Double Cut And Join Research Articles

The Floor Is Lava: Halving Natural Genomes with Viaducts, Piers, and Pontoons.

Whole Genome Duplications (WGDs) are events that double the content and structure of a genome. In some organisms, multiple WGD events have been observed while loss of genetic material is a typical occurrence following a WGD event. The requirement of classic rearrangement models that every genetic marker has to occur exactly two times in a given problem instance, therefore, poses a serious restriction in this context. The Double-Cut and Join (DCJ) model is a simple and powerful model for the analysis of large structural rearrangements. After being extended to the DCJ-Indel model, capable of handling gains and losses of genetic material, research has shifted in recent years toward enabling it to handle natural genomes, for which no assumption about the distribution of markers has to be made. The traditional theoretical framework for studying WGD events is the Genome Halving Problem (GHP). While the GHP is solved for the DCJ model for genomes without losses, there are currently no exact algorithms utilizing the DCJ-Indel model that are able to handle natural genomes. In this work, we present a general view on the DCJ-Indel model that we apply to derive an exact polynomial time and space solution for the GHP on genomes with at most two genes per family before generalizing the problem to an integer linear program solution for natural genomes.

Recombinations, chains and caps: resolving problems with the DCJ-indel model

One of the most fundamental problems in genome rearrangement studies is the (genomic) distance problem. It is typically formulated as finding the minimum number of rearrangements under a model that are needed to transform one genome into the other. A powerful multi-chromosomal model is the Double Cut and Join (DCJ) model.While the DCJ model is not able to deal with some situations that occur in practice, like duplicated or lost regions, it was extended over time to handle these cases. First, it was extended to the DCJ-indel model, solving the issue of lost markers. Later ILP-solutions for so called natural genomes, in which each genomic region may occur an arbitrary number of times, were developed, enabling in theory to solve the distance problem for any pair of genomes. However, some theoretical and practical issues remained unsolved. On the theoretical side of things, there exist two disparate views of the DCJ-indel model, motivated in the same way, but with different conceptualizations that could not be reconciled so far. On the practical side, while ILP solutions for natural genomes typically perform well on telomere to telomere resolved genomes, they have been shown in recent years to quickly loose performance on genomes with a large number of contigs or linear chromosomes. This has been linked to a particular technique, namely capping. Simply put, capping circularizes linear chromosomes by concatenating them during solving time, increasing the solution space of the ILP superexponentially. Recently, we introduced a new conceptualization of the DCJ-indel model within the context of another rearrangement problem. In this manuscript, we will apply this new conceptualization to the distance problem. In doing this, we uncover the relation between the disparate conceptualizations of the DCJ-indel model. We are also able to derive an ILP solution to the distance problem that does not rely on capping. This solution significantly improves upon the performance of previous solutions on genomes with high numbers of contigs while still solving the problem exactly and being competitive in performance otherwise. We demonstrate the performance advantage on simulated genomes as well as showing its practical usefulness in an analysis of 11 Drosophila genomes.

Investigating the complexity of the double distance problems

BackgroundTwo genomes mathbb {A} and mathbb {B} over the same set of gene families form a canonical pair when each of them has exactly one gene from each family. Denote by n_* the number of common families of mathbb {A} and mathbb {B}. Different distances of canonical genomes can be derived from a structure called breakpoint graph, which represents the relation between the two given genomes as a collection of cycles of even length and paths. Let c_i and p_j be respectively the numbers of cycles of length i and of paths of length j in the breakpoint graph of genomes mathbb {A} and mathbb {B}. Then, the breakpoint distance of mathbb {A} and mathbb {B} is equal to n_*-left( c_2+frac{p_0}{2}right). Similarly, when the considered rearrangements are those modeled by the double-cut-and-join (DCJ) operation, the rearrangement distance of mathbb {A} and mathbb {B} is n_*-left( c+frac{p_e }{2}right), where c is the total number of cycles and p_e is the total number of paths of even length.MotivationThe distance formulation is a basic unit for several other combinatorial problems related to genome evolution and ancestral reconstruction, such as median or double distance. Interestingly, both median and double distance problems can be solved in polynomial time for the breakpoint distance, while they are NP-hard for the rearrangement distance. One way of exploring the complexity space between these two extremes is to consider a sigma _k distance, defined to be n_*-left( c_2+c_4+ldots +c_k+frac{p_0+p_2+ldots +p_{k-2}}{2}right), and increasingly investigate the complexities of median and double distance for the sigma _4 distance, then the sigma _6 distance, and so on.ResultsWhile for the median much effort was done in our and in other research groups but no progress was obtained even for the sigma _4 distance, for solving the double distance under sigma _4 and sigma _6 distances we could devise linear time algorithms, which we present here.

Escape from Parsimony of a Double-Cut-and-Join Genome Evolution Process.

We analyze models of genome evolution based on both restricted and unrestricted double-cut-and-join (DCJ) operations. Not only do our models allow different types of operations generated by DCJs (including reversals, translocations, transpositions, fissions, and fusions) to take different weights during the course of evolution, but they also let these weights fluctuate over time. We compare the number of operations along the evolutionary trajectory with the DCJ distance of the genome from its ancestor at each step, and determine at what point they diverge: the process escapes from parsimony. Adapting the method developed by Berestycki and Durrett, we approximate the number of cycles in the breakpoint graph of a random genome at time t and its ancestral genome by the number of tree components in a random graph (not necessarily an Erdös-Rényi one) constructed from the model of evolution. In both models, the process on a genome of size n is bound to its parsimonious estimate up to steps.

Natural family-free genomic distance

BackgroundA classical problem in comparative genomics is to compute the rearrangement distance, that is the minimum number of large-scale rearrangements required to transform a given genome into another given genome. The traditional approaches in this area are family-based, i.e., require the classification of DNA fragments of both genomes into families. Furthermore, the most elementary family-based models, which are able to compute distances in polynomial time, restrict the families to occur at most once in each genome. In contrast, the distance computation in models that allow multifamilies (i.e., families with multiple occurrences) is NP-hard. Very recently, Bohnenkämper et al. (J Comput Biol 28:410–431, 2021) proposed an ILP formulation for computing the genomic distance of genomes with multifamilies, allowing structural rearrangements, represented by the generic double cut and join (DCJ) operation, and content-modifying insertions and deletions of DNA segments. This ILP is very efficient, but must maximize a matching of the genes in each multifamily, in order to prevent the free lunch artifact that would otherwise let empty or almost empty matchings give smaller distances.ResultsIn this paper, we adopt the alternative family-free setting that, instead of family classification, simply uses the pairwise similarities between DNA fragments of both genomes to compute their rearrangement distance. We adapted the ILP mentioned above and developed a model in which pairwise similarities are used to assign weights to both matched and unmatched genes, so that an optimal solution does not necessarily maximize the matching. Our model then results in a natural family-free genomic distance, that takes into consideration all given genes, without prior classification into families, and has a search space composed of matchings of any size. In spite of its bigger search space, our ILP seems to be boosted by a reduction of the number of co-optimal solutions due to the weights. Indeed, it converged faster than the original one by Bohnenkämper et al. for instances with the same number of multiple connections. We can handle not only bacterial genomes, but also fungi and insects, or sets of chromosomes of mammals and plants. In a comparison study of six fruit fly genomes, we obtained accurate results.

Computing the Inversion-Indel Distance.

The inversion distance, that is the distance between two unichromosomal genomes with the same content allowing only inversions of DNA segments, can be exactly computed thanks to a pioneering approach of Hannenhalli and Pevzner from 1995. In 2000, El-Mabrouk extended the inversion model to perform the comparison of unichromosomal genomes with unequal contents, combining inversions with insertions and deletions (indels) of DNA segments, giving rise to the inversion-indel distance. However, only a heuristic was provided for its computation. In 2005, Yancopoulos, Attie and Friedberg started a new branch of research by introducing the generic double cut and join (DCJ) operation, that can represent several genome rearrangements (including inversions). In 2006, Bergeron, Mixtacki and Stoye showed that the DCJ distance can be computed in linear time with a very simple procedure. As a consequence, in 2010 we gave a linear-time algorithm to compute the DCJ-indel distance. This result allowed the inversion-indel model to be revisited from another angle. In 2013, we could show that, when the diagram that represents the relation between the two compared genomes has no bad components, the inversion-indel distance is equal to the DCJ-indel distance. In the present work we complete the study of the inversion-indel distance by giving the first algorithm to compute it exactly even in the presence of bad components.

A general framework for genome rearrangement with biological constraints

This paper generalizes previous studies on genome rearrangement under biological constraints, using double cut and join (DCJ). We propose a model for weighted DCJ, along with a family of optimization problems called varphi-MCPS (Minimum Cost Parsimonious Scenario), that are based on labeled graphs. We show how to compute solutions to general instances of varphi-MCPS, given an algorithm to compute varphi-MCPS on a circular genome with exactly one occurrence of each gene. These general instances can have an arbitrary number of circular and linear chromosomes, and arbitrary gene content. The practicality of the framework is displayed by presenting polynomial-time algorithms that generalize the results of Bulteau, Fertin, and Tannier on the Sorting by wDCJs and indels in intergenes problem, and that generalize previous results on the Minimum Local Parsimonious Scenario problem.

Finding local genome rearrangements

BackgroundThe double cut and join (DCJ) model of genome rearrangement is well studied due to its mathematical simplicity and power to account for the many events that transform gene order. These studies have mostly been devoted to the understanding of minimum length scenarios transforming one genome into another. In this paper we search instead for rearrangement scenarios that minimize the number of rearrangements whose breakpoints are unlikely due to some biological criteria. One such criterion has recently become accessible due to the advent of the Hi-C experiment, facilitating the study of 3D spacial distance between breakpoint regions.ResultsWe establish a link between the minimum number of unlikely rearrangements required by a scenario and the problem of finding a maximum edge-disjoint cycle packing on a certain transformed version of the adjacency graph. This link leads to a 3/2-approximation as well as an exact integer linear programming formulation for our problem, which we prove to be NP-complete. We also present experimental results on fruit flies, showing that Hi-C data is informative when used as a criterion for rearrangements.ConclusionsA new variant of the weighted DCJ distance problem is addressed that ignores scenario length in its objective function. A solution to this problem provides a lower bound on the number of unlikely moves necessary when transforming one gene order into another. This lower bound aids in the study of rearrangement scenarios with respect to chromatin structure, and could eventually be used in the design of a fixed parameter algorithm with a more general objective function.

On the rank-distance median of 3 permutations

BackgroundRecently, Pereira Zanetti, Biller and Meidanis have proposed a new definition of a rearrangement distance between genomes. In this formulation, each genome is represented as a matrix, and the distance d is the rank distance between these matrices. Although defined in terms of matrices, the rank distance is equal to the minimum total weight of a series of weighted operations that leads from one genome to the other, including inversions, translocations, transpositions, and others. The computational complexity of the median-of-three problem according to this distance is currently unknown. The genome matrices are a special kind of permutation matrices, which we study in this paper.In their paper, the authors provide an Oleft (n^{3}right) algorithm for determining three candidate medians, prove the tight approximation ratio frac {4}{3}, and provide a sufficient condition for their candidates to be true medians. They also conduct some experiments that suggest that their method is accurate on simulated and real data.ResultsIn this paper, we extend their results and provide the following: Three invariants characterizing the problem of finding the median of 3 matricesA sufficient condition for uniqueness of medians that can be checked in O(n)A faster, Oleft (n^{2}right) algorithm for determining the median under this conditionA new heuristic algorithm for this problem based on compressed sensingA Oleft (n^{4}right) algorithm that exactly solves the problem when the inputs are orthogonal matrices, a class that includes both permutations and genomes as special cases.ConclusionsOur work provides the first proof that, with respect to the rank distance, the problem of finding the median of 3 genomes, as well as the median of 3 permutations, is exactly solvable in polynomial time, a result which should be contrasted with its NP-hardness for the DCJ (double cut-and-join) distance and most other families of genome rearrangement operations. This result, backed by our experimental tests, indicates that the rank distance is a viable alternative to the DCJ distance widely used in genome comparisons.

Implicit Transpositions in DCJ Scenarios.

Genome rearrangements are large-scale evolutionary events that shuffle genomic architectures. The minimal number of such events between two genomes is often used in phylogenomic studies to measure the evolutionary distance between the genomes. Double-Cut-and-Join (DCJ) operations represent a convenient model of most common genome rearrangements (reversals, translocations, fissions, and fusions), while other genome rearrangements, such as transpositions, can be modeled by pairs of DCJs. Since the DCJ model does not directly account for transpositions, their impact on DCJ scenarios is unclear. In the present work, we study implicit appearance of transpositions (as pairs of DCJs) in DCJ scenarios. We consider shortest DCJ scenarios satisfying the maximum parsimony assumption, as well as more general DCJ scenarios based on some realistic but less restrictive assumptions. In both cases, we derive a uniform lower bound for the rate of implicit transpositions, which depends only on the genomes but not a particular DCJ scenario between them. Our results imply that implicit appearance of transpositions in DCJ scenarios may be unavoidable or even abundant for some pairs of genomes. We estimate that for mammalian genomes implicit transpositions constitute at least 6% of genome rearrangements.

Sorting by reversals and the theory of 4-regular graphs

We show that the theory of sorting by reversals fits into the well-established theory of circuit partitions of 4-regular multigraphs (which also involves the combinatorial structures of circle graphs and delta-matroids). In this way, we expose strong connections between the two theories that have not been fully appreciated before. We also discuss a generalization of sorting by reversals involving the double-cut-and-join (DCJ) operation. Finally, we also show that the theory of sorting by reversals is closely related to that of gene assembly in ciliates.

Approximating the DCJ distance of balanced genomes in linear time

BackgroundRearrangements are large-scale mutations in genomes, responsible for complex changes and structural variations. Most rearrangements that modify the organization of a genome can be represented by the double cut and join (DCJ) operation. Given two balanced genomes, i.e., two genomes that have exactly the same number of occurrences of each gene in each genome, we are interested in the problem of computing the rearrangement distance between them, i.e., finding the minimum number of DCJ operations that transform one genome into the other. This problem is known to be NP-hard.ResultsWe propose a linear time approximation algorithm with approximation factor O(k) for the DCJ distance problem, where k is the maximum number of occurrences of any gene in the input genomes. Our algorithm works for linear and circular unichromosomal balanced genomes and uses as an intermediate step an O(k)-approximation for the minimum common string partition problem, which is closely related to the DCJ distance problem.ConclusionsExperiments on simulated data sets show that our approximation algorithm is very competitive both in efficiency and in quality of the solutions.

Genome rearrangements with indels in intergenes restrict the scenario space.

BackgroundGiven two genomes that have diverged by a series of rearrangements, we infer minimum Double Cut-and-Join (DCJ) scenarios to explain their organization differences, coupled with indel scenarios to explain their intergene size distribution, where DCJs themselves also alter the sizes of broken intergenes.ResultsWe give a polynomial-time algorithm that, given two genomes with arbitrary intergene size distributions, outputs a DCJ scenario which optimizes on the number of DCJs, and given this optimal number of DCJs, optimizes on the total sum of the sizes of the indels.ConclusionsWe show that there is a valuable information in the intergene sizes concerning the rearrangement scenario itself. On simulated data we show that statistical properties of the inferred scenarios are closer to the true ones than DCJ only scenarios, i.e. scenarios which do not handle intergene sizes.

Fast ancestral gene order reconstruction of genomes with unequal gene content.

BackgroundDuring evolution, genomes are modified by large scale structural events, such as rearrangements, deletions or insertions of large blocks of DNA. Of particular interest, in order to better understand how this type of genomic evolution happens, is the reconstruction of ancestral genomes, given a phylogenetic tree with extant genomes at its leaves. One way of solving this problem is to assume a rearrangement model, such as Double Cut and Join (DCJ), and find a set of ancestral genomes that minimizes the number of events on the input tree. Since this problem is NP-hard for most rearrangement models, exact solutions are practical only for small instances, and heuristics have to be used for larger datasets. This type of approach can be called event-based. Another common approach is based on finding conserved structures between the input genomes, such as adjacencies between genes, possibly also assigning weights that indicate a measure of confidence or probability that this particular structure is present on each ancestral genome, and then finding a set of non conflicting adjacencies that optimize some given function, usually trying to maximize total weight and minimizing character changes in the tree. We call this type of methods homology-based.ResultsIn previous work, we proposed an ancestral reconstruction method that combines homology- and event-based ideas, using the concept of intermediate genomes, that arise in DCJ rearrangement scenarios. This method showed better rate of correctly reconstructed adjacencies than other methods, while also being faster, since the use of intermediate genomes greatly reduces the search space. Here, we generalize the intermediate genome concept to genomes with unequal gene content, extending our method to account for gene insertions and deletions of any length. In many of the simulated datasets, our proposed method had better results than MLGO and MGRA, two state-of-the-art algorithms for ancestral reconstruction with unequal gene content, while running much faster, making it more scalable to larger datasets.ConclusionStuding ancestral reconstruction problems under a new light, using the concept of intermediate genomes, allows the design of very fast algorithms by greatly reducing the solution search space, while also giving very good results. The algorithms introduced in this paper were implemented in an open-source software called RINGO (ancestral Reconstruction with INtermediate GenOmes), available at https://github.com/pedrofeijao/RINGO.

Moments of genome evolution by Double Cut-and-Join.

We study statistical estimators of the number of genomic events separating two genomes under a Double Cut-and Join (DCJ) rearrangement model, by a method of moment estimation. We first propose an exact, closed, analytically invertible formula for the expected number of breakpoints after a given number of DCJs. This improves over the heuristic, recursive and computationally slower previously proposed one. Then we explore the analogies of genome evolution by DCJ with evolution of binary sequences under substitutions, permutations under transpositions, and random graphs. Each of these are presented in the literature with intuitive justifications, and are used to import results from better known fields. We formalize the relations by proving a correspondence between moments in sequence and genome evolution, provided substitutions appear four by four in the corresponding model. Eventually we prove a bounded error on two estimators of the number of cycles in the breakpoint graph after a given number of rearrangements, by an analogy with cycles in permutations and components in random graphs.

Sorting Linear Genomes with Rearrangements and Indels.

Rearrangements are mutations that can change the organization of a genome, but not its content. Examples are inversions of DNA segments, translocations of chromosome ends, fusions and fissions of chromosomes. All mentioned rearrangements can be represented by the generic Double Cut and Join (DCJ) operation. However, the DCJ operation also allows circular chromosomes to be created at intermediate steps, even if the compared genomes are linear. In this case it is more plausible to consider a restriction in which the reincorporation of a circular chromosome has to be done immediately after its creation. We call these two consecutive operations an ER composition. It has been shown that an ER composition mimics either an internal block interchange (when two segments in the same chromosome exchange their positions), or an internal transposition (the special case of a block interchange when the two segments are adjacent). The DCJ distance of two genomes is the same, regardless of this restriction, and can be computed in linear time. For comparing two genomes with unequal contents, in addition to rearrangements we have to allow insertions and deletions of DNA segments-named indels. It is already known that the distance in the model combining DCJ and indel operations can be exactly computed. Again, for linear genomes it would be more plausible to adopt a restricted version with ER compositions. This model was studied recently by da Silva et al. (BMC Bioinformatics 13, Suppl. 19, S14, 2012), but only an upper bound for the restricted DCJ-indel distance was provided. Here we first solve an open problem posed in that paper and present a very simple proof showing that the distance, which can be computed in linear time, is the same for both the unrestricted and the restricted DCJ-indel models. We then give a simpler algorithm for computing an optimal restricted DCJ-indel sorting scenario in O(n log n) time. We also relate the DCJ-indel distance to the restricted DCJ-substitution distance, which instead of indels considers a more powerful operation that allows the substitution of a DNA segment by another DNA segment. We show that the DCJ-indel distance is a 2-approximation for the restricted DCJ-substitution distance.

The Complete Chloroplast and Mitochondrial Genomes of the Green Macroalga Ulva sp. UNA00071828 (Ulvophyceae, Chlorophyta)

Sequencing mitochondrial and chloroplast genomes has become an integral part in understanding the genomic machinery and the phylogenetic histories of green algae. Previously, only three chloroplast genomes (Oltmannsiellopsis viridis, Pseudendoclonium akinetum, and Bryopsis hypnoides) and two mitochondrial genomes (O. viridis and P. akinetum) from the class Ulvophyceae have been published. Here, we present the first chloroplast and mitochondrial genomes from the ecologically and economically important marine, green algal genus Ulva. The chloroplast genome of Ulva sp. was 99,983 bp in a circular-mapping molecule that lacked inverted repeats, and thus far, was the smallest ulvophycean plastid genome. This cpDNA was a highly compact, AT-rich genome that contained a total of 102 identified genes (71 protein-coding genes, 28 tRNA genes, and three ribosomal RNA genes). Additionally, five introns were annotated in four genes: atpA (1), petB (1), psbB (2), and rrl (1). The circular-mapping mitochondrial genome of Ulva sp. was 73,493 bp and follows the expanded pattern also seen in other ulvophyceans and trebouxiophyceans. The Ulva sp. mtDNA contained 29 protein-coding genes, 25 tRNA genes, and two rRNA genes for a total of 56 identifiable genes. Ten introns were annotated in this mtDNA: cox1 (4), atp1 (1), nad3 (1), nad5 (1), and rrs (3). Double-cut-and-join (DCJ) values showed that organellar genomes across Chlorophyta are highly rearranged, in contrast to the highly conserved organellar genomes of the red algae (Rhodophyta). A phylogenomic investigation of 51 plastid protein-coding genes showed that Ulvophyceae is not monophyletic, and also placed Oltmannsiellopsis (Oltmannsiellopsidales) and Tetraselmis (Chlorodendrophyceae) closely to Ulva (Ulvales) and Pseudendoclonium (Ulothrichales).

On the family-free DCJ distance and similarity.

Structural variation in genomes can be revealed by many (dis)similarity measures. Rearrangement operations, such as the so called double-cut-and-join (DCJ), are large-scale mutations that can create complex changes and produce such variations in genomes. A basic task in comparative genomics is to find the rearrangement distance between two given genomes, i.e., the minimum number of rearragement operations that transform one given genome into another one. In a family-based setting, genes are grouped into gene families and efficient algorithms have already been presented to compute the DCJ distance between two given genomes. In this work we propose the problem of computing the DCJ distance of two given genomes without prior gene family assignment, directly using the pairwise similarities between genes. We prove that this new family-free DCJ distance problem is APX-hard and provide an integer linear program to its solution. We also study a family-free DCJ similarity and prove that its computation is NP-hard.

An Exact Algorithm to Compute the Double-Cut-and-Join Distance for Genomes with Duplicate Genes.

Computing the edit distance between two genomes is a basic problem in the study of genome evolution. The double-cut-and-join (DCJ) model has formed the basis for most algorithmic research on rearrangements over the last few years. The edit distance under the DCJ model can be computed in linear time for genomes without duplicate genes, while the problem becomes NP-hard in the presence of duplicate genes. In this article, we propose an integer linear programming (ILP) formulation to compute the DCJ distance between two genomes with duplicate genes. We also provide an efficient preprocessing approach to simplify the ILP formulation while preserving optimality. Comparison on simulated genomes demonstrates that our method outperforms MSOAR in computing the edit distance, especially when the genomes contain long duplicated segments. We also apply our method to assign orthologous gene pairs among human, mouse, and rat genomes, where once again our method outperforms MSOAR.

A unifying model of genome evolution under parsimony.

BackgroundParsimony and maximum likelihood methods of phylogenetic tree estimation and parsimony methods for genome rearrangements are central to the study of genome evolution yet to date they have largely been pursued in isolation.ResultsWe present a data structure called a history graph that offers a practical basis for the analysis of genome evolution. It conceptually simplifies the study of parsimonious evolutionary histories by representing both substitutions and double cut and join (DCJ) rearrangements in the presence of duplications. The problem of constructing parsimonious history graphs thus subsumes related maximum parsimony problems in the fields of phylogenetic reconstruction and genome rearrangement. We show that tractable functions can be used to define upper and lower bounds on the minimum number of substitutions and DCJ rearrangements needed to explain any history graph. These bounds become tight for a special type of unambiguous history graph called an ancestral variation graph (AVG), which constrains in its combinatorial structure the number of operations required. We finally demonstrate that for a given history graph G, a finite set of AVGs describe all parsimonious interpretations of G, and this set can be explored with a few sampling moves.ConclusionThis theoretical study describes a model in which the inference of genome rearrangements and phylogeny can be unified under parsimony.