Abstract
BackgroundThe 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.ResultsIn 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.ConclusionsWe 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.
Highlights
The double-cut-and-join (DCJ) is a model that is able to efficiently sort a genome into another, generalizing the typical mutations to which genomes are subject, but allowing the existence of circular chromosomes at the intermediate steps
No algorithm was provided up to now and even the question whether the distance is the same for both the general and the restricted DCJ-indel models remains open. We address this issue and give a sorting algorithm and a tight upper bound for the restricted DCJ-indel distance, allowing the assignment of distinct costs to indel and DCJ operations and with the restriction that the indel cost is upper bounded by the DCJ cost
Many different recombinations can occur, it is possible to explore the space of recombinations in linear time and compute the maximum deduction that we can obtain with respect to the upper bound of Lemma 1 [6]
Summary
We develop a restricted DCJ-indel sorting algorithm, from which we can derive an upper bound for the restricted DCJ-indel distance. We can always reincorporate the circular chromosome with a DCJ applied to any short-link (v1, v2), except if r(v1, v2) splits a run r that is already inversely split and r(v1, v2) cannot be chained with a previous inverted-split of r. In this case, r will be separated alone in a cycle (each run is immediately separated after its first invertedsplit). This algorithm results in a sorting sequence in the restricted model that has exactly the same cost given by the upper bound of Lemma 1
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