SCJ: a breakpoint-like distance that simplifies several rearrangement problems

The breakpoint distance is one of the most straightforward genome comparison measures. Surprisingly, when it comes to defining it precisely for multichromosomal genomes with both linear and circular chromosomes, there is more than one way to go about it. Pevzner and Tesler gave a definition in a 2003 paper, Tannier et al. defined it differently in 2008, and in this paper we provide yet another alternative, calling it SCJ for single-cut-or-join, in analogy to the popular double cut and join (DCJ) measure. We show that several genome rearrangement problems, such as median and halving, become easy for SCJ, and provide linear and higher polynomial time algorithms for them. For the multichromosomal linear genome median problem, this is the first polynomial time algorithm described, since for other distances this problem is NP-hard. In addition, we show that small parsimony under SCJ is also easy, and can be solved by a variant of Fitch's algorithm. In contrast, big parsimony is NP-hard under SCJ. This new distance measure may be of value as a speedily computable, first approximation to distances based on more realistic rearrangement models.     

Scaffold filling under the breakpoint and related distances

Motivated by the trend of genome sequencing without completing the sequence of the whole genomes, a problem on filling an incomplete multichromosomal genome (or scaffold) I with respect to a complete target genome G was studied. The objective is to minimize the resulting genomic distance between I' and G, where I' is the corresponding filled scaffold. We call this problem the onesided scaffold filling problem. In this paper, we conduct a systematic study for the scaffold filling problem under the breakpoint distance and its variants, for both unichromosomal and multichromosomal genomes (with and without gene repetitions). When the input genome contains no gene repetition (i.e., is a fragment of a permutation), we show that the two-sided scaffold filling problem (i.e., G is also incomplete) is polynomially solvable for unichromosomal genomes under the breakpoint distance and for multichromosomal genomes under the genomic (or DCJ--Double-Cut-and-Join) distance. However, when the input genome contains some repeated genes, even the one-sided scaffold filling problem becomes NP-complete when the similarity measure is the maximum number of adjacencies between two sequences. For this problem, we also present efficient constant-factor approximation algorithms: factor-2 for the general case and factor 1.33 for the one-sided case.     

Breakpoint medians and breakpoint phylogenies: a fixed-parameter approach

With breakpoint distance, the genome rearrangement field delivered one of the currently most popular measures in phylogenetic studies for related species. Here, BREAKPOINT MEDIAN, which is NP-complete already for three given species (whose genomes are represented as signed orderings), is the core basic problem. For the important special case of three species, approximation (ratio 7/6) and exact heuristic algorithms were developed. Here, we provide an exact, fixed-parameter algorithm with provable performance bounds. For instance, a breakpoint median for three signed orderings over nelements that causes at most d breakpoints can be computed in time O((2.15)(d).n). We show the algorithm's practical usefulness through experimental studies. In particular, we demonstrate that a simple implementation of our algorithm combined with a new tree construction heuristic allows for a new approach to breakpoint phylogeny, yielding evolutionary trees that are competitive in comparison with known results developed in a recent series of papers that use clever algorithm engineering methods.     

Multichromosomal median and halving problems under different genomic distances

Background  

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.     

Restricted DCJ model: rearrangement problems with chromosome reincorporation

We study three classical problems of genome rearrangement--sorting, halving, and the median problem--in a restricted double cut and join (DCJ) model. In the DCJ model, introduced by Yancopoulos et al., we can represent rearrangement events that happen in multichromosomal genomes, such as inversions, translocations, fusions, and fissions. Two DCJ operations can mimic transpositions or block interchanges by first extracting an appropriate segment of a chromosome, creating a temporary circular chromosome, and then reinserting it in its proper place. In the restricted model, we are concerned with multichromosomal linear genomes and we require that each circular excision is immediately followed by its reincorporation. Existing linear-time DCJ sorting and halving algorithms ignore this reincorporation constraint. In this article, we propose a new algorithm for the restricted sorting problem running in O(n log n) time, thus improving on the known quadratic time algorithm. We solve the restricted halving problem and give an algorithm that computes a multilinear halved genome in linear time. Finally, we show that the restricted median problem is NP-hard as conjectured.     

Algorithms for sorting unsigned linear genomes by the DCJ operations

MOTIVATION: The double cut and join operation (abbreviated as DCJ) has been extensively used for genomic rearrangement. Although the DCJ distance between signed genomes with both linear and circular (uni- and multi-) chromosomes is well studied, the only known result for the NP-complete unsigned DCJ distance problem is an approximation algorithm for unsigned linear unichromosomal genomes. In this article, we study the problem of computing the DCJ distance on two unsigned linear multichromosomal genomes (abbreviated as UDCJ). RESULTS: We devise a 1.5-approximation algorithm for UDCJ by exploiting the distance formula for signed genomes. In addition, we show that UDCJ admits a weak kernel of size 2k and hence an FPT algorithm running in O(2(2k)n) time.     

GRIMM: genome rearrangements web server

SUMMARY: Genome Rearrangements In Man and Mouse (GRIMM) is a tool for analyzing rearrangements of gene orders in pairs of unichromosomal and multichromosomal genomes, with either signed or unsigned gene data. Although there are several programs for analyzing rearrangements in unichromosomal genomes, this is the first to analyze rearrangements in multichromosomal genomes. GRIMM also provides a new algorithm for analyzing comparative maps for which gene directions are unknown. AVAILABILITY: A web server, with instructions and sample data, is available at http://www-cse.ucsd.edu/groups/bioinformatics/GRIMM.     

Poisson adjacency distributions in genome comparison: multichromosomal, circular, signed and unsigned cases

The number of common adjacencies of genetic markers, as a measure of the similarity of two genomes, has been widely used as indicator of evolutionary relatedness and as the basis for inferring phylogenetic relationships. Its probability distribution enables statistical tests in detecting whether significant evolutionary signal remains in the marker order. In this article, we derive the probability distributions of the number of adjacencies for a number of types of genome--signed or unsigned, circular or linear, single-chromosome or multichromosomal. Generating functions are found for singlechromosome cases, from which exact counts can be calculated. Probability approaches are adopted for multichromosomal cases, where we.nd the exact values for expectations and variances. In both cases, the limiting distributions are derived in term of numbers of adjacencies. For all unsigned cases, the limiting distribution is Poisson with parameter 2; for all signed cases, the limiting distribution is Poisson with parameter (1/2).     

The zero exemplar distance problem

Given two genomes with duplicate genes, Zero Exemplar Distance is the problem of deciding whether the two genomes can be reduced to the same genome without duplicate genes by deleting all but one copy of each gene in each genome. Blin, Fertin, Sikora, and Vialette recently proved that Zero Exemplar Distance for monochromosomal genomes is NP-hard even if each gene appears at most two times in each genome, thereby settling an important open question on genome rearrangement in the exemplar model. In this article, we give a very simple alternative proof of this result. We also study the problem Zero Exemplar Distance for multichromosomal genomes without gene order, and prove the analogous result that it is also NP-hard even if each gene appears at most two times in each genome. For the positive direction, we show that both variants of Zero Exemplar Distance admit polynomial-time algorithms if each gene appears exactly once in one genome and at least once in the other genome. In addition, we present a polynomial-time algorithm for the related problem Exemplar Longest Common Subsequence in the special case that each mandatory symbol appears exactly once in one input sequence and at least once in the other input sequence. This answers an open question of Bonizzoni et al. We also show that Zero Exemplar Distance for multichromosomal genomes without gene order is fixed-parameter tractable in the general case if the parameter is the maximum number of chromosomes in each genome.     

Duplication and inversion history of a tandemly repeated genes family.

Given a phylogenetic tree for a family of tandemly repeated genes and their signed order on the chromosome, we aim to find the minimum number of inversions compatible with an evolutionary history of this family. This is the first attempt to account for inversions in an evolutionary model of tandemly repeated genes. We present a branch-and-bound algorithm that finds the exact solution, and a polynomial-time heuristic based on the breakpoint distance. We show, on simulated data, that those algorithms can be used to improve phylogenetic inference of tandemly repeated gene families. An application on a published phylogeny of KRAB zinc finger genes is presented.     

Early eukaryote evolution based on mitochondrial gene order breakpoints.

The comparison of the gene orders in a set of genomes can be used to infer their phylogenetic relationships and to reconstruct ancestral gene orders. For three genomes this is done by solving the "median problem for breakpoints"; this solution can then be incorporated into a routine for estimating optimal gene orders for all the ancestral genomes in a fixed phylogeny. For the difficult (and most prevalent) case where the genomes contain partially different sets of genes, we present a general heuristic for the median problem for induced breakpoints. A fixed-phylogeny optimization based on this is applied in a phylogenetic study of a set of completely sequenced protist mitochondrial genomes, confirming some of the recent sequence-based groupings which have been proposed and, conversely, confirming the usefulness of the breakpoint method as a phylogenetic tool even for small genomes.     

Genome aliquoting revisited

We prove that the genome aliquoting problem, the problem of finding a recent polyploid ancestor of a genome, with breakpoint distance can be solved in polynomial time. We propose an aliquoting algorithm that is a 2-approximation for the genome aliquoting problem with double cut and join distance, improving upon the previous best solution to this problem, Feij?o and Meidanis' 4-approximation algorithm.     

On sorting by translocations.

The study of genome rearrangements is an important tool in comparative genomics. This paper revisits the problem of sorting a multichromosomal genome by translocations, i.e., exchanges of chromosome ends. We give an elementary proof of the formula for computing the translocation distance in linear time, and we give a new algorithm for sorting by translocations, correcting an error in a previous algorithm by Hannenhalli.     

Colored de Bruijn graphs and the genome halving problem

Breakpoint graph analysis is a key algorithmic technique in studies of genome rearrangements. However, breakpoint graphs are defined only for genomes without duplicated genes, thus limiting their applications in rearrangement analysis. We discuss a connection between the breakpoint graphs and de Bruijn graphs that leads to a generalization of the notion of breakpoint graph for genomes with duplicated genes. We further use the generalized breakpoint graphs to study the genome halving problem (first introduced and solved by Nadia El-Mabrouk and David Sankoff). The El-Mabrouk-Sankoff algorithm is rather complex, and, in this paper, we present an alternative approach that is based on generalized breakpoint graphs. The generalized breakpoint graphs make the El-Mabrouk-Sankoff result more transparent and promise to be useful in future studies of genome rearrangements     

Polarity Related Influence Maximization in Signed Social Networks

Influence maximization in social networks has been widely studied motivated by applications like spread of ideas or innovations in a network and viral marketing of products. Current studies focus almost exclusively on unsigned social networks containing only positive relationships (e.g. friend or trust) between users. Influence maximization in signed social networks containing both positive relationships and negative relationships (e.g. foe or distrust) between users is still a challenging problem that has not been studied. Thus, in this paper, we propose the polarity-related influence maximization (PRIM) problem which aims to find the seed node set with maximum positive influence or maximum negative influence in signed social networks. To address the PRIM problem, we first extend the standard Independent Cascade (IC) model to the signed social networks and propose a Polarity-related Independent Cascade (named IC-P) diffusion model. We prove that the influence function of the PRIM problem under the IC-P model is monotonic and submodular Thus, a greedy algorithm can be used to achieve an approximation ratio of 1-1/e for solving the PRIM problem in signed social networks. Experimental results on two signed social network datasets, Epinions and Slashdot, validate that our approximation algorithm for solving the PRIM problem outperforms state-of-the-art methods.     

Double cut and join with insertions and deletions

Many approaches to compute the genomic distance are still limited to genomes with the same content, without duplicated markers. However, differences in the gene content are frequently observed and can reflect important evolutionary aspects. While duplicated markers can hardly be handled by exact models, when duplicated markers are not allowed, a few polynomial time algorithms that include genome rearrangements, insertions and deletions were already proposed. In an attempt to improve these results, in the present work we give the first linear time algorithm to compute the distance between two multichromosomal genomes with unequal content, but without duplicated markers, considering insertions, deletions and double cut and join (DCJ) operations. We derive from this approach algorithms to sort one genome into another one also using DCJ operations, insertions and deletions. The optimal sorting scenarios can have different compositions and we compare two types of sorting scenarios: one that maximizes and one that minimizes the number of DCJ operations with respect to the number of insertions and deletions. We also show that, although the triangle inequality can be disrupted in the proposed genomic distance, it is possible to correct this problem adopting a surcharge on the number of non-common markers. We use our method to analyze six species of Rickettsia, a group of obligate intracellular parasites, and identify preliminary evidence of clusters of deletions.     

On the distribution of the number of cycles in the breakpoint graph of a random signed permutation

We use the finite Markov chain embedding technique to obtain the distribution of the number of cycles in the breakpoint graph of a random uniform signed permutation. This further gives a very good approximation of the distribution of the reversal distance between two random genomes.     

Perfect sorting by reversals is not always difficult

We propose new algorithms for computing pairwise rearrangement scenarios that conserve the combinatorial structure of genomes. More precisely, we investigate the problem of sorting signed permutations by reversals without breaking common intervals. We describe a combinatorial framework for this problem that allows us to characterize classes of signed permutations for which one can compute, in polynomial time, a shortest reversal scenario that conserves all common intervals. In particular, we define a class of permutations for which this computation can be done in linear time with a very simple algorithm that does not rely on the classical Hannenhalli-Pevzner theory for sorting by reversals. We apply these methods to the computation of rearrangement scenarios between permutations obtained from 16 synteny blocks of the X chromosomes of the human, mouse, and rat     

The gene family-free median of three

Background

The gene family-free framework for comparative genomics aims at providing methods for gene order analysis that do not require prior gene family assignment, but work directly on a sequence similarity graph. We study two problems related to the breakpoint median of three genomes, which asks for the construction of a fourth genome that minimizes the sum of breakpoint distances to the input genomes.

Methods

We present a model for constructing a median of three genomes in this family-free setting, based on maximizing an objective function that generalizes the classical breakpoint distance by integrating sequence similarity in the score of a gene adjacency. We study its computational complexity and we describe an integer linear program (ILP) for its exact solution. We further discuss a related problem called family-free adjacencies for k genomes for the special case of \(k \le 3\) and present an ILP for its solution. However, for this problem, the computation of exact solutions remains intractable for sufficiently large instances. We then proceed to describe a heuristic method, FFAdj-AM, which performs well in practice.

Results

The developed methods compute accurate positional orthologs for genomes comparable in size of bacterial genomes on simulated data and genomic data acquired from the OMA orthology database. In particular, FFAdj-AM performs equally or better when compared to the well-established gene family prediction tool MultiMSOAR.

Conclusions

We study the computational complexity of a new family-free model and present algorithms for its solution. With FFAdj-AM, we propose an appealing alternative to established tools for identifying higher confidence positional orthologs.

     

Assaying chromosomal inversions by single-molecule haplotyping

Inversions are an important form of structural variation, but they are difficult to characterize, as their breakpoints often fall within inverted repeats. We have developed a method called 'haplotype fusion' in which an inversion breakpoint is genotyped by performing fusion PCR on single molecules of human genomic DNA. Fusing single-copy sequences bracketing an inversion breakpoint generates orientation-specific PCR products, exemplified by a genotyping assay for the int22 hemophilia A inversion on Xq28. Furthermore, we demonstrated that inversion events with breakpoints embedded within long (>100 kb) inverted repeats can be genotyped by haplotype-fusion PCR followed by bead-based single-molecule haplotyping on repeat-specific markers bracketing the inversion breakpoint. We illustrate this method by genotyping a Yp paracentric inversion sponsored by >300-kb-long inverted repeats. The generality of our methods to survey for, and genotype chromosomal inversions should help our understanding of the contribution of inversions to genomic variation, inherited diseases and cancer.