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