On the distribution of cycles and paths in multichromosomal breakpoint graphs and the expected value of rearrangement distance

Feijão P, Martinez F, Thévenin A. On the distribution of cycles and paths in multichromosomal breakpoint graphs and the expected value of rearrangement distance. BMC Bioinformatics. 2015;16(Suppl 19): S1.Finding the smallest sequence of operations to transform one genome into another is an important problem in comparative genomics. The breakpoint graph is a discrete structure that has proven to be effective in solving distance problems, and the number of cycles in a cycle decomposition of this graph is one of the remarkable parameters to help in the solution of related problems. For a fixed k, the number of linear unichromosomal genomes (signed or unsigned) with n elements such that the induced breakpoint graphs have k disjoint cycles, known as the Hultman number, has been already determined. In this work we extend these results to multichromosomal genomes, providing formulas to compute the number of multichromosal genomes having a fixed number of cycles and/or paths. We obtain an explicit formula for circular multichromosomal genomes and recurrences for general multichromosomal genomes, and discuss how these series can be used to calculate the distribution and expected value of the rearrangement distance between random genomes

Sobre modelos de rearranjo de genomas

Orientador: João MeidanisTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Rearranjo de genomas é o nome dado a eventos onde grandes blocos de DNA trocam de posição durante o processo evolutivo. Com a crescente disponibilidade de sequências completas de DNA, a análise desse tipo de eventos pode ser uma importante ferramenta para o entendimento da genômica evolutiva. Vários modelos matemáticos de rearranjo de genomas foram propostos ao longo dos últimos vinte anos. Nesta tese, desenvolvemos dois novos modelos. O primeiro foi proposto como uma definição alternativa ao conceito de distância de breakpoint. Essa distância é uma das mais simples medidas de rearranjo, mas ainda não há um consenso quanto à sua definição para o caso de genomas multi-cromossomais. Pevzner e Tesler deram uma definição em 2003 e Tannier et al. a definiram de forma diferente em 2008. Nesta tese, nós desenvolvemos uma outra alternativa, chamada de single-cut-or-join (SCJ). Nós mostramos que, no modelo SCJ, além da distância, vários problemas clássicos de rearranjo, como a mediana de rearranjo, genome halving e pequena parcimônia são fáceis, e apresentamos algoritmos polinomiais para eles. O segundo modelo que apresentamos é o formalismo algébrico por adjacências, uma extensão do formalismo algébrico proposto por Meidanis e Dias, que permite a modelagem de cromossomos lineares. Esta era a principal limitação do formalismo original, que só tratava de cromossomos circulares. Apresentamos algoritmos polinomiais para o cálculo da distância algébrica e também para encontrar cenários de rearranjo entre dois genomas. Também mostramos como calcular a distância algébrica através do grafo de adjacências, para facilitar a comparação com outras distâncias de rearranjo. Por fim, mostramos como modelar todas as operações clássicas de rearranjo de genomas utilizando o formalismo algébricoAbstract: Genome rearrangements are events where large blocks of DNA exchange places during evolution. With the growing availability of whole genome data, the analysis of these events can be a very important and promising tool for understanding evolutionary genomics. Several mathematical models of genome rearrangement have been proposed in the last 20 years. In this thesis, we propose two new rearrangement models. The first was introduced as an alternative definition of the breakpoint distance. The breakpoint distance is one of the most straightforward genome comparison measures, but when it comes to defining it precisely for multichromosomal genomes, there is more than one way to go about it. Pevzner and Tesler gave a definition in a 2003 paper, and Tannier et al. defined it differently in 2008. In this thesis we provide yet another alternative, calling it single-cut-or-join (SCJ). We show that several genome rearrangement problems, such as genome median, genome halving and small parsimony, become easy for SCJ, and provide polynomial time algorithms for them. The second model we introduce is the Adjacency Algebraic Theory, an extension of the Algebraic Formalism proposed by Meidanis and Dias that allows the modeling of linear chromosomes, the main limitation of the original formalism, which could deal with circular chromosomes only. We believe that the algebraic formalism is an interesting alternative for solving rearrangement problems, with a different perspective that could complement the more commonly used combinatorial graph-theoretic approach. We present polynomial time algorithms to compute the algebraic distance and find rearrangement scenarios between two genomes. We show how to compute the rearrangement distance from the adjacency graph, for an easier comparison with other rearrangement distances. Finally, we show how all classic rearrangement operations can be modeled using the algebraic theoryDoutoradoCiência da ComputaçãoDoutor em Ciência da Computaçã

On the PATHGROUPS approach to rapid small phylogeny

We present a data structure enabling rapid heuristic solution to the ancestral genome reconstruction problem for given phylogenies under genomic rearrangement metrics. The efficiency of the greedy algorithm is due to fast updating of the structure during run time and a simple priority scheme for choosing the next step. Since accuracy deteriorates for sets of highly divergent genomes, we investigate strategies for improving accuracy and expanding the range of data sets where accurate reconstructions can be expected. This includes a more refined priority system, and a two-step look-ahead, as well as iterative local improvements based on a the median version of the problem, incorporating simulated annealing. We apply this to a set of yeast genomes to corroborate a recent gene sequence-based phylogeny

The rise and fall of breakpoint reuse depending on genome resolution

<p>Abstract</p> <p>Background</p> <p>During evolution, large-scale genome rearrangements of chromosomes shuffle the order of homologous genome sequences ("synteny blocks") across species. Some years ago, a controversy erupted in genome rearrangement studies over whether rearrangements recur, causing breakpoints to be reused.</p> <p>Methods</p> <p>We investigate this controversial issue using the synteny block's for human-mouse-rat reported by Bourque <it>et al</it>. and a series of synteny blocks we generated using Mauve at resolutions ranging from coarse to very fine-scale. We conducted analyses to test how resolution affects the traditional measure of the breakpoint reuse rate<it>.</it></p> <p>Results</p> <p>We found that the inversion-based breakpoint reuse rate is low at fine-scale synteny block resolution and that it rises and eventually falls as synteny block resolution decreases. By analyzing the cycle structure of the breakpoint graph of human-mouse-rat synteny blocks for human-mouse and comparing with theoretically derived distributions for random genome rearrangements, we showed that the implied genome rearrangements at each level of resolution become more “random” as synteny block resolution diminishes. At highest synteny block resolutions the Hannenhalli-Pevzner inversion distance deviates from the Double Cut and Join distance, possibly due to small-scale transpositions or simply due to inclusion of erroneous synteny blocks. At synteny block resolutions as coarse as the Bourque <it>et al</it>. blocks, we show the breakpoint graph cycle structure has already converged to the pattern expected for a random distribution of synteny blocks.</p> <p>Conclusions</p> <p>The inferred breakpoint reuse rate depends on synteny block resolution in human-mouse genome comparisons. At fine-scale resolution, the cycle structure for the transformation appears less random compared to that for coarse resolution. Small synteny blocks may contain critical information for accurate reconstruction of genome rearrangement history and parameters.</p

Parking functions, labeled trees and DCJ sorting scenarios

In genome rearrangement theory, one of the elusive questions raised in recent years is the enumeration of rearrangement scenarios between two genomes. This problem is related to the uniform generation of rearrangement scenarios, and the derivation of tests of statistical significance of the properties of these scenarios. Here we give an exact formula for the number of double-cut-and-join (DCJ) rearrangement scenarios of co-tailed genomes. We also construct effective bijections between the set of scenarios that sort a cycle and well studied combinatorial objects such as parking functions and labeled trees.Comment: 12 pages, 3 figure

Guided genome halving: hardness, heuristics and the history of the Hemiascomycetes

Motivation: Some present day species have incurred a whole genome doubling event in their evolutionary history, and this is reflected today in patterns of duplicated segments scattered throughout their chromosomes. These duplications may be used as data to ‘halve’ the genome, i.e. to reconstruct the ancestral genome at the moment of doubling, but the solution is often highly nonunique. To resolve this problem, we take account of outgroups, external reference genomes, to guide and narrow down the search

Genome dedoubling by DCJ and reversal

<p>Abstract</p> <p>Background</p> <p>Segmental duplications in genomes have been studied for many years. Recently, several studies have highlighted a biological phenomenon called <it>breakpoint-duplication</it> that apparently associates a significant proportion of segmental duplications in Mammals, and the Drosophila species group, to breakpoints in rearrangement events.</p> <p>Results</p> <p>In this paper, we introduce and study a combinatorial problem, inspired from the breakpoint-duplication phenomenon, called the <it>Genome Dedoubling Problem.</it> It consists of finding a minimum length rearrangement scenario required to transform a genome with duplicated segments into a non-duplicated genome such that duplications are caused by rearrangement breakpoints. We show that the problem, in the Double-Cut-and-Join (DCJ) and the reversal rearrangement models, can be reduced to an APX-complete problem, and we provide algorithms for the Genome Dedoubling Problem with 2-approximable parts. We apply the methods for the reconstruction of a non-duplicated ancestor of <it>Drosophila yakuba.</it></p> <p>Conclusions</p> <p>We present the <it>Genome Dedoubling Problem</it>, and describe two algorithms solving the problem in the DCJ model, and the reversal model. The usefulness of the problems and the methods are showed through an application to real Drosophila data.</p

Whole Genome Duplications and Contracted Breakpoint Graphs

The genome halving problem, motivated by the whole genome duplication events in molecular evolution, was solved by El-Mabrouk and Sankoff in the pioneering paper [SIAM J. Comput., 32 (2003), pp. 754–792]. The El-Mabrouk–Sankoff algorithm is rather complex, inspiring a quest for a simpler solution. An alternative approach to the genome halving problem based on the notion of the contracted breakpoint graph was recently proposed in [M. A. Alekseyev and P. A. Pevzner, IEEE/ACM Trans. Comput. Biol. Bioinformatics, 4 (2007), pp. 98–107]. This new technique reveals that while the El-Mabrouk–Sankoff result is correct in most cases, it does not hold in the case of unichromosomal genomes. This raises a problem of correcting a flaw in the El- Mabrouk–Sankoff analysis and devising an algorithm that deals adequately with all genomes. In this paper we efficiently classify all genomes into two classes and show that while the El-Mabrouk–Sankoff theorem holds for the first class, it is incorrect for the second class. The crux of our analysis is a new combinatorial invariant defined on duplicated permutations. Using this invariant we were able to come up with a full proof of the genome halving theorem and a polynomial algorithm for the genome halving problem

Streaming Breakpoint Graph Analytics for Accelerating and Parallelizing the Computation of DCJ Median of Three Genomes

AbstractThe problem of finding the median of three genomes is the key process in building the most parsimonious phylogenetic trees from genome rearrangement data. The median problem using Double-Cut-and-Join (DCJ) distance is NP-hard and the best exact algorithm is based on a branch-and-bound best-first search strategy to explore sub-graph patterns in Multiple BreakPoint Graph (MBG). In this paper, by taking advantage of the “streaming” property of MBG, we introduce the “footprint-based” data structure to reduce the space requirement of a single search nodes from O(v2) to O(v); minimize the redundant computation in counting cycles/paths to update bounds, which leads to dramatically decrease of workload of a single search node. Additional heuristic of branching strategy is introduced to help reducing the searching space. Last but not least, the introduction of a multi-thread shared memory parallel algorithm with two load balancing strategies bring in additional benefit by distributing search work efficiently among different processors. We conduct extensive experiments on simulated datasets and our results show significant improvement on all datasets. And we test our DCJ median algorithm with GASTS, a state of the art software phylogenetic tree construction package. On the real high resolution Drosophila data set, our exact algorithm run as fast as the heuristic algorithm and help construct a better phylogenetic tree