 Proceedings
 Open Access
Genome dedoubling by DCJ and reversal
 Antoine Thomas^{1},
 JeanStéphane Varré^{1}Email author and
 Aïda Ouangraoua^{1}Email author
https://doi.org/10.1186/1471210512S9S20
© Thomas et al; licensee BioMed Central Ltd. 2011
 Published: 5 October 2011
Abstract
Background
Segmental duplications in genomes have been studied for many years. Recently, several studies have highlighted a biological phenomenon called breakpointduplication that apparently associates a significant proportion of segmental duplications in Mammals, and the Drosophila species group, to breakpoints in rearrangement events.
Results
In this paper, we introduce and study a combinatorial problem, inspired from the breakpointduplication phenomenon, called the Genome Dedoubling Problem. It consists of finding a minimum length rearrangement scenario required to transform a genome with duplicated segments into a nonduplicated genome such that duplications are caused by rearrangement breakpoints. We show that the problem, in the DoubleCutandJoin (DCJ) and the reversal rearrangement models, can be reduced to an APXcomplete problem, and we provide algorithms for the Genome Dedoubling Problem with 2approximable parts. We apply the methods for the reconstruction of a nonduplicated ancestor of Drosophila yakuba.
Conclusions
We present the Genome Dedoubling Problem, and describe two algorithms solving the problem in the DCJ model, and the reversal model. The usefulness of the problems and the methods are showed through an application to real Drosophila data.
Keywords
 Segmental Duplication
 Ancestral Genome
 Duplicate Segment
 Rearrangement Event
 Linear Time Complexity
Introduction
Gene duplication is an important source of variations in genomes. Recently, several studies have highlighted biological evidence for abundant segmental duplications that occur around breakpoints of rearrangement events in the evolution of eukaryotes.
In mammals, an evidence for a strong association between duplications, genomic instability and largescale chromosomal rearrangements in primate evolution was first reported in [1]. Later in [2], a study of all evolutionary rearrangement breakpoints between human and mouse genomes reported that 53% of the breakpoints were associated with segmental duplications, as compared to 18% expected in a random assignment of breaks. In [3], a first study of the humanmouse rearrangement breakpoints, considering only synteny blocks of length larger than 100Kb and duplicated sequences of length larger than 10Kb, showed that 25% (122/461) of the breakpoints contained duplicated sequences of length greater than 10kb.
The association between segmental duplications and regions of breaks of synteny was also reported in the Drosophila species group. In [4], an analysis of the breakpoints of Drosophila yakuba compared to two related species, Drosophila simulans and Drosophila melanogaster, revealed that the breakpoint regions of 59% of the reversals (17/29) were associated with inverted duplications of genes or other nonrepetitive sequences. Further evidences of the recurrent presence of repetitive sequences near breakpoints of rearrangement in the evolution of Drosophila were also reported in [5–7].
A rearrangement event is an operation that modifies the organization of a given genome by cutting the genome at some points called breakpoints to glue the exposed extremities in a different way. The biological phenomenon called breakpointduplication results in the presence of the same genomic segment on both extremities of a breakpoint in a rearrangement. Several biological models have been presented to explain the presence of duplicated sequences at rearrangement breakpoint regions. These models are based on DNA breaks repairs that produce duplicated segments because of staggered SingleStrandBreaks [3, 4], or nonreciprocal genetic exchange in DoubleStrandBreaks [5]. Most of these biological models support a nonrandom model of chromosomal evolution that implicates a predominance of recurrent smallscale duplications and largescale evolutionary rearrangements within specific fragile regions. Moreover, the genetic instability of such regions is often suggested to be the cause rather than the consequence of duplicated genomic architecture [3, 8]. Interestingly, a growing number of the breakpointduplications detected in Supraprimates evolution have also been linked to recurrent chromosomal rearrangements associated with common diseases in the human population [1–3, 8, 9]. In [10], breakpointduplications were also identified in humam sex chromosomes, allowing to order rearrangement events in time, based on the degree of divergence of the breakpointduplicated sequences.
In this paper, we are interested in using the segmental duplications of a given presentday genome that has undergone breakpointduplication rearrangements, in order to reconstruct a nonduplicated ancestral genome. We formally define the breakpointduplication phenomenon, and introduce a combinatorial problem called the Genome Dedoublíng problem. Given a genome that has undergone breakpointduplication rearrangements, possibly with other rearrangements events, the problem asks to find an ancestral genome such that the number of rearrangement events needed to tranform the ancestor into the given genome is minimal. Note that the Genome Dedoubling problem asks to find a nondupliated ancestor of a given duplicated genomes, as the Genome Halving problem introduced in [11] that consists of, given a genome that has undergone a wholegenome duplication followed by rearrangement events, finding the ancestral genome that was present right before the wholegenome duplication event. However, the two problems and their solutions are different as they aim at recovering different types of duplication events, breakpointduplications and wholegenome duplications. As the Genome Halving problem is motivated by the wholegenome duplication events in molecular evolution, the Genome Dedoubling problem is motivated by breakpointduplication events in molecular evolution. Both problems are useful for the comparison of genomes with duplicated segments.
In the following, we study the Genome Dedoubling problem under the DoubleCutandJoin (DCJ) and the reversal rearrangement models. In Section Methods, we formally present breakpointduplication (BD) rearrangements and the Genome Dedoubling Problem. We show that the problem can always be regarded as a Dedoubling Problem on totally duplicated genomes. In Section Genome dedoubling by DCJ, we study the problem under the DCJ model, on multichromosomal then unichromosomal genomes. We prove the NPcompleteness of the problems by reduction to an APXcomplete problem, and provide algorithms with a linear time complexity, except for an APXcomplete part that is 2approximable. In Section Genome dedoubling by reversal, we study the problem under the reversal model on oriented genomes, making use of some results of the HannenhalliPevzner (HP) theory [12] on sorting by reversal described in [13, 14]. We provide an algorithm with a quadratic time complexity, except for an APXcomplete part that is 2approximabe. In Section Application, an application for the reconstruction of a nonduplicated ancestor of Drosophila yakuba, using data from [4], is presented.
Methods
In this section we give the main definitions and notations of duplicated genomes and rearrangements. Next, we generalize the definitions of rearrangements in order to introduce a formal definition of breakpointduplication rearrangements, and the Genome Dedoubling Problem studied in the paper.
Duplicated genomes
A genome consists of linear or circular chromosomes that are composed of genomic markers. Markers are represented by signed integers such that the sign indicates the orientations of markers in chromosomes. By convention, – –x = x. A linear chromosome is represented by an ordered sequence of signed integers surrounded by the unsigned marker ○ at each end indicating the telomeres. A circular chromosome is represented by a circularly ordered sequence of signed integers. For example, (1 2 –3) (○ 4 –5 ○) is a genome composed of one circular and one linear chromosome.
Each genome contains at most two occurrences of each marker. Two copies of a same marker in a genome are called paralogs. If a marker x is present twice, one of the paralogs is represented by . By convention, . Here, such markers represent segments duplicated by a breakpointduplication rearrangement.
Definition 1 A duplicated genome is a genome in which a subset of the markers are duplicated.
For example, is a duplicated genome where markers 1, 2, and 5 are duplicated. A nonduplicated genome is a genome in which no marker is duplicated. A totally duplicated genome is a duplicated genome in which all markers are duplicated. For example, is a totally duplicated genome.
An adjacency in a genome is a pair of consecutive markers. Since a genome can be read in two directions, the adjacencies (x y) and (–y –x) are equivalent. For example, the genome has seven adjacencies, , and .
Definition 2 A dedoubled genome is a duplicated genome G such that for any duplicated marker x in G, either , or is an adjacency of G.
For example, is a dedoubled genome. The reduction of a dedoubled genome G, denoted by G^{ R }, is the genome obtained from G by replacing every pair , or by a single marker x. For example the reduction of is G^{ R } = (1 –2) (○ –3 4 –5 ○).
Rearrangement
A rearrangement operation on a given genome cuts a set of adjacencies of the genome called breakpoints and forms new adjacencies with the exposed extremities, while altering no other adjacency. In this paper, we consider two types of rearrangement operation called doublecutandjoin (DCJ) and reversal. In the sequel, the breakpoints of a rearrangement operation are indicated in the genome by the symbol _{▲}, and the new adjacencies are indicated in the genome by dots.
A DCJ (resp. reversal) scenario between two genomes A and B is a sequence of DCJ (resp. reversal) operations allowing to transform A into B. The length of a scenario is the number of rearrangement operations composing the scenario.
The DCJ (resp. reversal) distance between two genomes A and B is the minimum length of a DCJ (resp. reversal) scenario between A and B.
Breakpointduplication rearrangements
We now generalize the definitions of rearrangement operations to account for possible duplications at their breakpoints.
A 1breakpointduplication DCJ (1BDDCJ) operation on a genome G is a rearrangement operation that alters two different adjacencies (a b) and (c d) of G, by:

first adding marker at the appropriate position to produce segment ,

then applying a DCJ operation that cuts adjacencies and (c d) to produce either (a d) and , or (a –c) and .
A 2breakpointduplication DCJ (2BDDCJ) operation on a genome G is a rearrangement operation that alters two different adjacencies (a b) and (c d) of G, by:

first adding markers and at the appropriate positions to produce segments and ,

then applying a DCJ operation that cuts adjacencies and to produce either and , or (a –c) and .
Definition 3 A breakpointduplication DCJ (BDDCJ) operation on a genome G is either a 1BDDCJ operation, or a 2BDDCJ operation.
To summarize, a BDDCJ operation consists of a first step in which one or two markers are duplicated, followed by a second step where a DCJ operation is applied. Similarly, we now define a breakpointduplication reversal (BDreversal) operation.
Definition 4 A breakpointduplication reversal (BDreversal) operation on a genome G is a BDDCJ operation such that the DCJ operation applied in the second step of the BDDCJ operation is a reversal.
A BDDCJ scenario (resp. BDreversal scenario) between a nonduplicated genome A and a duplicated genome B is a sequence composed of BDDCJ (resp. BDreversal) operations and possibly DCJ (resp. reversal) operations allowing to transform A into B.
Definition 5 Given a nonduplicated genome A and a duplicated genome B, the BDDCJ distance (resp. BDreversal distance) between A and B is the minimal length of a BDDCJ (resp. BDreversal) scenario between A and B.
We now give an obvious, but useful property allowing to reduce a BDDCJ scenario to a DCJ scenario.
Proposition 1 Given a nonduplicated genome A and a duplicated genome B, for any a BDDCJ (resp. BDreversal) scenario between A and B, there exists a DCJ (resp. reversal) scenario of same length between a dedoubled genome D and B such that the reduction of D is A (D^{ R } = A).
Proof. Let S be a BDDCJ (resp. BDreversal) scenario between A and B. D is the genome obtained from A, by adding, for any marker x duplicated by a BDDCJ operation in S, the marker in a way to produce either adjacency , or as done in S. Thus, D^{ R } = A. The DCJ (resp. reversal) scenario between D^{ R } and B having the same length as S, is the sequence of DCJ (resp. reversal) contained in S or in BDDCJ (resp. BDreversal) operations of S, with the same order as in S. ■
Genome dedoubling problem
We now state the genome dedoubling problems considered in this paper.
Genome dedoubling problem: Given a duplicated genome G, the DCJ (resp. reversal) genome dedoubling problem consists of finding a nonduplicated genome H such that the BDDCJ (resp. BDreversal) distance between H and G is minimal.
Given a duplicated genome G, we denote by d_{ dcj }(G) (resp. d_{ rev }(G)), the minimum BDDCJ (resp. BDreversal) distance between any nonduplicated genome and G. From Proposition 1, the following proposition is straightforward.
Proposition 2 Given a duplicated genome G, the DCJ (resp. reversal) genome dedoubling problem on G is equivalent to finding a dedoubled genome D such that the DCJ (resp. reversal) distance between D and G is minimal.
The next proposition describes a further reduction of the genome dedoubling problem on a duplicated genome G.
Proposition 3 Given a duplicated genome G, the DCJ (resp. reversal) genome dedoubling problem on G is equivalent to the DCJ (resp. reversal) genome dedoubling problem on the totally duplicated genome G^{ T } obtained from G by replacing every maximal subsequence of nonduplicated markers beginning with a marker x by the pair .
Proof. See proof in Additional file 1 (Supplemental proofs). ■
For example, solving the DCJ (resp. reversal) genome dedoubling problem on is equivalent to solving it on . The transformations applied on G to obtain G^{ T } are indicated in bold font.
In the sequel, G will always denote a totally duplicated genome, and we focus in Sections Genome dedoubling by DCJ and Genome dedoubling by reversal on the problem of finding a dedoubled genome D such that the DCJ (resp. reversal) distance between D and G is minimal.
Results
In this section, we first study the Genome Dedoubling Problem under the DCJ model. Next, we study the problem under the reversal model on oriented genomes described in the HannenhalliPevzner (HP) theory on sorting by reversal [12–14].
Genome dedoubling by DCJ
In this section, G denotes a totally duplicated genome. In order to give a formula for the DCJ dedoubling distance of G, d_{ dcj }(G), we use a graph called the dedoubled adjacency graph of G.
Dedoubled adjacency graph
Definition 6 The dedoubled adjacency graph of G, denoted by , is the graph whose vertices are the adjacencies of G, and for any marker x there is one edge between the vertices (x u) and , and one edge between the vertices (y x) and .
Note that all vertices in have degree one or two. Thus, the connected components of are only paths and cycles. These paths and cycles are called elements of .
Given a couple of paralogous markers , an element of the graph is said to contain the couple if it contains the edge linking vertices (x u) and , or the edge linking vertices (y x) and . By definition, a couple can possibly be contained in only one element A of if element A contains both edges and . In this case, A is said to contain twice the couple , and A is called a duplicated element of . If an element A contains no couple twice, then it is called a nonduplicated element of . If the two edges and belong to two different elements A and B of , then A and B will both contain . In this case, we say that A and B intersect. If two elements A and B do not intersect, then we say that A and B are independent. For example in Fig. 1 the two paths of the adjacency graph are duplicated, while the three cycles are nonduplicated. The leftmost path and the leftmost cycle intersect because they both contain the couple , while the two paths are independent. Given an element A of , the set induced by A is the set of couples contained in A.
General sorting
In this section, we prove the following theorem:
Theorem 1 Let n be the number of couples of paralogous markers in G. Let C_{ i } be the maximum size of a subset of nonduplicated pairwise independent cycles in . The DCJ dedoubling distance of G is d_{ dcj }(G) = n – C_{ i }.
For example, in Fig. 1, the maximum size of a subset of nonduplicated pairwise independent cycles is 2 as there are three cycles, and the two rightmost cycles intersect. The distance would then be d_{ dcj }(G) = 8 – 2 = 6. To prove Theorem 1, we use the following properties:
Property 1 Let n be the number of couples of paralogous markers in G.
1. The maximum size C _{ i } of a set of nonduplicated pairwise independent cycles in the graph is n.
2. If G is dedoubled genome, then contains n nonduplicated pairwise independent cycles, each containing a single couple of paralogous markers, plus possibly other cycles. In this case, C_{ i } = n.
3. A DCJ operation can only alter the maximum size C_{ i } of a set of nonduplicated pairwise independent cycles by –1, 0 or +1.
Proof. See proof in Additional file 1 (Supplemental proofs). ■
We now have all the prerequisites to give the proof of Theorem 1. The proof can be found in Additional file 1 (Supplemental proofs).
Lemma 1 Choosing a maximum size set of nonduplicated pairwise independent cycles in is an APXcomplete problem, approximable with an approximation ratio of 2.
Proof. See proof in Additional file 1 (Supplemental proofs). ■
From Lemma 1, the complexity of the Genome Dedoubling problem by DCJ follows immediately.
Corollary 1 The Genome Dedoubling problem by DCJ is NPcomplete. Algorithm 1 solves the problem in linear time complexity, except for the computation of the set of cycles S_{ i } that is 2approximable.
Sorting between linear unichromosomal genomes
In this section, we search for a minimum length DCJ scenario that transforms a duplicated genome consisting of a single linear chromosome into a dedoubled genome consisting of a single linear chromosome. The results of this section will then be used in the next section for the study of the Genome Dedoubling problem under the reversal model.
In this section and the sequel, G denotes a totally duplicated genome consisting of a single linear chromosome. In this case, the graph contains exactly one path, and possibly several cycles.
Definition 7 The path in is said to be valid if it contains every couple of paralogous markers in G.
A DCJ operation that merges a cycle c of in the path p is a DCJ operation that acts on an adjacency of c and an adjacency of p, thus gathering c and p into a longer path.
Note that if G is a dedoubled genome, then the path in is necessarily valid. We call such a genome a dedoubled linear genome. So, if the path in is not valid, then any DCJ scenario transforming G into a dedoubled linear genome will merge, in the path, cycles containing the couples that are not contained in the path.
In the following, we always denote by m the minimum number of cycles required to make the path of valid. We also denote by C_{ i } the maximum size of a subset of nonduplicated pairwise independent cycles. First, we have the following property:
Property 2 Let C be the number of cycles in . We have C_{ i } = C – m.
Proof. See proof in Additional file 1 (Supplemental proofs). ■
From Property 2, we then have the following lemma.
Lemma 2 Let n be the number of couples of paralogous markers in G. Let C be the number of cycles in . The minimum length d of a DCJ scenario transforming G into a dedoubled genome consisting of a single linear chromosome equals d = n – C + 2m.
Proof. See proof in Additional file 1 (Supplemental proofs). ■
From Property 2 and Lemma 2, we immediately have the following complexity.
Corollary 2 The problem of finding a DCJ scenario transforming G into a dedoubled genome consisting of a single linear chromosome is NPcomplete. Algorithm 1, in which we add the line (2’: Merge in the path all the cycles that are not in S_{ i }) between lines 2 and 3, solves the problem in linear time complexity, except for the computation of the set of cycles S_{ i } that is 2approximable.
Genome dedoubling by reversal
We build and use a graph that behaves like the arc overlap graph used in [13] for the HannenhalliPevzner theory of sorting by reversal [12]. The genome G consists of a single linear chromosome.
Dedoubled overlap graph
For any couple of paralogous markers in G, the segment of is the smallest segment of G containing both markers x and . For example, in genome , the segment of is , and the segment of is .
Definition 8 The dedoubled overlap graph of G, denoted by , is the graph whose vertices are all the couples of paralogous markers of G, and there is an edge between two vertices and if the segments of u and v overlap.
The overlap graph of G behaves like arc overlap graphs used in [13] for the HannenhaliPevzner theory of sorting by reversal [12]. Indeed, given an oriented vertex of the graph , performing the reversal or complements the subgraph induced by and all its neighbouring vertices, and changes the orientation of all vertices in this subgraph (see Fig. 4.b).
A connected component of the graph is oriented if it contains at least one oriented vertex, otherwise it is unoriented. A genome is oriented if all connected components of the graph are oriented, otherwise it is unoriented.
Given an oriented vertex of the graph , the score of is the number of oriented vertices in the overlap graph of the genome obtained after applying on G. Note that the same number of oriented vertices is obtained after applying on G.
Property 3 Let be an oriented vertex of of maximum score. Performing or does not create new unoriented connected components in the overlap graph of the genome obtained.
Proof. See proof in Additional file 1 (Supplemental proofs). ■
In the sequel, we focus on sorting oriented genomes using reversal dedoubling scenarios. A totally duplicated genome G consisting of a single linear chromosome is called a validpath genome if the single path in is valid. Otherwise, it is called a nonvalidpath genome.
Sorting an oriented validpath genome
In this section, we consider an oriented validpath genome G. With n being the number of couples of paralogous markers in G, we have the following theorem:
Theorem 2 Let G be an oriented validpath genome. Let C be the number of cycles in . The reversal dedouhling distance of G is d_{ rev }(G) = n – C.
Sorting an oriented nonvalidpath genome
In this section, G denotes an oriented nonvalid path genome. At least m cycles of have to be merged in the path to make it valid.
An edge or of the adjacency graph is called oriented if markers x and have different signs. Note that extracting a cycle from any element of the graph requires this element to contain oriented edges. It is easy to see that given two adjacencies picked in a given element, a reversal acting on these adjacencies will extract a cycle if and only if the path linking these adjacencies contains an odd number of oriented edges. Thus, we have the following lemma:
Lemma 3 Let G be an oriented nonvalidpath genome. Merging a cycle of in its path never creates unoriented connected components in the overlap graph of the genome obtained.
Proof. See proof in Additional file 1 (Supplemental proofs). ■
Theorem 3 Let G be an oriented nonvalidpath genome. Let C be the number of cycles in the graph and m be the minimum number of cycles to merge in the path to make it valid. The reversal dedoubling distance of G is d_{ rev }(G) = n – C + 2m.
Proof. See proof in Additional file 1 (Supplemental proofs). ■
Prom Lemma 1 and Property 2, the complexity of the Genome Dedoubling problem by reversal on oriented genomes follows immediately.
Corollary 3 The Genome Dedoubling problem by reversal on oriented genomes is NPcomplete. Algorithm 2 solves the problem in quadratic time complexity, except for the computation of S_{ i } that is 2approximable.
Application
We applied Algorithm 2 to reconstruct an ancestral chromosome for the chromosome 2 of Drosophila yakuba using a dataset obtained from [4] with Drosophila melanogaster used as the outgroup. The results obtained are in good agreement with the biological results explaining the evolution of the chromosome 2 from Drosophila yakuba to Drosophila melanogaster in the litterature [4, 15]. See Additional file 2 (Experimental results) for a description of the dataset and the results of the application.
Conclusion
In this paper, we introduced the genome dedoubling problem in the DCJ rearrangement model, NPcomplete in both the multichromosomal and the linear unichromosomal case, by reduction to an APXcomplete problem. For both cases, we described an algorithm solving the problems in linear time complexity, except for an APXcomplete part that is 2approximable. We also presented some results on the Genome Dedoubling problem by reversal, providing an algorithm solving the problem on oriented genomes in quadratic time complexity, except for an APXcomplete part that is 2approximable. The case of unoriented genomes in the reversal model will be treated in a future paper. Unsurprisingly, partial results obtained so far tend to show that the general distance formula can be written as d_{ rev }(G) = n – C + 2m + t, with t corresponding to the cost of genome orientation. However, the cost t here differs from the orientation cost described in the classical reversal theory based on the unoriented component tree [14]. In our case, the structure of the graph allows to orient components while not decreasing the number of cycles, or even increasing it in some cases. This requires proper identification of different kinds of merging reversals and further extensions on the data structures presented in this paper.
The second obvious extension of the present work, as in the the Genome Halving problem theory [16], is to generalize the Genome Dedoubling problem defined on a single genome, to the Guided Genome Dedoubüng problem, that asks to find a nonduplicated genome that minimizes the breakpointduplication distance to a given duplicated genome, plus the distance to a given nonduplicated genome. A further extension of this work consists of taking account of the degree of divergence of the breakpointduplicated sequences to order the rearrangement operations in time as done in [10].
Declarations
Acknowledgements
We would like to thank Anne Bergeron for her useful comments on the breakpointduplication phenomenon, and the anonymous reviewers of the paper for their useful comments on the first version of the document.
This article has been published as part of BMC Bioinformatics Volume 12 Supplement 9, 2011: Proceedings of the Ninth Annual Research in Computational Molecular Biology (RECOMB) Satellite Workshop on Comparative Genomics. The full contents of the supplement are available online at http://www.biomedcentral.com/14712105/12?issue=S9.
Authors’ Affiliations
References
 Samonte R, Eichler E: Segmental duplications and the evolution of the primate genome. Nature Reviews Genetics 2002, 6: 65–72.View ArticleGoogle Scholar
 Armengol L, Pujana M, Cheung J, Scherer S, Estivill X: Enrichment of segmental duplications in regions of breaks of synteny between the human and mouse genomes suggest their involvement in evolutionary rearrangements. Human Molecular Genetics 2003, 12(17):2201–2208. 10.1093/hmg/ddg223PubMedView ArticleGoogle Scholar
 Bailey J, Baertsch R, Kent W, Haussler D, Eichler E: Hotspots of mammalian chromosomal evolution. Genome Biology 2004, 5(4):R23. 10.1186/gb200454r23PubMedPubMed CentralView ArticleGoogle Scholar
 Ranz J, Maurin D, Chan Y, Von Grotthuss M: Principles of genome evolution in the Drosophila melanogaster species group. PLoS biology 2007, 5(6):e152+.PubMedPubMed CentralView ArticleGoogle Scholar
 Matzkin L, Merritt T, Zhu CT, Eanes W: The structure and population genetics of the breakpoints associated with the cosmopolitan chromosomal inversion In(3R)Payne in Drosophila melanogaster. Genetics 2005, 170: 1143–1152. 10.1534/genetics.104.038810PubMedPubMed CentralView ArticleGoogle Scholar
 Richards S, Liu Y, Bettencourt B, Hradecky P, Letovsky S: Comparative genome sequencing of Drosophila pseudoobscura: Chromosomal, gene, and ciselement evolution. Genome Research 2005, 15: 1–18. 10.1101/gr.3059305PubMedPubMed CentralView ArticleGoogle Scholar
 Meisel R: Repeat mediated gene duplication in the Drosophila pseudoobscura genome. Gene 2009, 438(1–2):1–7. 10.1016/j.gene.2009.02.019PubMedView ArticleGoogle Scholar
 Quinlan A, Clark R, Sokolova S, Leibowitz M, Zhang Y, Hurles M, Mell J, Hall I: Genomewide mapping and assembly of structural variant breakpoints in the mouse genome. Genome Research 2010, 20: 623–635. 10.1101/gr.102970.109PubMedPubMed CentralView ArticleGoogle Scholar
 Howarth KD, Pole JCM, Beavis JC, Batty EM, Newman S, Bignell GR, Edwards PAW: Large duplications at reciprocal translocation breakpoints that might be the counterpart of large deletions and could arise from stalled replication bubbles. Genome Research 2011, 21(4):525–534. 10.1101/gr.114116.110PubMedPubMed CentralView ArticleGoogle Scholar
 Lemaitre C, Braga M, Gautier C, Sagot MF, Tannier E, Marais G: Footprints of Inversions at Present and Past Pseudoautosomal Boundaries in Human Sex Chromosomes. Genome Biology 2009, 1: 56–66.View ArticleGoogle Scholar
 ElMabrouk N, Nadeau J, Sankoff D: Genome Halving. In In proc. of Combinatorial Pattern Matching. LNCS 1448. Springer; 1998:235–250.View ArticleGoogle Scholar
 Hannenhalli S, Pevzner PA: Transforming Men into Mice (Polynomial Algorithm for Genomic Distance Problem). In In proc. of FOCS 1995. IEEE Press; 1995:581–592.Google Scholar
 Bergeron A: A very elementary presentation of the HannenhalliPevzner theory. Discrete Applied Mathematics 2005, 146(2):134–145. 10.1016/j.dam.2004.04.010View ArticleGoogle Scholar
 Bergeron A, Mixtacki J, Stoye J: Reversal distance without hurdles and fortresses. In In proc. of Combinatorial Pattern Matching. LNCS 3109. SpringerVerlag; 2004:388–399.View ArticleGoogle Scholar
 Lemeunier F, Ashburner M: Relationships within the melanogaster species subgroup of the genus Drosophila (Sophophora). II. Phylogenetic relationships between six species based upon polytene chromosome banding sequences. Proceedings of the Royal Society of London, Series B, Biological Science 1976, 193(1112):275–294. 10.1098/rspb.1976.0046View ArticleGoogle Scholar
 Zheng C, Zhu Q, Sankoff D: Genome halving with an outgroup. Evolutionary Bioinformatics 2006, 2: 319–326.Google Scholar
 Berman P, Fujito T: Approximating independent sets in degree 3 graphs. In In proc. of Workshop on Algorithms and Data Structures. LNCS 955. SpringerVerlag; 1995:449–460.View ArticleGoogle Scholar
 Hochbaum DS: Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics 2004, 6: 243–254.View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.