Reconstruction Codes for DNA Sequences with Uniform Tandem-Duplication Errors

DNA as a data storage medium has several advantages, including far greater data density compared to electronic media. We propose that schemes for data storage in the DNA of living organisms may benefit from studying the reconstruction problem, which is applicable whenever multiple reads of noisy data are available. This strategy is uniquely suited to the medium, which inherently replicates stored data in multiple distinct ways, caused by mutations. We consider noise introduced solely by uniform tandem-duplication, and utilize the relation to constant-weight integer codes in the Manhattan metric. By bounding the intersection of the cross-polytope with hyperplanes, we prove the existence of reconstruction codes with greater capacity than known error-correcting codes, which we can determine analytically for any set of parameters.Comment: 11 pages, 2 figures, Latex; version accepted for publicatio

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)

Evolutionary synthesis of analog networks

The significant increase in the available computational power that took place in recent decades has been accompanied by a growing interest in the application of the evolutionary approach to the synthesis of many kinds of systems and, in particular, to the synthesis of systems like analog electronic circuits, neural networks, and, more generally, autonomous systems, for which no satisfying systematic and general design methodology has been found to date. Despite some interesting results in the evolutionary synthesis of these kinds of systems, the endowment of an artificial evolutionary process with the potential for an appreciable increase of complexity of the systems thus generated appears still an open issue. In this thesis the problem of the evolutionary growth of complexity is addressed taking as starting point the insights contained in the published material reporting the unfinished work done in the late 1940s and early 1950s by John von Neumann on the theory of self-reproducing automata. The evolutionary complexity-growth conditions suggested in that work are complemented here with a series of auxiliary conditions inspired by what has been discovered since then relatively to the structure of biological systems, with a particular emphasis on the workings of genetic regulatory networks seen as the most elementary, full-fledged level of organization of existing living organisms. In this perspective, the first chapter is devoted to the formulation of the problem of the evolutionary growth of complexity, going from the description of von Neumann's complexity-growth conditions to the specification of a set of auxiliary complexity-growth conditions derived from the analysis of the operation of genetic regulatory networks. This leads to the definition of a particular structure for the kind of systems that will be evolved and to the specification of the genetic representation for them. A system with the required structure — for which the name analog network is suggested — corresponds to a collection of devices whose terminals are connected by links characterized by a scalar value of interaction strength. One of the specificities of the evolutionary system defined in this thesis is the way these values of interaction strength are determined. This is done by associating with each device terminal of the evolving analog network a sequence of characters extracted from the sequences that constitute the genome representing the network, and by defining a map from pairs of sequences of characters to values of interaction strength. Whereas the first chapter gives general prescriptions for the definition of an evolutionary system endowed with the desired complexity-growth potential, the second chapter is devoted to the specification of all the details of an actual implementation of those prescriptions. In this chapter the structure of the genome and of the corresponding genetic operators are defined. A technique for the genetic encoding of the devices constituting the analog network is described, along with a way to implement the map that specifies the interaction between the devices of the evolved system, and between them and the devices constituting the external environment of the evolved system. The proposed implementation of the interaction map is based on the local alignment of sequences of characters. It is shown how the parameters defining the local alignment can be chosen, and what strategies can be adopted to prevent the proliferation of unwanted interactions. The third chapter is devoted to the application of the evolutionary system defined in the second chapter to problems aimed at assessing the suitability in an evolutionary context of the local alignment technique and to problems aimed at assessing the evolutionary potential of the complete evolutionary system when applied to the synthesis of analog networks. Finally, the fourth chapter briefly considers some further questions that are relevant to the proposed approach but could not be addressed in the context of this thesis. A series of appendixes is devoted to some complementary issues: the definition of a measure of diversity for an evolutionary population employing the genetic description introduced in this thesis; the choice of the quantizer for the values of interaction strength between the devices constituting the evolved analog network; the modifications required to use the analog electronic circuit simulator SPICE as a simulation engine for an evolutionary or an optimization process

Mathematical models for evolution of genome structure

The structure of a genome can be characterized by its gene content. Evolution of genome structure in closely related species can be studied by examining their synteny or conserved gene order and content. A variety of evolutionary rearrangements like polyploidy, inversions, transpositions, translocations, gene duplication and gene loss degrade synteny over time. In this dissertation, I approach the problem of understanding synteny in genomes and how far back its evolutionary history can be traced in multiple ways. First, I present a probabilistic model of the rearrangements gene loss and transposition (gain) and apply it to the problem of estimating the relative contribution of these rearrangements within a set of syntenic genome segments. This model can be used to predict gene content in syntenic regions of unsequenced genomes. Next, I use optimization methods to recover syntenic segments between genomes based on reconstructions of their parent ancestry. I examine how these reconstructions can be used as input to programs that identify syntenic regions in genomes to reveal more synteny than was previously detected. I use simulations that incorporate each of the evolutionary rearrangements described above to evaluate the models presented in this dissertation. Finally, I apply these models to genomic data from yeast and flowering plants, two eukaryotic systems that are known to have experienced polyploidy. This application is of particular relevance in flowering plants, in which a lot of economically and scientifically important polyploid species have incompletely sequenced genomes

Gene family-free genome comparison

Dörr D. Gene family-free genome comparison. Bielefeld: Universität Bielefeld; 2016.Computational comparative genomics offers valuable insights into the shared and individual evolutionary histories of living and extinct species and expands our understanding of cellular processes in living cells. Comparing genomes means identifying differences that originated from mutational modifications in their evolutionary past. In studying genome evolution, one differentiates between point mutations, genome rearrangements, and content modifications. Point mutations affect one or few consecutive nucleotide bases in the DNA sequence, whereas genome rearrangements operate on larger genomic regions, thereby altering the order and composition of genes in chromosomal sequences. Lastly, content modifications are a result of gene family evolution that causes gene duplications and losses. Genome rearrangement studies commonly assume that evolutionary relationships between all pairs of genes are resolved. Based on the biological concept of homology, the set of genes can be partitioned into gene families. All genes in a gene family are homologous, i.e., they evolved from the same ancestral sequence. Homology information is generally not given, hence gene families are commonly predicted computationally on the basis of sequence similarity or higher order features of their gene products. These predictions are often unreliable, leading to errors in subsequent genome rearrangement studies. In an attempt to avoid errors resulting from incorrect or incomplete gene family assignments, we develop new methods for genome rearrangement studies that do not require prior knowledge of gene family assignments of genes. Our approach, called gene family-free genome comparison, is innovative in that we account for differences between genes caused by point mutations while studying their order and composition in chromosomes. In lieu of gene family assignments, our proposed methods rely on pairwise similarities between genes. In practice, we obtain gene similarities from the conservation of their protein sequences. Two genes that are located next to each other on a chromosome are said to be adjacent, their adjoining extremities form an adjacency. The number of conserved adjacencies, i.e., those adjacencies that are common to two genomes, gives rise to a measure for gene~order-based genome similarity. If the gene content of both genomes is identical, the number of conserved adjacencies is the dual measure of the well-known breakpoint distance. We study the problem of computing the number of conserved adjacencies in a family-free setting, which relies on pairwise similarities between genes. We analyze its computational complexity and develop exact and heuristic algorithms for its solution in pairwise comparisons. We then advance to the problem of reconstructing ancestral sequences. Given three genomes, we study the problem of constructing a fourth genome, called the median, which maximizes a family-free, pairwise measure of conserved adjacencies between the median and each of the three given genomes. Our model is a family-free generalization of the well-studied mixed multichromosomal breakpoint median. We show that this problem is NP-hard and devise an exact algorithm for its solution. Gene orders become increasingly scrambled over longer evolutionary periods of time. In distant genomes, gene order analyses based on identifying pairs of conserved adjacencies might no longer be informative. Yet, relaxed constraints of gene order conservation are still able to capture weaker, but nonetheless existing remnants of common ancestral gene order, which leads to the problem of identifying syntenic blocks in two or more genomes. Knowing the evolutionary relationships between genes, one can assign a unique character to each gene family and represent a chromosome by a string drawn from the alphabet of gene family characters. Two intervals from two strings are called common intervals if the sets of characters within these intervals are identical. We extend this concept to indeterminate strings, which are a class of strings that have at every position a non-empty set of characters. We propose several models of common intervals in indeterminate strings and devise efficient algorithms for their corresponding discovery problems. Subsequently, we use the concept of common intervals in indeterminate strings to identify syntenic regions in a gene family-free setting. We evaluate all our proposed models and algorithms on simulated or biological datasets and assess their performance and applicability in gene family-free genome analyses

Festparameter-Algorithmen fuer die Konsens-Analyse Genomischer Daten

Fixed-parameter algorithms offer a constructive and powerful approach to efficiently obtain solutions for NP-hard problems combining two important goals: Fixed-parameter algorithms compute optimal solutions within provable time bounds despite the (almost inevitable) computational intractability of NP-hard problems. The essential idea is to identify one or more aspects of the input to a problem as the parameters, and to confine the combinatorial explosion of computational difficulty to a function of the parameters such that the costs are polynomial in the non-parameterized part of the input. This makes especially sense for parameters which have small values in applications. Fixed-parameter algorithms have become an established algorithmic tool in a variety of application areas, among them computational biology where small values for problem parameters are often observed. A number of design techniques for fixed-parameter algorithms have been proposed and bounded search trees are one of them. In computational biology, however, examples of bounded search tree algorithms have been, so far, rare. This thesis investigates the use of bounded search tree algorithms for consensus problems in the analysis of DNA and RNA data. More precisely, we investigate consensus problems in the contexts of sequence analysis, of quartet methods for phylogenetic reconstruction, of gene order analysis, and of RNA secondary structure comparison. In all cases, we present new efficient algorithms that incorporate the bounded search tree paradigm in novel ways. On our way, we also obtain results of parameterized hardness, showing that the respective problems are unlikely to allow for a fixed-parameter algorithm, and we introduce integer linear programs (ILP's) as a tool for classifying problems as fixed-parameter tractable, i.e., as having fixed-parameter algorithms. Most of our algorithms were implemented and tested on practical data.Festparameter-Algorithmen bieten einen konstruktiven Ansatz zur Loesung von kombinatorisch schwierigen, in der Regel NP-harten Problemen, der zwei Ziele beruecksichtigt: innerhalb von beweisbaren Laufzeitschranken werden optimale Ergebnisse berechnet. Die entscheidende Idee ist dabei, einen oder mehrere Aspekte der Problemeingabe als Parameter der Problems aufzufassen und die kombinatorische Explosion der algorithmischen Schwierigkeit auf diese Parameter zu beschraenken, so dass die Laufzeitkosten polynomiell in Bezug auf den nicht-parametrisierten Teil der Eingabe sind. Gibt es einen Festparameter-Algorithmus fuer ein kombinatorisches Problem, nennt man das Problem festparameter-handhabbar. Die Entwicklung von Festparameter-Algorithmen macht vor allem dann Sinn, wenn die betrachteten Parameter im Anwendungsfall nur kleine Werte annehmen. Festparameter-Algorithmen sind zu einem algorithmischen Standardwerkzeug in vielen Anwendungsbereichen geworden, unter anderem in der algorithmischen Biologie, wo in vielen Anwendungen kleine Parameterwerte beobachtet werden koennen. Zu den bekannten Techniken fuer den Entwurf von Festparameter-Algorithmen gehoeren unter anderem groessenbeschraenkte Suchbaeume. In der algorithmischen Biologie gibt es bislang nur wenige Beispiele fuer die Anwendung von groessenbeschraenkten Suchbaeumen. Diese Arbeit untersucht den Einsatz groessenbeschraenkter Suchbaeume fuer NP-harte Konsens-Probleme in der Analyse von DNS- und RNS-Daten. Wir betrachten Konsens-Probleme in der Analyse von DNS-Sequenzdaten, in der Analyse von sogenannten Quartettdaten zur Erstellung von phylogenetischen Hypothesen, in der Analyse von Daten ueber die Anordnung von Genen und beim Vergleich von RNS-Strukturdaten. In allen Faellen stellen wir neue effiziente Algorithmen vor, in denen das Paradigma der groessenbeschraenkten Suchbaeume auf neuartige Weise realisiert wird. Auf diesem Weg zeigen wir auch Ergebnisse parametrisierter Haerte, die zeigen, dass fuer die dabei betrachteten Probleme ein Festparameter-Algorithmus unwahrscheinlich ist. Ausserdem fuehren wir ganzzahliges lineares Programmieren als eine neue Technik ein, um die Festparameter-Handhabbarkeit eines Problems zu zeigen. Die Mehrzahl der hier vorgestellten Algorithmen wurde implementiert und auf Anwendungsdaten getestet