Genome Rearrangement

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Genome Rearrangement

• By
• Ghada Badr
• Part I

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Genome, chromosome, gene, gene order

• The entire complement of genetic material
  carried by an individual is called the genome.
• Each genome contains one or more DNA molecules,
  one per chromosome

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(No Transcript)
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Genome, chromosome, gene, gene order

• A gene is a segment of DNA sequence with a
  specific function

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Genome, chromosome, gene, gene order
A
C
D
F
5
3
3
5
B
E

• Genes can be ordered by their DNA sequence
  location.
• DNA consists of two complementary strands
  twisted around each other to form a right-handed
  double helix.
• A sign (/-) is usually used to indicate on
  which strand a gene is located.

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Genome, chromosome, gene, gene order
A
B
C
D
F
E
J
The DNA molecule (chromosome) may be circular or
linear
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Genome Rearrangement

• The genome is structurally specific to each
  species, and it changes only slowly over time.
  Therefore genome comparison among different
  species can provide us with much evidence about
  evolution.
• Genome rearrangements are an important aspect of
  the evolution of species. Even when the gene
  content of two genomes is almost identical, gene
  order can be quite different.

A
-B
C
D
F
-E
Genome 1
B
-E
F
-D
C
A
Genome 2
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Genome Rearrangement

• Gene order analysis on a set of organisms is
  a powerful technique for genomic comparison
  phylogenetic inference.

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Genome Rearrangement

• General Definition for the problem
• Given a set of genomes and a set of possible
  evolutionary events (operations), find a shortest
  set of events transforming (sorting) those
  genomes into one another.

What genome means and what events are, makes the
diversity of the problem.
Since these events are rare, scenarios minimizing
their number are more likely close to reality.
Many models have been proposed.
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Genome Models

• Genes (or blocks of contiguous genes) are a good
  example of homologous markers, segments of
  genomes, that can be found in several species.
• The simplest possible model is
• The order of genes in each genome is known,
• All the genomes share the same set of genes,
• All genomes contain a single copy of each gene,
  and
• All genomes consist of a single chromosome.

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Genome Models

• Genomes can be modeled by
  each gene can be assigned a unique number and is
  exactly found once in the genome.

permutations

• Signed Permutation Each gene may be assigned
  or - sign to indicate the strand it resides on.
• Unsigned Permutation If the corresponding strand
  is unknown.

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Permutaions

• Genes (markers) are represented by integers
• 1, 2, . . . . , n,
• with ,- sign to indicate the strand they
  lie on.
• The order and orientation of genes of one genome
  in relation to the other is represented by a
  signed permutation ?.
• ? (???? ?2???????????? ?n-1?? ?n) of size n
  over -n, ... , -1, 1, ... , n, such that for
  each i from 1 to n, either i or -i is mandatory
  represented, but not both.

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Permutaions
Identity permutation

• The identity permutation ?n (1, 2, 3, . . . . ,
  n).
• When multiple genomes with the same gene content
  are compared, one of them is chosen as a base
  (reference), i.e, represented as ?n, and all
  other identical genes are given the same integer
  values.

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Permutaions
Sorted/unsorted permutation

• In order to sort a permutation ???this means that
  we want to apply some operations on ??to change
  it to ?n.
• If (?1 ?2) We say that ???is sorted with
  respect to ??.
• If (?1 ? ?2) We say that ???is unsorted with
  respect to ??.

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Permutaions
Example Mitochondrial Genomes of 6 Arthropoda
?1 (1 , 2 , 3 , 4 , 5 , 6 , 7 ,
8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17)
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16 17
?2 (1 , 2 , 3 , 4 , 5 , 6 , 8 ,
7 , 9 ,-10 , 11 , 12 , 13 , 14 , 15 , 16 , 17)
1 2 3 4 5 6 8 7 9
-10 11 12 13 14 15 16 17
?3 (1 , 2 , 3 , 4 , 5 , 6 , 7 ,
8 , 9 , 10 , 11 , 12 , 14 , 13 , 15 , 16 , 17)
1 2 3 4 5 6 7 8 9
10 11 12 14 13 15 16 17
?4 (1 , 2 , 3 , 5 , 4 , 6 , 7 ,
8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17)
1 2 3 5 4 6 7 8 9
10 11 12 13 14 15 16 17
?5 (1 , 3 , 4 , 5 , 6 , 7 , 8 , 9
, 10 , 11 , -2 , 12 , 13 , 14 , 15 , 16 , 17)
1 3 4 5 6 7 8 9
10 11 -2 12 13 14 15 16 17
?6 (1 , 3 , 4 , 5 , 6 , 7 , 8 , 9
, 10 , 11 , -2 , 12 , 16 , 13 , 14 , 15 , 17)
1 3 4 5 6 7 8 9
10 11 -2 12 16 13 14 15 17
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Permutaions
Example Mitochondrial Genomes of 6 Arthropoda
?1 (1 , 2 , 3 , 4 , 5 , 6 , 7 ,
8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17)
?2 (1 , 2 , 3 , 4 , 5 , 6 , 8 ,
7 , 9 ,-10 , 11 , 12 , 13 , 14 , 15 , 16 , 17)
?3 (1 , 2 , 3 , 4 , 5 , 6 , 7 ,
8 , 9 , 10 , 11 , 12 , 14 , 13 , 15 , 16 , 17)
?4 (1 , 2 , 3 , 5 , 4 , 6 , 7 ,
8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17)
?5 (1 , 3 , 4 , 5 , 6 , 7 , 8 , 9
, 10 , 11 , -2 , 12 , 13 , 14 , 15 , 16 , 17)
?6 (1 , 3 , 4 , 5 , 6 , 7 , 8 , 9
, 10 , 11 , -2 , 12 , 16 , 13 , 14 , 15 , 17)
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Permutaions
Linear and circular permutation

• ? is linear when it represents a linear
  chromosome, or circular when it represents a
  circular chromosome.
• When ? (???? ?2???????????? ?n-1?? ?n) is
  circular
• ? (-?n?? ??n-1???????????? ??2??
  ??1)
• all permutations obtained by shifts on
  ???or ?
• shift( ?, i) (?n-i1??
  ?n-i2?????????n-1?? ?n???1????????? ?n-i??
• are all equivalent.
• Example (-3,2,1,-4) (-1,-2,3,4)

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Permutaions
Points in permutations

• For a given permutation ? (???? ?2????????????
  ?n-1?? ?n), there is a point between each pair of
  consecutive values ?i and ?i1 in ?.
• If ??is linear there are two additional points,
  one before ???and one after ?n.
• If ??is circular there is one additional point
  between ?n?and ?1.
• Pts(?) n1 if linear, and pts(?) n if
  circular.

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Permutaions
Linear extension of a permutation

• For a given ? (???? ?2???????????? ?n-1?? ?n)
• If ? is linear a linear extension of ??is
• ? (0, ???? ?2????????????
  ?n-1?? ?n, n1)
• If ? is circular a linear extension of ??is
• ? (0, ???? ?2????????????
  ?n-1?? ?n-1, n)

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Permutaions

• Example
• ????????????? (4,8,9,7,6,5,1,3,2)
• ? (0,4,8,9,7,6,5,1,3,2,10)
• ? (0.4.8.9.7.6.5.1.3.2.10)
• Then Pts(?) 10

• Now we want to compare our genomes.

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Permutations - similarity/distance
Problem Given two genomes, How do we measure
their similarity and/or distance?
?
A Related Problem Given two permutations, How do
we measure their similarity and/or distance?
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Permutations - similarity/distance

• A distance measure should be a metric on the set
  of genomes.
• A Metric d on a set S (d S ? S ? R) satisfies
  the following three axioms

• Positivity for all s, t in S, d(s,t) ? 0, and
  d(s,t)0 iff s t.
• Symmetry for all s, t in S, d(s,t) d(t,s).
• Triangular inequality for all s, t, u in S,
• d(s,u) ? d(s,t) d(t,u).

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Permutations - similarity/distance

• Measures of similarity between permutations that
  are used in computational biology are numerous in
  literature.
• First measures used are (will be useful later
  on)
• Breakpoints (Introduced by Sankoff and Blanchette
  (1997))
• Common intervals

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Permutations-distance - Breakpoints

• When analyze ? with respect to ??, each point in
  ? can be an adjacency or a breakpoint.
• A point (pair of consecutive values) (?i, ?i1)
  in ? is an adjacency between ? and ?? when
  either (?i, ?i1) or (-?I1, -?i) are consecutive
  in ??.
• If ? is linear we have adjacency before ?? if ??
  is also the first value in ??, and an adjacency
  after ?n, if ?n is also last value in ??.
• If ? is circular we assume that ?n is also last
  value in ?? and (?n, ?1) is an adjacency if ?? is
  also the first value in ??.

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Permutations-distance - Breakpoints

• Breakpoint distance counts the lost adjacencies
  between genomes.
• The breakpoint distance between ? and ?? is
  

• brp(?) pts(?) - adj(?)
• where
• pts(?) is the number of points in ?.
• adj(?) is the number of adjacencies.
• If ? is sorted (? ??) ? has only adjacencies
  and no breakpoints (brp(?) 0).
• If ? is unsorted (? ? ??) ? has at least one
  breakpoint (brp(?) ? 0).

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Permutations-distance - Breakpoints

• Back to our Example?
• ???????????? (4,8,9,7,6,5,1,3,2)
• ? (0,4,8,9,7,6,5,1,3,2,10)
• ? (0.4.8.9.7.6.5.1.3.2.10)
• Then Pts(?) 10, brp(?)?

Adjacencies? ?n (0.1.2.3.4.5.6.7.8.9.10) (8,9)
(7,6) (6,5) (3,2) ? adj(?) 4
? brp(?) pts(?) - adj(?)
10 - 4
6
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Permutations-distance - Breakpoints

• Breakpoint distance is based on the notion of
  conserved adjacencies and can be defined on a set
  of more than two genomes.
• It is easy to compute.
• It always fails to capture more global relations
  between genomes.
• The first generalization of adjacencies is the
  notion of common intervals.

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Permutations-distance - Common Intervals

• Common intervals subsets of genes that appear
  consecutively together in two or more genomes,
  where genes are the same in each interval but may
  be not in the same order or orientation.

Example (circular chromosomes)
?1 (1 , 2 , 3 , 4 , 5 , 6 , 7 ,
8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17)
?2 (1 , 2 , 3 , 4 , 5 , 6 , 8 ,
7 , 9 ,-10 , 11 , 12 , 13 , 14 , 15 , 16 , 17)
?3 (1 , 2 , 3 , 4 , 5 , 6 , 7 ,
8 , 9 , 10 , 11 , 12 , 14 , 13 , 15 , 16 , 17)
?4 (1 , 2 , 3 , 5 , 4 , 6 , 7 ,
8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17)
?5 (1 , 3 , 4 , 5 , 6 , 7 , 8 ,
9 , 10 , 11 , -2 , 12 , 13 , 14 , 15 , 16 , 17)
?6 (1 , 3 , 4 , 5 , 6 , 7 , 8 ,
9 , 10 , 11 , -2 , 12 , 16 , 13 , 14 , 15 , 17)
If compare the first 4 species they share 6
adjacencies 1,2, 2,3,11.12,15,16,16,17,
17,1
If compare all 6 species they share only 1
adjacency 17,1
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Permutations-distance - Common Intervals

• Common intervals subsets of genes that appear
  consecutively together in two or more genomes,
  where genes are the same in each interval but may
  be not in the same order or orientation.

Example (circular chromosomes)
?1 (1 , 2 , 3 , 4 , 5 , 6 , 7 ,
8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17)
?2 (1 , 2 , 3 , 4 , 5 , 6 , 8 ,
7 , 9 ,-10 , 11 , 12 , 13 , 14 , 15 , 16 , 17)
?3 (1 , 2 , 3 , 4 , 5 , 6 , 7 ,
8 , 9 , 10 , 11 , 12 , 14 , 13 , 15 , 16 , 17)
?4 (1 , 2 , 3 , 5 , 4 , 6 , 7 ,
8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17)
?5 (1 , 3 , 4 , 5 , 6 , 7 , 8 ,
9 , 10 , 11 , -2 , 12 , 13 , 14 , 15 , 16 , 17)
?6 (1 , 3 , 4 , 5 , 6 , 7 , 8 ,
9 , 10 , 11 , -2 , 12 , 16 , 13 , 14 , 15 , 17)
The six permutations are very similar.
The genes in the interval 1,12 are all the
same, as genes in the intervals 3,6,
6,9,9,11, and 12,17.
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Permutations-distance - Common Intervals

• We can use common intervals as a measure of
  similarity between species.

Disadvantage All these measures do not reflect
rearrangement operations or explain what happened
to the genome over time.
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Rearrangement operations (events)
Back to our original problem Given a
set of genomes and a set of possible evolutionary
events (operations), find a shortest set of
events transforming those genomes into one
another.
What are the Rearrangement events (Operation)?
These events (Operation) could be applied to a
single gene or to a group of genes, intervals.
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Rearrangement operations
Example Mitochondrial Genomes of 6 Arthropoda
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16 17
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Rearrangement Operations

• Rearrangement operations affect gene order
• and gene content. There are various types
• In case of single-chromosome genome
• Inversions
• Transpositions
• Reverse transpositions
• Gene Duplications
• Gene loss
• In case of multiple-chromosomes genomes we add
• Translocations
• fusions
• fissions

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Rearrangement Operations - Single Chro.
Inversion
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Rearrangement Operations - Single Chro.
Inversion
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Rearrangement Operations - Single Chro.
Inversion
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Rearrangement Operations - Single Chro.
Example Mitochondrial Genomes of 6 Arthropoda
An inversion.
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Rearrangement Operations - Single Chro.
Transposition
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Rearrangement Operations - Single Chro.
Transposition
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Rearrangement Operations - Single Chro.
Transposition
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Rearrangement Operations - Single Chro.
Example Mitochondrial Genomes of 6 Arthropoda
A transposition
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Rearrangement Operations - Single Chro.
Reverse Transposition
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Rearrangement Operations - Single Chro.
Reverse Transposition
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Rearrangement Operations - Single Chro.
Reverse Transposition
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Rearrangement Operations - Single Chro.
Example Mitochondrial Genomes of 6 Arthropoda
A reverse transposition
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Rearrangement Operations - Multiple Chro.
Translocation
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Rearrangement Operations - Multiple Chro.
Translocation
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Rearrangement Operations - Multiple Chro.
Translocation
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Rearrangement Operations - Multiple Chro.
Translocation
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Rearrangement Operations - Multiple Chro.
Translocation
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Rearrangement Operations - Multiple Chro.
Translocation
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Rearrangement Operations - Multiple Chro.
Fusion
Fission
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Rearrangement Operations - Multiple Chro.
Fusion
Fission
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Rearrangement Operations - Multiple Chro.
Fusion
Fission
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Rearrangement Operations - Multiple Chro.
Fusion
Fission
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Rearrangement Operations - Multiple Chro.
Fusion
Fission
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Rearrangement Operations - Multiple Chro.
Fusion
Fission
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Rearrangement Operations - Multiple Chro.
From 24 chromosomes
To 21 chromosomes
Source Linda Ashworth, LLNL DOE Human Genome
Program Report
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Rearrangement Problems

• Back to our original problem
• Given a set of genomes and a set of
  possible evolutionary events (operations), find a
  shortest set of events transforming those genomes
  into one another.

Any set of operations yields a distance between
genomes, by counting the minimum number of
operations needed to transform one genome into
the other.
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Rearrangement Problems

• Back to our original problem
• Given a set of genomes and a set of
  possible evolutionary events (operations), find a
  shortest set of events transforming those genomes
  into one another.

Two classical problems

• Computing the distance d(?)?
• Computing one optimal sorting sequence of events.

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Reversal Distance - Sorting by Reversals

• Given a permutation ?, calculate reversal
  distance d(?) and find one optimal sequence of
  reversals sorting ?.
• Assumption
• Only reversals are allowed.
• No duplication in genes.
• Genomes are unichromosomal.

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Reversal Distance - Sorting by Reversals
A reversal ??is represented as a set of genes
appearing together in the given genome.
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Reversal Distance - Sorting by Reversals
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Reversal Distance - Sorting by Reversals
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Reversal Distance - Sorting by Reversals
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Reversal Distance - Sorting by Reversals
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Reversal Distance - Sorting by Reversals
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Reversal Distance - Sorting by Reversals
This approach is symmetric
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Reversal Distance - Sorting by Reversals
Reversal graph for n 3
Vertices all permutations of n 3. Edges
connect an edge between ?1 and ?2 if reversal
distance d(?1, ?2) 1.
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Reversal Distance - Sorting by Reversals
Reversal graph for n 3

• Reversal distance d(?i, ?k) length of shortest
  path between vi and vk.

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Reversal Distance - Sorting by Reversals
Reversal graph for n 3

• The graph is huge V n!.2n
• A feasible graph-search algorithm is not possible!

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Reversal Distance - Sorting by Reversals

• The classical approach for solving these two
  problems in polynomial time was developed by
  Hannenhalli and Pevzner. (1995)
• The reversal distance can be computed in O(n)
  time by Bader et. al. (2000)
• The fastest algorithm to find an optimal sorting
  sequence is lt O(n2) by Tannier et. al. (2007)
• Most approaches are based on a special structure
  called the breakpoint graph.

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Reversal Distance - Sorting by Reversals

• Breakpoint Graph edges are black or gray.
• Given ? (?????????????????n-1???n)
• If ? is linear we add the values 0, and n1, the
  represents the extremities of the chromosome
  obtaining
• ? (0, ?????????????????n-1???n,
  n1)
• If ? is circular assume ?n n and add only the
  value 0, obtaining
• ? (0, ?????????????????n-1???n-1,
  n)

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Reversal Distance - Sorting by Reversals

• Black edge Links each pair of consecutive value
  in ? by a horizontal (a point in ?).
• Gray edges Link the extremities of black edges
  such that the values will be in order.
• Graph collection of cycles, where black and gray
  edges alternate.
• Trivial cycle one black and one gray edge
  (adjacency)
• Long Cycle four or more edges (? 2 breakpoints)

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Reversal Distance - Sorting by Reversals
0
5
When sorted
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Reversal Distance - Sorting by Reversals
0
5
When sorted
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Reversal Distance - Sorting by Reversals
0
5
When sorted
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Reversal Distance - Sorting by Reversals
0
5
When sorted
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Reversal Distance - Sorting by Reversals
0
5
When sorted
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Reversal Distance - Sorting by Reversals
0
5
When sorted
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Reversal Distance - Sorting by Reversals
0
5
When sorted
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Reversal Distance - Sorting by Reversals
0
5
When sorted
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Reversal Distance - Sorting by Reversals
? (-3 , 2 , 1 , -4)
Linear
Circular

• Linear and circular permutations are different in
  breakpoint graph construction.
• Same analyses.

84
Reversal Distance - Sorting by Reversals
0
5
When sorted
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Reversal Distance - Sorting by Reversals
0
5
When sorted
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Reversal Distance - Sorting by Reversals
sorted
? (-3 , 2 , 1 , -4)

• If ??is sorted
• Only adjacencies, no breakpoints.
• Breakpoint graph is a collection of trivial
  cycles.
• cycles in sorted graph cyc(?) pts(?)

87
Reversal Distance - Sorting by Reversals
sorted
? (-3 , 2 , 1 , -4)

• If ??is unsorted
• At least one breakpoint, at least one long cycle.
• cycles cyc(?) is at most pts(?) - 1

Observation To sort a permutation ?, we would
like to increase the number of cycles in its
breakpoint graph.
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Reversal Distance - Sorting by Reversals

• The effects of a reversal ? over a breakpoint
  graph ?.

Split reversal
Joint reversal
cyc(??? ?)??? cyc(?) ?????
cyc(??? ?)??? cyc(?) ?????
Neutral reversal
cyc(??? ?)??? cyc(?)?
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Reversal Distance - Sorting by Reversals

• The effects of a reversal ? over a breakpoint
  graph ?.

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Reversal Distance - Sorting by Reversals
Observation To sort ?, we must maximize the
number of split reversals in the sorting
sequence s.
If s has only split reversals what will be the
reversal distance d(?)????(Hint in terms of
pts(?) and cyc(?))
d(?)????????pts(?) - cyc(?)
Are we done?
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Reversal Distance - Sorting by Reversals
A split reversal does not always exist.
For example, if all black edges in the graph have
the same direction.
In this case, we need to add some joint and/or
neutral reversals in the sorting sequence s.
d(?)????????pts(?) - cyc(?)
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Reversal Distance - Sorting by Reversals

• It is always possible to calculate the number of
  non-split reversals in a sorting sequence.

• It will be the number of non-split reversals to
  sort some hard components in the graph with no
  orientation, unoriented components.

• In practice, split reversals are sufficient to
  sort the permutation.

93
Reversal Distance - Sorting by Reversals
Can we choose any split reversal? only safe
reversals.
Safe reversal a split reversal not producing
hurdles.
Unsafe reversal
Safe reversal
There is always a safe reversal for any oriented
?.
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Reversal Distance - Sorting by Reversals
The final formula for the reversal distance d(?)
is d(?)????????pts(?) - cyc(?)
hrd(?) frt(?)

• Where
• frt(?) 1, if ? is a fortress, and 0 otherwise.
• pts(?) n1, if ??is linear, and n if ??is
  circular.

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Reversal Distance - Sorting by Reversals
Algorithm Get optimal sorting sequence s that
sorts ??

• Input A signed permutation ?.
• Output An optimal sequence of reversals sorting
  ?.
• Construct the breakpoint graph of ?.
• S ? ? empty
• If frt(?) 1 then
• choose a reversal ? to eliminate the
  fortress
• ? ? ??????
• ? s ? s . ???concatenate the reversal ? to
  s
• End if
• While there is hurdles in ? do
• choose a reversal ? to eliminate the hurdle
• ? ? ??????
• ? s ? s . ???concatenate the reversal ? to
  s
• End while
• While ? is not sorted do
• choose a safe split reversal ? to ?
• ? ? ??????
• ? s ? s . ???concatenate the reversal ? to
  s
• End while

96
Reversal Distance - Sorting by Reversals
Algorithm Get optimal sorting sequence s that
sorts ??

• Input A signed permutation ?.
• Output An optimal sequence of reversals sorting
  ?.
• Construct the breakpoint graph of ?.
• S ? ? empty
• If frt(?) 1 then
• choose a reversal ? to eliminate the
  fortress
• ? ? ??????
• ? s ? s . ???concatenate the reversal ? to
  s
• End if
• While there is hurdles in ? do
• choose a reversal ? to eliminate the hurdle
• ? ? ??????
• ? s ? s . ???concatenate the reversal ? to
  s
• End while
• While ? is not sorted do
• choose a safe split reversal ? to ?
• ? ? ??????
• ? s ? s . ???concatenate the reversal ? to
  s
• End while

97
Reversal Distance - Sorting by Reversals
Algorithm Get optimal sorting sequence s that
sorts ??

• Input A signed permutation ?.
• Output An optimal sequence of reversals sorting
  ?.
• Construct the breakpoint graph of ?.
• S ? ? empty
• If frt(?) 1 then
• choose a reversal ? to eliminate the
  fortress
• ? ? ??????
• ? s ? s . ???concatenate the reversal ? to
  s
• End if
• While there is hurdles in ? do
• choose a reversal ? to eliminate the hurdle
• ? ? ??????
• ? s ? s . ???concatenate the reversal ? to
  s
• End while
• While ? is not sorted do
• choose a safe split reversal ? to ?
• ? ? ??????
• ? s ? s . ???concatenate the reversal ? to
  s
• End while

98
Reversal Distance - Sorting by Reversals
Algorithm Get optimal sorting sequence s that
sorts ??

• Input A signed permutation ?.
• Output An optimal sequence of reversals sorting
  ?.
• Construct the breakpoint graph of ?.
• S ? ? empty
• If frt(?) 1 then
• choose a reversal ? to eliminate the
  fortress
• ? ? ??????
• ? s ? s . ???concatenate the reversal ? to
  s
• End if
• While there is hurdles in ? do
• choose a reversal ? to eliminate the hurdle
• ? ? ??????
• ? s ? s . ???concatenate the reversal ? to
  s
• End while
• While ? is not sorted do
• choose a safe split reversal ? to ?
• ? ? ??????
• ? s ? s . ???concatenate the reversal ? to
  s
• End while

ComplexityO(n5)
Tools GRIMM GRAPPA
99
Reversal Distance - Sorting by Reversals
We can have more than one optimal solution
100
conclusions

• Represented linear and circular genomes as
  permutations in our simple model.
• Described first measures for similarity between
  permutation were breakpoint and common intervals
  --gt has no biological interpretation.
• Used genome rearrangement events to describe
  similarity/distances between genomes --gt has more
  biological meaning.
• Described in details one distance measure
  (reversal distance) and events (reversals) to
  sort genomes.

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Thank you
Questions?
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