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Abstract Background Classical approaches to compute the genomic distance are usually limited to genomes with the same content, without duplicated markers. However, differences in the gene content are frequently observed and can reflect important evolutionary aspects. A few polynomial time algorithms that include genome rearrangements, insertions and deletions (or substitutions) were already proposed. These methods often allow a block of contiguous markers to be inserted, deleted or substituted at once but result in distance functions that do not respect the triangular inequality and hence do not constitute metrics. Results In the present study we discuss the disruption of the triangular inequality in some of the available methods and give a framework to establish an efficient correction for two models recently proposed, one that includes insertions, deletions and double cut and join (DCJ) operations, and one that includes substitutions and DCJ operations. Conclusions We show that the proposed framework establishes the triangular inequality in both distances, by summing a surcharge on indel operations and on substitutions that depends only on the number of markers affected by these operations. This correction can be applied a posteriori, without interfering with the already available formulas to compute these distances. We claim that this correction leads to distances that are biologically more plausible.

Abstract Background The distance between two genomes is often computed by comparing only the common markers between them. Some approaches are also able to deal with non-common markers, allowing the insertion or the deletion of such markers. In these models, a deletion and a subsequent insertion that occur at the same position of the genome count for two sorting steps. Results Here we propose a new model that sorts non-common markers with substitutions, which are more powerful operations that comprehend insertions and deletions. A deletion and an insertion that occur at the same position of the genome can be modeled as a substitution, counting for a single sorting step. Conclusions Comparing genomes with unequal content, but without duplicated markers, we give a linear time algorithm to compute the genomic distance considering substitutions and double-cut-and-join (DCJ) operations. This model provides a parsimonious genomic distance to handle genomes free of duplicated markers, that is in practice a lower bound to the real genomic distances. The method could also be used to refine orthology assignments, since in some cases a substitution could actually correspond to an unannotated orthology.

Structural variation in genomes can be revealed by many (dis)similarity measures. Rearrangement operations, such as the so called double-cut-and-join (DCJ), are large-scale mutations that can create complex changes and produce such variations in genomes. A basic task in comparative genomics is to find the rearrangement distance between two given genomes, i.e., the minimum number of rearragement operations that transform one given genome into another one. In a family-based setting, genes are grouped into gene families and efficient algorithms have already been presented to compute the DCJ distance between two given genomes. In this work we propose the problem of computing the DCJ distance of two given genomes without prior gene family assignment, directly using the pairwise similarities between genes. We prove that this new family-free DCJ distance problem is APX-hard and provide an integer linear program to its solution. We also study a family-free DCJ similarity and prove that its computation is NP-hard. [ABSTRACT FROM AUTHOR]

Background: The inversion distance, that is the distance between two unichromosomal genomes with the same content allowing only inversions of DNA segments, can be computed thanks to a pioneering approach of Hannenhalli and Pevzner in 1995. In 2000, El-Mabrouk extended the inversion model to allow the comparison of unichromosomal genomes with unequal contents, thus insertions and deletions of DNA segments besides inversions. However, an exact algorithm was presented only for the case in which we have insertions alone and no deletion (or vice versa), while a heuristic was provided for the symmetric case, that allows both insertions and deletions and is called the inversion-indel distance. In 2005, Yancopoulos, Attie and Friedberg started a new branch of research by introducing the generic double cut and join (DCJ) operation, that can represent several genome rearrangements (including inversions). Among others, the DCJ model gave rise to two important results. First, it has been shown that the inversion distance can be computed in a simpler way with the help of the DCJ operation. Second, the DCJ operation originated the DCJ-indel distance, that allows the comparison of genomes with unequal contents, considering DCJ, insertions and deletions, and can be computed in linear time. Results: In the present work we put these two results together to solve an open problem, showing that, when the graph that represents the relation between the two compared genomes has no bad components, the inversion-indel distance is equal to the DCJ-indel distance. We also give a lower and an upper bound for the inversion-indel distance in the presence of bad components. [ABSTRACT FROM AUTHOR]

Background: Classical approaches to compute the genomic distance are usually limited to genomes with the same content and take into consideration only rearrangements that change the organization of the genome (i.e. positions and orientation of pieces of DNA, number and type of chromosomes, etc.), such as inversions, translocations, fusions and fissions. These operations are generically represented by the double-cut and join (DCJ) operation. The distance between two genomes, in terms of number of DCJ operations, can be computed in linear time. In order to handle genomes with distinct contents, also insertions and deletions of fragments of DNA – named indels – must be allowed. More powerful than an indel is a substitution of a fragment of DNA by another fragment of DNA. Indels and substitutions are called content-modifying operations. It has been shown that both the DCJ-indel and the DCJ-substitution distances can also be computed in linear time, assuming that the same cost is assigned to any DCJ or content-modifying operation. Results: In the present study we extend the DCJ-indel and the DCJ-substitution models, considering that the content-modifying cost is distinct from and upper bounded by the DCJ cost, and show that the distance in both models can still be computed in linear time. Although the triangular inequality can be disrupted in both models, we also show how to efficiently fix this problem a posteriori. [ABSTRACT FROM AUTHOR]

The computation of genomic distances has been a very active field of computational comparative genomics over the past 25 years. Substantial results include the polynomial-time computability of the inversion distance by Hannenhalli and Pevzner in 1995 and the introduction of the double cut and join distance by Yancopoulos et al. in 2005. Both results, however, rely on the assumption that the genomes under comparison contain the same set of unique markers (syntenic genomic regions, sometimes also referred to as genes). In 2015, Shao et al. relax this condition by allowing for duplicate markers in the analysis. This generalized version of the genomic distance problem is NP-hard, and they give an integer linear programming (ILP) solution that is efficient enough to be applied to real-world datasets. A restriction of their approach is that it can be applied only to balanced genomes that have equal numbers of duplicates of any marker. Therefore, it still needs a delicate preprocessing of the input data in which excessive copies of unbalanced markers have to be removed. In this article, we present an algorithm solving the genomic distance problem for natural genomes, in which any marker may occur an arbitrary number of times. Our method is based on a new graph data structure, the multi-relational diagram, that allows an elegant extension of the ILP by Shao et al. to count runs of markers that are under- or over-represented in one genome with respect to the other and need to be inserted or deleted, respectively. With this extension, previous restrictions on the genome configurations are lifted, for the first time enabling an uncompromising rearrangement analysis. Any marker sequence can directly be used for the distance calculation. The evaluation of our approach shows that it can be used to analyze genomes with up to a few 10,000 markers, which we demonstrate on simulated and real data.

Rearrangements are mutations that can change the organization of a genome, but not its content. Examples are inversions of DNA segments, translocations of chromosome ends, fusions and fissions of chromosomes. All mentioned rearrangements can be represented by the generic Double Cut and Join (DCJ) operation. However, the DCJ operation also allows circular chromosomes to be created at intermediate steps, even if the compared genomes are linear. In this case it is more plausible to consider a restriction in which the reincorporation of a circular chromosome has to be done immediately after its creation. We call these two consecutive operations an ER composition. It has been shown that an ER composition mimics either an internal block interchange (when two segments in the same chromosome exchange their positions), or an internal transposition (the special case of a block interchange when the two segments are adjacent). The DCJ distance of two genomes is the same, regardless of this restriction, and can be computed in linear time. For comparing two genomes with unequal contents, in addition to rearrangements we have to allow insertions and deletions of DNA segments-named indels. It is already known that the distance in the model combining DCJ and indel operations can be exactly computed. Again, for linear genomes it would be more plausible to adopt a restricted version with ER compositions. This model was studied recently by da Silva et al. (BMC Bioinformatics 13, Suppl. 19, S14, 2012), but only an upper bound for the restricted DCJ-indel distance was provided. Here we first solve an open problem posed in that paper and present a very simple proof showing that the distance, which can be computed in linear time, is the same for both the unrestricted and the restricted DCJ-indel models. We then give a simpler algorithm for computing an optimal restricted DCJ-indel sorting scenario in O(n log n) time. We also relate the DCJ-indel distance to the restricted DCJ-substitution distance, which instead of indels considers a more powerful operation that allows the substitution of a DNA segment by another DNA segment. We show that the DCJ-indel distance is a 2-approximation for the restricted DCJ-substitution distance.

Background: The double-cut-and-join (DCJ) is a model that is able to efficiently sort a genome into another, generalizing the typical mutations (inversions, fusions, fissions, translocations) to which genomes are subject, but allowing the existence of circular chromosomes at the intermediate steps. In the general model many circular chromosomes can coexist in some intermediate step. However, when the compared genomes are linear, it is more plausible to use the so-called restricted DCJ model, in which we proceed the reincorporation of a circular chromosome immediately after its creation. These two consecutive DCJ operations, which create and reincorporate a circular chromosome, mimic a transposition or a block-interchange. When the compared genomes have the same content, it is known that the genomic distance for the restricted DCJ model is the same as the distance for the general model. If the genomes have unequal contents, in addition to DCJ it is necessary to consider indels, which are insertions and deletions of DNA segments. Linear time algorithms were proposed to compute the distance and to find a sorting scenario in a general, unrestricted DCJ-indel model that considers DCJ and indels., Results: In the present work we consider the restricted DCJ-indel model for sorting linear genomes with unequal contents. We allow DCJ operations and indels with the following constraint: if a circular chromosome is created by a DCJ, it has to be reincorporated in the next step (no other DCJ or indel can be applied between the creation and the reincorporation of a circular chromosome). We then develop a sorting algorithm and give a tight upper bound for the restricted DCJ-indel distance., Conclusions: We have given a tight upper bound for the restricted DCJ-indel distance. The question whether this bound can be reduced so that both the general and the restricted DCJ-indel distances are equal remains open.