Genome Rearrangement SORTING BY REVERSALS

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Genome RearrangementSORTING BY REVERSALS
Ankur Jain Hoda Mokhtar CS290I SPRING 2003
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Comparative Genomics
The practice of analyzing and comparing the
genetic material of different species for the
purpose of studying evolution, the function of
genes and inherited diseases. Chromosome
breakage and mistakes in repair, along with a
number of other processes, give rise to changes
in gene order.  These have important
consequences for the evolution of species. 
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Problem Definition

• During biological evolution, inter- and
  intra-chromosomal  exchanges of chromosomal
  fragments disrupt the order of genes on a 
  chromosome.
• The genome rearrangements approach, is the use
  of  combinatorial optimization techniques, to
  infer a sequence of rearrangement events to
  account for the differences among the  genomes. 

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Outline

• Problem definition
• Genome Comparison
• Possible chromosomal changes
• Sorting by reversals - Previous work
• - Definitions
• - Duality Theorem
• Our technique - Bit Vector Method
• - Experimental results - Synthetic
  datasets
• - Real datasets
• - Breakpoints Technique
• Conclusions and Future work

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Genome Comparison

• In the late 1980 was discovered remarkable and
  novel
• pattern of evolutionary change in plant
  organelles.
• Jeffrey Palmer and his collegues compared the
  mitochondrial genomes of cabbage and turnip,
  which are very closely related. Molecules which
  are almost identical in gene sequences, differ
  dramatically in gene order. Sridhar, Pevzner
  1995
• This discovery and many other studies proved that
  genome rearrangements represent a common mode of
  molecular evolution.

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Cabbage and Turnip
Gene orientation
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Single Chromosome Operations

• Reversal A section of a chromosome is excised,
  reversed in orientation, and re-inserted.
• (abc1c2c3c4de -gt ab-c4-c3-c2-c1de)
• Transposition A section of a chromosome is
  excised and inserted at new position in the
  chromosome, without changing orientation. (abcd
  -gt cdab)
• Inverted transposition Exactly like
  transposition, except that the transposed segment
  changes orientation. (abcd -gt -c-dab)
• Gene duplication A section of a chromosome is
  duplicated, so that multiple copies exist of
  every gene in that section.
• (abc -gt abcb, abc -gt abbc)
• Gene loss A section of a chromosome is excised
  and lost.
• (abc-gtac )

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Operations on 2 Chromosomes

• Translocation The end of one chromosome is
  broken and attached to the end of another
  chromosome.
• Fusion two chromosomes merge.
• Fission one chromosome splits up into two
  chromosomes.

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Genomic Sorting Problem

• Given genomes the genomic
  sorting problem is to find a series of reversals
  where and
  t is minimal.
• We call t the genomic distance between
  

and
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Sorting by Reversals

• Genome rearrangements can be modelled by a
  combinatorical problem of sorting by reversals.

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Sorting by Reversals (Cont.)
Minimum Sorting by Reversals Given a permutation
?, what is the shortest sequence (?1?2.?t ) of
reversals that sorts ? ?Complexity remains
open. (NP-Hard) Caprara 97 Minimum Signed
Sorting by Reversals Given a signed permutation
?, what is the shortest sequence (?1?2.?t ) of
reversals that sorts ?? ?Solvable in polynomial
time.
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Sorting of Signed Permutations

• Transforming cabbage into turnip. Hannenhalli,
  S., and Pevzner, P. 95 - Polynomial algorithm
  for sorting signed permutations by reversals

• A Very Elementray Presentation of the
  Hannenhalli-Pevzner Theory, A. Bergeron95
  Polynomial algorithm for sorting signed
  permutations, efficiently implemented using bit
  vectors.

• A Very Elementray Presentation of the
  Hannenhalli-Pevzner Theory, A. Bergeron95
  Polynomial algorithm for sorting signed
  permutations, efficiently implemented using bit
  vectors.

• Experiments in Computing Sequences of Reversals,
  A. Bergeron and F. Strasbourg95 Polynomial
  algorithm for sorting signed permutations.

• Fast Sorting by Reversal, Berman, P.,
  Hannenhalli, S. 96. - exploit a few
  combinatorial properties of the cycle graph of a
  permutation and provided a polynomial algorithm.

• A Faster and Simpler Algorithm for Sorting Signed
  Permutations by Reversals, Kaplan, H., Shamir,
  R., and Tarjan, R. 99. O(n2) using hurdles,
  cycles and fortress.

• A Linear-Time Algorithm for Computing Inversion
  Distance between Signed Permutations with an
  Experimental Study, Moret, and Yan 00 -
  Computes reversal distance (without actually
  sorting) in O(n) time. Computes the connected
  components using stack rather than Union-Find.
  Hannenhalli-Pevzner 96 (GRAPPA program)

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Outline

• Problem definition
• Genome Comparison
• Possible chromosomal changes
• Sorting by reversals - Previous work
• - Definitions
• Our technique - Bit Vector Method
• Experimental results - Synthetic
  datasets
• - Real datasets
• - Breakpoints Technique
• Conclusions and Future work

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What is a Permutation?

• Permutation (?) an ordered arrangement of the
  set 1,2,,n
• Signed Permutation (?) a permutation where the
  elements are oriented a reversal switches element
  orientation
• 3 -4 7 -6 1 -5 2
• ?(7,-5) 3 -4 5 -1 6 -7 2

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BreakPoint

• Let i j if i j 1. Extend
  permutation
  
  by adding
  0 and n 1.
• We call pair of elements ,
  0 i n,
• of an adjacency if
• and a breakpoint if is not (
  )

0
n1


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What is breakpoint graph?
The breakpoint graph of a permutation
is a
edge-colored graph
with 2n2 vertices
by a black edge
We join vertices
and
and
by a gray edge if
We join vertices
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Breakpoint graph signed case
Straight edges every other pair of consecutive
elements Curved edges - every other pair of
consecutive integers
Every connected component of the graph is a cycle
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Correlation between the breakpoints and reversal
distance

• Correlations exists between the reversal distance
  and the number of breakpoints
• Sorting by reversals corresponds to eliminating
  breakpoints
• Every resersal can eliminate at most 2
  breakpoints

Shamir, 95
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Outline

• Problem definition
• Genome Comparison
• Possible chromosomal changes
• Sorting by reversals - Previous work
• - Definitions
• - Duality Theorem (Hurdles !!)
• Our technique - Vector-Method
• Experimental results - Synthetic
  datasets
• - Real datasets
• -Breakpoints Technique
• Conclusions and Future work

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Hurdle
Hurdle - an unoriented component whose elements
are consecutive Simple hurdle - a hurdle whose
deletion decreases the number of hurdles Super
hurdles - hurdles that are not simple
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Duality Theorem for Sorting Signed Permutations
Hannenhalli and Pevzner, 1995.
For every signed permutation
if
is a fortress

otherwise
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Safe reversal
C3, h1
C 5, h 2
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Outline

• Problem definition
• Genome Comparison
• Possible chromosomal changes
• Sorting by reversals - Previous work
• - Definitions
• - Duality Theorem (Hurdles !!)
• Our technique - Bit Vector Method
• Experimental results - Synthetic
  datasets
• - Real datasets
• - Breakpoints Technique
• Conclusions and Future work

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Our Approach

• Finding hurdles and fortresses in a graph are
  difficult and expensive Kaplan, H., Shamir, R.,
  and Tarjan, R. 99.
• Use oriented sort to remove the oriented
  components in a graph and then apply the
  breakpoint approach to perform the remaining
  reversals
• We used the bit-vector approach to perform the
  oriented sort

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Oriented Sort

• Choose among the several candidates, a
• safe reversal, that is a reversal that decreases
  the reversal distance.
• Theorem The reversal that maximizes the number
  of oriented vertices is safe A. Bergeron95

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Basic Sorting oriented pair

• An oriented pair is a pair of consecutive
  integers, that is
• with opposite signs
• Example
• (0 3 1 6 5 -2 -4 7)
• Oriented pairs are (1,-2) , (3, -4)

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Reversal score
The number of oriented pairs in the resulting
permutation as a result of a reversal Example
( 0 3 1 6 5 -2 4 7 )
( 0 3 1 6 5 -2 4 7 )
( 0 3 1 6 5 -2 4 7 )
(1, -2)
(3, -2)
( 0 -5 -6 -1 -3 -2 4 7 ) ( 0 3 1 2
-5 -6 4 7 )
Score 4
Score 2
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Algorithm

• As long as has an oriented pair choose the
  oriented reversal that has maximal score
• (0 3 1 6 5 2 4 7)

( 0 -5 -6 -1 -3 -2 4 7 ) (-3, 4)
( 0 -5 -6 -1 2 3 4 7 ) (-1,2)
( 0 -5 -6 1 2 3 4 7 ) (-6,7)
( 0 -5 -4 -3 -2 -1 6 7 ) (-5,6)
( 0 1 2 3 4 5 6 7 )
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Oriented edge
Let
be a gray edge incident to
black edges
and
Then
.
i k j - l
is oriented if and only if
Edge 20-21 is oriented (contains 3 odd number
of vertices). I 20, j21, k22, l23 I-k -2
j-l -2
Bergeron
Pevzner
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Oriented reversals

• Reversals induces by an oriented pair will be

, and
, if
, if
Reversals that create consecutive integers are
always induced by oriented pairs. Such
reversals are called oriented reversal.
Example The pair (1, -2) induces the
reversal (0 3 1 6 5 2 4 7) (0 3 1 2 5 6 4
7)
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Interleaving Graph
C
Every 2 components are adjacent if there is an
overlap between them but neither of them
contains the other.
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Constructing the Bit Matrix
Consider the sequence P 3 1 6 5 2 4
7 Represent Pi by
2i-1, 2i if Pi is ve and
2i, 2i-1 otherwise Pi is -ve 3 1
6 5 -2 4 7 0 5
6 1 2 11 12 9 10 4 3 7 8 13 14 15
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The Algorithm
Step 1. Select the vertex vi with the maximum
score and perform the these operations until we
reach a situation when parity of all the vertices
is zero Step 2. If the sequence is not sorted
completely apply the breakpoint technique to
complete the sorting
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Outline

• Problem definition
• Genome Comparison
• Possible chromosomal changes
• Sorting by reversals - Previous work
• - Definitions
• - Duality Theorem (Hurdles !!)
• Our technique - Bit Vector Method
• Experimental results - Synthetic
  datasets
• - Real datasets
• - Breakpoints Technique
• Conclusions and Future work

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Experimental Settings

• 1- Synthetic Datasets
• generated random signed permutation of different
  lengths and evolution rate using GRIMM
  permutation generation module
• 2- Real Datasets
• Used GRAPPA test sets for different species of
  Campanulaceae (flower plant)
• MGR (multiple genome rearrangement) human-mouse
  gene order data
• Genome.org Herpes Virus that affects human

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Experiment 1 - Synthetic
1- Generated files of random permutations of
different lengths (50, 100, 200, 400, 800, 1600)
each file with 50 permutations. 2- We computed
the number of correctly sorted permutations. 3-
Evolution rate varies 20,30,40
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Experiment 2 - Synthetic
1- Generated files of random permutations of
different lengths (50, 100, 200, 400, 800, 1600)
each file with 50 permutations. 2- We computed
the time needed to obtain the correctly sorted
permutations. 3- Evolution rate varies
20,30,40
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Experiment 3 - Synthetic
1- Generated files of random permutations of
length 1000 2- We computed the time needed to
obtain the correctly sorted permutations. 3-
Evolution rate varies in increments of 100.
Observation Saturation state is reached as
evolution rate approaches 1000
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Experiment 1 - Real
Considered Herpes simplex virus (HSV),
Epstein-Barr virus (EBV), and Cytomegalovirus
(CMV) gene orders (Hannenhalli et al. 1995) as
well as the identity gene order
(A) ObservationsOur reversal results matched
those obtained in optimal evolutionary scenario
recovered by MGR-MEDIAN.
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Experiment 2 - Real
1- Considered Campanulaceae species 2- Obtained
reversals for Cyanathus (11 reversals), Triodanus
(13 reversals), and Symphanra (12 reversals)
versus Tobacco but failed to sort Platyncodon,
Legousia and Codonopsis Observation The ones we
sorted were sorted with same number of reversals
as GRIMM
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Experiment 3 - Real
1- Considered Human-Mouse gene order from MGR
12 13 14 15 -9 -8 -7 -6 47 48 -46 -45 -44 -11 -10
-58 -57 -56 92 93 -95 -94 -21 -20 -5 -4 -3 -2 -1
34 35 41 42 43 36 37 38 -64 -63 61 62 65 66 67 68
90 91 -55 -54 51 52 53 39 40 -60 -59 -77 -76 -19
-18 16 17 -97 -96 -75 -74 -73 24 25 78 79 -83 -82
-81 -80 84 85 86 87 -28 -27 -26 22 23 98 99 69 70
-72 -71 -33 -32 -31 -30 -29 88 89 -50 -49 -105
-104 106 107 108 114 115 -117 -116 -103 -102 109
110 111 112 113 -101 -100 118 119 120 121 122 123
(mouse genome and human is identity)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
35 36 37 38 71 72 73 74 75 76 77 78 79 80 81 82
83 84 85 86 87 88 89 90 91 92 93 39 40 41 42 43
44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69 70 94 95 96 97 98
99 100 101 102 103 104 105 106 107 108 109 110
111 112 113 114 115 116 117 118 119 120 121 122
123 124
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
35 36 37 38 71 72 73 74 75 76 77 78 79 80 81 82
83 84 85 86 87 88 89 90 91 92 93 39 40 41 42 43
44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69 70 94 95 96 97 98
99 100 101 102 103 104 105 106 107 108 109 110
111 112 113 114 115 116 117 118 119 120 121 122
123 124
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
35 36 37 38 71 72 73 74 75 76 77 78 79 80 81 82
83 84 85 86 87 88 89 90 91 92 93 39 40 41 42 43
44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69 70 94 95 96 97 98
99 100 101 102 103 104 105 106 107 108 109 110
111 112 113 114 115 116 117 118 119 120 121 122
123 124
Identity
GRIMM sorts the permutation in 41 reversals
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Conclusions

• We implemented a technique that integrates the
  bit-matrix oriented sorting technique together
  with the greedy breakpoint reversal technique.
• The technique proposed was tested on both real
  and synthetic data and was able to sort signed
  permutations in a fair number of the test data
• We think that such integration can yield good
  results beside being a simple and relatively fast
  technique
• However, the oriented sort algorithm fails to
  sort permutations that have hurdles, in those
  cases we have to apply the breakpoint approach

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Future Work

• We really think that the technique we
  implemented can provide good results, we think
  that further experiments can strengthen our claim
• We started implementing the algorithm proposed
  in Kaplan, H., Shamir, R., and Tarjan R. 99 but
  didnt succeed to complete the implementation. We
  think that having this technique implemented
  under that same conditions as ours can provide a
  good source of comparative results, and can give
  a better confidence about what we propose.
• Applying the technique in different datasets
  including exon order rather than gene order
• Considering different species and trying to
  compute reversal distance and use it to confirm
  phylogenetic trees

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Oriented Pairs
(0 )

• An oriented pair ( , ) is a pair of
  consecutive integers, that is
• with opposite signs
• Example
• (0 3 1 6 5 2 4 7)
• Oriented pairs are

(1,-2)
(3, 2)
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Reversal Distance Estimation
This reversal distance is very in-accurate.
Bafna and Pevzner, 1996 showed that another
hidden parameter hurdles estimated reversal
distance with much greater accuracy.
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Proper reversal
For every permutation
and reversal
Given an arbitary reversal
denote
(increase in the size of cycle decomposition)
Then for every permutation
and reversal
We call reversal proper if
1
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Oriented pairs

• Oriented pairs are useful because they indicate
  reversals that create consecutive elements of the
  permutation.
• Example
• The pair (1, -2) induces the reversal
• (0 3 1 6 5 2 4 7)
• (0 3 1 2 5 6 4 7)