Sequence AlignmentCont’d

Linear-space alignment

• Iterate this procedure to the left and right!

N-k*

M/2M/2

k*

The Four-Russian Algorithm

Main structure of the algorithm:• Divide NN DP matrix into KK

log2N-blocks that overlap by 1 column & 1 row

• For i = 1……K• For j = 1……K• Compute Di,j as a function of

Ai,j, Bi,j, Ci,j, x[li…l’i], y[rj…r’j]

Time: O(N2 / log2N) times the cost of step 4 t t t

Heuristic Local Aligners

BLAST, WU-BLAST, BlastZ, MegaBLAST, BLAT, PatternHunter, ……

State of biological databases

Sequenced Genomes:

Human 3109 Yeast 1.2107

Mouse 2.7109 12 different strainsRat 2.6109 Neurospora 4107

14 more fungi within next yearFugu fish 3.3108

Tetraodon 3108 ~250 bacteria/viruses

Mosquito 2.8108 Next year: Drosophila 1.2108 Dog, Chimpanzee, ChickenWorm 1.0108

2 sea squirts 1.6108 Current rate of sequencing:Rice 1.0109 4 big labs 3 109 bp /year/labArabidopsis 1.2108 10s small labs

State of biological databases

• Number of genes in these genomes:

Vertebrate: ~30,000 Insects: ~14,000 Worm: ~17,000 Fungi: ~6,000-10,000

Small organisms: 100s-1,000s

• Each known or predicted gene has an associated protein sequence

• >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

104

1010 - 1011

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

3109

1010 - 1011

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

BLAST Original Version

Dictionary:All words of length k (~11)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

BLAST Original VersionA 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,T = 4

The matching word GGTC initiates an alignment

Extension to the left and right with no gaps until alignment falls < 50%

Output:GTAAGGTCCGTTAGGTCC

Gapped BLASTA 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

Added features:

• Pairs of words can initiate alignment

• Extensions with gaps in a band around anchor

Output:

GTAAGGTCCAGTGTTAGGTC-AGT

Gapped BLASTA 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

Added features:

• Pairs of words can initiate alignment

• Nearby alignments are merged

• Extensions with gaps until score < T below best score so far

Output:

GTAAGGTCCAGTGTTAGGTC-AGT

Variants of BLAST

• MEGABLAST: Optimized to align very similar sequences

• Works best when k = 4i 16• Linear gap penalty

• PSI-BLAST: BLAST produces many hits Those are aligned, and a pattern is extracted Pattern is used for next search; above steps iterated

• WU-BLAST: (Wash U BLAST) Optimized, added features

• BlastZ Combines BLAST/PatternHunter methodology

ExampleQuery: 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

BLAT: Blast-Like Alignment Tool

Differences with BLAST:

1. Dictionary of the DB (BLAST: query)

2. Initiate alignment on any # of matching hits (BLAST: 1 or 2 hits)

3. More aggressive stitching-together of alignments

4. Special code to align mature mRNA to DNA

Result: very efficient for:• Aligning many seqs to a DB• Aligning two genomes

PatternHunter

Main features: • Non-consecutive position words• Highly optimized

5 hits3 hits

3 hits

7 hits

7 hits

Consecutive Positions Non-Consecutive Positions

6 hits

On a 70% conserved region: Consecutive Non-consecutive

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

Advantage of Non-Consecutive Words

11 positions11 positions

10 positions

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

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

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

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

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

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, x1, x2, …, xt) =

P(t+1 = k | t)

K

1

…

2

A parse of a sequence

Given a sequence x = x1……xN,

A parse of x is a sequence of states = 1, ……, N

1

2

K

…

1

2

K

…

1

2

K

…

…

…

…

1

2

K

…

x1 x2 x3 xK

2

1

K

2

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) =

a01 a12……aN-1N e1(x1)……eN(xN)

1

2

K…

1

2

K…

1

2

K…

…

…

…

1

2

K…

x1

x2 x3 xK

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 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

Example: the dishonest casino

So, the likelihood the die is fair in all this runis 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 = 7.9 10-10

Therefore, it is after all 6.59 times 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 = .00000049238235134735 = 0.5 10-7

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

GIVEN a HMM M, and a sequence x,FIND the sequence of states that maximizes P[ x, | M ]

3. Learning

GIVEN a HMM M, with unspecified transition/emission probs.,and a sequence x,

FIND parameters = (ei(.), aij) 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 = aij, ei(.)

So, P[ x | ], and P[ x ] are the same, when the architecture, and the entire model, respectively, are implied

Similarly, P[ x, | M ] and P[ x, ] are the same

In the LEARNING problem we always write P[ x | ] to emphasize that we are seeking the that maximizes P[ x | ]