Group-theoretic models of the inversion process in bacterial genomes

The variation in genome arrangements among bacterial taxa is largely due to the process of inversion. Recent studies indicate that not all inversions are equally probable, suggesting, for instance, that shorter inversions are more frequent than longer, and those that move the terminus of replication are less probable than those that do not. Current methods for establishing the inversion distance between two bacterial genomes are unable to incorporate such information. In this paper we suggest a group-theoretic framework that in principle can take these constraints into account. In particular, we show that by lifting the problem from circular permutations to the affine symmetric group, the inversion distance can be found in polynomial time for a model in which inversions are restricted to acting on two regions. This requires the proof of new results in group theory, and suggests a vein of new combinatorial problems concerning permutation groups on which group theorists will be needed to collaborate with biologists. We apply the new method to inferring distances and phylogenies for published Yersinia pestis data.Comment: 19 pages, 7 figures, in Press, Journal of Mathematical Biolog

A fast algorithm for the multiple genome rearrangement problem with weighted reversals and transpositions

<p>Abstract</p> <p>Background</p> <p>Due to recent progress in genome sequencing, more and more data for phylogenetic reconstruction based on rearrangement distances between genomes become available. However, this phylogenetic reconstruction is a very challenging task. For the most simple distance measures (the breakpoint distance and the reversal distance), the problem is NP-hard even if one considers only three genomes.</p> <p>Results</p> <p>In this paper, we present a new heuristic algorithm that directly constructs a phylogenetic tree w.r.t. the weighted reversal and transposition distance. Experimental results on previously published datasets show that constructing phylogenetic trees in this way results in better trees than constructing the trees w.r.t. the reversal distance, and recalculating the weight of the trees with the weighted reversal and transposition distance. An implementation of the algorithm can be obtained from the authors.</p> <p>Conclusion</p> <p>The possibility of creating phylogenetic trees directly w.r.t. the weighted reversal and transposition distance results in biologically more realistic scenarios. Our algorithm can solve today's most challenging biological datasets in a reasonable amount of time.</p

Multichromosomal median and halving problems under different genomic distances

<p>Abstract</p> <p>Background</p> <p>Genome median and genome halving are combinatorial optimization problems that aim at reconstructing ancestral genomes as well as the evolutionary events leading from the ancestor to extant species. Exploring complexity issues is a first step towards devising efficient algorithms. The complexity of the median problem for unichromosomal genomes (permutations) has been settled for both the breakpoint distance and the reversal distance. Although the multichromosomal case has often been assumed to be a simple generalization of the unichromosomal case, it is also a relaxation so that complexity in this context does not follow from existing results, and is open for all distances.</p> <p>Results</p> <p>We settle here the complexity of several genome median and halving problems, including a surprising polynomial result for the breakpoint median and guided halving problems in genomes with circular and linear chromosomes, showing that the multichromosomal problem is actually easier than the unichromosomal problem. Still other variants of these problems are NP-complete, including the DCJ double distance problem, previously mentioned as an open question. We list the remaining open problems.</p> <p>Conclusion</p> <p>This theoretical study clears up a wide swathe of the algorithmical study of genome rearrangements with multiple multichromosomal genomes.</p

Dynamics of Genome Rearrangement in Bacterial Populations

Genome structure variation has profound impacts on phenotype in organisms ranging from microbes to humans, yet little is known about how natural selection acts on genome arrangement. Pathogenic bacteria such as Yersinia pestis, which causes bubonic and pneumonic plague, often exhibit a high degree of genomic rearrangement. The recent availability of several Yersinia genomes offers an unprecedented opportunity to study the evolution of genome structure and arrangement. We introduce a set of statistical methods to study patterns of rearrangement in circular chromosomes and apply them to the Yersinia. We constructed a multiple alignment of eight Yersinia genomes using Mauve software to identify 78 conserved segments that are internally free from genome rearrangement. Based on the alignment, we applied Bayesian statistical methods to infer the phylogenetic inversion history of Yersinia. The sampling of genome arrangement reconstructions contains seven parsimonious tree topologies, each having different histories of 79 inversions. Topologies with a greater number of inversions also exist, but were sampled less frequently. The inversion phylogenies agree with results suggested by SNP patterns. We then analyzed reconstructed inversion histories to identify patterns of rearrangement. We confirm an over-representation of “symmetric inversions”—inversions with endpoints that are equally distant from the origin of chromosomal replication. Ancestral genome arrangements demonstrate moderate preference for replichore balance in Yersinia. We found that all inversions are shorter than expected under a neutral model, whereas inversions acting within a single replichore are much shorter than expected. We also found evidence for a canonical configuration of the origin and terminus of replication. Finally, breakpoint reuse analysis reveals that inversions with endpoints proximal to the origin of DNA replication are nearly three times more frequent. Our findings represent the first characterization of genome arrangement evolution in a bacterial population evolving outside laboratory conditions. Insight into the process of genomic rearrangement may further the understanding of pathogen population dynamics and selection on the architecture of circular bacterial chromosomes

BeamGA Median: A Hybrid Heuristic Search Approach

The median problem is significantly applied to derive the most reasonable rearrangement phylogenetic tree for many species. More specifically, the problem is concerned with finding a permutation that minimizes the sum of distances between itself and a set of three signed permutations. Genomes with equal number of genes but different order can be represented as permutations. In this paper, an algorithm, namely BeamGA median, is proposed that combines a heuristic search approach (local beam) as an initialization step to generate a number of solutions, and then a Genetic Algorithm (GA) is applied in order to refine the solutions, aiming to achieve a better median with the smallest possible reversal distance from the three original permutations. In this approach, any genome rearrangement distance can be applied. In this paper, we use the reversal distance. To the best of our knowledge, the proposed approach was not applied before for solving the median problem. Our approach considers true biological evolution scenario by applying the concept of common intervals during the GA optimization process. This allows us to imitate a true biological behavior and enhance genetic approach time convergence. We were able to handle permutations with a large number of genes, within an acceptable time performance and with same or better accuracy as compared to existing algorithms

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)

SoRT2: a tool for sorting genomes and reconstructing phylogenetic trees by reversals, generalized transpositions and translocations

SoRT2 is a web server that allows the user to perform genome rearrangement analysis involving reversals, generalized transpositions and translocations (including fusions and fissions), and infer phylogenetic trees of genomes being considered based on their pairwise genome rearrangement distances. It takes as input two or more linear/circular multi-chromosomal gene (or synteny block) orders in FASTA-like format. When the input is two genomes, SoRT2 will quickly calculate their rearrangement distance, as well as a corresponding optimal scenario by highlighting the genes involved in each rearrangement operation. In the case of multiple genomes, SoRT2 will also construct phylogenetic trees of these genomes based on a matrix of their pairwise rearrangement distances using distance-based approaches, such as neighbor-joining (NJ), unweighted pair group method with arithmetic mean (UPGMA) and Fitch–Margoliash (FM) methods. In addition, if the function of computing jackknife support values is selected, SoRT2 will further perform the jackknife analysis to evaluate statistical reliability of the constructed NJ, UPGMA and FM trees. SoRT2 is available online at http://bioalgorithm.life.nctu.edu.tw/SORT2/