Whole-Genome
Optical Mapping
Michael Waterman
University of Southern California
Human Genome Variation

Types of Variation
Substitutions
Insertion/deletions
Duplications
Rearrangements
SNPs: single nucleotide polymorphisms
Optical Mapping
A single-molecular restriction mapping
technology
 Developed by D. Schwartz (University of
Wisconsin-Madison)

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
DNA Imaging
Lambda
DNA
individual
fragments
Estimated
sizes of
fragments
cuts
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 0.3-1.5 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:
10.23 54.32 32.43 12.43 9.54 0.45 3.98 2.76 3.45 19.23 27.81 92.12 0.65 4.22
Map #2:
23.12 68.42 28.12 15.43 12.92 32.90 0.34 0.78 5.43 54.22 29.69 27.12 14.23
13.08 0.54 12.35 22.19 1.34
...
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)
Optical Mapping: Pros and Cons

Pros:
 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
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

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

Overview: Assembly for Sequences
(Overlap – Layout – Consensus Paradigm)
Genomic
region
cloning, sequencing
Overlap
Piles of
sequence reads
(~600Bp each)
Physical overlaps
between reads are
captured by means of
filtration
GTTGA
GTTGA
ATGATCC
ATGATCC
Filtration
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
Assembly for Optical Maps:
Overlap-Layout-Consensus
Overlap
Mutual overlaps are detected by
finding similar size patterns
Filtration
Filtration significantly speeds up the
computation of overlaps
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)

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
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
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
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
91010 pair-wise overlaps
To calculate all pair-wise overlaps:



At the rate of 5 overlaps per second or 1.8 10 4 overlaps per hour
510 6 computer hours
4.5 years on the 128 node cluster like hto-g.
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)
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)

Optical Mapping: Calculation of
Alignments


Alignments are computed using standard DP
algorithm for map comparison (due to
Waterman et al, 1984)
Time complexity: m2  n 2, but can be
approximated by a restricted   m n version
n is the size of the reference map
m is the size of the optical map
Optical Mapping: Data Models

Sizing errors   X  Y (about 10-15% of the
fragment size)
 Modeled as normal r.v.
2
  ~ N (0, Y ) for fragments longer than 4 Kb (CLT idea)
2

~
N
(
0
,

) 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
Why normal error model?
Fluorescent dye
DNA


Let X i be the # of photons captured from the i-th base
The total registered fluorescence from the DNA fragment
is L   X (n DNA bases)
After applying CLT for an unbiased measurement Lm,
L ~ N ( L,   n)  N ( L,  L) since L is proportional to n
Hence for the measurement error   ( L  L) ~ N (0,  L)
n
m

i 1
2
i
2
m

2
m
Testing the Error model:
Scatter Plot
X Y
e
~ N (0, 2 )
Y
e vs Y
e
Y , ( Kb)
Data collected from 10-mers.
Histogram of e.
Error model: qqnorm
e
X Y
~ N (0, 2 )
Y
Data collected from 10-mers.
Alignment Score: Key Idea
Define two competing hypothesis
• under
maps
similarity)
and
and
:
are independent (have no
• under
maps and are related (e.g. optical map
comes from the genomic region
)
• write the likelihood ratio under
and
:
Alignment Score: Key Idea
• define an alignment score as the –log(LR) to make it
additive:
Two Alignment Types: Fit and Overlap

Fit alignment: to find genomic regions of origin for
optical maps
Sizing errors
Aligned pairs
of sites

Missed cut
Reference
restriction map
False cut
Overlap alignment: to detect overlaps between optical
maps
Aligned pairs
of sites
Optical maps
Optical Mapping: Alignment Scores
Matching
regions
R
Score =

R2
score(R1) + score(R2) +
Rd
...
+
score(R d )
Score of the matching region is composed of two parts:
score for the sizing error and score for extra/missing cut
sites
score( Rk )  score( sizek )  score(cutsk )
Some Mathematical Facts
Fit Alignment Score
Overlap Alignment Score
Overlap Alignment Score
Example of Fit Alignment
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

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
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 (haploid, @12x)
 limphoblastoid control (diploid white blood cells, @8x)
Selected variations


By p-value (<0.05)
Discovered:
 Mole
(Haploid
tumor), 12x, 93%
cov:
728 indels (>5Kb)
 394 EC
 489 MC

 Lymphoblastoid
(normal
white blood cell), 8x,
63% cov:
131 indel (>5Kb)
 491 EC
 609 MC

Mole: indels
501 out of 728 indels are
5-10 Kb deletions
Mole indels:
600
500
300
Frequency
200
100
Bin
50
40
30
20
10
0
-1
0
-2
0
-3
0
-4
0
0
-5
0
Frequency
400
Control: Lymphoblastoid indels:
Lymphoblastoid indels:
45
40
35
25
Frequency
20
15
10
5
Bin
More
40
35
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
-35
0
-40
Frequency
30
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
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)