Double Cut And Join Distance Research Articles

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.

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.

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.

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.

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.

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.

Distribution of genome rearrangement distance under double cut and join

Using the double-cut-and-join (DCJ) model for genome rearrangement we use combinatorial techniques to analyze the distribution of genomes under DCJ distance. We present an exponential generating function for the number of genomes that are maximally distant from a given genome and provide a formula for the number of genomes that are any given distance from an arbitrary starting genome. Many mathematical models have been developed to aid biologists and bioinformaticians in their study of the genome rearrangement problem, whose goal is to find the optimal sequence of mutations for the transformation of one genome into another. Using the double-cut-and-join (DCJ) model, Bergeron, Mixtacki, and Stoye [Bergeron et al. 2006] found that the distance between two genomes is completely determined by a bipartite graph created from the genomes. We utilize their data structure to find the distribution of genomes that are distance d from a given genome under DCJ. In Section 2, we introduce genome rearrangement, DCJ, and an important result of the same authors. In Section 3, we present a generating function for the number of maximally distant genomes from a given genome, and in Section 4, we obtain the distribution of all genomes by distance from a given genome.

AliquotG: An Improved Heuristic Algorithm for Genome Aliquoting

An extant genome can be the descendant of an ancient polyploid genome. The genome aliquoting problem is to reconstruct the latter from the former such that the rearrangement distance (i.e., the number of genome rearrangements necessary to transform the former into the latter) is minimal. Though several heuristic algorithms have been published, here, we sought improved algorithms for the problem with respect to the double cut and join (DCJ) distance. The new algorithm makes use of partial and contracted partial graphs, and locally minimizes the distance. Our test results with simulation data indicate that it reliably recovers gene order of the ancestral polyploid genome even when the ancestor is ancient. We also compared the performance of our method with an earlier method using simulation data sets and found that our algorithm has higher accuracy. It is known that vertebrates had undergone two rounds of whole-genome duplication (2R-WGD) during early vertebrate evolution. We used the new algorithm to calculate the DCJ distance between three modern vertebrate genomes and their 2R-WGD ancestor and found that the rearrangement rate might have slowed down significantly since the 2R-WGD. The software AliquotG implementing the algorithm is available as an open-source package from our website (http://mosas.sysu.edu.cn/genome/download_softwares.php).