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L3: Blast: Keyword match basics

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Silly Quiz

• TRUE or FALSE: • In New York City at any moment, there are 2

people (not bald) with exactly the same number of hairs!

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Assignment 1 is online

• Due 10/6 (Thursday) in class.

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An O(nm) algorithm for score computation

• The iteration ensures that all values on the right are computed in earlier steps.

• How much space do you need?

€

S[i, j] = maxS[i −1, j −1] + C(si, t j )

S[i −1, j] + C(si,−)S[i, j −1] + C(−, t j )

⎧ ⎨ ⎪

⎩ ⎪

For i = 1 to n For j = 1 to m

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Alignment?

• Is O(nm) space too much?• What if the query and database are each

1Mbp?€

S[i, j] = maxS[i −1, j −1] + C(si, t j )

S[i −1, j] + C(si,−)S[i, j −1] + C(−, t j )

⎧ ⎨ ⎪

⎩ ⎪

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Alignment (Linear Space)

• Score computation

€

i2 = i%2; i1 = (i −1)%2;

S[i2, j] = max

S[i1, j −1] + C(si, t j )

S[i1, j] + C(si,−)

S[i2, j −1] + C(−, t j )

⎧

⎨ ⎪

⎩ ⎪

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

For i = 1 to n For j = 1 to m

€

S[i, j] = max

S[i −1, j −1] + C(si, t j )

S[i −1, j] + C(si,−)

S[i, j −1] + C(−, t j )

⎧

⎨ ⎪

⎩ ⎪

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Linear Space

• In Linear Space, we can do each row of the D.P.

• We need to compute the optimum path from the origin (0,0) to (m,n)

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Linear Space (cont’d)

• At i=n/2, we know scores of all the optimal paths ending at that row.

• Define F[j] = S[n/2,j]• One of these j is on

the true path. Which one?

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Backward alignment

• Let S’[i,j] be the optimal score of aligning s[i..n] with t[j..m]

• Define B[j] = S’[n/2,j]• One of these j is on

the true path. Which one?

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Forward, Backward computation

• At the optimal coordinate, j– F[j]+B[j]=S[n,m]

• In O(nm) time, and O(m) space, we can compute one of the coordinates on the optimum path.

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Linear Space Alignment

• Align(1..n,1..m)– For all 1<=j <= m

• Compute F[j]=S(n/2,j)

– For all 1<=j <= m• Compute B[j]=Sb(n/2,j)

– j* = maxj {F[j]+B[j] }

– X = Align(1..n/2,1..j*)– Y = Align(n/2..n,j*..m)– Return X,j*,Y

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Linear Space complexity

• T(nm) = c.nm + T(nm/2) = O(nm)• Space = O(m)

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Why is Blast Fast?

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Large database search

Query (m)

Database (n)

Database size n=10M, Querysize m=300.O(nm) = 3. 109 computations

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Observations

• Much of the database is random from the query’s perspective

• Consider a random DNA string of length n. – Pr[A]=Pr[C] = Pr[G]=Pr[T]=0.25

• Assume for the moment that the query is all 1’s (length m).

• What is the probability that an exact match to the query can be found?

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Basic probability

• Probability that there is a match starting at a fixed position i = 0.25m

• What is the probability that some position i has a match.

• Dependencies confound probability estimates.

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Basic Probability:Expectation

• Q: Toss a coin: each time it comes up heads, you get a dollar– What is the money you expect to get after n

tosses?

– Let Xi be the amount earned in the i-th toss

€

E(X i) =1.p + 0.(1− p) = p

Total money you expect to earn

€

E( X i) = E(X i) =i

∑ npi

∑

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Expected number of matches

• Expected number of matches can still be computed.

Let Xi=1 if there is a match starting at position i, Xi=0 otherwise

€

Pr(Match at Position i) = pi = 0.25m

E(X i) = pi = 0.25m

Expected number of matches =

€

E( X i) = E(X i)i∑

i∑ = n 1

4( )m

i

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Expected number of exact Matches is small!

• Expected number of matches = n*0.25m

– If n=107, m=10, • Then, expected number of matches = 9.537

– If n=107, m=11• expected number of hits = 2.38

– n=107,m=12, • Expected number of hits = 0.5 < 1

• Bottom Line: An exact match to a substring of the query is unlikely just by chance.

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Observation 2

• What is the pigeonhole principle?

Suppose we are looking for a database string with greater than 90% identity to the query (length 100) Partition the query into size 10 substrings. At least one much match the database string exactly

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Why is this important?

• Suppose we are looking for sequences that are 80% identical to the query sequence of length 100.

• Assume that the mismatches are randomly distributed.• What is the probability that there is no stretch of 10 bp, where the query and

the subject match exactly?

• Rough calculations show that it is very low. Exact match of a short query substring to a truly similar subject is very high.

– The above equation does not take dependencies into account– Reality is better because the matches are not randomly distributed

€

€

≅ 1− 810( )

10 ⎛ ⎝ ⎜ ⎞

⎠ ⎟90

= 0.000036

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Just the Facts

• Consider the set of all substrings of the query string of fixed length W.– Prob. of exact match to a random database

string is very low.– Prob. of exact match to a true homolog is

very high.– Keyword Search (exact matches) is MUCH

faster than sequence alignment

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BLAST

• Consider all (m-W) query words of size W (Default = 11)• Scan the database for exact match to all such words• For all regions that hit, extend using a dynamic programming alignment.• Can be many orders of magnitude faster than SW over the entire string

Database (n)

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Why is BLAST fast?• Assume that keyword searching does not consume any

time and that alignment computation the expensive step.• Query m=1000, random Db n=107, no TP• SW = O(nm) = 1000*107 = 1010 computations• BLAST, W=11

• E(#11-mer hits)= 1000* (1/4)11 * 107=2384• Number of computations = 2384*100*100=2.384*107

• Ratio=1010/(2.384*107)=420• Further speed improvements are possible

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Keyword Matching• How fast can we

match keywords?• Hash table/Db index?

What is the size of the hash table, for m=11

• Suffix trees? What is the size of the suffix trees?

• Trie based search. We will do this in class.

AATCA 567

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Related notes

• How to choose the alignment region?– Extend greedily until the score falls below a certain

threshold• What about protein sequences?

– Default word size = 3, and mismatches are allowed.• Like sequences, BLAST has been evolving continuously

– Banded alignment– Seed selection– Scanning for exact matches, keyword search versus

database indexing

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P-value computation• How significant is a score? What happens to

significance when you change the score function• A simple empirical method:

• Compute a distribution of scores against a random database.

• Use an estimate of the area under the curve to get the probability.

• OR, fit the distribution to one of the standard distributions.

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Z-scores for alignment• Initial assumption was that the scores followed a

normal distribution.• Z-score computation:

– For any alignment, score S, shuffle one of the sequences many times, and recompute alignment. Get mean and standard deviation

– Look up a table to get a P-value

€

ZS =S − μ

σ

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Blast E-value

• Initial (and natural) assumption was that scores followed a Normal distribution• 1990, Karlin and Altschul showed that ungapped local alignment scores follow

an exponential distribution• Practical consequence:

– Longer tail. – Previously significant hits now not so significant

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Exponential distribution

• Random Database, Pr(1) = p • What is the expected number of hits to a sequence of k 1’s

• Instead, consider a random binary Matrix. Expected # of diagonals of k 1s

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(n − k)pk ≅ nek ln p = ne−k ln

1

p

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Λ=(n − k)(m − k) pk ≅ nmek ln p = nme−k ln

1

p

⎛

⎝ ⎜

⎞

⎠ ⎟

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• As you increase k, the number decreases exponentially.• The number of diagonals of k runs can be approximated by a Poisson

process

• In ungapped alignments, we replace the coin tosses by column scores, but the behaviour does not change (Karlin & Altschul).

• As the score increases, the number of alignments that achieve the score decreases exponentially €

Pr[u] =Λue−Λ

u!

Pr[u > 0] =1− e−Λ

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Blast E-value• Choose a score such that the expected score between a pair of

residues < 0• Expected number of alignments with a particular score

• For small values, E-value and P-value are the same

€

E = Kmne−λS = mn2−

λS−ln K

ln 2

⎛

⎝ ⎜

⎞

⎠ ⎟

Pr(S ≥ x) =1− e−Kmne −λx

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Blast Variants

1. What is mega-blast?2. What is discontiguous

mega-blast?3. Phi-Blast/Psi-Blast?4. BLAT?5. PatternHunter?

Longer seeds.Seeds with don’t care valuesLaterDatabase pre-processingSeeds with don’t care values

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Keyword Matching

P O T A S T P O T A T O

P O T A T O

T UIS M

S ETA

l c c c cc c c c c cc cl cl

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