Physical Mapping

1

• Physical Mapping
• --An Algorithm and An Approximation for
  Hybridization Mapping
• Shi Chen
• CSE497
• 04Mar2004

2
Introduction

• Why physical mapping?
• -Physical mapping is a central in Molecular
  Biology.
• -DNA is cut into small fragments for replicate
  and study, and information on the ordering is
  lost.
• -The goal of physical mapping is to reconstruct
  the relative ordering of the clones.

3
Introduction

• Two Popular ways of obtaining fingerprints
• Restriction site analysis.
• Measure fragments length which is its
  fingerprint.
• Hybridization.
• Check whether a small sequence known as a probe
  binds or hybridizes to the clone which is DNA
  fragment.
• Most often a probe is a STS (sequence tagged
  sites) DNA string of 200-300 bp whose ends
  occur only once in the entire genome.
  

4
Models for Hybridization Mapping

• -Interval Graph Models
• Vertices represent clones and edges represent
  overlap information between clones.
• -Disadvantage complexity NP-hard.

5
Models for Hybridization Mapping-C1P definition

• Definition
• A binary matrix is said to have the
• consecutive ones property (C1P)
• if a permutation of its columns can be found such
  that all 1s in each row are consecutive.

A B C D
1 1 0 0 1
2 0 1 0 1
3 1 0 1 0
C A D B
1 0 1 1 0
2 0 0 1 1
3 1 1 0 0
6
Models for Hybridization Mapping C1P

• Assumptions for Consecutive Ones Property (C1P)
  Model
• a. Probes are unique a probe can bind to a
  clone in at most one place use STS (sequence
  tagged sites)
• b. No errors (C1P permutation exists)
• c. All clonesprobes hybridization experiments
  have been done difficult to achieve.
• Advantage
• Polynomial-time solvable.

7
Models for Hybridization Mapping C1P model

• n clones and m probes
• n m binary matrix M built from experimental
  data
• Mij 1 probe j hybridized to clone i
• Mij 0 probe j not hybridized to clone i

c1 c2 c3 c4 c5 c6 c7 c8 c9
l1 1 1 0 1 1 0 1 0 1
l2 0 1 1 1 1 1 1 1 1
l3 0 1 0 1 1 0 1 0 1
l4 0 0 1 0 0 0 0 1 0
l5 0 0 1 0 0 1 0 0 0
l6 0 0 0 1 0 0 1 0 0
l7 0 1 0 0 0 0 1 0 0
l8 0 0 0 1 1 0 0 0 1
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Algorithm for C1P - Introduction

• Goal Find a permutation of the columns such
  that in each row all 1s are consecutive.
• Assumptions
• All rows are different, i.e. no two clones have
  the same fingerprint.
• No row is all zeros, i.e. every clone is
  hybridized by at least one probe.

9
Algorithm for C1P Algorithm sketch

• Separation of the rows into components (subsets
  of rows).
• Permutation of the columns of each component.
• Join of the components together.

10
Algorithm for C1P Row relations
Definition " row iÎM, Sicolumns k Mi,k1

• Given two rows i and j
• Si Ç Sj Æ or
• Si Í Sj or Sj Í Si or
• Si Ç Sj ¹ Æ and none is a subset of the other.

• First case i and j have no conflicts - they can
  be dealt with separately.
• Second case i and j are compatible - any
  solution for the row with
• fewer 1s is acceptable.
• Third case i and j have to be treated
  simultaneously - they are
• connected.

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Algorithm for C1PTaking care of a component
ß
c1 c2 c3 c4 c5 c6 c7 c8 c9
l1 1 1 0 1 1 0 1 0 1
l2 0 1 1 1 1 1 1 1 1
l3 0 1 0 1 1 0 1 0 1
l4 0 0 1 0 0 0 0 1 0
l5 0 0 1 0 0 1 0 0 0
l6 0 0 0 1 0 0 1 0 0
l7 0 1 0 0 0 0 1 0 0
l8 0 0 0 1 1 0 0 0 1

• l3

• l2

a

• l4

• l1

?

• l5

d

• l8

• l6

• l7

TABLE 5.1 A binary matrix.
Figure 5.7 Graph Gc corresponding to the matrix
of Table 5.1
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Algorithm for C1PExample Matrix
A section of a binary matrix
c1 c2 c3 c4 c5 c6 c7 c8
l1 0 1 0 0 0 0 1 1
l2 0 1 0 0 1 0 1 0
l3 1 0 0 1 0 0 1 1
l1
l2
l3
2,7,8 2,7,82,7,8
l1? 0 1 1
1 0
5 2,7 2,7 8
l1? 0 0 1 1 1 0
l2? 0 1 1 1 0 0
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Algorithm for C1PExample Matrix
c1 c2 c3 c4 c5 c6 c7 c8
l1 0 1 0 0 0 0 1 1
l2 0 1 0 0 1 0 1 0
l3 1 0 0 1 0 0 1 1
l1
l2
l3
What will happen if we place 5 on the right?
8 7,2 7,2 5
l1? 0 1 1 1 0 0
l2? 0 0 1 1 1 0
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Algorithm for C1PExample Matrix
c1 c2 c3 c4 c5 c6 c7 c8
l1 0 1 0 0 0 0 1 1
l2 0 1 0 0 1 0 1 0
l3 1 0 0 1 0 0 1 1
l1
l2
l3

• How to place l3?
• Consider the number of elements in the
  intersections between S1, S2 and S3.
• Definition Let xy SxnSy be the internal
  product of rows x and y.
• -If l1l3 lt min(l1l2, l2l3), place l3 in the
  same direction that l2 was placed with respect to
  l1.
• -If l1l3 gt min(l1l2, l2l3), place l3 in the
  opposite direction that l2 was placed with
  respect to l1.

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Algorithm for C1PExample Matrix
c1 c2 c3 c4 c5 c6 c7 c8
l1 0 1 0 0 0 0 1 1
l2 0 1 0 0 1 0 1 0
l3 1 0 0 1 0 0 1 1
l1
l2
l3

• In our case
• S3 1,4,7,8, Then
• l1l3 2, l1l2 2, l3l2 1.
• So, place l3 to the right of l2.

52781,41,4 l1? 0
0 1 1 1 0 0 0 l2? 0 1
1 1 0 0 0 0 l3? 0 0 0
1 1 1 1 0
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Algorithm for C1PComplexity

• Building Graph Gc takes O(nm) time.
• Process n rows, spending O(m) per row to check
  consistency of column sets.
• Total time is O(nm).

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Algorithm for C1PJoining Components Together
c1 c2 c3 c4 c5 c6 c7 c8 c9
l1 1 1 0 1 1 0 1 0 1
l2 0 1 1 1 1 1 1 1 1
l3 0 1 0 1 1 0 1 0 1
l4 0 0 1 0 0 0 0 1 0
l5 0 0 1 0 0 1 0 0 0
l6 0 0 0 1 0 0 1 0 0
l7 0 1 0 0 0 0 1 0 0
l8 0 0 0 1 1 0 0 0 1
ß
a
?
d
Figure 5.9 Graph GM corresponding to the
components of the matrix from Table 5.1.
TABLE 5.1 A binary matrix.
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Algorithm for C1PJoining Components Together

• Process GM in topological ordering
• -Process first components that have sets that are
  not contained anywhere else.
• -Suppose following edge (a,ß), find reference
  column in component athat will tell us how to
  place the rows of ß.
• a. Choose row l fromßthat has the leftmost 1,
  and call the column where this 1 is cß.
• b. Find all rows fromathat contain Sl, and
  find the leftmost column where all such rows have
  1s, this column cais the reference column.

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Algorithm for C1PJoining Components Together

• 1 2,4,5,7,9 3,6,8
• l1? 1 1 1 1 1 0 0 0
• l2? 0 1 1 1 1 1 1 1

a
2,4,5,7,9 l3? 1 1 1 1 1
ß
1 2,4,5,7,9 3,6,8 l1? 1 1 1
1 1 1 0 0 0 l2? 0 1 1 1 1 1 1 1 1
l3? 0 1 1 1 1 1 0 0 0
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Algorithm for C1PJoining Components Together

9,5 4 7 2 l6? 0 0
1 1 0 l7? 0 0 0 1 1
l8? 1 1 1 0 0
d
1 9,5 4 7 2 3,6,8
l1? 1 1 1 1 1 1 0 0 0
l2? 0 1 1 1 1 1 1 1 1
l3? 0 1 1 1 1 1 0 0 0
l6? 0 0 0 1 1 0 0 0 0
l7? 0 0 0 0 1 1 0 0 0
l8? 0 1 1 1 0 0 0 0 0
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Algorithm for C1PJoining Components Together
6 3 8 l4? 0 1 1
l5? 1 1 0
?
1 9,5 4 7 2 6 3
8 l1? 1 1 1 1 1 1 0
0 0 l2? 0 1 1 1 1 1
1 1 1 l3? 0 1 1 1 1
1 0 0 0 l6? 0 0 0 1
1 0 0 0 0 l7? 0 0 0 0
1 1 0 0 0 l8? 0 1 1
1 0 0 0 0 0 l4? 0
0 0 0 0 0 0 1 1 l5?
0 0 0 0 0 0 1 1 0
a
ß
d
?
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Algorithm for C1PJoining Components Together

• Complexity
• Topological sorting O(nm)
• Preprocessing takes at most O(nm),
• e.g. store for each row the column where its
  leftmost 1 is
• Total time O(nm).

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Approximation for Hybridization Mapping with
Errors

• 0 1 1 0 1 1 1 1 0 0
• a false negation
• separate two blocks of 1s, creating another gap
• Approach find a permutation where the total
  number of gaps in the matrix is minimum.

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Approximation - Graph Model
Gap minimization is equivalent to solving
traveling salesman problem (TSP).
TABLE 5.3 A clonesprobes matrix with added
column p6.
p1 p2 p3 p4 p5 P6
c1 1 1 1 0 0 0
c2 0 1 1 1 0 0
c3 1 0 0 1 1 0
c4 1 1 1 1 0 0
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Approximation - Graph Model
p1 p2 p3 p4 p5 P6
c1 1 1 1 0 0 0
c2 0 1 1 1 0 0
c3 1 0 0 1 1 0
c4 1 1 1 1 0 0
p1
3
2
2
2
3
P6
p2
2
1
0
3
4
p5
p3
The weight on each edge of G is the number of
rows where the two corresponding columns differ.
4
2
3
2
2
p4
FIGURE 5.10 TSP graph for matrix of Table 5.3.
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Approximation - Graph Model

• a gap a transition from 1 to 0 and further on a
  transition from 0 to 1.
• -two transitions for each gap, each gap
  contributes 2 to the weight of the cycle.
• extremal transitions transitions between
  elements in extremal (1 or m) column.
• -include an extra column of zeros in column
  m1 to ensure every row has a pair of extremal
  transitions. prevent consecutive 1s to wrap
  around in each row.

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Approximation - Graph Model

• -Relationship between cycles and permutations
• Cycle weight number of gap transitions 2n
• For a given n, minimizing cycle weight is the
  same as minimizing the number of gaps.
• -Drawback one or a few rows may have many gaps,
  while others may have none. One clone was subject
  to many more errors than other clones, and this
  contradicts laboratory experience.
• -Solution minimizing the number of gaps per row.

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Approximation - Guarantee
-Assumptions a. The number of probes is
sufficiently large. b. The mapping process obeys
a certain mathematical model. -Features a. Each
clones position is an independent random
variable, clone locators are distributed
uniformly over 0, N-1. b. Occurrences of a
given probe obey a Poisson process with rate ?.
Pra given probe occurs k times in a given
clone e-??k/k!.
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Approximation - Guarantee
TSP permutation is a good approximation to the
true permutation. Prove in terms of graph
weights or clone distances. tij lj li
rj-ri 2lj-li tij true distance clones
coordinates l (left), r (right) hij Hamming
distance between clones i and j. Given any four
clones i, j, r, and s, hij lt hrs implies tij lt
trs tij lt trs implies hij lt hrs.
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Approximation Computational Practice

• Define hybridization graph H as a bipartite graph
  (U, V, E)
• Clones are the vertices of the U partition
• Probes are the vertices of the V partition
• There is an edge between two vertices if the
  corresponding probe hybridized to the
  corresponding clone.

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Approximation Computational Practice
TABLE 5.3 A clonesprobes matrix with added
column p6.
p1
p2
p3
p4
p5
p1 p2 p3 p4 p5 P6
c1 1 1 1 0 0 0
c2 0 1 1 1 0 0
c3 1 0 0 1 1 0
c4 1 1 1 1 0 0
c1
c2
c3
c4
FIGURE 5.11 Hybridization graph H corresponding
to hybridization matrix from Table 5.3, without
the added column.
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Approximation Computational Practice

• Observations
• a. H may not be connected, not be able to tell
  the relative order between probes that belong to
  different components.
• b. Connected component may be as simple as a
  singleton vertex. No hybridization - 0 in Column.
• c. Redundant probes, or probes that hybridize to
  exactly the same set of clones - same 1s and 0s
  in columns.

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Approximation Computational Practice

• Evaluation of a mapping algorithm is a difficult
  task. The fraction of strong adjacencies is used
  to measure a mapping algorithm.
• -Strong adjacencies the number b of blocks of
  consecutive 1s present in a hybridization matrix
  with a given probe permutation p p1, p2, , pm.
• -Translocations operations that reverse the
  order of a set of consecutive probes.
• Two adjacent probes pi and pi1 represent a
  strong adjacency if placing these probes apart by
  any translocation increases b in each row.

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Approximation Computational Practice
Strong adjacency cost 100(1/m-1?di) di 1, if
pi and pi1 is a strong adjacency in the true
permutation but these probes are not adjacent in
the proposed permutation. d i 0, otherwise.

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Approximation Computational Practice
TABLE 5.4 Strong adjacency costs for two
algorithms on matrices with different kinds of
errors. Error rates are indicates in the heading
of each column (only one type of error per
column). Coverage in all cases is 10, where
coverage is the ratio between the total length
of all clones and target DNA length.
C1P 0 Chimerism 0.5 False Positives 0.04 False Negatives 0.32
Greedy TSP 1.9 0.9 16.0 28.3
Random 86.4 89.7 94.4 94.9
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REFERENCES 1. Sections 5.3 and 5.4 in our
textbook Introduction to Computational
Molecualar Biology, Setubal/Meidanis, 1997. 2. On
the Complexity of DNA Physical Mapping, Martin
Charles Golumbic, Haim Kaplan and Ron Shamir,
Advances in Applied Mathematics 15, 251-261
(1994).
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