MOTIVATION: Reconciliation between a host and its symbiont phylogenies or between a species and a gene phylogenies is a prevalent approach in evolution, however no simple generic tool (i.e. virtually usable by all reconciliation software, from host/symbiont to species/gene comparisons) is available to visualize reconciliation results. Moreover there is no tool to visualize 3-levels reconciliations, i.e. to visualize 2 nested reconciliations as for example in a host/symbiont/gene complex. RESULTS: Thirdkind is a light and easy to install command line software producing svg files displaying reconciliations, including 3-levels reconciliations. It takes a standard format recPhyloXML as input, and is thus usable with most reconciliation software. AVAILABILITY AND IMPLEMENTATION: https://github.com/simonpenel/thirdkind/wiki. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

In phylogenetics, reconciliation is an approach to connect the history of two or more coevolving biological entities.The general idea of reconciliation is that a phylogenetic tree representing the evolution of an entity (e.g.homologous genes, symbionts. ..) can be drawn within another phylogenetic tree representing an encompassing entity (respectively, species, hosts) to reveal their interdependence and the evolutionary events that have marked their shared history (Fig 1).The development of reconciliation approaches started in the 1980s, mainly to depict the coevolution of a gene and a genome, and of a host and a symbiont, which can be mutualist, commensalist or parasitic.It has also been used for example to detect horizontal gene transfer, or understand the dynamics of genome evolution.Phylogenetic reconciliation can account for a diversity of evolutionary trajectories of what makes life's history, intertwined with each other at all scales that can be considered, from molecules to populations or cultures.A recent avatar of the importance of interactions between levels of organization is the holobiont concept, where a macro-organism is seen as a complex partnership of diverse species.Modeling the evolution of such complex entities are one of the challenging and exciting direction of current research on reconciliation. Phylogenetic trees as matryoshka dollsPhylogenies have been used for representing the diversification of life at many levels of organization: macro-organisms [1], their cells throughout development [2], micro-organisms through marker genes [3], chromosomes [4], proteins [5], protein domains [6], and can also be helpful to understand the evolution of human culture elements such as languages [7] or folktales [8]. At each of these levels, phylogenetic trees describe different stories made of specific diversification events, which may or may not be shared among levels.Yet because they are structurally nested or functionally dependent, the evolution at a particular level is bound to others.Phylogenetic reconciliation is the identification of the links between levels through the comparison of at least two associated trees.Originally developed for two trees, reconciliations for more than two levels have been recently constructed.As such, reconciliation provides evolutionary scenarios that reveal conflict and cooperation among evolving entities.These links may be unintuitive, for instance, genes present in the same genome may show uncorrelated evolutionary histories while some genes present in the genome of a symbiont may show a

MOTIVATION: Gene and species tree reconciliation methods are used to interpret gene trees, root them and correct uncertainties that are due to scarcity of signal in multiple sequence alignments. So far, reconciliation tools have not been integrated in standard phylogenetic software and they either lack performance on certain functions, or usability for biologists. RESULTS: We present Treerecs, a phylogenetic software based on duplication-loss reconciliation. Treerecs is simple to install and to use. It is fast and versatile, has a graphic output, and can be used along with methods for phylogenetic inference on multiple alignments like PLL and Seaview. AVAILABILITY AND IMPLEMENTATION: Treerecs is open-source. Its source code (C++, AGPLv3) and manuals are available from https://project.inria.fr/treerecs/.

SUMMARY: Here we present Zombi, a tool to simulate the evolution of species, genomes and sequences in silico, that considers for the first time the evolution of genomes in extinct lineages. It also incorporates various features that have not to date been combined in a single simulator, such as the possibility of generating species trees with a pre-defined variation of speciation and extinction rates through time, simulating explicitly intergenic sequences of variable length and outputting gene tree-species tree reconciliations. AVAILABILITY AND IMPLEMENTATION: Source code and manual are freely available in https://github.com/AADavin/ZOMBI/. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Motivation: A reconciliation is an annotation of the nodes of a gene tree with evolutionary events-for example, speciation, gene duplication, transfer, loss, etc.-along with a mapping onto a species tree. Many algorithms and software produce or use reconciliations but often using different reconciliation formats, regarding the type of events considered or whether the species tree is dated or not. This complicates the comparison and communication between different programs. Results: Here, we gather a consortium of software developers in gene tree species tree reconciliation to propose and endorse a format that aims to promote an integrative-albeit flexible-specification of phylogenetic reconciliations. This format, named recPhyloXML, is accompanied by several tools such as a reconciled tree visualizer and conversion utilities. Availability and implementation: http://phylariane.univ-lyon1.fr/recphyloxml/.

Summary: Gene trees reconstructed from sequence alignments contain poorly supported branches when the phylogenetic signal in the sequences is insufficient to determine them all. When a species tree is available, the signal of gains and losses of genes can be used to correctly resolve the unsupported parts of the gene history. However finding a most parsimonious binary resolution of a non-binary tree obtained by contracting the unsupported branches is NP-hard if transfer events are considered as possible gene scale events, in addition to gene origination, duplication and loss. We propose an exact, parameterized algorithm to solve this problem in single-exponential time, where the parameter is the number of connected branches of the gene tree that show low support from the sequence alignment or, equivalently, the maximum number of children of any node of the gene tree once the low-support branches have been collapsed. This improves on the best known algorithm by an exponential factor. We propose a way to choose among optimal solutions based on the available information. We show the usability of this principle on several simulated and biological datasets. The results are comparable in quality to several other tested methods having similar goals, but our approach provides a lower running time and a guarantee that the produced solution is optimal. Availability and Implementation: Our algorithm has been integrated into the ecceTERA phylogeny package, available at http://mbb.univ-montp2.fr/MBB/download_sources/16__ecceTERA and which can be run online at http://mbb.univ-montp2.fr/MBB/subsection/softExec.php?soft=eccetera . Contact: [email protected]. Supplementary information: Supplementary data are available at Bioinformatics online.

BACKGROUND: Given two genomes that have diverged by a series of rearrangements, we infer minimum Double Cut-and-Join (DCJ) scenarios to explain their organization differences, coupled with indel scenarios to explain their intergene size distribution, where DCJs themselves also alter the sizes of broken intergenes. RESULTS: We give a polynomial-time algorithm that, given two genomes with arbitrary intergene size distributions, outputs a DCJ scenario which optimizes on the number of DCJs, and given this optimal number of DCJs, optimizes on the total sum of the sizes of the indels. CONCLUSIONS: We show that there is a valuable information in the intergene sizes concerning the rearrangement scenario itself. On simulated data we show that statistical properties of the inferred scenarios are closer to the true ones than DCJ only scenarios, i.e. scenarios which do not handle intergene sizes.

We study statistical estimators of the number of genomic events separating two genomes under a Double Cut-and Join (DCJ) rearrangement model, by a method of moment estimation. We first propose an exact, closed, analytically invertible formula for the expected number of breakpoints after a given number of DCJs. This improves over the heuristic, recursive and computationally slower previously proposed one. Then we explore the analogies of genome evolution by DCJ with evolution of binary sequences under substitutions, permutations under transpositions, and random graphs. Each of these are presented in the literature with intuitive justifications, and are used to import results from better known fields. We formalize the relations by proving a correspondence between moments in sequence and genome evolution, provided substitutions appear four by four in the corresponding model. Eventually we prove a bounded error on two estimators of the number of cycles in the breakpoint graph after a given number of rearrangements, by an analogy with cycles in permutations and components in random graphs.

Most models of genome evolution concern either genetic sequences, gene content or gene order. They sometimes integrate two of the three levels, but rarely the three of them. Probabilistic models of gene order evolution usually have to assume constant gene content or adopt a presence/absence coding of gene neighborhoods which is blind to complex events modifying gene content. We propose a probabilistic evolutionary model for gene neighborhoods, allowing genes to be inserted, duplicated or lost. It uses reconciled phylogenies, which integrate sequence and gene content evolution. We are then able to optimize parameters such as phylogeny branch lengths, or probabilistic laws depicting the diversity of susceptibility of syntenic regions to rearrangements. We reconstruct a structure for ancestral genomes by optimizing a likelihood, keeping track of all evolutionary events at the level of gene content and gene synteny. Ancestral syntenies are associated with a probability of presence. We implemented the model with the restriction that at most one gene duplication separates two gene speciations in reconciled gene trees. We reconstruct ancestral syntenies on a set of 12 drosophila genomes, and compare the evolutionary rates along the branches and along the sites. We compare with a parsimony method and find a significant number of results not supported by the posterior probability. The model is implemented in the Bio++ library. It thus benefits from and enriches the classical models and methods for molecular evolution.

Abstract Motivations: Recent progress in ancient DNA sequencing technologies and protocols has lead to the sequencing of whole ancient bacterial genomes, as illustrated by the recent sequence of the Yersinia pestis strain that caused the Black Death pandemic. However, sequencing ancient genomes raises specific problems, because of the decay and fragmentation of ancient DNA among others, making the scaffolding of ancient contigs challenging. Results: We show that computational paleogenomics methods aimed at reconstructing the organization of ancestral genomes from the comparison of extant genomes can be adapted to correct, order and orient ancient bacterial contigs. We describe the method FPSAC (fast phylogenetic scaffolding of ancient contigs) and apply it on a set of 2134 ancient contigs assembled from the recently sequenced Black Death agent genome. We obtain a unique scaffold for the whole chromosome of this ancient genome that allows to gain precise insights into the structural evolution of the Yersinia clade. Availability and Implementation: Code, data and results are available at http://paleogenomics.irmacs.sfu.ca/FPSAC. Contact: [email protected] Supplementary information: Supplementary data are available at Bioinformatics online.

BACKGROUND: Reconciled gene trees yield orthology and paralogy relationships between genes. This information may however contradict other information on orthology and paralogy provided by other footprints of evolution, such as conserved synteny. RESULTS: We explore a way to include external information on orthology in the process of gene tree construction. Given an initial gene tree and a set of orthology constraints on pairs of genes or on clades, we give polynomial-time algorithms for producing a modified gene tree satisfying the set of constraints, that is as close as possible to the original one according to the Robinson-Foulds distance. We assess the validity of the modifications we propose by computing the likelihood ratio between initial and modified trees according to sequence alignments on Ensembl trees, showing that often the two trees are statistically equivalent. AVAILABILITY: Software and data available upon request to the corresponding author.

BACKGROUND: Models of ancestral gene order reconstruction have progressively integrated different evolutionary patterns and processes such as unequal gene content, gene duplications, and implicitly sequence evolution via reconciled gene trees. These models have so far ignored lateral gene transfer, even though in unicellular organisms it can have an important confounding effect, and can be a rich source of information on the function of genes through the detection of transfers of clusters of genes. RESULT: We report an algorithm together with its implementation, DeCoLT, that reconstructs ancestral genome organization based on reconciled gene trees which summarize information on sequence evolution, gene origination, duplication, loss, and lateral transfer. DeCoLT optimizes in polynomial time on the number of rearrangements, computed as the number of gains and breakages of adjacencies between pairs of genes. We apply DeCoLT to 1099 gene families from 36 cyanobacteria genomes. CONCLUSION: DeCoLT is able to reconstruct adjacencies in 35 ancestral bacterial genomes with a thousand gene families in a few hours, and detects clusters of co-transferred genes. DeCoLT may also be used with any relationship between genes instead of adjacencies, to reconstruct ancestral interactions, functions or complexes. AVAILABILITY: http://pbil.univ-lyon1.fr/software/DeCoLT/

MOTIVATION: Most models of genome evolution integrating gene duplications, losses and chromosomal rearrangements are computationally intract able, even when comparing only two genomes. This prevents large-scale studies that consider different types of genome structural variations. RESULTS: We define an 'adjacency phylogenetic tree' that describes the evolution of an adjacency, a neighborhood relation between two genes, by speciation, duplication or loss of one or both genes, and rearrangement. We describe an algorithm that, given a species tree and a set of gene trees where the leaves are connected by adjacencies, computes an adjacency forest that minimizes the number of gains and breakages of adjacencies (caused by rearrangements) and runs in polynomial time. We use this algorithm to reconstruct contiguous regions of mammalian and plant ancestral genomes in a few minutes for a dozen species and several thousand genes. We show that this method yields reduced conflict between ancestral adjacencies. We detect duplications involving several genes and compare the different modes of evolution between phyla and among lineages. AVAILABILITY: C++ implementation using BIO++ package, available upon request to Sèverine Bérard. CONTACT: [email protected] or [email protected] SUPPLEMENTARY INFORMATION: Supplementary material is available at Bioinformatics online.

SUMMARY: ANGES is a suite of Python programs that allows reconstructing ancestral genome maps from the comparison of the organization of extant-related genomes. ANGES can reconstruct ancestral genome maps for multichromosomal linear genomes and unichromosomal circular genomes. It implements methods inspired from techniques developed to compute physical maps of extant genomes. Examples of cereal, amniote, yeast or bacteria ancestral genomes are provided, computed with ANGES. AVAILABILITY: ANGES is freely available for download at http://paleogenomics.irmacs.sfu.ca/ANGES/. Documentation and examples are available together with the package. CONTACT: [email protected].

BACKGROUND: Recovering the structure of ancestral genomes can be formalized in terms of properties of binary matrices such as the Consecutive-Ones Property (C1P). The Linearization Problem asks to extract, from a given binary matrix, a maximum weight subset of rows that satisfies such a property. This problem is in general intractable, and in particular if the ancestral genome is expected to contain only linear chromosomes or a unique circular chromosome. In the present work, we consider a relaxation of this problem, which allows ancestral genomes that can contain several chromosomes, each either linear or circular. RESULT: We show that, when restricted to binary matrices of degree two, which correspond to adjacencies, the genomic characters used in most ancestral genome reconstruction methods, this relaxed version of the Linearization Problem is polynomially solvable using a reduction to a matching problem. This result holds in the more general case where columns have bounded multiplicity, which models possibly duplicated ancestral genes. We also prove that for matrices with rows of degrees 2 and 3, without multiplicity and without weights on the rows, the problem is NP-complete, thus tracing sharp tractability boundaries. CONCLUSION: As it happened for the breakpoint median problem, also used in ancestral genome reconstruction, relaxing the definition of a genome turns an intractable problem into a tractable one. The relaxation is adapted to some biological contexts, such as bacterial genomes with several replicons, possibly partially assembled. Algorithms can also be used as heuristics for hard variants. More generally, this work opens a way to better understand linearization results for ancestral genome structure inference.

MOTIVATION: Ancestral genomes provide a better way to understand the structural evolution of genomes than the simple comparison of extant genomes. Most ancestral genome reconstruction methods rely on universal markers, that is, homologous families of DNA segments present in exactly one exemplar in every considered species. Complex histories of genes or other markers, undergoing duplications and losses, are rarely taken into account. It follows that some ancestors are inaccessible by these methods, such as the proto-monocotyledon whose evolution involved massive gene loss following a whole genome duplication. RESULTS: We propose a mapping approach based on the combinatorial notion of 'sandwich consecutive ones matrix', which explicitly takes gene losses into account. We introduce combinatorial optimization problems related to this concept, and propose a heuristic solver and a lower bound on the optimal solution. We use these results to propose a configuration for the proto-chromosomes of the monocot ancestor, and study the accuracy of this configuration. We also use our method to reconstruct the ancestral boreoeutherian genomes, which illustrates that the framework we propose is not specific to plant paleogenomics but is adapted to reconstruct any ancestral genome from extant genomes with heterogeneous marker content. AVAILABILITY: Upon request to the authors. CONTACT: [email protected]; [email protected].

MOTIVATION: The ancestor of birds and mammals lived approximately 300 million years ago. Inferring its genome organization is key to understanding the differentiated evolution of these two lineages. However, detecting traces of its chromosomal organization in its extant descendants is difficult due to the accumulation of molecular evolution since birds and mammals lineages diverged. RESULTS: We address several methodological issues for the detection and assembly of ancestral genomic features of ancient vertebrate genomes, which encompass adjacencies, contiguous segments, syntenies and double syntenies in the context of a whole genome duplication. Using generic, but stringent, methods for all these problems, some of them new, we analyze 15 vertebrate genomes, including 12 amniotes and 3 teleost fishes, and infer a high-resolution genome organization of the amniote ancestral genome, composed of 39 ancestral linkage groups at a resolution of 100 kb. We extensively discuss the validity and robustness of the method to variations of data and parameters. We introduce a support value for each of the groups, and show that 36 out of 39 have maximum support. CONCLUSIONS: Single methodological principle cannot currently be used to infer the organization of the amniote ancestral genome, and we demonstrate that it is possible to gather several principles into a computational paleogenomics pipeline. This strategy offers a solid methodological base for the reconstruction of ancient vertebrate genomes. AVAILABILITY: Source code, in C++ and Python, is available at http://www.cecm.sfu.ca/~cchauve/SUPP/AMNIOTE2010/ CONTACT: [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

SUMMARY: Genomes undergo large structural changes that alter their organization. The chromosomal regions affected by these rearrangements are called breakpoints, while those which have not been rearranged are called synteny blocks. Lemaitre et al. presented a new method to precisely delimit rearrangement breakpoints in a genome by comparison with the genome of a related species. Receiving as input a list of one2one orthologous genes found in the genomes of two species, the method builds a set of reliable and non-overlapping synteny blocks and refines the regions that are not contained into them. Through the alignment of each breakpoint sequence against its specific orthologous sequences in the other species, we can look for weak similarities inside the breakpoint, thus extending the synteny blocks and narrowing the breakpoints. The identification of the narrowed breakpoints relies on a segmentation algorithm and is statistically assessed. Here, we present the package Cassis that implements this method of precise detection of genomic rearrangement breakpoints. AVAILABILITY: Perl and R scripts are freely available for download at http://pbil.univ-lyon1.fr/software/Cassis/. Documentation with methodological background, technical aspects, download and setup instructions, as well as examples of applications are available together with the package. The package was tested on Linux and Mac OS environments and is distributed under the GNU GPL License.

MOTIVATION: When comparing the organization of two genomes, it is important not to draw conclusions on their modes of evolution from a single most parsimonious scenario explaining their differences. Better estimations can be obtained by sampling many different genomic rearrangement scenarios. For this problem, the Double Cut and Join (DCJ) model, while less relevant, is computationally easier than the Hannenhalli-Pevzner (HP) model. Indeed, in some special cases, the total number of DCJ sorting scenarios can be analytically calculated, and uniformly distributed random DCJ scenarios can be drawn in polynomial running time, while the complexity of counting the number of HP scenarios and sampling from the uniform distribution of their space is unknown, and conjectured to be #P-complete. Statistical methods, like Markov chain Monte Carlo (MCMC) for sampling from the uniform distribution of the most parsimonious or the Bayesian distribution of all possible HP scenarios are required. RESULTS: We use the computational facilities of the DCJ model to draw a sampling of HP scenarios. It is based on a parallel MCMC method that cools down DCJ scenarios to HP scenarios. We introduce two theorems underlying the theoretical mixing properties of this parallel MCMC method. The method was tested on yeast and mammalian genomic data, and allowed us to provide estimates of the different modes of evolution in diverse lineages. AVAILABILITY: The program implemented in Java 1.5 programming language is available from http://www.renyi.hu/~miklosi/DCJ2HP/.

BACKGROUND: To understand the evolutionary role of Lateral Gene Transfer (LGT), accurate methods are needed to identify transferred genes and infer their timing of acquisition. Phylogenetic methods are particularly promising for this purpose, but the reconciliation of a gene tree with a reference (species) tree is computationally hard. In addition, the application of these methods to real data raises the problem of sorting out real and artifactual phylogenetic conflict. RESULTS: We present Prunier, a new method for phylogenetic detection of LGT based on the search for a maximum statistical agreement forest (MSAF) between a gene tree and a reference tree. The program is flexible as it can use any definition of "agreement" among trees. We evaluate the performance of Prunier and two other programs (EEEP and RIATA-HGT) for their ability to detect transferred genes in realistic simulations where gene trees are reconstructed from sequences. Prunier proposes a single scenario that compares to the other methods in terms of sensitivity, but shows higher specificity. We show that LGT scenarios carry a strong signal about the position of the root of the species tree and could be used to identify the direction of evolutionary time on the species tree. We use Prunier on a biological dataset of 23 universal proteins and discuss their suitability for inferring the tree of life. CONCLUSIONS: The ability of Prunier to take into account branch support in the process of reconciliation allows a gain in complexity, in comparison to EEEP, and in accuracy in comparison to RIATA-HGT. Prunier's greedy algorithm proposes a single scenario of LGT for a gene family, but its quality always compares to the best solutions provided by the other algorithms. When the root position is uncertain in the species tree, Prunier is able to infer a scenario per root at a limited additional computational cost and can easily run on large datasets.Prunier is implemented in C++, using the Bio++ library and the phylogeny program Treefinder. It is available at: http://pbil.univ-lyon1.fr/software/prunier.

BACKGROUND: Genome median and genome halving are combinatorial optimization problems that aim at reconstructing ancestral genomes as well as the evolutionary events leading from the ancestor to extant species. Exploring complexity issues is a first step towards devising efficient algorithms. The complexity of the median problem for unichromosomal genomes (permutations) has been settled for both the breakpoint distance and the reversal distance. Although the multichromosomal case has often been assumed to be a simple generalization of the unichromosomal case, it is also a relaxation so that complexity in this context does not follow from existing results, and is open for all distances. RESULTS: We settle here the complexity of several genome median and halving problems, including a surprising polynomial result for the breakpoint median and guided halving problems in genomes with circular and linear chromosomes, showing that the multichromosomal problem is actually easier than the unichromosomal problem. Still other variants of these problems are NP-complete, including the DCJ double distance problem, previously mentioned as an open question. We list the remaining open problems. CONCLUSION: This theoretical study clears up a wide swathe of the algorithmical study of genome rearrangements with multiple multichromosomal genomes.

BACKGROUND: Genomes undergo large structural changes that alter their organisation. The chromosomal regions affected by these rearrangements are called breakpoints, while those which have not been rearranged are called synteny blocks. We developed a method to precisely delimit rearrangement breakpoints on a genome by comparison with the genome of a related species. Contrary to current methods which search for synteny blocks and simply return what remains in the genome as breakpoints, we propose to go further and to investigate the breakpoints themselves in order to refine them. RESULTS: Given some reliable and non overlapping synteny blocks, the core of the method consists in refining the regions that are not contained in them. By aligning each breakpoint sequence against its specific orthologous sequences in the other species, we can look for weak similarities inside the breakpoint, thus extending the synteny blocks and narrowing the breakpoints. The identification of the narrowed breakpoints relies on a segmentation algorithm and is statistically assessed. Since this method requires as input synteny blocks with some properties which, though they appear natural, are not verified by current methods for detecting such blocks, we further give a formal definition and provide an algorithm to compute them. The whole method is applied to delimit breakpoints on the human genome when compared to the mouse and dog genomes. Among the 355 human-mouse and 240 human-dog breakpoints, 168 and 146 respectively span less than 50 Kb. We compared the resulting breakpoints with some publicly available ones and show that we achieve a better resolution. Furthermore, we suggest that breakpoints are rarely reduced to a point, and instead consist in often large regions that can be distinguished from the sequences around in terms of segmental duplications, similarity with related species, and transposable elements. CONCLUSION: Our method leads to smaller breakpoints than already published ones and allows for a better description of their internal structure. In the majority of cases, our refined regions of breakpoint exhibit specific biological properties (no similarity, presence of segmental duplications and of transposable elements). We hope that this new result may provide some insight into the mechanism and evolutionary properties of chromosomal rearrangements.

The reconstruction of ancestral genome architectures and gene orders from homologies between extant species is a long-standing problem, considered by both cytogeneticists and bioinformaticians. A comparison of the two approaches was recently investigated and discussed in a series of papers, sometimes with diverging points of view regarding the performance of these two approaches. We describe a general methodological framework for reconstructing ancestral genome segments from conserved syntenies in extant genomes. We show that this problem, from a computational point of view, is naturally related to physical mapping of chromosomes and benefits from using combinatorial tools developed in this scope. We develop this framework into a new reconstruction method considering conserved gene clusters with similar gene content, mimicking principles used in most cytogenetic studies, although on a different kind of data. We implement and apply it to datasets of mammalian genomes. We perform intensive theoretical and experimental comparisons with other bioinformatics methods for ancestral genome segments reconstruction. We show that the method that we propose is stable and reliable: it gives convergent results using several kinds of data at different levels of resolution, and all predicted ancestral regions are well supported. The results come eventually very close to cytogenetics studies. It suggests that the comparison of methods for ancestral genome reconstruction should include the algorithmic aspects of the methods as well as the disciplinary differences in data acquisition.

In comparative genomics, algorithms that sort permutations by reversals are often used to propose evolutionary scenarios of rearrangements between species. One of the main problems of such methods is that they give one solution while the number of optimal solutions is huge, with no criteria to discriminate among them. Bergeron et al. started to give some structure to the set of optimal solutions, in order to be able to deliver more presentable results than only one solution or a complete list of all solutions. However, no algorithm exists so far to compute this structure except through the enumeration of all solutions, which takes too much time even for small permutations. Bergeron et al. state as an open problem the design of such an algorithm. We propose in this paper an answer to this problem, that is, an algorithm which gives all the classes of solutions and counts the number of solutions in each class, with a better theoretical and practical complexity than the complete enumeration method. We give an example of how to reduce the number of classes obtained, using further constraints. Finally, we apply our algorithm to analyse the possible scenarios of rearrangement between mammalian sex chromosomes.

In comparative genomics, gene order data is often modeled as signed permutations. A classical problem for genome comparison is to detect common intervals in permutations, that is, genes that are colocalized in several species, indicating that they remained grouped during evolution. A second largely studied problem related to gene order is to compute a minimum scenario of reversals that transforms a signed permutation into another. Several studies began to mix the two problems and it was observed that their results are not always compatible: Often, parsimonious scenarios of reversals break common intervals. If a scenario does not break any common interval, it is called perfect. In two recent studies, Bérard et al. defined a class of permutations for which building a perfect scenario of reversals sorting a permutation was achieved in polynomial time and stated as an open question whether it is possible to decide, given a permutation, if there exists a minimum scenario of reversals that is perfect. In this paper, we give a solution to this problem and prove that this widens the class of permutations addressed by the aforementioned studies. We implemented and tested this algorithm on gene order data of chromosomes from several mammal species and we compared it to other methods. The algorithm helps to choose among several possible scenarios of reversals and indicates that the minimum scenario of reversals is not always the most plausible.