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’ = xi…xj, for some 1 i j |x|)

A string x’ is a subsequence of a string xif 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 = abracadabray = cadabr; substringz = brcdbr; subseqence, not substring

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

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) = …

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…xr

i & yr1…yr

j

same as aligning xM-i+1…xM & yN-j+1…yN

Linear-space alignment

Lemma: (assume M is even)

F(M, N) = maxk=0…N( F(M/2, k) + Fr(M/2, N – k) )

x

y

M/2

k*

F(M/2, k) Fr(M/2, N – k)

Example:ACC-GGTGCCCAGGACTG--CATACCAGGTG----GGACTGGGCAG

k* = 8

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

y1

xM

yN

x1 … xM/2+1 xM

…

y1

yN

…

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*

k*

k*+1

0 1 …… M/2 M/2+1 …… M M+1

Linear-space alignment

• Iterate this procedure to the left and right!

N-k*

M/2M/2

k*

Linear-space alignment

Hirschberg’s Linear-space algorithm:

MEMALIGN(l, l’, r, r’): (aligns xl…xl’ with yr…yr’)

1. Let h = (l’-l)/22. Find (in Time O((l’ – l) (r’ – r)), Space O(r’ – r))

the optimal path, Lh, entering column h – 1, exiting column hLet 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

5. MEMALIGN(h + 1, l’, k2, r’)

Top level call: MEMALIGN(1, M, 1, N)

Linear-space alignment

Time, Space analysis of Hirschberg’s algorithm: To compute optimal path at middle column,

For box of size M N,Space: 2NTime: 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

Heuristic Local Alignerers

1. The basic indexing & extension technique

2. Indexing: techniques to improve sensitivityPairs of Words, Patterns

3. Systems for local alignment

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

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:

GTAAGGTCCAGTGTTAGGTC-AGT

Sensitivity-Speed Tradeoff

long words(k = 15)

short words(k = 7)

Sensitivity Speed

Kent WJ, Genome Research 2002

Sens.

Speed

X%

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

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 Positions Non-Consecutive Positions

6 common

On a 100-long 70% conserved region: Consecutive Non-consecutive

Expected # hits: 1.07 0.97Prob[at least one hit]: 0.30 0.47

Advantage of Patterns

11 positions

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 2003Sun & Buhler RECOMB 2004

How long does it take to search the query?

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

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

Example

Query: Human atoh enhancer, 179 letters [1.5 min]

Result: 57 blast hits1. gi|7677270|gb|AF218259.1|AF218259 Homo sapiens ATOH1 enhanc... 355 1e-95 2. gi|22779500|gb|AC091158.11| Mus musculus Strain C57BL6/J ch... 264 4e-68 3. gi|7677269|gb|AF218258.1|AF218258 Mus musculus Atoh1 enhanc... 256 9e-66 4. gi|28875397|gb|AF467292.1| Gallus gallus CATH1 (CATH1) gene... 78 5e-12 5. gi|27550980|emb|AL807792.6| Zebrafish DNA sequence from clo... 54 7e-05 6. gi|22002129|gb|AC092389.4| Oryza sativa chromosome 10 BAC O... 44 0.068 7. gi|22094122|ref|NM_013676.1| Mus musculus suppressor of Ty ... 42 0.27 8. 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/

Hidden Markov Models

1

2

K

…

1

2

K

…

1

2

K

…

…

…

…

1

2

K

…

x1 x2 x3 xK

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) = 1/10P(6) = 1/2

Casino player switches back-&-forth between fair and loaded die once every 20 turns

Game:1. You bet $12. 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

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

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

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%

The dishonest casino model

FAIR LOADED

0.05

0.05

0.950.95

P(1|F) = 1/6P(2|F) = 1/6P(3|F) = 1/6P(4|F) = 1/6P(5|F) = 1/6P(6|F) = 1/6

P(1|L) = 1/10P(2|L) = 1/10P(3|L) = 1/10P(4|L) = 1/10P(5|L) = 1/10P(6|L) = 1/2