Sequence Alignment

CS262 Lecture 3, Win06, Batzoglou Sequence Alignment -AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC Definition Given two strings x = x 1 x 2...x M, y = y 1 y 2 …y N, an alignment is an assignment of gaps to positions 0,…, N in x, and 0,…, N in y, so as to line up each letter in one sequence with either a letter, or a gap in the other sequence AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGGTCGATTTGCCCGAC

CS262 Lecture 3, Win06, Batzoglou The Needleman-Wunsch Algorithm 1.Initialization. F(0, 0) = F(0, j) = F(i, 0) = 0 2.Main Iteration. a.For each i = 1……M For eachj = 1……N F(i-1,j-1) + s(x i, y j ) [case 1] F(i, j) = max F(i-1, j) – d [case 2] F(i, j-1) – d [case 3] DIAG, if [case 1] Ptr(i,j) = LEFT,if [case 2] UP,if [case 3] 3.Termination. F(M, N) is the optimal score, and from Ptr(M, N) can trace back optimal alignment

CS262 Lecture 3, Win06, Batzoglou The Smith-Waterman algorithm Idea: Ignore badly aligning regions Modifications to Needleman-Wunsch: Initialization:F(0, j) = F(i, 0) = 0 0 Iteration:F(i, j) = max F(i – 1, j) – d F(i, j – 1) – d F(i – 1, j – 1) + s(x i, y j )

CS262 Lecture 3, Win06, Batzoglou Scoring the gaps more accurately Simple, linear gap model: Gap of length n incurs penaltyn  d However, gaps usually occur in bunches Convex gap penalty function:  (n): for all n,  (n + 1) -  (n)   (n) -  (n – 1) Algorithm: O(N 3 ) time, O(N 2 ) space  (n)

CS262 Lecture 3, Win06, Batzoglou Compromise: affine gaps  (n) = d + (n – 1)  e || gap gap open extend To compute optimal alignment, At position i, j, need to “remember” best score if gap is open best score if gap is not open F(i, j):score of alignment x 1 …x i to y 1 …y j if if x i aligns to y j if G(i, j):score if x i aligns to a gap after y j if H(i, j): score if y j aligns to a gap after x i V(i, j) = best score of alignment x 1 …x i to y 1 …y j d e  (n)

CS262 Lecture 3, Win06, Batzoglou Needleman-Wunsch with affine gaps Why do we need matrices F, G, H? x i aligns to y j x 1 ……x i-1 x i x i+1 y 1 ……y j-1 y j - 2.x i aligns to a gap x 1 ……x i-1 x i x i+1 y 1 ……y j …- - Add -d Add -e G(i+1, j) = V(i, j) – d G(i+1, j) = G(i, j) – e Because, perhaps G(i, j) < V(i, j) (it is best to align x i to y j if we were aligning only x 1 …x i to y 1 …y j and not the rest of x, y), but on the contrary G(i, j) – e > V(i, j) – d (i.e., had we “fixed” our decision that x i aligns to y j, we could regret it at the next step when aligning x 1 …x i+1 to y 1 …y j )

CS262 Lecture 3, Win06, Batzoglou 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: V(i, j) has the best alignment Time? Space?

CS262 Lecture 3, Win06, Batzoglou To generalize a little… … think of how you would compute optimal alignment with this gap function ….in time O(MN)  (n)

CS262 Lecture 3, Win06, Batzoglou Bounded Dynamic Programming Assume we know that x and y are very similar Assumption: # gaps(x, y) < k(N) xixi Then,|implies | i – j | < k(N) yj yj We can align x and y more efficiently: Time, Space: O(N  k(N)) << O(N 2 )

CS262 Lecture 3, Win06, Batzoglou Bounded Dynamic Programming Initialization: F(i,0), F(0,j) undefined for i, j > k Iteration: For i = 1…M For j = max(1, i – k)…min(N, i+k) F(i – 1, j – 1)+ s(x i, y j ) F(i, j) = maxF(i, j – 1) – d, if j > i – k(N) F(i – 1, j) – d, if j < i + k(N) Termination:same Easy to extend to the affine gap case x 1 ………………………… x M y 1 ………………………… y N k(N)

CS262 Lecture 3, Win06, Batzoglou Linear-Space Alignment

CS262 Lecture 3, Win06, Batzoglou 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 if x’ can be obtained from x by deleting 0 or more letters (x’ = x i1 …x ik, for some 1  i 1  …  i k  |x|) Note: a substring is always a subsequence Example: x = abracadabra y = cadabr; substring z = brcdbr;subseqence, not substring

CS262 Lecture 3, Win06, Batzoglou 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 = x 1 x 2 … x M, y = y 1 y 2 … y N,  Find longest common subsequence u = u 1 … u k Algorithm: F(i – 1, j) F(i, j) = maxF(i, j – 1) F(i – 1, j – 1) + [1, if x i = y j ; 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 algorithm solves this in linear space

CS262 Lecture 3, Win06, Batzoglou 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) = …

CS262 Lecture 3, Win06, Batzoglou Linear-space alignment To compute both the optimal score and the optimal alignment: Divide & Conquer approach: Notation: x r, y r : reverse of x, y E.g.x = accgg; x r = ggcca F r (i, j): optimal score of aligning x r 1 …x r i & y r 1 …y r j same as aligning x M-i+1 …x M & y N-j+1 …y N

CS262 Lecture 3, Win06, Batzoglou Linear-space alignment Lemma: (assume M is even) F(M, N) = max k=0…N ( F(M/2, k) + F r (M/2, N-k) ) x y M/2 k*k* F(M/2, k) F r (M/2, N-k)

CS262 Lecture 3, Win06, Batzoglou Linear-space alignment Now, using 2 columns of space, we can compute for k = 1…M, F(M/2, k), F r (M/2, N-k) PLUS the backpointers

CS262 Lecture 3, Win06, Batzoglou Linear-space alignment Now, we can find k * maximizing F(M/2, k) + F r (M/2, N-k) Also, we can trace the path exiting column M/2 from k * k*k* k * …… M/2 M/2+1 …… M M+1

CS262 Lecture 3, Win06, Batzoglou Linear-space alignment Iterate this procedure to the left and right! N-k * M/2 k*k*

CS262 Lecture 3, Win06, Batzoglou Linear-space alignment Hirschberg’s Linear-space algorithm: MEMALIGN(l, l’, r, r’):(aligns x l …x l’ with y r …y r’ ) 1.Let h =  (l’-l)/2  2.Find (in Time O((l’ – l)  (r’-r)), Space O(r’-r)) the optimal path,L h, entering column h-1, exiting column h Let k 1 = pos’n at column h – 2 where L h enters k 2 = pos’n at column h + 1 where L h exits 3.MEMALIGN(l, h-2, r, k 1 ) 4.Output L h 5.MEMALIGN(h+1, l’, k 2, r’) Top level call: MEMALIGN(1, M, 1, N)

CS262 Lecture 3, Win06, Batzoglou 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

CS262 Lecture 3, Win06, Batzoglou Heuristic Local Alignerers 1.The basic indexing & extension technique 2.Indexing: techniques to improve sensitivity Pairs of Words, Patterns 3.Systems for local alignment

CS262 Lecture 3, Win06, Batzoglou State of biological databases ~10x per 3 years

CS262 Lecture 3, Win06, Batzoglou State of biological databases Number of genes in these genomes:  Mammals: ~24,000  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

CS262 Lecture 3, Win06, Batzoglou 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

CS262 Lecture 3, Win06, Batzoglou 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 

CS262 Lecture 3, Win06, Batzoglou 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

CS262 Lecture 3, Win06, Batzoglou 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

CS262 Lecture 3, Win06, Batzoglou 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 < C below best so far Output: GTAAGGTCC GTTAGGTCC

CS262 Lecture 3, Win06, Batzoglou 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

CS262 Lecture 3, Win06, Batzoglou 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 < C below best score so far Output: GTAAGGTCCAGT GTTAGGTC-AGT

CS262 Lecture 3, Win06, Batzoglou Sensitivity-Speed Tradeoff long words (k = 15) short words (k = 7) Sensitivity Speed Kent WJ, Genome Research 2002 Sens. Speed X%

CS262 Lecture 3, Win06, Batzoglou Sensitivity-Speed Tradeoff 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

CS262 Lecture 3, Win06, Batzoglou Measured improvement Kent WJ, Genome Research 2002

CS262 Lecture 3, Win06, Batzoglou 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]:

CS262 Lecture 3, Win06, Batzoglou Advantage of Patterns 11 positions 10 positions

CS262 Lecture 3, Win06, Batzoglou 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?

CS262 Lecture 3, Win06, Batzoglou 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 Typhon  Uses multiple alignments to improve sensitivity/speed tradeoff

CS262 Lecture 3, Win06, Batzoglou 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

CS262 Lecture 3, Win06, Batzoglou 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

CS262 Lecture 3, Win06, Batzoglou The Four-Russian Algorithm brief overview A (not so useful) speedup of Dynamic Programming [ Arlazarov, Dinic, Kronrod, Faradzev 1970]

CS262 Lecture 3, Win06, Batzoglou Main Observation Within a rectangle of the DP matrix, values of D depend only on the values of A, B, C, and substrings x l...l’, y r…r’ Definition: A t-block is a t  t square of the DP matrix Idea: Divide matrix in t-blocks, Precompute t-blocks Speedup: O(t) A B C D xlxl x l’ yryr y r’ t

CS262 Lecture 3, Win06, Batzoglou The Four-Russian Algorithm Main structure of the algorithm: 1.Divide N  N DP matrix into K  K log 2 N-blocks that overlap by 1 column & 1 row 2.For i = 1……K 3. For j = 1……K 4. Compute D i,j as a function of A i,j, B i,j, C i,j, x[l i …l’ i ], y[r j …r’ j ] Time: O(N 2 / log 2 N) times the cost of step 4 t t t

CS262 Lecture 3, Win06, Batzoglou The Four-Russian Algorithm t t t