1. Whole-Genome Optical Mapping
Michael Waterman
University of Southern California

3. Human Genome Variation
Types of Variation
Substitutions
Insertion/deletions
Duplications
Rearrangements
SNPs: single nucleotide polymorphisms

4. Optical Mapping A single-molecular restriction mapping technology
Developed by D. Schwartz (University of Wisconsin-Madison)

6. Optical Mapping: Overview+
DNA
extract
Silicon bed with embedded grooves
Molecules attached to the surface and straightened within the grooves
Restriction enzymes are added in the solution
DNA is fluorescently dyed and the chip is photographed.
DNA is digested and cuts are formed by shrinking ends

7. DNA Imaging Lambda DNA individual fragments cuts
Estimated sizes of fragments

8. Optical Mapping: Data
Each optical map is represented by an array of DNA sizes in the order they appear on imaged DNA molecules.
Individual maps correspond to different DNA molecules of length Mb.
Each number in the map corresponds to size of the restriction fragment (in Kb) on the molecule.
Order information of restriction fragments is preserved within each map.
Map #1:
Map #2:
...

9. Optical Mapping: Errors
Sizing errors (sizes of individual restriction fragments are measured with errors)
Missing cuts (due to underdigestion)
False cuts (random DNA breaks)
Missing fragments (unable to attach to the surface)
Chimeras (due to concatenation of maps during imaging)

10. Optical Mapping: Pros and Cons
No cloning, no amplification, hence no PCR related errors.
Deep (~100x and more) coverage
Reads span very large portions of chromosomes
~(up to 4Mb).
Cons:
Resolution at the restriction site level
Maps contain many errors

11. Optical Mapping: Goals
Assembly of restriction maps for target organisms (before sequencing)
Variation studies (cancer analysis)
Mapping of methylation patterns
Mapping of transcription factor binding sites

12. Map Making
We are confronted with many relatively short somewhat inaccurate maps and want to piece together a genome map
The problem was approached by a sophisticated statistical sampling model by Mishra et al.
We try another quite simple approach

13. Overview: Assembly for Sequences (Overlap – Layout – Consensus Paradigm)
Genomic region
cloning, sequencing
Overlap
GTTGA
Piles of sequence reads (~600Bp each)
Physical overlaps between reads are captured by means of filtration
GTTGA
ATGATCC
Filtration
ATGATCC
Overlapping sequence reads are put together to produce the scaffold of the reference genomic region (Layout)
Layout
Consensus map is inferred by means of multiple sequence alignment, Euler assembler, etc.
Consensus

14. Assembly for Optical Maps: Overlap-Layout-Consensus
Mutual overlaps are detected by finding similar size patterns
Overlap
Filtration significantly speeds up the computation of overlaps
Filtration
Overlaps are computed according to our new probabilistic score
Layout is produced similar to sequence layout
Consensus is inferred by refinement of the layout (HMM)

15. Assembly for Optical Maps: Overlap Detection
Huge number of false positive overlaps
False negatives (missing overlaps) are not a problem for layout construction
Many optical maps, hence all pairwise overlaps are expensive to calculate ( n(n-1) overlaps, if n optical maps )
Filtration is needed to speed up the search for overlaps

16. Assembly for Optical Maps: Filtration
Filtration is used to find:
Potential overlaps of optical maps
Possible fit locations against the reference
Filtration is based on finding matching tuples of fragments for optical maps:
Matching tuples are calculated to form matching diagonal stretches in the alignment matrix
Matching diagonal stretches in the alignment matrix are chained to find alignments and calculate the score (FASTA idea)
Full dynamic programming is applied for candidate overlaps to calculate the overlap

17. Filtration continued
In sequences overlapping reads are expected to have several matching 20-tuples
In Optical Mapping filtration is challenging because of the sizing error and presence of missing/false cuts

18. Assembly for Optical Maps: Why Things are Hard
Consider a human size genome (3 000 K bp)
Av. rf size 30K (8-cutter), hence 100K restriction fragments in 1 genome
With maps of 33 rf (1 Mbp) there is
1x – 3K maps
100x – 300K maps
pair-wise overlaps
To calculate all pair-wise overlaps:
At the rate of 5 overlaps per second or overlaps per hour
computer hours
4.5 years on the 128 node cluster like hto-g.

19. Alignment Score: Problem Description
Account for features specific for optical mapping:
Sizing error distribution
False cut distribution
Missing cut distribution
Design a score as a –log(LR) for testing: true matching vs. random matching:
True match assumes direct dependence between maps
Random match assumes independence between maps
The optimal alignment has the lowest LR-test value (maximum score)

20. Previous work on the subject
Heuristic alignment score and DP for restriction map alignments (Waterman et al, 1984)
Alignment score and DP for restriction maps with local rearrangements (Huang et al, 1992)
Extensive Bayesian models for map assemblies (Ananthraman et al, 1997)

21. Optical Mapping: Calculation of Alignments
Alignments are computed using standard DP algorithm for map comparison (due to Waterman et al, 1984)
Time complexity: , but can be approximated by a restricted version
is the size of the reference map
is the size of the optical map

22. Optical Mapping: Data Models
Sizing errors (about 10-15% of the fragment size)
Modeled as normal r.v.
for fragments longer than 4 Kb (CLT idea)
for fragments shorter than 4 Kb
About 20% of cuts are missing (80% digestion)
Modeled as Bernoulli r.v.
False cuts occur at the rate of 5 per Mb
Modeled as Poisson Process with rate 0.005

23. Why normal error model?
Fluorescent dye
DNA
Let be the # of photons captured from the i-th base
The total registered fluorescence from the DNA fragment is (n DNA bases)
After applying CLT for an unbiased measurement , since L is proportional to n
Hence for the measurement error

24. Testing the Error model:
Scatter Plot vs
Histogram of
Data collected from 10-mers.

25. Error model: qqnorm
Data collected from 10-mers.

26. Alignment Score: Key Idea
Define two competing hypothesis and :
under maps and are independent (have no similarity)
under maps and are related (e.g. optical map comes from the genomic region )
write the likelihood ratio under and :

27. Alignment Score: Key Idea
define an alignment score as the –log(LR) to make it additive:

28. Two Alignment Types: Fit and Overlap
Fit alignment: to find genomic regions of origin for optical maps
Sizing errors
Reference restriction map
Aligned pairs of sites
Missed cut
False cut
Overlap alignment: to detect overlaps between optical maps
Aligned pairs of sites
Optical maps

29. Optical Mapping: Alignment Scores
Matching regions R R R 2 d
Score = score(R ) + score(R ) score(R ) 1 2 d
Score of the matching region is composed of two parts: score for the sizing error and score for extra/missing cut sites

30. Some Mathematical Facts

31. Fit Alignment Score

32. Overlap Alignment Score

33. Overlap Alignment Score

34. Example of Fit Alignment

35. Comparison of two alignment scores
M1: Our alignment score
M2: Alignment score due to Waterman et al 1984
P-values are consistently smaller for our new score

36. Comparison of two alignment scores
Generate a map from a 40 MB region of HS13.
Verify that optimal score places into a correct genomic location
Examine 19 next best scoring alignments
Study how sparsely populated are the neighborhoods of optimal alignments using M1 and M2 (using std of optimal score)
For our new score (M1): neighborhoods of optimal scores are very sparsely occupied
For the old score (M2): neighborhoods of optimal score are densely occupied

37. Tumor study: analysis of variations
Variations to find:
indels (5Kb or more)
extra or missing restriction cut sites (EC or MC)
Variations are relative to published DNA human sequence (build 35)
Data:
human hematadiform mole
limphoblastoid control (diploid white blood

38. Selected variations By p-value (<0.05) Discovered:
Lymphoblastoid (normal white blood cell), 8x, 63% cov:
131 indel (>5Kb)
491 EC
609 MC
Mole (Haploid tumor), 12x, 93% cov:
728 indels (>5Kb)
394 EC
489 MC

39. Mole: indels
501 out of 728 indels are 5-10 Kb deletions

40. Control: Lymphoblastoid indels:

41. Why such a difference? Hypothesis: L1 line elements:
6-8Kb retrotransposons:
Pop out in mole
Stay in place in normal cells
Hypothesis: EC, MC are due to SNPs at the restriction sites

42. Our Research Group: Michael Waterman (USC) Lei Li (USC) Yi Yang (USC)
Yu-Chi Liu (USC)
Yu Zhang (Harvard)
Anton Valouev (USC)
& Many many thanks to David Schwartz and his Lab (U.Wisconsin)