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Sequence Alignment Cont’d

Sequence Alignment Cont’d. Needleman-Wunsch with affine gaps. Initialization: V(i, 0) = d + (i – 1) e V(0, j) = d + (j – 1) e Iteration: V(i, j) = max{ F(i, j), G(i, j), H(i, j) } F(i, j) = V(i – 1, j – 1) + s(x i , y j ) V(i – 1, j) – d G(i, j) = max

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Sequence Alignment Cont’d

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  1. Sequence AlignmentCont’d

  2. Needleman-Wunsch with affine gaps Initialization: V(i, 0) = d + (i – 1)e V(0, j) = d + (j – 1)e Iteration: V(i, j) = max{ F(i, j), G(i, j), H(i, j) } F(i, j) = V(i – 1, j – 1) + s(xi, yj) V(i – 1, j) – d G(i, j) = max G(i – 1, j) – e V(i, j – 1) – d H(i, j) = max H(i, j – 1) – e Termination: similar HINT: With similar idea we can do arbitrary (non-convex) gaps

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

  4. The Four-Russian Algorithm t t t

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

  6. State of biological databases http://www.genome.gov/10005141 http://www.cbs.dtu.dk/databases/DOGS/

  7. State of biological databases • Number of genes in these genomes: • Mammals: ~25,000 • Insects: ~14,000 • Worms: ~17,000 • Fungi: ~6,000-10,000 • Small organisms: 100s-1,000s • Each known or predicted gene has one or more associated protein sequences • >1,000,000 known / predicted protein sequences

  8. Some useful applications of alignments • Given a newly discovered gene, • Does it occur in other species? • How fast does it evolve? • Assume we try Smith-Waterman: Our new gene 104 The entire genomic database 1010 - 1012

  9. Some useful applications of alignments • Given a newly sequenced organism, • Which subregions align with other organisms? • Potential genes • Other biological characteristics • Assume we try Smith-Waterman: Our newly sequenced mammal 3109 The entire genomic database 1010 - 1012

  10. Indexing-based local alignment (BLAST- Basic Local Alignment Search Tool) Main idea: • Construct a dictionary of all the words in the query • Initiate a local alignment for each word match between query and DB Running Time: O(MN) However, orders of magnitude faster than Smith-Waterman query DB

  11. 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

  12. Indexing-based local alignment—Extensions A C G A A G T A A G G T C C A G T Example: k = 4 The matching word GGTC initiates an alignment Extension to the left and right with no gaps until alignment falls < T below best so far Output: GTAAGGTCC GTTAGGTCC C C C T T C C T G G A T T G C G A

  13. Indexing-based local alignment—Extensions A C G A A G T A A G G T C C A G T Gapped extensions • Extensions with gaps in a band around anchor Output: GTAAGGTCCAGT GTTAGGTC-AGT C T G A T C C T G G A T T G C G A

  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 < T 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. Indexing-based local alignment—The index • Sensitivity/speed tradeoff Sens. Speed Kent WJ, Genome Research 2002

  16. Indexing-based local alignment—The index 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: • Optimized to align very similar sequences • Works best when k = 4i  16 • Linear gap penalty • WU-BLAST: (Wash U BLAST) • Very good optimizations • Good set of features & command line arguments • BLAT • Faster, less sensitive than BLAST • Good for aligning huge numbers of queries • CHAOS • Uses inexact k-mers, sensitive • PatternHunter • Uses patterns instead of k-mers • BlastZ • Uses patterns, good for finding genes

  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

  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

  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

  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

  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

  31. Definition of a hidden Markov model Definition: A hidden Markov model (HMM) • Alphabet = { b1, b2, …, bM } • Set of states Q = { 1, ..., K } • Transition probabilities between any two states aij = transition prob from state i to state j ai1 + … + aiK = 1, for all states i = 1…K • Start probabilities a0i a01 + … + a0K = 1 • Emission probabilities within each state ei(b) = P( xi = b | i = k) ei(b1) + … + ei(bM) = 1, for all states i = 1…K 1 2 K …

  32. A HMM is memory-less At each time step t, the only thing that affects future states is the current state t P(t+1 = k | “whatever happened so far”) = P(t+1 = k | 1, 2, …, t, x1, x2, …, xt) = P(t+1 = k | t) 1 2 K …

  33. 1 1 1 1 … 2 2 2 2 … … … … … K K K K … A parse of a sequence Given a sequence x = x1……xN, A parse of x is a sequence of states  = 1, ……, N 1 2 2 K x1 x2 x3 xK

  34. 1 2 2 1 1 1 1 … 2 2 2 2 … K … … … … x1 K x2 K x3 K K xK … Likelihood of a parse Given a sequence x = x1……xN and a parse  = 1, ……, N, To find how likely is the parse: (given our HMM) P(x, ) = P(x1, …, xN, 1, ……, N) = P(xN, N | N-1) P(xN-1, N-1 | N-2)……P(x2, 2 | 1) P(x1, 1) = P(xN | N)P(N | N-1) ……P(x2 | 2)P(2 | 1)P(x1 | 1)P(1) = a01 a12……aN-1Ne1(x1)……eN(xN)

  35. Example: the dishonest casino Let the sequence of rolls be: x = 1, 2, 1, 5, 6, 2, 1, 6, 2, 4 Then, what is the likelihood of  = Fair, Fair, Fair, Fair, Fair, Fair, Fair, Fair, Fair, Fair? (say initial probs a0Fair = ½, aoLoaded = ½) ½  P(1 | Fair) P(Fair | Fair) P(2 | Fair) P(Fair | Fair) … P(4 | Fair) = ½  (1/6)10  (0.95)9 = .00000000521158647211 = 0.5  10-9

  36. Example: the dishonest casino So, the likelihood the die is fair in all this run is just 0.521  10-9 OK, but what is the likelihood of = Loaded, Loaded, Loaded, Loaded, Loaded, Loaded, Loaded, Loaded, Loaded, Loaded? ½  P(1 | Loaded) P(Loaded, Loaded) … P(4 | Loaded) = ½  (1/10)8  (1/2)2 (0.95)9 = .00000000078781176215 = 0.79  10-9 Therefore, it somewhat more likely that the die is fair all the way, than that it is loaded all the way

  37. Example: the dishonest casino Let the sequence of rolls be: x = 1, 6, 6, 5, 6, 2, 6, 6, 3, 6 Now, what is the likelihood  = F, F, …, F? ½  (1/6)10  (0.95)9 = 0.5  10-9, same as before What is the likelihood  = L, L, …, L? ½  (1/10)4  (1/2)6 (0.95)9 = .00000049238235134735 = 0.5  10-7 So, it is 100 times more likely the die is loaded

  38. The three main questions on HMMs • Evaluation GIVEN a HMM M, and a sequence x, FIND Prob[ x | M ] • Decoding GIVEN a HMM M, and a sequence x, FIND the sequence  of states that maximizes P[ x,  | M ] • Learning GIVEN a HMM M, with unspecified transition/emission probs., and a sequence x, FIND parameters  = (ei(.), aij) that maximize P[ x |  ]

  39. Let’s not be confused by notation P[ x | M ]: The probability that sequence x was generated by the model The model is: architecture (#states, etc) + parameters  = aij, ei(.) So, P[x | M] is the same with P[ x |  ], and P[ x ], when the architecture, and the parameters, respectively, are implied Similarly, P[ x,  | M ], P[ x,  |  ] and P[ x,  ] are the same when the architecture, and the parameters, are implied In the LEARNING problem we always write P[ x |  ] to emphasize that we are seeking the * that maximizes P[ x |  ]

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