Sorting cancer karyotypes using double-cut- and-joins, duplications and deletions

Motivation: Problems of genome rearrangement are central in both evolution and cancer research. Most genome rearrangement models assume that the genome contains a single copy of each gene and the only changes in the genome are structural, i.e. reordering of segments. In contrast, tumor genomes also undergo numerical changes such as deletions and duplications, and thus the number of copies of genes varies. Dealing with unequal gene content is a very challenging task, addressed by few algorithms to date. More realistic models are needed to help trace genome evolution during tumorigenesis. Results: Here, we present a model for the evolution of genomes with multiple gene copies using the operation types double-cut-and-joins, duplications and deletions. The events supported by the model are reversals, translocations, tandem duplications, segmental deletions and chromosomal amplifications and deletions, covering most types of structural and numerical changes observed in tumor samples. Our goal is to find a series of operations of minimum length that transform one karyotype into the other. We show that the problem is NP-hard and give an integer linear programming formulation that solves the problem exactly under some mild assumptions. We test our method on simulated genomes and on ovarian cancer genomes. Our study advances the state of the art in two ways: It allows a broader set of operations than extant models, thus being more realistic and it is the first study attempting to re-construct the full sequence of structural and numerical events during cancer evolution.