Sampling solution traces for the problem of sorting permutations by signed reversals

International audienceBackgroundTraditional algorithms to solve the problem of sorting by signed reversals output just one optimal solution while the space of all optimal solutions can be huge. A so-called trace represents a group of solutions which share the same set of reversals that must be applied to sort the original permutation following a partial ordering. By using traces, we therefore can represent the set of optimal solutions in a more compact way. Algorithms for enumerating the complete set of traces of solutions were developed. However, due to their exponential complexity, their practical use is limited to small permutations. A partial enumeration of traces is a sampling of the complete set of traces and can be an alternative for the study of distinct evolutionary scenarios of big permutations. Ideally, the sampling should be done uniformly from the space of all optimal solutions. This is however conjectured to be ♯P-complete.ResultsWe propose and evaluate three algorithms for producing a sampling of the complete set of traces that instead can be shown in practice to preserve some of the characteristics of the space of all solutions. The first algorithm (RA) performs the construction of traces through a random selection of reversals on the list of optimal 1-sequences. The second algorithm (DFALT) consists in a slight modification of an algorithm that performs the complete enumeration of traces. Finally, the third algorithm (SWA) is based on a sliding window strategy to improve the enumeration of traces. All proposed algorithms were able to enumerate traces for permutations with up to 200 elements.ConclusionsWe analysed the distribution of the enumerated traces with respect to their height and average reversal length. Various works indicate that the reversal length can be an important aspect in genome rearrangements. The algorithms RA and SWA show a tendency to lose traces with high average reversal length. Such traces are however rare, and qualitatively our results show that, for testable-sized permutations, the algorithms DFALT and SWA produce distributions which approximate the reversal length distributions observed with a complete enumeration of the set of traces

Genome Rearrangement Problems

Various global rearrangements of permutations, such as reversals and transpositions, have recently become of interest because of their applications in computational molecular biology. A reversal is an operation that reverses the order of a substring of a permutation. A transposition is an operation that swaps two adjacent substrings of a permutation. The problem of determining the smallest number of reversals required to transform a given permutation into the identity permutation is called sorting by reversals. Similar problems can be defined for transpositions and other global rearrangements. Related to sorting by reversals is the problem of establishing the reversal diameter. The reversal diameter of Sn (the symmetric group on n elements) is the maximum number of reversals required to sort a permutation of length n. Of course, diameter problems can be posed for other global rearrangements. These various problems are of interest because the permutations can be used to represent sequences of genes in chromosomes, and the global rearrangements then represent evolutionary events. As a result, we call these problems genome rearrangement problems. Genome rearrangement problems seem to be unlike previously studied algorithmic problems on sequences, so new methods have had to be developed to deal with them. These methods predominantly employ graphs to model permutation structure. However, even using these methods, often a genome rearrangement problem has no obvious polynomial-time algorithm, and in some cases can be shown to be NP-hard. For example, the problem of sorting by reversals is NP-hard, whereas the computational complexity of sorting by transpositions is open. For problems like these, it is natural to seek polynomial-time approximation algorithms that achieve an approximation guarantee. In this thesis, we study several genome rearrangement problems as interesting and challenging algorithmic problems in their own right, including some problems for which the global rearrangement has no immediate biological equivalent. For example, we define a block-interchange to be a rearrangement that swaps any two substrings of the permutation. We examine, in particular, how the graph theoretic models relate to the genome rearrangement problems that we study. The major new results contained in this thesis are as follows: We present a 3/2-approximation algorithm for sorting by reversals. This is the best known approximation algorithm for the problem, and improves upon the 7/4 approximation bound of the previous best algorithm. We give a polynomial-time algorithm for a significant special case of sorting by reversals, thereby disproving a conjecture of Kececioglu and Sankoff, who had suggested that this special case was likely to be NP-hard. We analyse the structure of the so-called cpcle graph of a permutation in the context of sorting by transpositions, and thereby gain a deeper insight into this problem. Among the consequences are; a tighter lower bound for the problem, a simpler 3/2-aproximation algorithm than had previously been described, and algorithms that, in empirical tests, almost always find the exact transposition distance of random permutations. We introduce a natural generalisation of sorting by transpositions called sorting by block-interchanges, and present a polynomial-time algorithm for this problem. We initiate the study of analogous problems on strings over a fixed length alphabet. We establish upper and lower bounds and diameter results for the problems over a binary alphabet. We also prove that the problems analogous to sorting by reversals and sorting by block-interchanges are NP-hard. (Abstract shortened by ProQuest.)

Storage and Retrieval of Individual Genomes

A repetitive sequence collection is one where portions of a emph{base sequence} of length $n$ are repeated many times with small variations, forming a collection of total length $N$. Examples of such collections are version control data and genome sequences of individuals, where the differences can be expressed by lists of basic edit operations. Flexible and efficient data analysis on a such typically huge collection is plausible using suffix trees. However, suffix tree occupies $O(N log N)$ bits, which very soon inhibits in-memory analyses. Recent advances in full-text emph{self-indexing} reduce the space of suffix tree to $O(N log sigma)$ bits, where $sigma$ is the alphabet size. In practice, the space reduction is more than $10$-fold for example on suffix tree of Human Genome. However, this reduction remains a constant factor when more sequences are added to the collection We develop a new self-index suited for the repetitive sequence collection setting. Its expected space requirement depends only on the length $n$ of the base sequence and the number $s$ of variations in its repeated copies. That is, the space reduction is no longer constant, but depends on $N/n$. We believe the structure developed in this work will provide a fundamental basis for storage and retrieval of individual genomes as they become available due to rapid progress in the sequencing technologies

Storage and retrieval of individual genomes

Volume: 5541A repetitive sequence collection is one where portions of a base sequence of length n are repeated many times with small variations, forming a collection of total length N. Examples of such collections are version control data and genome sequences of individuals, where the differences can be expressed by lists of basic edit operations. Flexible and efficient data analysis on a such typically huge collection is plausible using suffix trees. However, suffix tree occupies O(N log N) bits, which very soon inhibits in-memory analyses. Recent advances in full-text self-indexing reduce the space of suffix tree to O(N log σ) bits, where σ is the alphabet size. In practice, the space reduction is more than 10-fold, for example on suffix tree of Human Genome. However, this reduction factor remains constant when more sequences are added to the collection. We develop a new family of self-indexes suited for the repetitive sequence collection setting. Their expected space requirement depends only on the length n of the base sequence and the number s of variations in its repeated copies. That is, the space reduction factor is no longer constant, but depends on N / n. We believe the structures developed in this work will provide a fundamental basis for storage and retrieval of individual genomes as they become available due to rapid progress in the sequencing technologies.Peer reviewe

Models and Algorithms for Sorting Permutations with Tandem Duplication and Random Loss

A central topic of evolutionary biology is the inference of phylogeny, i. e., the evolutionary history of species. A powerful tool for the inference of such phylogenetic relationships is the arrangement of the genes in mitochondrial genomes. The rationale is that these gene arrangements are subject to different types of mutations in the course of evolution. Hence, a high similarity in the gene arrangement between two species indicates a close evolutionary relation. Metazoan mitochondrial gene arrangements are particularly well suited for such phylogenetic studies as they are available for a wide range of species, their gene content is almost invariant, and usually free of duplicates. With these properties gene arrangements of mitochondrial genomes are modeled by permutations in which each element represents a gene, i. e., a specific genetic sequence. The mutations that shape the gene arrangement of genomes are then represented by operations that rearrange elements in permutations, so-called genome rearrangements, and thereby bridge the gap between evolutionary biology and optimization. Many problems of phylogeny inference can be formulated as challenging combinatorial optimization problems which makes this research area especially interesting for computer scientists. The most prominent examples of such optimization problems are the sorting problem and the distance problem. While the sorting problem requires a minimum length sequence of rearrangements that transforms one given permutation into another given permutation, i. e., it aims for a hypothetical scenario of gene order evolution, the distance problem intends to determine only the length of such a sequence. This minimum length is called distance and used as a (dis)similarity measure quantifying the evolutionary relatedness. Most evolutionary changes occurring in gene arrangements of mitochondrial genomes can be explained by the tandem duplication random loss (TDRL) genome rearrangement model. A TDRL consists of a duplication of a consecutive set of genes in tandem followed by a random loss of one copy of each duplicated gene. In spite of the importance of the TDRL genome rearrangement in mitochondrial evolution, its combinatorial properties have rarely been studied. In addition, models of genome rearrangements which include all types of rearrangement that are relevant for mitochondrial genomes, i. e., inversions, transpositions, inverse transpositions, and TDRLs, while admitting computational tractability are rare. Nevertheless, especially for metazoan gene arrangements the TDRL rearrangement should be considered for the reconstruction of phylogeny. Realizing that a better understanding of the TDRL model is indispensable for the study of mitochondrial gene arrangements, the central theme of this thesis is to broaden the horizon of TDRL genome rearrangements with respect to mitochondrial genome evolution. For this purpose, this thesis provides combinatorial properties of the TDRL model and its variants as well as efficient methods for a plausible reconstruction of rearrangement scenarios between gene arrangements. The methods that are proposed consider all types of genome rearrangements that predominately occur during mitochondrial evolution. More precisely, the main points contained in this thesis are as follows: The distance problem and the sorting problem for the TDRL model are further examined in respect to circular permutations, a formal concept that reflects the circular structure of mitochondrial genomes. As a result, a closed formula for the distance is provided. Recently, evidence for a variant of the TDRL rearrangement model in which the duplicated set of genes is additionally inverted have been found. Initiating the algorithmic study of this new rearrangement model on a certain type of permutations, a closed formula solving the distance problem is proposed as well as a quasilinear time algorithm that solves the corresponding sorting problem. The assumption that only one type of genome rearrangement has occurred during the evolution of certain gene arrangements is most likely unrealistic, e. g., at least three types of rearrangements on top of the TDRL rearrangement have to be considered for the evolution metazoan mitochondrial genomes. Therefore, three different biologically motivated constraints are taken into account in this thesis in order to produce plausible evolutionary rearrangement scenarios. The first constraint is extending the considered set of genome rearrangements to the model that covers all four common types of mitochondrial genome rearrangements. For this 4-type model a sharp lower bound and several close additive upper bounds on the distance are developed. As a byproduct, a polynomial-time approximation algorithm for the corresponding sorting problem is provided that guarantees the computation of pairwise rearrangement scenarios that deviate from a minimum length scenario by at most two rearrangement operations. The second biologically motivated constraint is the relative frequency of the different types of rearrangements occurring during the evolution. The frequency is modeled by employing a weighting scheme on the 4-type model in which every rearrangement is weighted with respect to its type. The resulting NP-hard sorting problem is then solved by means of a polynomial size integer linear program. The third biologically motivated constraint that has been taken into account is that certain subsets of genes are often found in close proximity in the gene arrangements of many different species. This observation is reflected by demanding rearrangement scenarios to preserve certain groups of genes which are modeled by common intervals of permutations. In order to solve the sorting problem that considers all three types of biologically motivated constraints, the exact dynamic programming algorithm CREx2 is proposed. CREx2 has a linear runtime for a large class of problem instances. Otherwise, two versions of the CREx2 are provided: The first version provides exact solutions but has an exponential runtime in the worst case and the second version provides approximated solutions efficiently. CREx2 is evaluated by an empirical study for simulated artificial and real biological mitochondrial gene arrangements

Étude de l’évolution des génomes par duplications, pertes et réarrangements

La duplication est un des évènements évolutifs les plus importants, car elle peut mener à la création de nouvelles fonctions géniques. Durant leur évolution, les génomes sont aussi affectés par des inversions, des translocations (incluant des fusions et fissions de chromosomes), des transpositions et des délétions. L'étude de l'évolution des génomes est importante, notamment pour mieux comprendre les mécanismes biologiques impliqués, les types d'évènements qui sont les plus fréquents et quels étaient les contenus en gènes des espèces ancestrales. Afin d'analyser ces différents aspects de l'évolution des génomes, des algorithmes efficaces doivent être créés pour inférer des génomes ancestraux, des histoires évolutives, des relations d'homologies et pour calculer les distances entre les génomes. Dans cette thèse, quatre projets reliés à l'étude et à l'analyse de l'évolution des génomes sont présentés : 1) Nous proposons deux algorithmes pour résoudre des problèmes reliés à la duplication de génome entier : un qui généralise le problème du genome halving aux pertes de gènes et un qui permet de calculer la double distance avec pertes. 2) Nous présentons une nouvelle méthode pour l'inférence d'histoires évolutives de groupes de gènes orthologues répétés en tandem. 3) Nous proposons une nouvelle approche basée sur la théorie des graphes pour inférer des gènes in-paralogues qui considère simultanément l'information provenant de différentes espèces afin de faire de meilleures prédictions. 4) Nous présentons une étude de l'histoire évolutive des gènes d'ARN de transfert chez 50 souches de Bacillus.Gene duplication is one of the most important types of events affecting genomes during their evolution because it can create novel gene function. During the evolution process, genomes are also affected by inversions, translocations (including chromosome fusions and fissions), transpositions and deletions. Studying the evolution of genomes is important to get a better understanding of the biological mechanisms involved, which types of events are more frequent than others and what was the gene content in the ancestral species just to name a few. In order to analyze these different aspects of genome evolution, efficient algorithms need to be developed to infer ancestral genomes, evolutionary histories, homology relationships between genes and to compute distances between genomes. In this thesis, four different projects related to the study and analysis of genome evolution are presented: 1) We developed two algorithms to solve problems related to whole genome duplication: one that generalizes the genome halving problem to gene losses, and one that allows to compute the double distance with losses. 2) We developed a new method to infer evolutionary histories of orthologous tandemly arrayed gene clusters. 3) We proposed a new graph-theoretic approach to infer inparalogs that simultaneously considers the information given by multiple species in order to make better inferences of inparalogous gene pairs. 4) We studied the evolutionary history of the tRNA genes of 50 Bacillus strains

Macroevolution: Explanation, Interpretation and Evidence

info:eu-repo/semantics/publishedVersio

Evolutionary genomics : statistical and computational methods

This open access book addresses the challenge of analyzing and understanding the evolutionary dynamics of complex biological systems at the genomic level, and elaborates on some promising strategies that would bring us closer to uncovering of the vital relationships between genotype and phenotype. After a few educational primers, the book continues with sections on sequence homology and alignment, phylogenetic methods to study genome evolution, methodologies for evaluating selective pressures on genomic sequences as well as genomic evolution in light of protein domain architecture and transposable elements, population genomics and other omics, and discussions of current bottlenecks in handling and analyzing genomic data. Written for the highly successful Methods in Molecular Biology series, chapters include the kind of detail and expert implementation advice that lead to the best results. Authoritative and comprehensive, Evolutionary Genomics: Statistical and Computational Methods, Second Edition aims to serve both novices in biology with strong statistics and computational skills, and molecular biologists with a good grasp of standard mathematical concepts, in moving this important field of study forward