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Genome rearrangement events are widely used to estimate a minimum-size sequence of mutations capable of transforming a genome into another. The length of this sequence is called distance, and determining it is the main goal in genome rearrangement distance problems. Problems in the genome rearrangement field differ regarding the set of rearrangement events allowed and the genome representation. In this work, we consider the scenario where the genomes share the same set of genes, gene orientation is known or unknown, and intergenic regions (structures between a pair of genes and at the extremities of the genome) are taken into account. We use two models, the first model allows only conservative events (reversals and moves), and the second model includes non-conservative events (insertions and deletions) in the intergenic regions. We show that both models result in NP-hard problems no matter if gene orientation is known or unknown. When the information regarding the orientation of genes is available, we present for both models an approximation algorithm with a factor of 2. For the scenario where this information is unavailable, we propose a 4-approximation algorithm for both models. [ABSTRACT FROM AUTHOR]

Genome Rearrangements are events that affect large stretches of genomes during evolution. Many mathematical models have been used to estimate the evolutionary distance between two genomes based on genome rearrangements. However, most of them focused on the (order of the) genes of a genome, disregarding other important elements in it. Recently, researchers have shown that considering regions between each pair of genes, called intergenic regions, can enhance distance estimation in realistic data. Two of the most studied genome rearrangements are the reversal, which inverts a sequence of genes, and the transposition, which occurs when two adjacent gene sequences swap their positions inside the genome. In this work, we study the transposition distance between two genomes, but we also consider intergenic regions, a problem we name Sorting by Intergenic Transpositions. We show that this problem is NP-hard and propose two approximation algorithms, with factors 3.5 and 2.5, considering two distinct definitions for the problem. We also investigate the signed reversal and transposition distance between two genomes considering their intergenic regions. This second problem is called Sorting by Signed Intergenic Reversals and Intergenic Transpositions. We show that this problem is NP-hard and develop two approximation algorithms, with factors 3 and 2.5. We check how these algorithms behave when assigning weights for genome rearrangements. Finally, we implemented all these algorithms and tested them on real and simulated data. [ABSTRACT FROM AUTHOR]

Genome rearrangements are mutations affecting large portions of a genome, and a reversal is one of the most studied genome rearrangements in the literature through the Sorting by Reversals (SbR) problem. SbR is solvable in polynomial time on signed permutations (i.e., the gene orientation is known), and it is NP-hard on unsigned permutations. This problem (and many others considering genome rearrangements) models genome as a list of its genes in the order they appear, ignoring all other information present in the genome. Recent works claimed that the incorporation of the size of intergenic regions, i.e., sequences of nucleotides between genes, may result in better estimators for the real distance between genomes. Here we introduce the Sorting Signed Permutations by Intergenic Reversals problem, that sorts a signed permutation using reversals both on gene order and intergenic sizes. We show that this problem is NP-hard by a reduction from the 3-partition problem. Then, we propose a 2-approximation algorithm for it. Finally, we also incorporate intergenic indels (i.e., insertions or deletions of intergenic regions) to overcome a limitation of sorting by conservative events (such as reversals) and propose two approximation algorithms. [ABSTRACT FROM AUTHOR]

During the evolutionary process, genomes are affected by various genome rearrangements, that is, events that modify large stretches of the genetic material. In the literature, a large number of models have been proposed to estimate the number of events that occurred during evolution; most of them represent a genome as an ordered sequence of genes, and, in particular, disregard the genetic material between consecutive genes. However, recent studies showed that taking into account the genetic material between consecutive genes can enhance evolutionary distance estimations. Reversal and transposition are genome rearrangements that have been widely studied in the literature. A reversal inverts a (contiguous) segment of the genome, while a transposition swaps the positions of two consecutive segments. Genomes also undergo nonconservative events (events that alter the amount of genetic material) such as insertions and deletions, in which genetic material from intergenic regions of the genome is inserted or deleted, respectively. In this article, we study a genome rearrangement model that considers both gene order and sizes of intergenic regions. We investigate the reversal distance, and also the reversal and transposition distance between two genomes in two scenarios: with and without nonconservative events. We show that these problems are NP-hard and we present constant ratio approximation algorithms for all of them. More precisely, we provide a 4-approximation algorithm for the reversal distance, both in the conservative and nonconservative versions. For the reversal and transposition distance, we provide a 4.5-approximation algorithm, both in the conservative and nonconservative versions. We also perform experimental tests to verify the behavior of our algorithms, as well as to compare the practical and theoretical results. We finally extend our study to scenarios in which events have different costs, and we present constant ratio approximation algorithms for each scenario. [ABSTRACT FROM AUTHOR]

In comparative genomics, rearrangements are mutations that affect a stretch of DNA sequences. Reversals and transpositions are well-known rearrangements, and each has a vast literature. The reversal and transposition distance, that is, the minimum number of reversals and transpositions needed to transform one genome into another is a relevant evolutionary distance. The problem of computing this distance when genomes are represented by permutations was proposed >20 years ago and received the name of sorting by reversals and transpositions problem. It has been the focus of a number of studies, but the computational complexity has remained open until now. We hereby solve this question and prove that it is NP-hard no matter whether genomes are represented by signed or unsigned permutations. In addition, we prove that a usual generalization of this problem, which assigns weights wρ for reversals and wτ for transpositions, is also NP-hard as long as wτ/wρ ≤ 1.5 for both signed and unsigned permutations. [ABSTRACT FROM AUTHOR]