Sorting by reversals, block interchanges, tandem duplications, and deletions

<p>Abstract</p> <p>Background</p> <p>Finding sequences of evolutionary operations that transform one genome into another is a classic problem in comparative genomics. While most of the genome rearrangement algorithms assume that there is exactly one copy of each gene in both genomes, this does not reflect the biological reality very well – most of the studied genomes contain duplicated gene content, which has to be removed before applying those algorithms. However, dealing with unequal gene content is a very challenging task, and only few algorithms allow operations like duplications and deletions. Almost all of these algorithms restrict these operations to have a fixed size.</p> <p>Results</p> <p>In this paper, we present a heuristic algorithm to sort an ancestral genome (with unique gene content) into a genome of a descendant (with arbitrary gene content) by reversals, block interchanges, tandem duplications, and deletions, where tandem duplications and deletions are of arbitrary size.</p> <p>Conclusion</p> <p>Experimental results show that our algorithm finds sorting sequences that are close to an optimal sorting sequence when the ancestor and the descendant are closely related. The quality of the results decreases when the genomes get more diverged or the genome size increases. Nevertheless, the calculated distances give a good approximation of the true evolutionary distances.</p

Representing and decomposing genomic structural variants as balanced integer flows on sequence graphs

The study of genomic variation has provided key insights into the functional role of mutations. Predominantly, studies have focused on single nucleotide variants (SNV), which are relatively easy to detect and can be described with rich mathematical models. However, it has been observed that genomes are highly plastic, and that whole regions can be moved, removed or duplicated in bulk. These structural variants (SV) have been shown to have significant impact on the phenotype, but their study has been held back by the combinatorial complexity of the underlying models. We describe here a general model of structural variation that encompasses both balanced rearrangements and arbitrary copy-numbers variants (CNV). In this model, we show that the space of possible evolutionary histories that explain the structural differences between any two genomes can be sampled ergodically

On Distance and Sorting of the Double Cut-and-Join and the Inversion-*indel* Model

Willing E. On Distance and Sorting of the Double Cut-and-Join and the Inversion-*indel* Model. Bielefeld: Universität Bielefeld; 2018.In der vergleichenden Genomik werden zwei oder mehrere Genome hinsichtlich ihres Verwandtschaftsgrades verglichen. Das Ziel dieser Arbeit ist die Erforschung von mathematischen Modellen, die zum einen die evolutionäre *Distanz*, zum anderen die evolutionären Vorgänge zwischen zwei Genomen bestimmen können. Neben Methoden, welche auf einer niedrigen Ebene, z. B. den Basen(paarungen), ansetzen, sind auch abstraktere Modelle, die auf einzelnen Genen oder noch größeren Abschnitten Genome vergleichen, etabliert. Handelt es sich auf niedrigerer Ebene um einzelne Basen, die eingefügt, gelöscht oder ersetzt werden, sind es auf höherer Ebene beispielsweise ganze Gene. Auf höherer Ebene können Ergebnisse sogenannter Umordnungsprozesse (*genome rearrangements*) beobachtet werden, welche in einem *Sortierszenario* beschrieben werden. Im Vergleich eines Genoms mit einem anderen können dies unter anderem Inversionen, Translokationen, aber auch Einfügungen oder Löschungen von großen Bereichen sein. Ein bekanntes Modell ist das *Inversionsmodell*, welches den Verwandtschaftsgrad zweier Genome ausschließlich durch Inversionen bestimmt. Ein weiteres ist das *double cut-and-join (DCJ)* Modell, welches neben Inversionen auch Translokationen, Chromosomenfusionen, bzw. -fissionen, sowie Integration und Extraktion von kleinen zirkulären Trägern erlaubt. Die Distanz ist hierbei die Anzahl Zwischenschritte eines Sortierszenarios von geringster Länge. Diese Dissertation ist in zwei Teile gegliedert. Der erste Teil beschäftigt sich mit dem zufälligen Ziehen eines Sortierszenarios innerhalb des DCJ-Modells. Neben einigen naiven Ansätzen interessieren wir uns im Wesentlichen dafür, jedes Szenario mit gleicher Wahrscheinlichkeit, also uniform verteilt, zu ziehen. Hierfür wird nicht nur der gesamte Sortierraum betrachtet, sondern auch Maßnahmen zur effizienten Berechnung aufgezeigt. Der vorgestellte Algorithmus ist in einer Software-suite implementiert und wird hinsichtlich seiner Erzeugung von zufälligen Szenarien evaluiert. Der zweite Teil der Arbeit beschäftigt sich mit dem Inversions-*indel* Modell. Dieses wenig erforschte Modell erlaubt Inversionen, sowie Einfügungen und Löschungen (kurz *indels*). Dessen Distanz soll in Abhängigkeit von der DCJ- bzw. der DCJ-*indel*-Distanz wiedergegeben werden. Wir erweitern altbekannte Datenstrukturen des Inversionsmodells um Einfügungen und Löschungen repräsentieren zu können. Hierfür benutzen wir unter anderem Ansätze aus zwei anderen Modellen: Die Erweiterung des DCJ-Modells um indels, sowie die Ermittlung der Abhängigkeit von DCJ- und Inversionsmodell. Um die minimale Anzahl an Inversionen, Einfügungen und Löschungen zu ermitteln muss beachtet werden, dass durch Inversionen zwei oder mehr getrennte Bereiche, die zur Löschung vorgesehen sind, verschmolzen werden. Diese können sodann in einem einzigen Schritt gelöscht werden. Ähnlich verhält es sich mit Einfügungen. Zunächst betrachten wir Instanzen in denen die DCJ-indel-Distanz und die Inversions-indel-Distanz identisch sind. Im Weiteren gehen wir dazu über, schwierige Instanzen, d.h. jene die mehr Schritte benötigen als die DCJ(-indel)-Distanz, zu berechnen. Zu diesen Zweck müssen die unterschiedlichen Eigenschaften der Instanzen und deren Auswirkungen ausgemacht werden. Durch geschickte Reduzierung des Lösungsraums gelangen wir zu einer Menge von Basisfällen, welche wir durch erschöpfende Aufzählung lösen können. Insgesamt bieten die unternommenen Schritte nicht nur die Lösung der Inversions-indel Distanz in Abhängigkeit zur DCJ-indel Distanz, sondern auch eine Möglichkeit des Sortierens. Die Suche nach einer exakten Lösung für das Distanz- und das Sortierproblem im Inversions-indel Modell blieb lange unbeantwortet. Der Hauptbeitrag dieser Arbeit liegt darin diese zwei Fragen zu klären

Évolution des génomes par mutations locales et globales : une approche d’alignement

Durant leur évolution, les génomes accumulent des mutations pouvant affecter d’un nucléotide à plusieurs gènes. Les modifications au niveau du nombre et de l’organisation des gènes dans les génomes sont dues à des mutations globales, telles que les duplications, les pertes et les réarrangements. En comparant les ordres de gènes des génomes, il est possible d’inférer les événements évolutifs les plus fréquents, le contenu en gènes des espèces ancestrales ainsi que les histoires évolutives ayant menées aux ordres observés. Dans cette thèse, nous nous intéressons au développement de nouvelles méthodes algorithmiques, par approche d’alignement, afin d’analyser ces différents aspects de l’évolution des génomes. Nous nous intéressons à la comparaison de deux ou d’un ensemble de génomes reliés par une phylogénie, en tenant compte des mutations globales. Pour commencer, nous étudions la complexité théorique de plusieurs variantes du problème de l’alignement de deux ordres de gènes par duplications et pertes, ainsi que de l’approximabilité de ces problèmes. Nous rappelons ensuite les algorithmes exacts, en temps exponentiel, existants, et développons des heuristiques efficaces. Nous proposons, dans un premier temps, DLAlign, une heuristique quadratique pour le problème d’alignement de deux ordres de gènes par duplications et pertes. Ensuite, nous présentons, OrthoAlign, une extension de DLAlign, qui considère, en plus des duplications et pertes, les réarrangements et les substitutions. Nous abordons également le problème de l’alignement phylogénétique de génomes. Pour commencer, l’heuristique OrthoAlign est adaptée afin de permettre l’inférence de génomes ancestraux au noeuds internes d’un arbre phylogénétique. Nous proposons enfin, MultiOrthoAlign, une heuristique plus robuste, basée sur la médiane, pour l’inférence de génomes ancestraux et d’histoires évolutives d’un ensemble de génomes représentés aux feuilles d’un arbre phylogénétique.During the evolution process, genomes accumulate mutations that may affect the genome at different levels, ranging from one base to the overall gene content. Global mutations affecting gene content and organization are mainly duplications, losses and rearrangements. By comparing gene orders, it is possible to infer the most frequent events, the gene content in the ancestral genomes and the evolutionary histories of the observed gene orders. In this thesis, we are interested in developing new algorithmic methods based on an alignment approach for comparing two or a set of genomes represented as gene orders and related through a phylogenetic tree, based on global mutations. We study the theoretical complexity and the approximability of different variants of the two gene orders alignment problem by duplications and losses. Then, we present the existing exact exponential time algorithms, and develop efficient heuristics for these problems. First, we developed DLAlign, a quadratic time heuristic for the two gene orders alignment problem by duplications and losses. Then, we developed OrthoAlign, a generalization of DLAlign, accounting for most genome-wide evolutionary events such as duplications, losses, rearrangements and substitutions. We also study the phylogenetic alignment problem. First, we adapt our heuristic OrthoAlign in order to infer ancestral genomes at the internal nodes of a given phylogenetic tree. Finally, we developed MultiOrthoAlign, a more robust heuristic, based on the median problem, for the inference of ancestral genomes and evolutionary histories of extent genomes labeling leaves of a phylogenetic tree

Genome reconstruction and combinatoric analyses of rearrangement evolution

Cancer is often associated with a high number of large-scale, structural rearrangements. In a highly selective environment, some `driver' mutations conferring clonal growth advantage will be positively selected, accounting for further cancer development. Clarifying their nature, as well as their contribution to the pathology is a major current focus of biomedical research. Next generation sequencing technologies can be used nowadays to generate high-resolution data-sets of these alterations in cancer genomes. This project has been developed along two main lines: 1) the reconstruction of cancer aberrant karyotypes, together with their underlying evolutionary history; 2) the elucidation of some combinatorial properties associated with gene duplications. We applied graph theory to the problem of reconstructing the final cancer genome sequence; additionally, we developed an algorithmic approach for the reconstruction of a multi-step evolution consistent with read coverage and paired end data, giving insights on the possible molecular mechanisms underlying rearrangements. Looking at the combinatorics of both tandem and inverted duplication, we developed an algebraic formalism for the representation of these processes. This allowed us to both explore the geometric properties of sequences arising by Tandem Duplication (TD), and obtain a recursion for the number of tandem duplications evolutions after n events. Such results are missing for inverted duplications, whose combinatorial properties have been nevertheless deeply elucidated. Our results have allowed: 1) the identification, through an original approach, of potential rearrangement mechanisms associated with cancer development, and 2) the definition and mathematical description of the complete evolutionary space of specific rearrangement classes

Gene order rearrangement methods for the reconstruction of phylogeny

The study of phylogeny, i.e. the evolutionary history of species, is a central problem in biology and a key for understanding characteristics of contemporary species. Many problems in this area can be formulated as combinatorial optimisation problems which makes it particularly interesting for computer scientists. The reconstruction of the phylogeny of species can be based on various kinds of data, e.g. morphological properties or characteristics of the genetic information of the species. Maximum parsimony is a popular and widely used method for phylogenetic reconstruction aiming for an explanation of the observed data requiring the least evolutionary changes. A certain property of the genetic information gained much interest for the reconstruction of phylogeny in recent time: the organisation of the genomes of species, i.e. the arrangement of the genes on the chromosomes. But the idea to reconstruct phylogenetic information from gene arrangements has a long history. In Dobzhansky and Sturtevant (1938) it was already pointed out that “a comparison of the different gene arrangements in the same chromosome may, in certain cases, throw light on the historical relationships of these structures, and consequently on the history of the species as a whole”. This kind of data is promising for the study of deep evolutionary relationships because gene arrangements are believed to evolve slowly (Rokas and Holland, 2000). This seems to be the case especially for mitochondrial genomes which are available for a wide range of species (Boore, 1999). The development of methods for the reconstruction of phylogeny from gene arrangement data has made considerable progress during the last years. Prominent examples are the computation of parsimonious evolutionary scenarios, i.e. a shortest sequence of rearrangements transforming one arrangement of genes into another or the length of such a minimal scenario (Hannenhalli and Pevzner, 1995b; Sankoff, 1992; Watterson et al., 1982); the reconstruction of parsimonious phylogenetic trees from gene arrangement data (Bader et al., 2008; Bernt et al., 2007b; Bourque and Pevzner, 2002; Moret et al., 2002a); or the computation of the similarities of gene arrangements (Bergeron et al., 2008a; Heber et al., 2009). 1 1 Introduction The central theme of this work is to provide efficient algorithms for modified versions of fundamental genome rearrangement problems using more plausible rearrangement models. Two types of modified rearrangement models are explored. The first type is to restrict the set of allowed rearrangements as follows. It can be observed that certain groups of genes are preserved during evolution. This may be caused by functional constraints which prevented the destruction (Lathe et al., 2000; Sémon and Duret, 2006; Xie et al., 2003), certain properties of the rearrangements which shaped the gene orders (Eisen et al., 2000; Sankoff, 2002; Tillier and Collins, 2000), or just because no destructive rearrangement happened since the speciation of the gene orders. It can be assumed that gene groups, found in all studied gene orders, are not acquired independently. Accordingly, these gene groups should be preserved in plausible reconstructions of the course of evolution, in particular the gene groups should be present in the reconstructed putative ancestral gene orders. This can be achieved by restricting the set of rearrangements, which are allowed for the reconstruction, to those which preserve the gene groups of the given gene orders. Since it is difficult to determine functionally what a gene group is, it has been proposed to consider common combinatorial structures of the gene orders as gene groups (Marcotte et al., 1999; Overbeek et al., 1999). The second considered modification of the rearrangement model is extending the set of allowed rearrangement types. Different types of rearrangement operations have shuffled the gene orders during evolution. It should be attempted to use the same set of rearrangement operations for the reconstruction otherwise distorted or even wrong phylogenetic conclusions may be obtained in the worst case. Both possibilities have been considered for certain rearrangement problems before. Restricted sets of allowed rearrangements have been used successfully for the computation of parsimonious rearrangement scenarios consisting of inversions only where the gene groups are identified as common intervals (Bérard et al., 2007; Figeac and Varré, 2004). Extending the set of allowed rearrangement operations is a delicate task. On the one hand it is unknown which rearrangements have to be regarded because this is part of the phylogeny to be discovered. On the other hand, efficient exact rearrangement methods including several operations are still rare, in particular when transpositions should be included. For example, the problem to compute shortest rearrangement scenarios including transpositions is still of unknown computational complexity. Currently, only efficient approximation algorithms are known (e.g. Bader and Ohlebusch, 2007; Elias and Hartman, 2006). Two problems have been studied with respect to one or even both of these possibilities in the scope of this work. The first one is the inversion median problem. Given the gene orders of some taxa, this problem asks for potential ancestral gene orders such that the corresponding inversion scenario is parsimonious, i.e. has a minimum length. Solving this problem is an essential component 2 of algorithms for computing phylogenetic trees from gene arrangements (Bourque and Pevzner, 2002; Moret et al., 2002a, 2001). The unconstrained inversion median problem is NP-hard (Caprara, 2003). In Chapter 3 the inversion median problem is studied under the additional constraint to preserve gene groups of the input gene orders. Common intervals, i.e. sets of genes that appear consecutively in the gene orders, are used for modelling gene groups. The problem of finding such ancestral gene orders is called the preserving inversion median problem. Already the problem of finding a shortest inversion scenario for two gene orders is NP-hard (Figeac and Varré, 2004). Mitochondrial gene orders are a rich source for phylogenetic investigations because they are known for more than 1 000 species. Four rearrangement operations are reported at least in the literature to be relevant for the study of mitochondrial gene order evolution (Boore, 1999): That is inversions, transpositions, inverse transpositions, and tandem duplication random loss (TDRL). Efficient methods for a plausible reconstruction of genome rearrangements for mitochondrial gene orders using all four operations are presented in Chapter 4. An important rearrangement operation, in particular for the study of mitochondrial gene orders, is the tandem duplication random loss operation (e.g. Boore, 2000; Mauro et al., 2006). This rearrangement duplicates a part of a gene order followed by the random loss of one of the redundant copies of each gene. The gene order is rearranged depending on which copy is lost. This rearrangement should be regarded for reconstructing phylogeny from gene order data. But the properties of this rearrangement operation have rarely been studied (Bouvel and Rossin, 2009; Chaudhuri et al., 2006). The combinatorial properties of the TDRL operation are studied in Chapter 5. The enumeration and counting of sorting TDRLs, that is TDRL operations reducing the distance, is studied in particular. Closed formulas for computing the number of sorting TDRLs and methods for the enumeration are presented. Furthermore, TDRLs are one of the operations considered in Chapter 4. An interesting property of this rearrangement, distinguishing it from other rearrangements, is its asymmetry. That is the effects of a single TDRL can (in the most cases) not be reversed with a single TDRL. The use of this property for phylogeny reconstruction is studied in Section 4.3. This thesis is structured as follows. The existing approaches obeying similar types of modified rearrangement models as well as important concepts and computational methods to related problems are reviewed in Chapter 2. The combinatorial structures of gene orders that have been proposed for identifying gene groups, in particular common intervals, as well as the computational approaches for their computation are reviewed in Section 2.2. Approaches for computing parsimonious pairwise rearrangement scenarios are outlined in Section 2.3. Methods for the computation genome rearrangement scenarios obeying biologically motivated constraints, as introduced above, are detailed in Section 2.4. The approaches for the inversion median problem are covered in Section 2.5. Methods for the reconstruction of phylogenetic trees from gene arrangement data are briefly outlined in Section 2.6.3 1 Introduction Chapter 3 introduces the new algorithms CIP, ECIP, and TCIP for solving the preserving inversion median problem. The efficiency of the algorithm is empirically studied for simulated as well as mitochondrial data. The description of algorithms CIP and ECIP is based on Bernt et al. (2006b). TCIP has been described in Bernt et al. (2007a, 2008b). But the theoretical foundation of TCIP is extended significantly within this work in order to allow for more than three input permutations. Gene order rearrangement methods that have been developed for the reconstruction of the phylogeny of mitochondrial gene orders are presented in the fourth chapter. The presented algorithm CREx computes rearrangement scenarios for pairs of gene orders. CREx regards the four types of rearrangement operations which are important for mitochondrial gene orders. Based on CREx the algorithm TreeREx for assigning rearrangement events to a given tree is developed. The quality of the CREx reconstructions is analysed in a large empirical study for simulated gene orders. The results of TreeREx are analysed for several mitochondrial data sets. Algorithms CREx and TreeREx have been published in Bernt et al. (2008a, 2007c). The analysis of the mitochondrial gene orders of Echinodermata was included in Perseke et al. (2008). Additionally, a new and simple method is presented to explore the potential of the CREx method. The new method is applied to the complete mitochondrial data set. The problem of enumerating and counting sorting TDRLs is studied in Chapter 5. The theoretical results are covered to a large extent by Bernt et al. (2009b). The missing combinatorial explanation for some of the presented formulas is given here for the first time. Therefor, a new method for the enumeration and counting of sorting TDRLs has been developed (Bernt et al., 2009a)

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