…yN

  6. Linear-space alignment Lemma: (assume M is even) F(M, N) = maxk=0…N( F(M/2, k) + Fr(M/2, N – k) ) M/2 x F(M/2, k) Fr(M/2, N – k) y k* Example: ACC-GGTGCCCAGGACTG--CAT ACCAGGTG----GGACTGGGCAG k* = 8

  7. Linear-space alignment • Now, using 2 columns of space, we can compute for k = 1…M, F(M/2, k), Fr(M/2, N – k) PLUS the backpointers x1 … xM/2 xM x1 … xM/2+1 xM y1 y1 … … yN yN

  8. Linear-space alignment • Now, we can find k* maximizing F(M/2, k) + Fr(M/2, N-k) • Also, we can trace the path exiting column M/2 from k* 0 1 …… M/2 M/2+1 …… M M+1 k* k*+1

  9. Linear-space alignment • Iterate this procedure to the left and right! k* N-k* M/2 M/2

  10. Linear-space alignment Hirschberg’s Linear-space algorithm: MEMALIGN(l, l’, r, r’): (aligns xl…xl’ with yr…yr’) • Let h = (l’-l)/2 • Find (in Time O((l’ – l)  (r’ – r)), Space O(r’ – r)) the optimal path, Lh, entering column h – 1, exiting column h Let k1 = pos’n at column h – 2 where Lh enters k2 = pos’n at column h + 1 where Lh exits 3. MEMALIGN(l, h – 2, r, k1) 4. Output Lh • MEMALIGN(h + 1, l’, k2, r’) Top level call: MEMALIGN(1, M, 1, N)

  11. Linear-space alignment Time, Space analysis of Hirschberg’s algorithm: To compute optimal path at middle column, For box of size M  N, Space: 2N Time: cMN, for some constant c Then, left, right calls cost c( M/2  k* + M/2  (N – k*) ) = cMN/2 All recursive calls cost Total Time: cMN + cMN/2 + cMN/4 + ….. = 2cMN = O(MN) Total Space: O(N) for computation, O(N + M) to store the optimal alignment

  12. Heuristic Local Alignerers • The basic indexing & extension technique • Indexing: techniques to improve sensitivity • Pairs of Words, Patterns • Systems for local alignment

  13. Indexing-based local alignment …… Dictionary: All words of length k (~10) Alignment initiated between words of alignment score  T (typically T = k) Alignment: Ungapped extensions until score below statistical threshold Output: All local alignments with score > statistical threshold query …… scan DB query

  14. Indexing-based local alignment—Extensions A C G A A G T A A G G T C C A G T Gapped extensions until threshold • Extensions with gaps until score < C below best score so far Output: GTAAGGTCCAGT GTTAGGTC-AGT C T G A T C C T G G A T T G C G A

  15. Sensitivity-Speed Tradeoff X% Sens. Speed Kent WJ, Genome Research 2002

  16. Sensitivity-Speed Tradeoff Methods to improve sensitivity/speed • Using pairs of words • Using inexact words • Patterns—non consecutive positions ……ATAACGGACGACTGATTACACTGATTCTTAC…… ……GGCACGGACCAGTGACTACTCTGATTCCCAG…… ……ATAACGGACGACTGATTACACTGATTCTTAC…… ……GGCGCCGACGAGTGATTACACAGATTGCCAG…… TTTGATTACACAGAT T G TT CAC G

  17. Measured improvement Kent WJ, Genome Research 2002

  18. Non-consecutive words—Patterns Patterns increase the likelihood of at least one match within a long conserved region Consecutive Positions Non-Consecutive Positions 6 common 5 common 7 common 3 common On a 100-long 70% conserved region: ConsecutiveNon-consecutive Expected # hits: 1.07 0.97 Prob[at least one hit]: 0.30 0.47

  19. Advantage of Patterns 11 positions 11 positions 10 positions

  20. Multiple patterns TTTGATTACACAGAT T G TT CAC G T G T C CAG TTGATT A G • K patterns • Takes K times longer to scan • Patterns can complement one another • Computational problem: • Given: a model (prob distribution) for homology between two regions • Find: best set of K patterns that maximizes Prob(at least one match) How long does it take to search the query? Buhler et al. RECOMB 2003 Sun & Buhler RECOMB 2004

  21. Variants of BLAST • NCBI BLAST: search the universe http://www.ncbi.nlm.nih.gov/BLAST/ • MEGABLAST: http://genopole.toulouse.inra.fr/blast/megablast.html • Optimized to align very similar sequences • Works best when k = 4i  16 • Linear gap penalty • WU-BLAST: (Wash U BLAST) http://blast.wustl.edu/ • Very good optimizations • Good set of features & command line arguments • BLAT http://genome.ucsc.edu/cgi-bin/hgBlat • Faster, less sensitive than BLAST • Good for aligning huge numbers of queries • CHAOS http://www.cs.berkeley.edu/~brudno/chaos • Uses inexact k-mers, sensitive • PatternHunter http://www.bioinformaticssolutions.com/products/ph/index.php • Uses patterns instead of k-mers • BlastZ http://www.psc.edu/general/software/packages/blastz/ • Uses patterns, good for finding genes • Typhon http://typhon.stanford.edu • Uses multiple alignments to improve sensitivity/speed tradeoff

  22. Example Query:gattacaccccgattacaccccgattaca (29 letters) [2 mins] Database: All GenBank+EMBL+DDBJ+PDB sequences (but no EST, STS, GSS, or phase 0, 1 or 2 HTGS sequences) 1,726,556 sequences; 8,074,398,388 total letters >gi|28570323|gb|AC108906.9|Oryza sativa chromosome 3 BAC OSJNBa0087C10 genomic sequence, complete sequence Length = 144487 Score = 34.2 bits (17), Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plus Query: 4 tacaccccgattacaccccga 24 ||||||| ||||||||||||| Sbjct: 125138 tacacccagattacaccccga 125158 Score = 34.2 bits (17), Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plus Query: 4 tacaccccgattacaccccga 24 ||||||| ||||||||||||| Sbjct: 125104 tacacccagattacaccccga 125124 >gi|28173089|gb|AC104321.7| Oryza sativa chromosome 3 BAC OSJNBa0052F07 genomic sequence, complete sequence Length = 139823 Score = 34.2 bits (17), Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plus Query: 4 tacaccccgattacaccccga 24 ||||||| ||||||||||||| Sbjct: 3891 tacacccagattacaccccga 3911

  23. Example Query: Human atoh enhancer, 179 letters [1.5 min] Result: 57 blast hits • gi|7677270|gb|AF218259.1|AF218259 Homo sapiens ATOH1 enhanc... 355 1e-95 • gi|22779500|gb|AC091158.11| Mus musculus Strain C57BL6/J ch... 264 4e-68 • gi|7677269|gb|AF218258.1|AF218258 Mus musculus Atoh1 enhanc... 256 9e-66 • gi|28875397|gb|AF467292.1| Gallus gallus CATH1 (CATH1) gene... 78 5e-12 • gi|27550980|emb|AL807792.6| Zebrafish DNA sequence from clo... 54 7e-05 • gi|22002129|gb|AC092389.4| Oryza sativa chromosome 10 BAC O... 44 0.068 • gi|22094122|ref|NM_013676.1| Mus musculus suppressor of Ty ... 42 0.27 • gi|13938031|gb|BC007132.1| Mus musculus, Similar to suppres... 42 0.27 gi|7677269|gb|AF218258.1|AF218258 Mus musculus Atoh1 enhancer sequence Length = 1517 Score = 256 bits (129), Expect = 9e-66 Identities = 167/177 (94%), Gaps = 2/177 (1%) Strand = Plus / Plus Query: 3 tgacaatagagggtctggcagaggctcctggccgcggtgcggagcgtctggagcggagca 62 ||||||||||||| ||||||||||||||||||| |||||||||||||||||||||||||| Sbjct: 1144 tgacaatagaggggctggcagaggctcctggccccggtgcggagcgtctggagcggagca 1203 Query: 63 cgcgctgtcagctggtgagcgcactctcctttcaggcagctccccggggagctgtgcggc 122 |||||||||||||||||||||||||| ||||||||| |||||||||||||||| ||||| Sbjct: 1204 cgcgctgtcagctggtgagcgcactc-gctttcaggccgctccccggggagctgagcggc 1262 Query: 123 cacatttaacaccatcatcacccctccccggcctcctcaacctcggcctcctcctcg 179 ||||||||||||| || ||| |||||||||||||||||||| ||||||||||||||| Sbjct: 1263 cacatttaacaccgtcgtca-ccctccccggcctcctcaacatcggcctcctcctcg 1318 http://www.ncbi.nlm.nih.gov/BLAST/

  24. 1 2 2 1 1 1 1 … 2 2 2 2 … K … … … … x1 K K K K x2 x3 xK … Hidden Markov Models

  25. Outline for our next topic • Hidden Markov models – the theory • Probabilistic interpretation of alignments using HMMs Later in the course: • Applications of HMMs to biological sequence modeling and discovery of features such as genes

  26. Example: The Dishonest Casino A casino has two dice: • Fair die P(1) = P(2) = P(3) = P(5) = P(6) = 1/6 • Loaded die P(1) = P(2) = P(3) = P(5) = 1/10 P(6) = 1/2 Casino player switches back-&-forth between fair and loaded die once every 20 turns Game: • You bet $1 • You roll (always with a fair die) • Casino player rolls (maybe with fair die, maybe with loaded die) • Highest number wins $2

  27. Question # 1 – Evaluation GIVEN A sequence of rolls by the casino player 1245526462146146136136661664661636616366163616515615115146123562344 QUESTION How likely is this sequence, given our model of how the casino works? This is the EVALUATION problem in HMMs Prob = 1.3 x 10-35

  28. Question # 2 – Decoding GIVEN A sequence of rolls by the casino player 1245526462146146136136661664661636616366163616515615115146123562344 QUESTION What portion of the sequence was generated with the fair die, and what portion with the loaded die? This is the DECODING question in HMMs FAIR LOADED FAIR

  29. Question # 3 – Learning GIVEN A sequence of rolls by the casino player 1245526462146146136136661664661636616366163616515615115146123562344 QUESTION How “loaded” is the loaded die? How “fair” is the fair die? How often does the casino player change from fair to loaded, and back? This is the LEARNING question in HMMs Prob(6) = 64%

  30. The dishonest casino model 0.05 0.95 0.95 FAIR LOADED P(1|F) = 1/6 P(2|F) = 1/6 P(3|F) = 1/6 P(4|F) = 1/6 P(5|F) = 1/6 P(6|F) = 1/6 P(1|L) = 1/10 P(2|L) = 1/10 P(3|L) = 1/10 P(4|L) = 1/10 P(5|L) = 1/10 P(6|L) = 1/2 0.05

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