Genome rearrangements with indels in intergenes restrict the scenario space

Reversal and Transposition Distance on Unbalanced Genomes Using Intergenic Information

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.

An improved approximation algorithm for the reversal and transposition distance considering gene order and intergenic sizes

BACKGROUND

In the comparative genomics field, one of the goals is to estimate a sequence of genetic changes capable of transforming a genome into another. Genome rearrangement events are mutations that can alter the genetic content or the arrangement of elements from the genome. Reversal and transposition are two of the most studied genome rearrangement events. A reversal inverts a segment of a genome while a transposition swaps two consecutive segments. Initial studies in the area considered only the order of the genes. Recent works have incorporated other genetic information in the model. In particular, the information regarding the size of intergenic regions, which are structures between each pair of genes and in the extremities of a linear genome.

RESULTS AND CONCLUSIONS

In this work, we investigate the SORTING BY INTERGENIC REVERSALS AND TRANSPOSITIONS problem on genomes sharing the same set of genes, considering the cases where the orientation of genes is known and unknown. Besides, we explored a variant of the problem, which generalizes the transposition event. As a result, we present an approximation algorithm that guarantees an approximation factor of 4 for both cases considering the reversal and transposition (classic definition) events, an improvement from the 4.5-approximation previously known for the scenario where the orientation of the genes is unknown. We also present a 3-approximation algorithm by incorporating the generalized transposition event, and we propose a greedy strategy to improve the performance of the algorithms. We performed practical tests adopting simulated data which indicated that the algorithms, in both cases, tend to perform better when compared with the best-known algorithms for the problem. Lastly, we conducted experiments using real genomes to demonstrate the applicability of the algorithms.

Incorporating intergenic regions into reversal and transposition distances with indels

Problems in the genome rearrangement field are often formulated in terms of pairwise genome comparison: given two genomes [Formula: see text] and [Formula: see text], 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 [Formula: see text]-approximation algorithm for transpositions and indels, and a 6-approximation for reversals, transpositions, and indels.

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.

Sorting Signed Permutations by Intergenic Reversals

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.

Sorting by Genome Rearrangements on Both Gene Order and Intergenic Sizes

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.

A 3.5-Approximation Algorithm for Sorting by Intergenic Transpositions

Genome Rearrangements affect large stretches of genomes during evolution. One of the most studied genome rearrangement is the transposition, which occurs when a sequence of genes is moved to another position inside the genome. Mathematical models have been used to estimate the evolutionary distance between two different genomes based on genome rearrangements. However, many of these models have focused only on the (order of the) genes of a genome, disregarding other important elements in it. Recently, researchers have shown that considering existing regions between each pair of genes, called intergenic regions, can enhance the distance estimation in realistic data. In this work, we study the transposition distance between two genomes, but we also consider intergenic regions, a problem we name Sorting Permutations by Intergenic Transpositions (SbIT). We show that this problem is NP-hard and propose a 3.5-approximation algorithm for it.

Super short operations on both gene order and intergenic sizes

Background

The evolutionary distance between two genomes can be estimated by computing a minimum length sequence of operations, called genome rearrangements, that transform one genome into another. Usually, a genome is modeled as an ordered sequence of genes, and most of the studies in the genome rearrangement literature consist in shaping biological scenarios into mathematical models. For instance, allowing different genome rearrangements operations at the same time, adding constraints to these rearrangements (e.g., each rearrangement can affect at most a given number of genes), considering that a rearrangement implies a cost depending on its length rather than a unit cost, etc. Most of the works, however, have overlooked some important features inside genomes, such as the presence of sequences of nucleotides between genes, called intergenic regions.

Results and conclusions

In this work, we investigate the problem of computing the distance between two genomes, taking into account both gene order and intergenic sizes. The genome rearrangement operations we consider here are constrained types of reversals and transpositions, called super short reversals (SSRs) and super short transpositions (SSTs), which affect up to two (consecutive) genes. We denote by super short operations (SSOs) any SSR or SST. We show 3-approximation algorithms when the orientation of the genes is not considered when we allow SSRs, SSTs, or SSOs, and 5-approximation algorithms when considering the orientation for either SSRs or SSOs. We also show that these algorithms improve their approximation factors when the input permutation has a higher number of inversions, where the approximation factor decreases from 3 to either 2 or 1.5, and from 5 to either 3 or 2.

A general framework for genome rearrangement with biological constraints

This paper generalizes previous studies on genome rearrangement under biological constraints, using double cut and join (DCJ). We propose a model for weighted DCJ, along with a family of optimization problems called \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document}φ-MCPS (Minimum Cost Parsimonious Scenario), that are based on labeled graphs. We show how to compute solutions to general instances of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document}φ-MCPS, given an algorithm to compute \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document}φ-MCPS on a circular genome with exactly one occurrence of each gene. These general instances can have an arbitrary number of circular and linear chromosomes, and arbitrary gene content. The practicality of the framework is displayed by presenting polynomial-time algorithms that generalize the results of Bulteau, Fertin, and Tannier on the Sorting by wDCJs and indels in intergenes problem, and that generalize previous results on the Minimum Local Parsimonious Scenario problem.

Sorting signed circular permutations by super short operations

Background

One way to estimate the evolutionary distance between two given genomes is to determine the minimum number of large-scale mutations, or genome rearrangements, that are necessary to transform one into the other. In this context, genomes can be represented as ordered sequences of genes, each gene being represented by a signed integer. If no gene is repeated, genomes are thus modeled as signed permutations of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi =(\pi _1 \pi _2 \ldots \pi _n)$$\end{document}π=(π1π2…πn), and in that case we can consider without loss of generality that one of them is the identity permutation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\iota _n =(1 2 \ldots n)$$\end{document}ιn=(12…n), and that we just need to sort the other (i.e., transform it into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\iota _n$$\end{document}ιn). The most studied genome rearrangement events are reversals, where a segment of the genome is reversed and reincorporated at the same location; and transpositions, where two consecutive segments are exchanged. Many variants, e.g., combining different types of (possibly constrained) rearrangements, have been proposed in the literature. One of them considers that the number of genes involved, in a reversal or a transposition, is never greater than two, which is known as the problem of sorting by super short operations (or SSOs).

Results and conclusions

All problems considering SSOs in permutations have been shown to be in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {P}$$\end{document}P, except for one, namely sorting signed circular permutations by super short reversals and super short transpositions. Here we fill this gap by introducing a new graph structure called cyclic permutation graph and providing a series of intermediate results, which allows us to design a polynomial algorithm for sorting signed circular permutations by super short reversals and super short transpositions.

Partially local three-way alignments and the sequence signatures of mitochondrial genome rearrangements

Background

Genomic DNA frequently undergoes rearrangement of the gene order that can be localized by comparing the two DNA sequences. In mitochondrial genomes different mechanisms are likely at work, at least some of which involve the duplication of sequence around the location of the apparent breakpoints. We hypothesize that these different mechanisms of genome rearrangement leave distinctive sequence footprints. In order to study such effects it is important to locate the breakpoint positions with precision.

Results

We define a partially local sequence alignment problem that assumes that following a rearrangement of a sequence F, two fragments L, and R are produced that may exactly fit together to match F, leave a gap of deleted DNA between L and R, or overlap with each other. We show that this alignment problem can be solved by dynamic programming in cubic space and time. We apply the new method to evaluate rearrangements of animal mitogenomes and find that a surprisingly large fraction of these events involved local sequence duplications.

Conclusions

The partially local sequence alignment method is an effective way to investigate the mechanism of genomic rearrangement events. While applied here only to mitogenomes there is no reason why the method could not be used to also consider rearrangements in nuclear genomes.

Electronic supplementary material

The online version of this article (doi:10.1186/s13015-017-0113-0) contains supplementary material, which is available to authorized users.

Algorithms for computing the double cut and join distance on both gene order and intergenic sizes

BACKGROUND

Combinatorial works on genome rearrangements have so far ignored the influence of intergene sizes, i.e. the number of nucleotides between consecutive genes, although it was recently shown decisive for the accuracy of inference methods (Biller et al. in Genome Biol Evol 8:1427-39, 2016; Biller et al. in Beckmann A, Bienvenu L, Jonoska N, editors. Proceedings of Pursuit of the Universal-12th conference on computability in Europe, CiE 2016, Lecture notes in computer science, vol 9709, Paris, France, June 27-July 1, 2016. Berlin: Springer, p. 35-44, 2016). In this line, we define a new genome rearrangement model called wDCJ, a generalization of the well-known double cut and join (or DCJ) operation that modifies both the gene order and the intergene size distribution of a genome.

RESULTS

We first provide a generic formula for the wDCJ distance between two genomes, and show that computing this distance is strongly NP-complete. We then propose an approximation algorithm of ratio 4/3, and two exact ones: a fixed-parameter tractable (FPT) algorithm and an integer linear programming (ILP) formulation.

CONCLUSIONS

We provide theoretical and empirical bounds on the expected growth of the parameter at the center of our FPT and ILP algorithms, assuming a probabilistic model of evolution under wDCJ, which shows that both these algorithms should run reasonably fast in practice.