Heuristics for the inversion median problem

Background: The study of genome rearrangements has become a mainstay of phylogenetics and comparative genomics. Fundamental in such a study is the median problem: given three gene arrangements, find a fourth that minimizes the sum of the evolutionary distances between itself and the given three. Many exact algorithms and heuristics have been developped for the inversion median problem, of which the best known is MGR. Results: We present a unifying framework for median heuristics, which enables us to clarify existing strategies and to place them in a partial ordering. Analysis of this framework leads to a new insight: the best strategies continue to refer to the input data rather than just to updated estimates. Using this insight, we develop a new heuristic for inversion medians that uses input data to the end of its computation and leverages our previous work with DCJ medians. Finally, we present the results of extensive experimentation showing that our new heuristic outperforms all others in accuracy and, especially, in running time: the heuristic typically returns solutions within 1 % of optimal and runs in seconds to minutes even on genomes with 25’000 genes—in contrast, MGR can take days on instances of 200 genes and cannot be used beyond 1’000 genes. Conclusions: Finding good rearrangement medians, in particular inversion medians, had long been regarded as the computational bottleneck in whole-genome studies. Our new heuristic for inversion medians, ASM, which dominates all others in our framework, puts that issue to rest by providing near-optimal solutions within seconds to minutes on even the largest genomes

Bridging Disparate Views on the DCJ-Indel Model for a Capping-Free Solution to the Natural Distance Problem

One of the most fundamental problems in genome rearrangement is the (genomic) distance problem. It is typically formulated as finding the minimum number of rearrangements under a model that are needed to transform one genome into the other. A powerful multi-chromosomal model is the Double Cut and Join (DCJ) model. While the DCJ model is not able to deal with some situations that occur in practice, like duplicated or lost regions, it was extended over time to handle these cases. First, it was extended to the DCJ-indel model, solving the issue of lost markers. Later ILP-solutions for so called natural genomes, in which each genomic region may occur an arbitrary number of times, were developed, enabling in theory to solve the distance problem for any pair of genomes. However, some theoretical and practical issues remained unsolved. On the theoretical side of things, there exist two disparate views of the DCJ-indel model, motivated in the same way, but with different conceptualizations that could not be reconciled so far. On the practical side, while the solutions for natural genomes typically perform well on telomere to telomere resolved genomes, they have been shown in recent years to quickly loose performance on genomes with a large number of contigs or linear chromosomes. This has been linked to a particular technique increasing the solution space superexponentially named capping. Recently, we introduced a new conceptualization of the DCJ-indel model within the context of another rearrangement problem. In this manuscript, we will apply this new conceptualization to the distance problem. In doing this, we uncover the relation between the disparate conceptualizations of the DCJ-indel model. We are also able to derive an ILP solution to the distance problem that does not rely on capping and therefore significantly improves upon the performance of previous solutions for genomes with high numbers of contigs while still solving the problem exactly. To the best of our knowledge, our approach is the first allowing for an exact computation of the DCJ-indel distance for natural genomes with large numbers of linear chromosomes. We demonstrate the performance advantage as well as limitations in comparison to an existing solution on simulated genomes as well as showing its practical usefulness in an analysis of 11 Drosophila genomes

On the weight of indels in genomic distances

Dias Vieira Braga M, Machado R, Ribeiro LC, Stoye J. On the weight of indels in genomic distances. BMC Bioinformatics. 2011;12(Suppl 9: RECOMB-CG 2011): S13.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

On the Inversion-Indel Distance

Willing E, Zaccaria S, Dias Vieira Braga M, Stoye J. On the Inversion-Indel Distance. BMC Bioinformatics. 2013;14(Suppl 15: Proc. of RECOMB-CG 2013): S3.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