An Exact Algorithm for Sorting by Weighted Preserving Genome Rearrangements

New algorithms for structure informed genome rearrangement

We define two new computational problems in the domain of perfect genome rearrangements, and propose three algorithms to solve them. The rearrangement scenarios modeled by the problems consider Reversal and Block Interchange operations, and a PQ-tree is utilized to guide the allowed operations and to compute their weights. In the first problem, [Formula: see text] ([Formula: see text]), we define the basic structure-informed rearrangement measure. Here, we assume that the gene order members of the gene cluster from which the PQ-tree is constructed are permutations. The PQ-tree representing the gene cluster is ordered such that the series of gene IDs spelled by its leaves is equivalent to that of the reference gene order. Then, a structure-informed genome rearrangement distance is computed between the ordered PQ-tree and the target gene order. The second problem, [Formula: see text] ([Formula: see text]), generalizes [Formula: see text], where the gene order members are not necessarily permutations and the structure informed rearrangement measure is extended to also consider up to [Formula: see text] and [Formula: see text] gene insertion and deletion operations, respectively, when modelling the PQ-tree informed divergence process from the reference gene order to the target gene order. The first algorithm solves [Formula: see text] in [Formula: see text] time and [Formula: see text] space, where [Formula: see text] is the maximum number of children of a node, n is the length of the string and the number of leaves in the tree, and [Formula: see text] and [Formula: see text] are the number of P-nodes and Q-nodes in the tree, respectively. If one of the penalties of [Formula: see text] is 0, then the algorithm runs in [Formula: see text] time and [Formula: see text] space. The second algorithm solves [Formula: see text] in [Formula: see text] time and [Formula: see text] space, where [Formula: see text] is the maximum number of children of a node, n is the length of the string, m is the number of leaves in the tree, [Formula: see text] and [Formula: see text] are the number of P-nodes and Q-nodes in the tree, respectively, and allowing up to [Formula: see text] deletions from the tree and up to [Formula: see text] deletions from the string. The third algorithm is intended to reduce the space complexity of the second algorithm. It solves a variant of the problem (where one of the penalties of [Formula: see text] is 0) in [Formula: see text] time and [Formula: see text] space. The algorithm is implemented as a software tool, denoted MEM-Rearrange, and applied to the comparative and evolutionary analysis of 59 chromosomal gene clusters extracted from a dataset of 1487 prokaryotic genomes.

Labeled Cycle Graph for Transposition and Indel Distance

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.

Sorting Signed Permutations by Inverse Tandem Duplication Random Losses

Gene order evolution of unichromosomal genomes, for example mitochondrial genomes, has been modelled mostly by four major types of genome rearrangements: inversions, transpositions, inverse transpositions, and tandem duplication random losses. Generalizing models that include all those rearrangements while admitting computational tractability are rare. In this paper, we study such a rearrangement model, namely the inverse tandem duplication random loss (iTDRL) model, where an iTDRL duplicates and inverts a continuous segment of a gene order followed by the random loss of one of the redundant copies of each gene. The iTDRL rearrangement has currently been proposed by several authors suggesting it to be a possible mechanisms of mitochondrial gene order evolution. We initiate the algorithmic study of this new model of genome rearrangement by proving that a shortest rearrangement scenario that transforms one given gene order into another given gene order can be obtained in quasilinear time. Furthermore, we show that the length of such a scenario, i.e., the minimum number of iTDRLs in the transformation, can be computed in linear time.

Sorting Permutations by Intergenic Operations

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.

Heuristics for the Reversal and Transposition Distance Problem

We present three heuristics-Sliding Window, Look Ahead, and Iterative Sliding Window-to improve solutions for the Sorting Signed Permutations by Reversals and Transpositions Problem. We investigate the classical version of the problem as well as versions restricted to prefix and prefix or suffix operations. To assess the heuristics based on its improvement, we implemented algorithms described in the literature to provide initial solutions. Although we have a limited number of problems, these heuristics can be applied to many others within the area of genome rearrangement. When time is a crucial factor, Sliding Window is a better choice because it runs in linear time. If the quality of the solution is a priority, Look Ahead should be preferred. Iterative Sliding Window is the most flexible heuristic and allows us to find a trade-off for specific scenarios where running time and solution quality are relevant.

EqualTDRL: illustrating equivalent tandem duplication random loss rearrangements

BACKGROUND

To study the differences between two unichromosomal circular genomes, e.g., mitochondrial genomes, under the tandem duplication random loss (TDRL) rearrangement it is important to consider the whole set of potential TDRL rearrangement events that could have taken place. The reason is that for two given circular gene orders there can exist different TDRL rearrangements that transform one of the gene orders into the other. Hence, a TDRL event cannot always be reconstructed only from the knowledge of the circular gene order before a TDRL event and the circular gene order after it.

RESULTS

We present the program EqualTDRL that computes and illustrates the complete set of TDRLs for pairs of circular gene orders that differ by only one TDRL. EqualTDRL considers the circularity of the given genomes and certain restrictions on the TDRL rearrangements. Examples for the latter are sequences of genes that have to be conserved during a TDRL or pairs of genes that frame intergenic regions which might represent remnants of duplicated genes. Additionally, EqualTDRL allows to determine the set of TDRLs that are minimum with respect to the number of duplicated genes.

CONCLUSION

EqualTDRL supports scientists to study the complete set of TDRLs that possibly could have taken place in the evolution of mitochondrial genomes. EqualTDRL is implemented in C++ using the ggplot2 package of the open source programming language R and is freely available from http://pacosy.informatik.uni-leipzig.de/equaltdrl .