Precomputing Edit-Distance Specificity of Short Oligonucleotides Nathan Edwards Center for Bioinformatics and Computational Biology University of Maryland, College Park

Polymerase Chain Reaction

Polymerase Chain Reaction

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

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

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

Inexact sequence match 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)) g ≠ = ≠ q

Example worst case alignments TCCCGC-TAGATTGAGATCT ||||||v||||||*|||||| TCCCGCCTAGATTTAGATCT ACTTGTCCACAGTGCTTAAG ||||||*||||||*|||||| ACTTGTGCACAGTCCTTAAG

Brute-force approach ACTTGTGCACAGTCCTTAAG 2-mer position table AA:18

Brute-force approach ACTTGTGCACAGTCCTTAAG ACTTGTGCACAGTCCTTAAG

Brute-force approach ACTTGTGCACAGTCCTTAAG ACTTGTGCACAGTCCTTAAG

Brute-force approach ACTTGTGCACAGTCCTTAAG ACTTGTGCACAGTCCTTAAG

Brute-force approach ACTTGTGCACAGTCCTTAAG ACTTGTGCACAGTCCTTAAG

Brute-force approach ACTTGTGCACAGTCCTTAAG ACTTGTGCACAGTCCTTAAG

Brute-force approach ACTTGTGCACAGTCCTTAAG ACTTGTGCACAGTCCTTAAG

Brute-force approach Divide the genome into 10 Mb blocks For all pairs of blocks: For all l-mer matches: Do all pair-wise DPs containing match If ≤ k edits, mark position non-unique 300 x 300 pairs of blocks For 20-mers: k=1 ) l=10; k=2 ) l=6; k=3 ) l=5 ; k=4 ) l=4.

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

In practice (edit-dist 4)

In practice (edit-dist 4)

In practice (edit-dist 4)

In practice (edit-dist 3)

In practice (edit-dist 3)

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

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

Edit distance 2 After seed size 12 After seed size 8 ~ 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!

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.

Solution for (20,2,8) 11111111, 111101111 TCCCGCGTAGATTGAGATCT ||||||*||||||*|||||| TCCCGCCTAGATTTAGATCT How can we find these spaced seed set solutions?

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!

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

O(1) spaced seed update ACGTACGTACGTACGTACGT A G A G C T C T G A G A T C T C ... Spaced seed 1010101 can be updated in 1 step!

O(1) spaced seed update “Periodic” spaced seeds can be updated in “constant” time 11011011011 2 steps 11001100110011 2 steps 1000010000100001 1 step Need to minimize the number of update steps, not the number of templates 11111111,111101111 has update cost 5.

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

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

Solution for (20,2,10) Query Templates: 1: 11111111110000000000 Cost: 1 2: 11111011111000000000 Cost: 5 27: 11111000001111100000 Cost: 5 42: 11111000000001111100 Cost: 5 Text Templates: 32: 11111000000111110000 Cost: 5 37: 11111000000011111000 Cost: 5 Pairs of templates: 1: 11111111110000000000 1: 11111111110000000000 Covers: 1274 2: 11111011111000000000 1: 11111111110000000000 Covers: 260 2: 11111011111000000000 2: 11111011111000000000 Covers: 1218 1: 11111111110000000000 2: 11111011111000000000 Covers: 309 42: 11111000000001111100 32: 11111000000111110000 Covers: 42 27: 11111000001111100000 32: 11111000000111110000 Covers: 319 42: 11111000000001111100 37: 11111000000011111000 Covers: 186 27: 11111000001111100000 37: 11111000000011111000 Covers: 51 42: 11111000000001111100 42: 11111000000001111100 Covers: 287

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

F. tularensis 20-mer signatures Exact match in all six strains No match to bacterial background at edit-distance k No 3-unique 20-mer signatures 263 2-unique 20-mer signatures 0.013% 1.3M 20-mer signatures (no background check) 1.2M 0-unique 20-mer signatures 580K 1-unique 20-mer signatures

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

Next Steps Publish! Adapt for Tm and/or hybridization model Convert to native BOINC-application Integrate with primer-design software