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The most common way to calculate the rearrangement distance between two genomes is to use the size of a minimum length sequence of rearrangements that transforms one of the two given genomes into the other, where the genomes are represented as permutations using only their gene order, based on the assumption that genomes have the same gene content. With the advance of research in genome rearrangements, new works extended the classical models by either considering genomes with different gene content (unbalanced genomes) or including more genomic characteristics to the mathematical representation of the genomes, such as the distribution of intergenic regions sizes. In this study, we study the Reversal, Transposition, and Indel (Insertion and Deletion) Distance using intergenic information, which allows comparing unbalanced genomes, because indels are included in the rearrangement model (i.e., the set of possible rearrangements allowed when we compute the distance). For the particular case of transpositions and indels on unbalanced genomes, we present a 4-approximation algorithm, improving a previous 4.5 approximation. This algorithm is extended so as to deal with gene orientation and to maintain the 4-approximation factor for the Reversal, Transposition, and Indel Distance on unbalanced genomes. Furthermore, we evaluate the proposed algorithms using experiments on simulated data. [ABSTRACT FROM AUTHOR]

In the comparative genomics field, one way to infer the evolutionary distance between two organisms of related species is by finding the minimum number of large-scale mutations, called genome rearrangements, that transform one genome into the other. This number is referred to as the rearrangement distance. Since problems in this area emerged in the mid-1990s, several genome rearrangements have been proposed. Rearrangements that do not alter the genome content are called conservative, and in this group we have the following: the reversal, which inverts a segment of the genome; the transposition, which exchanges two consecutive segments; and the double cut and join, which cuts two different pairs of adjacent blocks and joins them differently. Seminal works compared genomes sharing the same set of conserved blocks, but nowadays, researchers started looking at genomes with unequal gene content, by allowing the use of nonconservative rearrangements such as insertion and deletion (jointly called indel). The transposition distance and the transposition and indel distance are both NP-hard. We investigate the transposition and indel distance and present a structure called labeled cycle graph, representing an instance of rearrangement distance problems for genomes with unequal gene content. This structure is used to devise a lower bound and a 2-approximation algorithm for the transposition and indel distance. [ABSTRACT FROM AUTHOR]

Problems in the genome rearrangement field are often formulated in terms of pairwise genome comparison: given two genomes G 1 and G 2 , find the minimum number of genome rearrangements that may have occurred during the evolutionary process. This broad definition lacks at least two important considerations: the first being which features are extracted from genomes to create a useful mathematical model, and the second being which types of genome rearrangement events should be represented. Regarding the first consideration, seminal works in the genome rearrangement field solely used gene order to represent genomes as permutations of integer numbers, neglecting many important aspects like gene duplication, intergenic regions, and complex interactions between genes. Regarding the second consideration, some rearrangement events are widely studied such as reversals and transpositions. In this paper, we shed light on the first consideration and created a model that takes into account gene order and the number of nucleotides in intergenic regions. In addition, we consider events of reversals, transpositions, and indels (insertions and deletions) of genomic material. We present a 4-approximation algorithm for reversals and indels, a 4. 5 -approximation algorithm for transpositions and indels, and a 6-approximation for reversals, transpositions, and indels. [ABSTRACT FROM AUTHOR]

The rearrangement distance is a well-known problem in the field of comparative genomics. Given two genomes, the rearrangement distance is the minimum number of rearrangements in a set of allowed rearrangements (rearrangement model), which transforms one genome into the other. In rearrangement distance problems, a genome is modeled as a string, where each element represents a conserved region within the two genomes. When the orientation of the genes is known, it is represented by (plus or minus) signs assigned to the elements of the string. Two of the most studied rearrangements are reversals, which invert a segment of the genome, and transpositions, which exchange the relative positions of two adjacent segments of the genome. The first works in genome rearrangements considered that the genomes being compared had the same genetic material and that rearrangement events were restricted to reversals, transpositions, or both. El-Mabrouk extended the reversal model on signed strings to include the operations of insertion and deletion of segments in the genome, which allowed the comparison of genomes with different genetic material. Other studies also addressed this problem and, recently, this problem was proved to be solvable in polynomial time by Willing et al. For unsigned strings, we still observe a lack of results. That said, in this study we prove that computing the rearrangement distance for the following models is NP-Hard: reversals and indels on unsigned strings; transpositions and indels on unsigned strings; and reversals, transpositions, and indels on signed and unsigned strings. Along with the NP-hardness proofs, we present a 2-approximation algorithm for reversals on unsigned strings and 3-approximation algorithms for the other models. [ABSTRACT FROM AUTHOR]