PatternHunter: A Fast and Highly Sensitive Homology Search Method Bin Ma Department of Computer Science University of Western Ontario

GCNTACACGTCACCATCTGTGCCACCACNCATGTCTCTAGTGATCCCTCATAAGTTCCAACAAAGTTTGC || ||||| | ||| |||| || |||||||||||||||||| | |||||||| | | ||||| GCCTACACACCGCCAGTTGTG-TTCCTGCTATGTCTCTAGTGATCCCTGAAAAGTTCCAGCGTATTTTGC GAGTACTCAACACCAACATTGATGGGCAATGGAAAATAGCCTTCGCCATCACACCATTAAGGGTGA---- || ||||||||| |||||| | ||||| |||||||| ||| |||||||| | | | || GAATACTCAACAGCAACATCAACGGGCAGCAGAAAATAGGCTTTGCCATCACTGCCATTAAGGATGTGGG TGTTGAGGAAAGCAGACATTGACCTCACCGAGAGGGCAGGCGAGCTCAGGTA ||||||||||||| ||| ||||||||||| || ||||||| || |||| | TTGACAGTACACTCATAGTGTTGAGGAAAGCTGACGTTGACCTCACCAAGTGGGCAGGAGAACTCACTGA GGATGAGGTGGAGCATATGATCACCATCATACAGAACTCAC CAAGATTCCAGACTGGTTCTTG ||||||| |||| | | |||| ||||| || ||||| || |||||| ||||||||||||||| GGATGAGATGGAACGTGTGATGACCATTATGCAGAATCCATGCCAGTACAAGATCCCAGACTGGTTCTTG A homology between mouse and human genomes Smith-Waterman is the most accurate method. Time complexity ： O(mn).

GCNTACACGTCACCATCTGTGCCACCACNCATGTCTCTAGTGATCCCTCATAAGTTCCAACAAAGTTTGC || ||||| | ||| |||| || |||||||||||||||||| | |||||||| | | ||||| GCCTACACACCGCCAGTTGTG-TTCCTGCTATGTCTCTAGTGATCCCTGAAAAGTTCCAGCGTATTTTGC GAGTACTCAACACCAACATTGATGGGCAATGGAAAATAGCCTTCGCCATCACACCATTAAGGGTGA---- || ||||||||| |||||| | ||||| |||||||| ||| |||||||| | | | || GAATACTCAACAGCAACATCAACGGGCAGCAGAAAATAGGCTTTGCCATCACTGCCATTAAGGATGTGGG TGTTGAGGAAAGCAGACATTGACCTCACCGAGAGGGCAGGCGAGCTCAGGTA ||||||||||||| ||| ||||||||||| || ||||||| || |||| | TTGACAGTACACTCATAGTGTTGAGGAAAGCTGACGTTGACCTCACCAAGTGGGCAGGAGAACTCACTGA GGATGAGGTGGAGCATATGATCACCATCATACAGAACTCAC CAAGATTCCAGACTGGTTCTTG ||||||| |||| | | |||| ||||| || ||||| || |||||| ||||||||||||||| GGATGAGATGGAACGTGTGATGACCATTATGCAGAATCCATGCCAGTACAAGATCCCAGACTGGTTCTTG BLAST finds a “hit” and then extends Seed match = hit

Example of missing a target Fail: GAGTACTCAACACCAACATTAGTGGGCAATGGAAAAT || ||||||||| |||||| | |||||| |||||| GAATACTCAACAGCAACATCAATGGGCAGCAGAAAAT Dilemma Sensitivity – needs shorter seeds the success rate of finding a homology Speed – needs longer seeds Mega-BLAST uses seeds of length 28.

PatternHunter uses “spaced seeds” (called a model) Eleven required matches (weight=11) Seven “don’t care” positions GAGTACTCAACACCAACATTAGTGGCAATGGAAAAT… || ||||||||| ||||| || ||||| |||||| GAATACTCAACAGCAACACTAATGGCAGCAGAAAAT… Hit = all the required matches are satisfied. BLAST seed model =

Observations re. spaced seeds Seed models with different shapes can detect different homologies. Two consequences: Some models may detect more homologies than others More sensitive homology search PatternHunter I Can use several seed models simultaneously to hit more homologies Approaching 100% sensitive homology search PatternHunter II

Spaced Seed – PatternHunter I:

Weight of a seed Lemma: The expected number of hits of a weight W length M seed model within a length L region with similarity p is (L-M+1)p W Proof: There are (L-M+1) positions a hit can occur. At each position, p W hit is expected. Q.E.D. Seed models with the same weight generate approximately the same amount of hits. Speed is approximately the same. Sensitivity is not necessarily the same. num of hits v.s. num of regions that contain hits. GAGTACTCAACACCAACATTAGTGGCAATGGAAAAT || ||||||||| ||||| || ||||| |||||| GAATACTCAACAGCAACACTAATGGCAGCAGAAAAT

Simulated sensitivity curves

Why spaced seeds are better? TTGACCTCACC? |||||||||||? TTGACCTCACC? CAA?A??A?C??TA?TGG? |||?|??|?|??||?|||? CAA?A??A?C??TA?TGG? BLAST’s seed usually uses more than one hits to detect one homology (redundant) Spaced seeds uses fewer hits to detect one homology (efficient)

PH’s seed does not overlap heavily PH’s seed do not overlap heavily when shifts: The hits at different positions are independent. The probability of having the second hit is 5*p 6 + … compare to BLAST’s model p + p 2 + p 3 + p 4 + …

Indeed Indeed, under the condition that there is one hit in a length 64, 70% similar homology, the average number of hits in that region is 2.0 for PH’s weight-11 seed 3.6 for contiguous weight-11 seed.

A dynamic programming algorithm to compute sensitivity R[1..n]: Random homology, Pr(R[i]=1) = p; We want Pr(R is hit by a seed model x) DP[i,s] denotes Pr(R[1..i] is hit | R[1..i] ends with s) 1; |s|=|x| and s is hit DP[i,s]= DP[i-1,s[1..|s|-1]; |s|=|x| and s is not hit p*DP[i,(1s)] + (1-p)*DP[i,(0s)]; else O( n*2 |x| ). Better algorithm exists.

PatternHunter I performance Blastn MB28 PH E.coli (4.7M) v.s. H.inf (1.8M) 716s /158M 5s/561M 34s/78M Arabidopsis chr2 (19.6M) v.s. chr4 (17.5M) s/1087M 5020s/279M Human chr21 (26.2M) v.s. chr22 (35M) s/419M All used a 700MHZ PentiumIII PC with 1G byte memory. Human (3G) v.s. Mouse (3G)* Using 2-hit, weight 12 seed, PH used 6 days with a 1GHZ PentiumIII PC with 2G byte memory. With Blast, it would otherwise take months with parallel computers to finish.

Multiple Seeds – PatternHunter II:

PatternHunter II: Optimized Multiple seeds Basic Searching Algorithm 1. Select a group of spaced seed models 2. For each hit of each model, conduct extension to find a homology. Selecting optimal multiple seed is NP- hard.

Seed Selection Algorithm 1. Let A be an empty set. 2. Let s be the seed such that A ⋃ {s} has the highest hit probability. 3. A=A ⋃ {s}; if |A|<K go to 2. Approximation ratio 1-1/e Computing the hit probability of multiple seeds is NP-hard. Efficient algorithm when number of zeros is limited. PTAS to compute the probability approximately.

Randomly generate m homologies independently. Suppose n of them are hit by our seeds. Let p be the sensitivity of our seeds. If, then with probability 1-2/K, Can be proved by Chernoff’s bounds.

The seeds obtained under a simple homology distribution (homology identity = 0.7, homology length=64) , , , , ……

Simulated sensitivity curves: Solid curves: Multiple (1, 2, 4, 8, 16) weight-12 spaced seeds. Dashed curves: Optimal spaced seeds with weight = 11, 10, 9, 8. Typically, “Doubling the seed number” gains better sensitivity than “decreasing the weight by 1”. One weight-12 Two weight-12 One weight-11

Coding region seeds The first two bases of a codon is more conserved than the third base. Coding regions matches have patterns like …… The seeds trained under a coding region homology distribution are called the coding region seeds. PHII’s default seeds were trained under a simple distribution (0.8, 0.8, 0.5).

Experiments on real data About 30k mouse ESTs (25Mb) and 4k human ESTs (3Mb) downloaded from NCBI genbank. “low complexity” regions were filtered out. SSearch (Smith-Waterman method) finds “all” pairs of ESTs with significant local alignments. Check how many percents of those pairs can be “found” by BLAST and different configurations of PatternHunter.

Sensitivity curves:

Recent development Can 100% sensitivity be achieved with reasonable speed? Yes. When >=80% similarity, 100% sensitivity can be achieved with approximately 40 weight-9 seeds.

Open questions: Can the hit probability of one (or constant number of) seed be computed in polynomial time? Current: Polynomial time algorithms exist when num of 0s in one seed is O(log n). PTAS. Can the optimal seed (or set of seeds) be found in polynomial time? For general distributions of the homologies, these are NP-hard.

How the hits are found efficiently? Put all the seeds of database in a lookup table. For each seed in the query, find all the occurrences of the seed in the database by looking at the lookup table.