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

2. Polymerase Chain Reaction

3. Polymerase Chain Reaction

4. 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

5. 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

6. 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

7. 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

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

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

10. Brute-force approach
ACTTGTGCACAGTCCTTAAG
ACTTGTGCACAGTCCTTAAG

11. Brute-force approach
ACTTGTGCACAGTCCTTAAG
ACTTGTGCACAGTCCTTAAG

12. Brute-force approach
ACTTGTGCACAGTCCTTAAG
ACTTGTGCACAGTCCTTAAG

13. Brute-force approach
ACTTGTGCACAGTCCTTAAG
ACTTGTGCACAGTCCTTAAG

14. Brute-force approach
ACTTGTGCACAGTCCTTAAG
ACTTGTGCACAGTCCTTAAG

15. Brute-force approach
ACTTGTGCACAGTCCTTAAG
ACTTGTGCACAGTCCTTAAG

16. 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.

17. 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

18. In practice (edit-dist 4)

19. In practice (edit-dist 4)

20. In practice (edit-dist 4)

21. In practice (edit-dist 3)

22. In practice (edit-dist 3)

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

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

25. 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!

26. 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.

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

28. 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!

29. Dirty secret of spaced seeds
Spaced seeds take O(# cares) to update!
Contiguous seeds are O(1) to update vs
8 steps to update vs 1 step to update
Constant time update for spaced seeds?
Yes, if they have a certain structure

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

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

32. TCCCGC-TAGATTGAGATCT ||||||v||||||*|||||| TCCCGCCTAGATTTAGATCT
Edit-distance SS-SDP
Position of matching bases might shift!
Need ↓ to get CCGCTAGA
Need ↑ to get CCGCTAGA
Set cover formulation no longer works
TCCCGC-TAGATTGAGATCT
||||||v||||||*||||||
TCCCGCCTAGATTTAGATCT

33. r:TCCCGC-TAGATTGAGATCT ||||||v||||||*|||||| q:TCCCGCCTAGATTTAGATCT
Edit-Distance SS-SDP
Use a variation on set cover:
q: ,r: 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

34. Solution for (20,2,10) Query Templates:
1: Cost: 1
2: Cost: 5
27: Cost: 5
42: Cost: 5
Text Templates:
32: Cost: 5
37: Cost: 5
Pairs of templates:
1: : Covers: 1274
2: : Covers: 260
2: : Covers: 1218
1: : Covers: 309
42: : Covers: 42
27: : Covers: 319
42: : Covers: 186
27: : Covers: 51
42: : Covers: 287

35. 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
in total
about 1 every 2638 bases
Fast 2-uniquness oracle

36. 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

37. 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

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