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CSE182-L4: Keyword matching

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Backward scoring

• Defin Sb[i,j] : Best scoring alignment of the suffixes s[i+1..n] and t[j+1..m]

• Q: What is the score of the best alignment of the two strings s and t?

• HW: Write the recurrences for Sb

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Forward/Backward computations

• F[j] = Score of the best scoring alignment of s[1..n/2] and t[1..j]– F[j] = S[n/2,j]

• B[j] = Score of the best scoring alignment of s[n/2+1..n] and t[j+1..m]

– B[j] = Sb[n/2,j]

n/2

j1 m

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Forward/Backward computations

• 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.

n/2

j1 m

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Forward, Backward computation

• There exists a 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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Summary

• We considered the basics of sequence alignment– Opt score computation– Reconstructing alignments– Local alignments– Affine gap costs– Space saving measures

• Can we recreate Blast?

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Blast and local alignment

• Concatenate all of the database sequences to form one giant sequence.

• Do local alignment computation with the query.

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Large database search

Query (m)

Database (n)

Database size n=100M, Querysize m=1000.O(nm) = 1011 computations

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Why is Blast Fast?

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Silly Question!

• True or False: No two people in new york city have the same number of hair

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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 A’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

€

(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 values

Later

Database pre-processing

Seeds with don’t care values

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Silly Quiz

• Name a famous Bioinformatics Researcher

• Name a famous Bioinformatics Researcher who is a woman

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Scoring DNA

• DNA has structure.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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DNA scoring matrices

• So far, we considered a simple match/mismatch criterion.

• The nucleotides can be grouped into Purines (A,G) and Pyrimidines.

• Nucleotide substitutions within a group (transitions) are more likely than those across a group (transversions)

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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Scoring proteins• Scoring protein sequence alignments is

a much more complex task than scoring DNA– Not all substitutions are equal

• Problem was first worked on by Pauling and collaborators

• In the 1970s, Margaret Dayhoff created the first similarity matrices.– “One size does not fit all”– Homologous proteins which are

evolutionarily close should be scored differently than proteins that are evolutionarily distant

– Different proteins might evolve at different rates and we need to normalize for that

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PAM 1 distance

• Two sequences are 1 PAM apart if they differ in 1 % of the residues.

• PAM1(a,b) = Pr[residue b substitutes residue a, when the sequences are 1 PAM apart]

1% mismatch

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PAM1 matrix

• Align many proteins that are very similar– Is this a problem?

• PAM1 distance is the probability of a substitution when 1% of the residues have changed

• Estimate the frequency Pb|a of residue a being substituted by residue b.

• S(a,b) = log10(Pab/PaPb) = log10(Pb|a/Pb)

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PAM 1

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PAM distance

• Two sequences are 1 PAM apart when they differ in 1% of the residues.

• When are 2 sequences 2 PAMs apart?

1 PAM

1 PAM

2 PAM

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Higher PAMs

• PAM2(a,b) = ∑c PAM1(a,c). PAM1 (c,b)

• PAM2 = PAM1 * PAM1 (Matrix multiplication)

• PAM250

– = PAM1*PAM249

– = PAM1250

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Note: This is not the score matrix: What happens as you keep increasing the power?

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Scoring using PAM matrices

• Suppose we know that two sequences are 250 PAMs apart.

• S(a,b) = log10(Pab/PaPb)= log10(Pb|a/Pb) = log10(PAM250(a,b)/Pb)

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BLOSUM series of Matrices

• Henikoff & Henikoff: Sequence substitutions in evolutionarily distant proteins do not seem to follow the PAM distributions

• A more direct method based on hand-curated multiple alignments of distantly related proteins from the BLOCKS database.

• BLOSUM60 Merge all proteins that have greater than 60%. Then, compute the substitution probability.– In practice BLOSUM62 seems to work very well.

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PAM vs. BLOSUM

• What is the correspondence?

• PAM1 Blosum1• PAM2 Blosum2

• Blosum62

• PAM250 Blosum100

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Dictionary Matching, R.E. matching, and position specific scoring

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Keyword search

• Recall: In BLAST, we get a collection of keywords from the query sequence, and identify all db locations with an exact match to the keyword.

• Question: Given a collection of strings (keywords), find all occrrences in a database string where they keyword might match.

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Dictionary Matching

• Q: Given k words (si has length li), and a database of size n, find all matches to these words in the database string.

• How fast can this be done?

1:POTATO2:POTASSIUM3:TASTE

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

dictionary

database

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Dict. Matching & string matching

• How fast can you do it, if you only had one word of length m?– Trivial algorithm O(nm) time– Pre-processing O(m), Search O(n) time.

• Dictionary matching

– Trivial algorithm (l1+l2+l3…)n

– Using a keyword tree, lpn (lp is the length of the longest pattern)

– Aho-Corasick: O(n) after preprocessing O(l1+l2..)

• We will consider the most general case

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Direct Algorithm

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

Observations:• When we mismatch, we (should) know something about

where the next match will be.• When there is a mismatch, we (should) know something

about other patterns in the dictionary as well.

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P O T A T O

T UIS M

S ETA

The Trie Automaton

• Construct an automaton A from the dictionary– A[v,x] describes the transition from node v to a node w upon

reading x.– A[u,’T’] = v, and A[u,’S’] = w– Special root node r– Some nodes are terminal, and labeled with the index of the

dictionary word.

1:POTATO2:POTASSIUM3:TASTE

1

2

3

w

vu

S

r

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An O(lpn) algorithm for keyword matching

• Start with the first position in the db, and the root node.

• If successful transition– Increment current

pointer– Move to a new node– If terminal node

“success”• Else

– Retract ‘current’ pointer– Increment ‘start’ pointer– Move to root & repeat

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Illustration:

P O T A T O

T UIS M

S ETA

P O T A S T P O T A T Ol c

v

S

1

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Idea for improving the time

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

• Suppose we have partially matched pattern i (indicated by l, and c), but fail subsequently. If some other pattern j is to match– Then prefix(pattern j) = suffix [ first c-l characters of

pattern(i))

l c

1:POTATO2:POTASSIUM3:TASTE

P O T A S S I U MT A S T E

Pattern i

Pattern j

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Improving speed of dictionary matching

• Every node v corresponds to a string sv that is a prefix of some pattern.

• Define F[v] to be the node u such that su is the longest suffix of sv

• If we fail to match at v, we should jump to F[v], and commence matching from there

• Let lp[v] = |su|

P O T A T O

T UIS M

S ETA

1 2 3 4 5

67

89 10

11S

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An O(n) alg. For keyword matching

• Start with the first position in the db, and the root node.

• If successful transition– Increment current pointer– Move to a new node– If terminal node “success”

• Else (if at root)– Increment ‘current’ pointer– Mv ‘start’ pointer– Move to root

• Else – Move ‘start’ pointer forward– Move to failure node

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Illustration

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

P O T A T O

T UIS M

S ETA

lc

v S

1

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Time analysis

• In each step, either c is incremented, or l is incremented

• Neither pointer is ever decremented (lp[v] < c-l).

• l and c do not exceed n• Total time <= 2n

P O T A S T P O T A T Ol c

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Blast: Putting it all together

• Input: Query of length m, database of size n

• Select word-size, scoring matrix, gap penalties, E-value cutoff

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Blast Steps

1. Generate an automaton of all query keywords.2. Scan database using a “Dictionary Matching” algorithm

(O(n) time). Identify all hits.3. Extend each hit using a variant of “local alignment”

algorithm. Use the scoring matrix and gap penalties.4. For each alignment with score S, compute the bit-

score, E-value, and the P-value. Sort according to increasing E-value until the cut-off is reached.

5. Output results.

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Protein Sequence Analysis

• What can you do if BLAST does not return a hit?– Sometimes, homology (evolutionary similarity) exists at

very low levels of sequence similarity.

• A: Accept hits at higher P-value. – This increases the probability that the sequence similarity

is a chance event.– How can we get around this paradox?– Reformulated Q: suppose two sequences B,C have the

same level of sequence similarity to sequence A. If A& B are related in function, can we assume that A& C are? If not, how can we distinguish?