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 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

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

The Four-Russian Algorithm t t t

Heuristic Local Alignerers 1.The basic indexing & extension technique 2.Indexing: techniques to improve sensitivity Pairs of Words, Patterns 3.Systems for local alignment

State of biological databases

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

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: The entire genomic database Our new gene

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: The entire genomic database Our newly sequenced mammal 3 

Indexing-based local alignment (BLAST- Basic Local Alignment Search Tool) Main idea: 1.Construct a dictionary of all the words in the query 2.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

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 DB query scan

Indexing-based local alignment— Extensions A C G A A G T A A G G T C C A G T C C C T T C C T G G A T T G C G A 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

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

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

Indexing-based local alignment—The index Sensitivity/speed tradeoff long words (k = 15) short words (k = 7) Sensitivity Speed Kent WJ, Genome Research 2002 Sens. Speed

Indexing-based local alignment—The index Methods to improve sensitivity/speed 1.Using pairs of words 2.Using inexact words 3.Patterns—non consecutive positions ……ATAACGGACGACTGATTACACTGATTCTTAC…… ……GGCACGGACCAGTGACTACTCTGATTCCCAG…… ……ATAACGGACGACTGATTACACTGATTCTTAC…… ……GGCGCCGACGAGTGATTACACAGATTGCCAG…… TTTGATTACACAGAT T G TT CAC G

Measured improvement Kent WJ, Genome Research 2002

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

Advantage of Patterns 11 positions 10 positions

Multiple patterns 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) TTTGATTACACAGAT T G TT CAC G T G T C CAG TTGATT A G Buhler et al. RECOMB 2003 Sun & Buhler RECOMB 2004 How long does it take to search the query?

Variants of BLAST NCBI BLAST: search the universe 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

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| |gb|AC | Oryza sativa chromosome 3 BAC OSJNBa0087C10 genomic sequence, complete sequence Length = Score = 34.2 bits (17), Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plusgi| |gb|AC | Query: 4 tacaccccgattacaccccga 24 ||||||| ||||||||||||| Sbjct: tacacccagattacaccccga Score = 34.2 bits (17), Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plus Query: 4 tacaccccgattacaccccga 24 ||||||| ||||||||||||| Sbjct: tacacccagattacaccccga >gi| |gb|AC | Oryza sativa chromosome 3 BAC OSJNBa0052F07 genomic sequence, complete sequence Length = Score = 34.2 bits (17), Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plusgi| |gb|AC | Query: 4 tacaccccgattacaccccga 24 ||||||| ||||||||||||| Sbjct: 3891 tacacccagattacaccccga 3911

Example Query: Human atoh enhancer, 179 letters[1.5 min] Result: 57 blast hits 1. gi| |gb|AF |AF Homo sapiens ATOH1 enhanc e-95 gi| |gb|AF |AF gi| |gb|AC | Mus musculus Strain C57BL6/J ch e-68gi| |gb|AC |264 3.gi| |gb|AF |AF Mus musculus Atoh1 enhanc e-66gi| |gb|AF |AF gi| |gb|AF | Gallus gallus CATH1 (CATH1) gene e-12gi| |gb|AF |78 5.gi| |emb|AL | Zebrafish DNA sequence from clo e-05gi| |emb|AL |54 6.gi| |gb|AC | Oryza sativa chromosome 10 BAC O gi| |gb|AC |44 7.gi| |ref|NM_ | Mus musculus suppressor of Ty gi| |ref|NM_ |42 8.gi| |gb|BC | Mus musculus, Similar to suppres gi| |gb|BC |42 gi| |gb|AF |AF218258gi| |gb|AF |AF 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

Hidden Markov Models 1 2 K … 1 2 K … 1 2 K … … … … 1 2 K … x1x1 x2x2 x3x3 xKxK 2 1 K 2

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

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: 1.You bet $1 2.You roll (always with a fair die) 3.Casino player rolls (maybe with fair die, maybe with loaded die) 4.Highest number wins $2

Question # 1 – Evaluation GIVEN A sequence of rolls by the casino player QUESTION How likely is this sequence, given our model of how the casino works? This is the EVALUATION problem in HMMs

Question # 2 – Decoding GIVEN A sequence of rolls by the casino player 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

Question # 3 – Learning GIVEN A sequence of rolls by the casino player 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

The dishonest casino model FAIRLOADED 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

Definition of a hidden Markov model Definition: A hidden Markov model (HMM) Alphabet  = { b 1, b 2, …, b M } Set of states Q = { 1,..., K } Transition probabilities between any two states a ij = transition prob from state i to state j a i1 + … + a iK = 1, for all states i = 1…K Start probabilities a 0i a 01 + … + a 0K = 1 Emission probabilities within each state e i (b) = P( x i = b |  i = k) e i (b 1 ) + … + e i (b M ) = 1, for all states i = 1…K K 1 … 2

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, x 1, x 2, …, x t )= P(  t+1 =k |  t ) K 1 … 2

A parse of a sequence Given a sequence x = x 1 ……x N, A parse of x is a sequence of states  =  1, ……,  N 1 2 K … 1 2 K … 1 2 K … … … … 1 2 K … x1x1 x2x2 x3x3 xKxK 2 1 K 2

Likelihood of a parse Given a sequence x = x 1 ……x N and a parse  =  1, ……,  N, To find how likely is the parse: (given our HMM) P(x,  ) = P(x 1, …, x N,  1, ……,  N ) = P(x N,  N |  N-1 ) P(x N-1,  N-1 |  N-2 )……P(x 2,  2 |  1 ) P(x 1,  1 ) = P(x N |  N ) P(  N |  N-1 ) ……P(x 2 |  2 ) P(  2 |  1 ) P(x 1 |  1 ) P(  1 ) = a 0  1 a  1  2 ……a  N-1  N e  1 (x 1 )……e  N (x N ) 1 2 K … 1 2 K … 1 2 K … … … … 1 2 K … x1x1 x2x2 x3x3 xKxK 2 1 K 2

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 a 0Fair = ½, a oLoaded = ½) ½  P(1 | Fair) P(Fair | Fair) P(2 | Fair) P(Fair | Fair) … P(4 | Fair) = ½  (1/6) 10  (0.95) 9 = = 0.5  10 -9

Example: the dishonest casino So, the likelihood the die is fair in all this run is just  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 = = 0.79  Therefore, it somewhat more likely that the die is fair all the way, than that it is loaded all the way

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 = = 0.5  So, it is 100 times more likely the die is loaded

The three main questions on HMMs 1.Evaluation GIVEN a HMM M, and a sequence x, FIND Prob[ x | M ] 2.Decoding GIVENa HMM M, and a sequence x, FINDthe sequence  of states that maximizes P[ x,  | M ] 3.Learning GIVENa HMM M, with unspecified transition/emission probs., and a sequence x, FINDparameters  = (e i (.), a ij ) that maximize P[ x |  ]

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  = a ij, e i (.) 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 |  ]