Genome Rearrangement Problems Research Articles

A Barrier for Further Approximating Sorting by Transpositions.

The transposition distance problem is a classical problem in genome rearrangements, which seeks to determine the minimum number of transpositions needed to transform a linear chromosome into another represented by the permutations and , respectively. This article focuses on the equivalent problem of sorting by transpositions (SBT), where is the identity permutation . Specifically, we investigate palisades, a family of permutations that are "hard" to sort, as they require numerous transpositions above the celebrated lower bound devised by Bafna and Pevzner. By determining the transposition distance of palisades, we were able to provide the exact transposition diameter for 3-permutations (TD3), a special subset of the symmetric group Sn, essential for the study of approximate solutions for SBT using the simplification technique. The exact value for TD3 has remained unknown since Elias and Hartman showed an upper bound for it. Another consequence of determining the transposition distance of palisades is that, using as lower bound the one by Bafna and Pevzner, it is impossible to guarantee approximation ratios lower than 1.375 when approximating SBT. This finding has significant implications for the study of SBT, as this problem has been the subject of intense research efforts for the past 25 years.

Genome Rearrangement Distance With a Flexible Intergenic Regions Aspect.

Most mathematical models for genome rearrangement problems have considered only gene order. In this way, the rearrangement distance considering some set of events, such as reversal and transposition events, is commonly defined as the minimum number of rearrangement events that transform the gene order from a genome G1 into the gene order from a genome G2. Recent works initiate incorporating more information such as the sizes of the intergenic regions (i.e., number of nucleotides between pairs of consecutive genes), which yields good results for estimated distances on real data. In these models, besides transforming the gene order, the sequence of rearrangement events must transform the list of intergenic regions sizes from G1 into the list of intergenic regions sizes from G2 (target list). We study a new variation where the target list is flexible, in the sense that each target intergenic region size is in a range of acceptable values. This allows us to model scenarios where the main objective is still to transform the order of genes from the source genome into the target genome, allowing flexibility in the sizes of the intergenic regions, since the nucleotides in these regions tend to undergo more changes when compared to genes. We investigate the rearrangement distance considering three sets of events, two with the exclusive use of reversals or transpositions, and the other allowing both rearrangement events. We present approximation algorithms for the problems and an NP-hardness proof. Our results rely on the Flexible Weighted Cycle Graph, adapted from the breakpoint graph to deal with flexible intergenic regions sizes.

Approximation algorithms for sorting by bounded singleton moves

Sorting permutations by block moves is a fundamental combinatorial problem in genome rearrangements. The classic block move operation is called transposition, which switches two adjacent blocks, or equivalently, moves a block to some other position. But large blocks movement rarely occurs during real evolutionary events. A natural restriction of transposition is to bound the length of the blocks to be switched. In this paper, we investigate the problem called sorting by bounded singleton moves, where one of the switched blocks is exactly a singleton while the other is of length at most c. This problem generalizes the sorting by short block moves problem proposed by Heath and Vergara [11], which requires the total length of blocks switched bounded by 3. When c=3, we devise a 95-approximation algorithm for an arbitrary permutation, and a 127-approximation algorithm for a woven double-strip permutation. Our algorithms can be slightly extended to solve the sorting by c-bounded singleton moves problem for any constant c≥3, guaranteeing an approximation factor of 3c5 and 4c7 for arbitrary permutations and woven double-strip permutations respectively, just by exploiting a new lower bound of sorting by c-bounded singleton moves.

A new 1.375-approximation algorithm for sorting by transpositions

Backgroundsorting by transpositions (SBT) is a classical problem in genome rearrangements. In 2012, SBT was proven to be mathcal {NP}-hard and the best approximation algorithm with a 1.375 ratio was proposed in 2006 by Elias and Hartman (EH algorithm). Their algorithm employs simplification, a technique used to transform an input permutation pi into a simple permutation{hat{pi }}, presumably easier to handle with. The permutation {hat{pi }} is obtained by inserting new symbols into pi in a way that the lower bound of the transposition distance of pi is kept on {hat{pi }}. The simplification is guaranteed to keep the lower bound, not the transposition distance. A sequence of operations sorting {hat{pi }} can be mimicked to sort pi.Results and conclusionsFirst, using an algebraic approach, we propose a new upper bound for the transposition distance, which holds for all S_n. Next, motivated by a problem identified in the EH algorithm, which causes it, in scenarios involving how the input permutation is simplified, to require one extra transposition above the 1.375-approximation ratio, we propose a new approximation algorithm to solve SBT ensuring the 1.375-approximation ratio for all S_n. We implemented our algorithm and EH’s. Regarding the implementation of the EH algorithm, two other issues were identified and needed to be fixed. We tested both algorithms against all permutations of size n, 2le n le 12. The results show that the EH algorithm exceeds the approximation ratio of 1.375 for permutations with a size greater than 7. The percentage of computed distances that are equal to transposition distance, computed by the implemented algorithms are also compared with others available in the literature. Finally, we investigate the performance of both implementations on longer permutations of maximum length 500. From the experiments, we conclude that maximum and the average distances computed by our algorithm are a little better than the ones computed by the EH algorithm and the running times of both algorithms are similar, despite the time complexity of our algorithm being higher.

A Path-Deformation Framework for Determining Weighted Genome Rearrangement Distance.

Measuring the distance between two bacterial genomes under the inversion process is usually done by assuming all inversions to occur with equal probability. Recently, an approach to calculating inversion distance using group theory was introduced, and is effective for the model in which only very short inversions occur. In this paper, we show how to use the group-theoretic framework to establish minimal distance for any weighting on the set of inversions, generalizing previous approaches. To do this we use the theory of rewriting systems for groups, and exploit the Knuth–Bendix algorithm, the first time this theory has been introduced into genome rearrangement problems. The central idea of the approach is to use existing group theoretic methods to find an initial path between two genomes in genome space (for instance using only short inversions), and then to deform this path to optimality using a confluent system of rewriting rules generated by the Knuth–Bendix algorithm.

Permutation patterns in genome rearrangement problems: The reversal model

In the context of the genome rearrangement problem, we analyze two well known models, namely the reversal and the prefix reversal models, by exploiting the connection with the notion of permutation patterns. More specifically, for any k, we provide a characterization of the set of permutations having distance less than or equal to k from the identity (which is known to be a permutation class) in terms of what we call generating peg permutations and we describe some properties of its basis, which allow us to compute such a basis for small values of k.

The Deletion-Insertion model applied to the genome rearrangement problem

Abstract Permutations are frequently used in solving the genome rearrangement problem, whose goal is finding the shortest sequence of mutations transforming one genome into another. We introduce the Deletion-Insertion model (DI) to model small-scale mutations in species with linear chromosomes, such as humans. Applying one restriction to this model, we obtain the transposition model for genome rearrangement, which was shown to be NP-hard in [4]. We use combinatorial reasoning and permutation statistics to develop a polynomial-time algorithm to approximate the minimum number of transpositions required in the transposition model and to analyze the sharpness of several bounds on transpositions between genomes.

Genome Rearrangements on Multigenomic Models: Applications of Graph Convexity Problems.

Genome rearrangements are events where large blocks of DNA exchange pieces during evolution. The analysis of such events is a tool for understanding evolutionary genomics, in whose context many rearrangement distances have been proposed, based on finding the minimum number of rearrangements to transform one genome into another, using some predefined operation. However, when more than two genomes are considered, we have new challenging problems. Studying such problems from a combinatorial point of view has been shown to be a useful tool to approach such problems, for example, the reconstruction of phylogenetic trees. We focus on genome rearrangement problems related to graph convexity. Such an approach is in connection with some other well-known studies on multigenomic models, for example, those based on the median and on the closest string. We propose an association between graph convexities and genome rearrangements in such a way that graph convexity problems deal with input sets of vertices and try to answer questions concerning the closure of such inputs. The concept of closure is useful for studies on genome rearrangement by suggesting mechanisms to reduce the genomic search space. Regarding the computational complexity, and considering the Hamming distance on strings, we solve the following problems: decide if a given set is convex; compute the interval and the convex hull of a given set; and determine the convexity number, interval number, and hull number of a Hamming graph. All such problems are solved for three types of convexities: geodetic, monophonic, and P3. Considering the Cayley distance on permutations, we solve the convexity number and interval determination problems for the geodetic convexity.

An approximation algorithm for genome sorting by reversals to recover all adjacencies

Genome rearrangement problems have been extensively studied for more than two decades, intended to understand the species evolutionary relationships in terms of the long range genetic mutations at the genome level. While most earlier studies focus on the simplified genomes ignoring gene duplicates, thousands of whole genome sequencing projects reveal that a genome typically carries multiple gene duplicates distributed in various ways along the genome. Given a source genome and a target genome such that one is a re-ordering of the genes in the other, we measure the evolutionary distance by the minimum number of reversals applied on the source genome to recover all the gene adjacencies in the target genome. We define this optimization problem as sorting by reversals to recover all adjacencies, or SBR2RA in short. We show that SBR2RA is APX-hard and uncover some similarities and differences to the classic counterpart, the sorting by reversals problem. From the approximability perspective, we present a \(2 \alpha \)-approximation algorithm, where \(\alpha \in [1, 2]\) is the best approximation ratio for a related optimization problem which is suspected to be NP-hard.

Closure of genomic sets: applications of graph convexity to genome rearrangement problems

Genome rearrangement problems aim at finding the minimum number of mutational events required to transform a genome into another. Studying such problems from a combinatorial point of view has been proved to be useful for the reconstruction of phylogenetic trees. In this work we focus on the relations between genome rearrangement and graph convexity in the following way. Graph convexity problems deal with input sets of vertices and try to understand properties on the closure of such inputs. In turn, the concept of closure is useful for studies on genome rearrangement by suggesting mechanisms to reduce the genomic search space. Computational complexity studies are widely developed in graph convexity problems, since there are several types of convexities, each capturing a distinct way of defining what is meant by “closure”. In this regard, considering the Hamming distance of strings, we solve the following problems: decide whether a given set is convex; compute the interval and the convex hull of a given set; and determine the convexity number of a given graph. All such problems are solved for the geodesic and monophonic convexities. On the other hand, for the Cayley distance of permutations, we study convexity of sets and interval determination of sets, in the context of the geodesic convexity.

Sorting by Cuts, Joins, and Whole Chromosome Duplications.

Genome rearrangement problems have been extensively studied due to their importance in biology. Most studied models assumed a single copy per gene. However, in reality, duplicated genes are common, most notably in cancer. In this study, we make a step toward handling duplicated genes by considering a model that allows the atomic operations of cut, join, and whole chromosome duplication. Given two linear genomes, [Formula: see text] with one copy per gene and [Formula: see text] with two copies per gene, we give a linear time algorithm for computing a shortest sequence of operations transforming [Formula: see text] into [Formula: see text] such that all intermediate genomes are linear. We also show that computing an optimal sequence with fewest duplications is NP-hard.

A factor-(1.408 + ε) approximation for sorting unsigned genomes by reciprocal translocations

Sorting genomes by translocations is a classic combinatorial problem in genome rearrangements. The translocation distance for signed genomes can be computed exactly in polynomial time, but for unsigned genomes the problem becomes NP-hard and the current best approximation ratio is 1.5+ε. In this paper, we investigate the problem of sorting unsigned genomes by translocations. Firstly, we propose a tighter lower bound of the optimal solution by analyzing some special sub-permutations; then, by exploiting the two well-known algorithms for approximating the maximum independent set on graphs with a bounded degree and for set packing with sets of bounded size, we devise a new polynomial-time approximation algorithm, improving the approximation ratio to 1.408+ε, where ε=O(1/log⁡n).

Counting and sampling SCJ small parsimony solutions

The Single Cut or Join (SCJ) operation on genomes, generalizing chromosome evolution by fusions and fissions, is the computationally simplest known model of genome rearrangement. While most genome rearrangement problems are already hard when comparing three genomes, it is possible to compute in polynomial time a most parsimonious SCJ scenario for an arbitrary number of genomes related by a binary phylogenetic tree.Here we consider the problems of sampling and counting the most parsimonious SCJ scenarios. We show that both the sampling and counting problems are easy for two genomes, and we relate SCJ scenarios to alternating permutations. However, for an arbitrary number of genomes related by a binary phylogenetic tree, the counting and sampling problems become hard. We prove that if a Fully Polynomial Randomized Approximation Scheme or a Fully Polynomial Almost Uniform Sampler exist for the most parsimonious SCJ scenario, then RP=NP.The proof has a wider scope than genome rearrangements: the same result holds for parsimonious evolutionary scenarios on any set of discrete characters.

A general heuristic for genome rearrangement problems.

In this paper, we present a general heuristic for several problems in the genome rearrangement field. Our heuristic does not solve any problem directly, it is rather used to improve the solutions provided by any non-optimal algorithm that solve them. Therefore, we have implemented several algorithms described in the literature and several algorithms developed by ourselves. As a whole, we implemented 23 algorithms for 9 well known problems in the genome rearrangement field. A total of 13 algorithms were implemented for problems that use the notions of prefix and suffix operations. In addition, we worked on 5 algorithms for the classic problem of sorting by transposition and we conclude the experiments by presenting results for 3 approximation algorithms for the sorting by reversals and transpositions problem and 2 approximation algorithms for the sorting by reversals problem. Another algorithm with better approximation ratio can be found for the last genome rearrangement problem, but it is purely theoretical with no practical implementation. The algorithms we implemented in addition to our heuristic lead to the best practical results in each case. In particular, we were able to improve results on the sorting by transpositions problem, which is a very special case because many efforts have been made to generate algorithms with good results in practice and some of these algorithms provide results that equal the optimum solutions in many cases. Our source codes and benchmarks are freely available upon request from the authors so that it will be easier to compare new approaches against our results.

Distribution of genome rearrangement distance under double cut and join

Using the double-cut-and-join (DCJ) model for genome rearrangement we use combinatorial techniques to analyze the distribution of genomes under DCJ distance. We present an exponential generating function for the number of genomes that are maximally distant from a given genome and provide a formula for the number of genomes that are any given distance from an arbitrary starting genome. Many mathematical models have been developed to aid biologists and bioinformaticians in their study of the genome rearrangement problem, whose goal is to find the optimal sequence of mutations for the transformation of one genome into another. Using the double-cut-and-join (DCJ) model, Bergeron, Mixtacki, and Stoye [Bergeron et al. 2006] found that the distance between two genomes is completely determined by a bipartite graph created from the genomes. We utilize their data structure to find the distribution of genomes that are distance d from a given genome under DCJ. In Section 2, we introduce genome rearrangement, DCJ, and an important result of the same authors. In Section 3, we present a generating function for the number of maximally distant genomes from a given genome, and in Section 4, we obtain the distribution of all genomes by distance from a given genome.

The Application of Immune Genetic Algorithm in Parallel Sorting

Taking an immune operator added to the framework of the genetic algorithm, individuals vaccinated to further enhance the survival ability of the individual, the immune genetic algorithm as a set of immune mechanisms and evolutionary mechanisms in one of the new evolutionary algorithm is applied to reverse genome rearrangement problem is a new attempt, the genetic algorithm can be avoided premature shortcomings, while the immune genetic algorithm parallelization can improve convergence speed.

Approximating the number of Double Cut-and-Join scenarios

The huge number of solutions in genome rearrangement problems calls for algorithms for counting and sampling in the space of solutions, rather than drawing one arbitrary scenario. A closed formula exists for counting the number of DCJ scenarios between co-tailed genomes, but no polynomial result has been published so far for arbitrary genomes. We prove here that it admits a Fully Polynomial time Randomized Approximation Scheme. We use an MCMC almost uniform sampler and prove that it converges to the uniform distribution in fully polynomial time. The MCMC can be used to quickly draw a sample of DCJ scenarios from a prescribed distribution and test some hypotheses on genome evolution.

Sorting by Transpositions Is Difficult

In comparative genomics, a transposition is an operation that exchanges two consecutive sequences of genes in a genome. The transposition distance between two genomes, that is, the minimum number of transpositions needed to transform a genome into another, is, according to numerous studies, a relevant evolutionary distance. The problem of computing this distance when genomes are represented by permutations is called the Sorting by Transpositions problem, and has been introduced by Bafna and Pevzner in [Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 1995, pp. 614--623]. It has naturally been the focus of a number of studies (see, for instance, [G. Fertin, A. Labarre, I. Rusu, É. Tannier, and S. Vialette, Combinatorics of Genome Rearrangements, The MIT Press, Cambridge, MA, 2009]), but the computational complexity of this problem has remained undetermined for 15 years. In this paper, we answer this long-standing open question by proving that the Sorting by Transpositions problem is \sf NP-hard. As a corollary of our result, we also prove that the following problem, first described in [D. A. Christie, Genome Rearrangement Problems, Ph.D. thesis, University of Glasgow, Glasgow, Scotland, 1998], is \sf NP-hard: given a permutation $\pi$, is it possible to sort $\pi$ using exactly $d_b(\pi)/3$ transpositions, where $d_b(\pi)$ is the number of breakpoints of $\pi$?

Efficient solutions for the far from most string problem

Computational molecular biology has emerged as one of the most exciting interdisciplinary fields. It has currently benefited from concepts and theoretical results obtained by different scientific research communities, including genetics, biochemistry, and computer science. In the past few years it has been shown that a large number of molecular biology problems can be formulated as combinatorial optimization problems, including sequence alignment problems, genome rearrangement problems, string selection and comparison problems, and protein structure prediction and recognition. This paper provides a detailed description of string selection and string comparison problems. For finding good-quality solutions of a particular class of string comparison molecular biology problems, known as the far from most string problem, we propose new heuristics, including a Greedy Randomized Adaptive Search Procedure (GRASP) and a Genetic Algorithm (GA). Computational results indicate that these randomized heuristics find better quality solutions compared with results produced by the best state-of-the-art heuristic approach.

SCJ: A Breakpoint-Like Distance that Simplifies Several Rearrangement Problems

The breakpoint distance is one of the most straightforward genome comparison measures. Surprisingly, when it comes to defining it precisely for multichromosomal genomes with both linear and circular chromosomes, there is more than one way to go about it. Pevzner and Tesler gave a definition in a 2003 paper, Tannier et al. defined it differently in 2008, and in this paper we provide yet another alternative, calling it SCJ for single-cut-or-join, in analogy to the popular double cut and join (DCJ) measure. We show that several genome rearrangement problems, such as median and halving, become easy for SCJ, and provide linear and higher polynomial time algorithms for them. For the multichromosomal linear genome median problem, this is the first polynomial time algorithm described, since for other distances this problem is NP-hard. In addition, we show that small parsimony under SCJ is also easy, and can be solved by a variant of Fitch's algorithm. In contrast, big parsimony is NP-hard under SCJ. This new distance measure may be of value as a speedily computable, first approximation to distances based on more realistic rearrangement models.