L3: Blast: Keyword match basics

Transcription

1 L3: Blast: Keyword match basics Fa05 CSE 182

2 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!

3 Assignment 1 is online Due 10/6 (Thursday) in class.

4 An O(nm) algorithm for score computation For i = 1 to n For j = 1 to m S[i 1, j 1] + C(s i,t j ) S[i, j] = max S[i 1, j] + C(s i, ) S[i, j 1] + C(,t j ) The iteration ensures that all values on the right are computed in earlier steps. How much space do you need?

5 Alignment? S[i 1, j 1] + C(s i,t j ) S[i, j] = max S[i 1, j] + C(s i, ) S[i, j 1] + C(,t j ) Is O(nm) space too much? What if the query and database are each 1Mbp?

6 Alignment (Linear Space) Score computation For i = 1 to n For j = 1 to m S[i 1, j 1] + C(s i,t j ) S[i, j] = max S[i 1, j] + C(s i, ) S[i, j 1] + C(,t j ) i 2 = i%2; i 1 = (i 1)%2; S[i 1, j 1] + C(s i,t j ) S[i 2, j] = max S[i 1, j] + C(s i, ) S[i 2, j 1] + C(,t j )

7 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)

8 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?

9 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?

10 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.

11 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]=S b (n/2,j) j* = max j {F[j]+B[j] } X = Align(1..n/2,1..j*) Y = Align(n/2..n,j*..m) Return X,j*,Y

12 Linear Space complexity T(nm) = c.nm + T(nm/2) = O(nm) Space = O(m)

13 Why is Blast Fast?

14 Large database search Database (n) Query (m) Database size n=10m, Querysize m=300. O(nm) = computations

15 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?

16 Basic probability Probability that there is a match starting at a fixed position i = 0.25 m What is the probability that some position i has a match. Dependencies confound probability estimates.

17 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 X i be the amount earned in the i-th toss E(X i ) =1.p + 0.(1 p) = p Total money you expect to earn i i E( X i ) = E(X i ) = np

18 Expected number of matches Expected number of matches can still be computed. i Let X i =1 if there is a match starting at position i, X i =0 otherwise Pr(Match at Position i) = p i = 0.25 m E(X i ) = p i = 0.25 m Expected number of matches = E( X i ) = E(X i ) = n 1 i i 4 ( ) m

19 Expected number of exact Matches is small! Expected number of matches = n*0.25 m If n=10 7, m=10, Then, expected number of matches = If n=10 7, m=11 expected number of hits = 2.38 n=10 7,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.

20 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

21 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? ( ) 10 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 90 =

22 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

23 BLAST Database (n) 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

24 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=10 7, no TP SW = O(nm) = 1000*10 7 = computations BLAST, W=11 E(#11-mer hits)= 1000* (1/4) 11 * 10 7 =2384 Number of computations = 2384*100*100=2.384*10 7 Ratio=10 10 /(2.384*10 7 )=420 Further speed improvements are possible

25 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

26 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

27 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.

28 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 Z S = S µ σ Look up a table to get a P-value

29 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

30 Exponential distribution Random Database, Pr(1) = p What is the expected number of hits to a sequence of k 1 s (n k)p k ne k ln p = ne k ln Instead, consider a random binary Matrix. Expected # of diagonals of k 1s Λ = (n k)(m k) p k nme k ln p = nme 1 p k ln 1 p

31 As you increase k, the number decreases exponentially. The number of diagonals of k runs can be approximated by a Poisson process Pr[u] = Λu e Λ u! Pr[u > 0] =1 e Λ 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

32 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 E = Kmne λs = mn2 λx Kmne Pr(S x) =1 e λs ln K ln 2 For small values, E-value and P-value are the same

33 Blast Variants 1. What is mega-blast? 2. What is discontiguous megablast? 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

34 Keyword Matching P O T A S T P O T A T O l c cl c c c cl c c c c c c P O T A T O T S I U M A S T E

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