Precomputing Edit-Distance

Specificity of Short

Oligonucleotides

Precomputing Edit-Distance

Specificity of Short

OligonucleotidesNathan EdwardsCenter for Bioinformatics and Computational BiologyUniversity of Maryland, College Park

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Polymerase Chain Reaction

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Polymerase Chain Reaction

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Primer Specificity

• Need to ensure that primers hybridize to a specific (specified) locus only• Require exactly one occurrence of specified

sequence• Require no (potential) mis-hybridization loci

• Bottleneck computation in primer-design• Design / check iteration is problematic

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k-unique 20-mers

• Edit-distance as a surrogate for mis-hybridization potential

• k-unique loci:• All non-self genomic loci are require more

than k edits in (global) alignment• Closest non-self genomic loci requires

(k+1) edits in (global) alignment

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Find all k-unique 20-mers

• Naïve algorithm: O(n2km)• Quadratic in size of genome.

• 0-unique (exact match) 20-mers• (Expected) linear time algorithm

• Achieve expected linear time using a hybrid approach (blastn):• Use partial exact match to “seed” expensive

dynamic programming alignment• Large chunks ) Fast, but miss occurrences• Small chunks ) Slow, but correct

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Baeza-Yates Perleberg: • Correct and O(n) for small k

• At least 1 chunk is observed with no error.• Small k → Large chunks → Fast and correct• Largest correct chunk: floor(m/(k+1))

Inexact sequence match

≠ = ≠q

g

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Example worst case alignments

TCCCGC-TAGATTGAGATCT||||||v||||||*||||||TCCCGCCTAGATTTAGATCT

ACTTGTCCACAGTGCTTAAG||||||*||||||*||||||ACTTGTGCACAGTCCTTAAG

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Brute-force approach

ACTTGTGCACAGTCCTTAAG

AA:18AC:1,9AG:11,19CA:8,10CC:14CT:2,15

GC:7GT:5,12TA:17TC:13TG:4,6 TT:3,16

2-mer position table

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Brute-force approach

ACTTGTGCACAGTCCTTAAG

ACTTGTGCACAGTCCTTAAG

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Brute-force approach

ACTTGTGCACAGTCCTTAAG

ACTTGTGCACAGTCCTTAAG

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Brute-force approach

ACTTGTGCACAGTCCTTAAG

ACTTGTGCACAGTCCTTAAG

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Brute-force approach

ACTTGTGCACAGTCCTTAAG

ACTTGTGCACAGTCCTTAAG

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Brute-force approach

ACTTGTGCACAGTCCTTAAG

ACTTGTGCACAGTCCTTAAG

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Brute-force approach

ACTTGTGCACAGTCCTTAAG

ACTTGTGCACAGTCCTTAAG

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Brute-force approach

Divide the genome into 10 Mb blocksFor all pairs of blocks:

For all l-mer matches:Do all pair-wise DPs containing matchIf ≤ k edits, mark position non-unique

300 x 300 pairs of blocksFor 20-mers:

k=1 ) l=10; k=2 ) l=6; k=3 ) l=5 ; k=4 ) l=4.

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Brute-force approach

Things are looking really, really, bad:• Seeds are too short• 90,000 pair-wise block comparisons

Actually quite good (seed size 12):• Non-uniqueness certificates are dense• Almost all positions eliminated early• Behaves more like linear time than

quadratic

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In practice (edit-dist 4)

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In practice (edit-dist 4)

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In practice (edit-dist 4)

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In practice (edit-dist 3)

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In practice (edit-dist 3)

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In practice (edit-dist 4,3,2)

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In practice (edit-dist 4,3,2)

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Edit distance 2

• After seed size 12• ~ 27K (0.288%) positions have no match

• After seed size 8• ~ 3K (0.029%) positions have no match

• Using seed size 6 is still too slow• Need a more sophisticated hashing strategy• 6-mers match in too many places!

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Spaced seed-set design problem

• Given:• mer-size: m ( = 20 )• # errors: k ( = 1,2,3)• # cares: l ( = 10,12,14 )

• Find the smallest set of spaced seeds that will find all alignments.

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Solution for (20,2,8)

• 11111111, 111101111

TCCCGCGTAGATTGAGATCT ||||||*||||||*|||||| TCCCGCCTAGATTTAGATCT

• How can we find these spaced seed set solutions?• One/two table? 2 x false positives

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Spaced seed set design set-cover formulation

• Set cover instance:• Ground set: all possible placements of the

k errors (alignments)• Covering sets: all possible placements of

the l care positions

• For (m=20,k=2,l=10),• 190 elements, 184,756 sets!• Need to reduce the number of sets!

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Dirty secret of spaced seeds

• Spaced seeds take O(# cares) to update!• Contiguous seeds are O(1) to update

• 101010101010101 vs 11111111• 8 steps to update vs 1 step to update

• Constant time update for spaced seeds?• Yes, if they have a certain structure(s)• Restrict spaced seeds to small update cost

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O(1) spaced seed update

ACGTACGTACGTACGTACGT1: A G A G2: C T C T ...Spaced seed 1010101 can be updated

in 1 step!

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O(1) spaced seed update

ACGTACGTACGTACGTACGTACGTACG -> ACGACG CGTACGT -> CGTCGT...Spaced seed 1110111 can be updated

in 1 step!

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O(1) spaced seed update

• “Period” step update• 11001100110011 2 steps• 1000010000100001 1 step

• “Runs” step update• 11100111111 1 step• 11101110111 2 steps

• Minimize the number of update steps• Weighted set-cover instance…

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Edit-distance SS-SDP

• Position of matching bases might shift!• Need 11111111 ↓ to get CCGCTAGA

• Need 111101111 ↑ to get CCGCTAGA

• Set cover formulation no longer works

TCCCGC-TAGATTGAGATCT||||||v||||||*||||||TCCCGCCTAGATTTAGATCT

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Edit-Distance SS-SDP

• Use a variation on set cover:• q:111101111,r:11111111 covers:

• Pay for query & reference update costs separately

• Control size of problem by only enumerating templates with small update cost

r:TCCCGC-TAGATTGAGATCT ||||||v||||||*||||||q:TCCCGCCTAGATTTAGATCT

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Correct solutions for 1-unique 20-mers107 random sequence x 107 random sequence

• Seed: 1111111111 (Best single seed solution, weight 10)• ~ 9.5 expensive dynamic programs per locus

• Seed set: 11111111111;111110111111 (weight 11)• ~ 8.9 DP/locus

• Seed set (weight 11)

• ~ 7.8 DP/locus

• Seed set: 111111111111;1111101111111 (weight 12)• ~ 2.2 DP/locus

11111111111 ~ 111101111111111101111111 ~ 11111111111111101111111 ~ 111101111111

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Correct solutions for 1-unique 20-mers

107 random sequence x 107 random sequence

• Seed set: 111111111111;1111101111111 (weight 12)• ~ 2.5 DP/locus

• Seed set (weight 12)

• ~ 1.8 DP/locus

• Seed set: 1111111111111;11111101111111 (weight 13)• ~ 0.56 DP/locus (same specificity as contiguous seed weight 12)

• Seed set (weight 14)• ~ 0.20

1111110111111 x 111111111111 1111110111111

111111001111111

11111111111111 ~ 11111111111111 111110111111111 ~ 11111111111111 111111111110111 ~ 11111111111111 111111111110111 ~ 111111111110111 11111111111111 ~ 111111101111111111110111111111 ~ 111110111111111

Correct solutions for 2-unique 20-mers

• Seed: 111111 (Best single seed solution, weight 6)• ~ 2439 DP/locus

• Weight 10 – 73 DP/locus (specificity of 8/9 contig seed)

• Weight 12 – 10 DP/locus (specificity of 10 contig seed)

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1111111111 ~ 1111111111 11111011111 ~ 1111111111 11111011111 ~ 11111011111 1111100000011111 ~ 1111100000011111 11111000000011111 ~ 1111100000011111 111110000000011111 ~ 1111100000011111 111110000011111 ~ 1111100000011111 111110000000011111 ~ 11111000000000011111

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k-unique human 20-mers

• No 4-unique 20-mers• No 3-unique 20-mers

• 0. 038% of (forward) human 20-mers are 2-unique• 1088322 in total• about 1 every 2638 bases• Fast 2-uniquness oracle

Genome Browser Track

Edit Distance:

•UCSC Track: 1-unique 20-mers

•UCSC Track: 2-unique 20-mers

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Integration with High Performance Computing

• Break sequence into chunks of size 107.• Remember which positions have been eliminated.

• Integrated with (UMIACS) Condor• Too unreliable for very large sequences• NFS filesystem is unreliable• Simultaneous jobs capped at ~ 300

• Integrated with hadoop/map-reduce on 80 nodes (Edwards Lab)• Reliability improved, DFS solves (some) filesystem

woes• Much better scalability (in theory, yet to be tested)• Explicit synchronization of reduce step is undesirable.

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Other improvements

• Other groupings are possible:• Species designation on FASTA defline can be

any suitable partition• Constraints on the position of edits:

• False positive due to mishybridization at 3’ end is unlikely to be observed with some technologies

• Constraint on valid Tm range:• Computed as in Primer3• Can eliminate undesirable mers early

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Conclusions

• Precompute of human k-unique 20-mers is now feasible!• Faster for large edit-distance!• Need spaced seed-set designs

• Constant time update for spaced seeds• Good integer programming formulation of SS-

SDP• Limited template enumeration based on update cost• Work with integer programming experts to solve

effectively