sequence alignment cont’d. cs262 lecture 4, win06, batzoglou indexing-based local alignment...

Sequence AlignmentCont’d

CS262 Lecture 4, Win06, Batzoglou

Indexing-based local alignment

(BLAST- Basic Local Alignment Search Tool)

1.1. SEEDSEED

Construct a dictionary of all the words in query (database) & search database (query) in linear time for word matches

2.2. EXTENDEXTEND

Initiate fast local alignment procedures to the left and right of each word match

3.3. REPORTREPORT

Retain all alignments that score above a threshold and report them

……

……

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 < C 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 until threshold

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

Reference: Zhang, Bergman, Miller, RECOMB ‘98

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 different patterns Construct K distinct dictionaries, one for each pattern 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

http://www.ncbi.nlm.nih.gov/BLAST/



http://genopole.toulouse.inra.fr/blast/megablast.html

http://blast.wustl.edu/

http://genome.ucsc.edu/cgi-bin/hgBlat

http://www.cs.berkeley.edu/~brudno/chaos

http://www.bioinformaticssolutions.com/products/ph/index.php

http://www.psc.edu/general/software/packages/blastz/

http://typhon.stanford.edu/


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/

http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=Nucleotide&list_uids=07677270&dopt=GenBank

http://www.ncbi.nlm.nih.gov/blast/Blast.cgi#7677270











The Four-Russian Algorithmbrief overview

A (not so useful) speedup of Dynamic Programming

[Arlazarov, Dinic, Kronrod, Faradzev 1970]


Main Observation

Within a rectangle of the DP matrix,values of D depend onlyon the values of A, B, C,

and substrings xl...l’, yr…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

xl xl’

yr

yr’

t


The Four-Russian Algorithm

Main structure of the algorithm:

1. Divide NN DP matrix into KK log2N-blocks that overlap by 1 column & 1 row

2. For i = 1……K

3. For j = 1……K

4. 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 4t t t


The Four-Russian Algorithm

t t t

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


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


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


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

A compact way to write a01 a12……aN-1N e1(x1)……eN(xN)

Number all parameters aij and ei(b); n paramsExample: a0Fair : 1; a0Loaded : 2; … eLoaded(6) = 18

Then, count in x and the # of times eachparameter j = 1, …, n occurs

F(j, x, ) = # parameter j occurs in (x, )

(call F(.,.,.) the feature counts) Then,

P(x, ) = j=1…n jF(j, x, ) =

= exp[j=1…n log(j)F(j, x, )]


Example: the dishonest casino

Let the sequence of rolls be:

x = 1, 2, 1, 5, 6, 2, 1, 5, 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



So, the likelihood the die is fair in 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)9 (1/2)1 (0.95)9 = .00000000015756235243 ~= 0.16 10-9

Therefore, it somewhat more likely that all the rolls are done with the fair die, than that they are all done with the loaded die



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 | 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 | ]


Problem 1: Decoding

Find the best parse of a sequence


Decoding

GIVEN x = x1x2……xN

We want to find = 1, ……, N,such that P[ x, ] is maximized

* = argmax P[ x, ]

We can use dynamic programming!

Let Vk(i) = max{1… i-1} P[x1…xi-1, 1, …, i-1, xi, i = k] = Probability of most likely sequence of states ending at state i = k

1

2

K

…

1

2

K

…

1

2

K

…

…

…

…

1

2

K

…

x1

x2 x3 xK

2

1

K

2


Decoding – main idea

Given that for all states k, and for a fixed position i,

Vk(i) = max{1… i-1} P[x1…xi-1, 1, …, i-1, xi, i = k]

What is Vl(i+1)?

From definition,

Vl(i+1) = max{1… i}P[ x1…xi, 1, …, i, xi+1, i+1 = l ]

= max{1… i}P(xi+1, i+1 = l | x1…xi,1,…, i) P[x1…xi, 1,…, i]

= max{1… i}P(xi+1, i+1 = l | i ) P[x1…xi-1, 1, …, i-1, xi, i]

= maxk [P(xi+1, i+1 = l | i = k) max{1… i-1}P[x1…xi-1,1,…,i-1, xi,i=k]] = el(xi+1) maxk akl Vk(i)


The Viterbi Algorithm

Input: x = x1……xN

Initialization:V0(0) = 1 (0 is the imaginary first position)Vk(0) = 0, for all k > 0

Iteration:Vj(i) = ej(xi) maxk akj Vk(i – 1)

Ptrj(i) = argmaxk akj Vk(i – 1)

Termination:P(x, *) = maxk Vk(N)

Traceback: N* = argmaxk Vk(N) i-1* = Ptri (i)


The Viterbi Algorithm

Similar to “aligning” a set of states to a sequence

Time:

O(K2N)

Space:

O(KN)

x1 x2 x3 ………………………………………..xN

State 1

2

K

Vj(i)


Viterbi Algorithm – a practical detail

Underflows are a significant problem

P[ x1,…., xi, 1, …, i ] = a01 a12……ai e1(x1)……ei(xi)

These numbers become extremely small – underflow

Solution: Take the logs of all values

Vl(i) = log ek(xi) + maxk [ Vk(i-1) + log akl ]


Example

Let x be a long sequence with a portion of ~ 1/6 6’s, followed by a portion of ~ ½ 6’s…

x = 123456123456…12345 6626364656…1626364656

Then, it is not hard to show that optimal parse is (exercise):

FFF…………………...F LLL………………………...L

6 characters “123456” parsed as F, contribute .956(1/6)6 = 1.610-5

parsed as L, contribute .956(1/2)1(1/10)5 = 0.410-5

“162636” parsed as F, contribute .956(1/6)6 = 1.610-5

parsed as L, contribute .956(1/2)3(1/10)3 = 9.010-

5

sequence alignment cont’d. cs262 lecture 4, win06, batzoglou indexing-based local alignment...

Documents