The Distance and Median Problems in the Single-Cut-Or-Join Model with Single-Gene Duplications

Background. In the field of genome rearrangement algorithms, models accounting for gene duplication lead often to hard problems. For example, while computing the pairwise distance is tractable in most duplication-free models, the problem is NP-complete for most extensions of these models accounting for duplicated genes. Moreover, problems involving more than two genomes, such as the genome median and the Small Parsimony problem, are intractable for most duplication-free models, with some exceptions, for example the Single-Cut-or-Join (SCJ) model. Results. We introduce a variant of the SCJ distance that accounts for duplicated genes, in the context of directed evolution from an ancestral genome to a descendant genome where orthology relations between ancestral genes and their descendant are known. Our model includes two duplication mechanisms: single-gene tandem duplication and the creation of single-gene circular chromosomes. We prove that in this model, computing the directed distance and a parsimonious evolutionary scenario in terms of SCJ and single-gene duplication events can be done in linear time. We also show that the directed median problem is tractable for this distance, while the rooted median problem, where we assume that one of the given genomes is ancestral to the median, is NP-complete. We also describe an Integer Linear Program for solving this problem. We evaluate the directed distance and rooted median algorithms on simulated data. Conclusion. Our results provide a simple genome rearrangement model, extending the SCJ model to account for single-gene duplications, for which we prove a mix of tractability and hardness results. For the NP-complete rooted median problem, we design a simple Integer Linear Program. Our publicly available implementation of these algorithms for the directed distance and median problems allow to solve efficiently these problems on large instances

The Rooted SCJ Median with Single Gene Duplications

The median problem is a classical problem in genome rearrangements. It aims to compute a gene order that minimizes the sum of the genomic distances to  k>=3  given gene orders. This problem is intractable except in the related Single-Cut-or-Join and breakpoint rearrangement models. Here we consider the rooted median problem, where we assume one of the given genomes to be ancestral to the median, which is itself ancestral to the other genomes. We show that in the Single-Cut-or-Join model with single gene duplications, the rooted median problem is NP-hard. We also describe an Integer Linear Program for solving this problem, which we apply to simulated data, showing high accuracy of the reconstructed medians

Using Genetic Algorithm to solve Median Problem and Phylogenetic Inference

Genome rearrangement analysis has attracted a lot of attentions in phylogenetic com- putation and comparative genomics. Solving the median problems based on various distance definitions has been a focus as it provides the building blocks for maximum parsimony analysis of phylogeny and ancestral genomes. The Median Problem (MP) has been proved to be NP-hard and although there are several exact or heuristic al- gorithms available, these methods all are difficulty to compute distant three genomes containing high evolution events. Such as current approaches, MGR[1] and GRAPPA [2], are restricted on small collections of genomes and low-resolution gene order data of a few hundred rearrangement events. In my work, we focus on heuristic algorithms which will combine genomic sorting algorithm with genetic algorithm (GA) to pro- duce new methods and directions for whole-genome median solver, ancestor inference and phylogeny reconstruction. In equal median problem, we propose a DCJ sorting operation based genetic algorithms measurements, called GA-DCJ. Following classic genetic algorithm frame, we develop our algorithms for every procedure and substitute for each traditional genetic algorithm procedure. The final results of our GA-based algorithm are optimal median genome(s) and its median score. In limited time and space, especially in large scale and distant datasets, our algorithm get better results compared with GRAPPA and AsMedian. Extending the ideas of equal genome median solver, we develop another genetic algorithm based solver, GaDCJ-Indel, which can solve unequal genomes median prob- lem (without duplication). In DCJ-Indel model, one of the key steps is still sorting operation[3]. The difference with equal genomes median is there are two sorting di- rections: minimal DCJ operation path or minimal indel operation path. Following different sorting path, in each step scenario, we can get various genome structures to fulfill our population pool. Besides that, we adopt adaptive surcharge-triangle inequality instead of classic triangle inequality in our fitness function in order to fit unequal genome restrictions and get more efficient results. Our experiments results show that GaDCJ-Indel method not only can converge to accurate median score, but also can infer ancestors that are very close to the true ancestors. An important application of genome rearrangement analysis is to infer ancestral genomes, which is valuable for identifying patterns of evolution and for modeling the evolutionary processes. However, computing ancestral genomes is very difficult and we have to rely on heuristic methods that have various limitations. We propose a GA-Tree algorithm which adapts meta-population [4], co-evolution and repopulation pool methods In this paper, we describe and illuminate the first genetic algorithm for ancestor inference step by step, which uses fitness scores designed to consider co- evolution and uses sorting-based methods to initialize and evolve populations. Our extensive experiments show that compared with other existing tools, our method is accurate and can infer ancestors that are much closer to true ancestors

A Unifying Model of Genome Evolution Under Parsimony

We 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.Comment: 52 pages, 24 figure

Phylogeny and Ancestral Genome Reconstruction from Gene Order Using Maximum Likelihood and Binary Encoding

Over the long history of genome evolution, genes get rearranged under events such as rearrangements, losses, insertions and duplications, which in all change the ordering and content along the genome. Recent progress in genome-scale sequencing renews the challenges in the reconstructions of phylogeny and ancestral genomes with gene-order data. Such problems have been proved so interesting that a large number of algorithms have been developed rigorously over the past few years in attempts to tackle these problems following various principles. However, difficulties and limitations in performance and scalability largely prevent us from analyzing emerging modern whole-genome data, our study presented in this dissertation focuses on developing appropriate evolutionary models and robust algorithms for solving the phylogenetic and ancestral inference problems using gene-order data under the whole-genome evolution, along with their applications. To reconstruct phylogenies from gene-order data, we developed a collection of closely-related methods following the principle of likelihood maximization. To the best of our knowledge, it was the first successful attempt to apply maximum likelihood optimization technique into the analysis of gene-order phylogenetic problem. Later we proposed MLWD (in collaboration with Lin and Moret) in which we described an effective transition model to account for the transitions between presence and absence states of an gene adjacency. Besides genome rearrangements, other evolutionary events modify gene contents such as gene duplications and gene insertion/deletion (indels) can be naturally processed as well. We present our results from extensive testing on simulated data showing that our approach returns very accurate results very quickly. With a known phylogeny, a subsequent problem is to reconstruct the gene-order of ancestral genomes from their living descendants. To solve this problem, we adopted an adjacency-based probabilistic framework, and developed a method called PMAG. PMAG decomposes gene orderings into a set of gene adjacencies and then infers the probability of observing each adjacency in the ancestral genome. We conducted extensive simulation experiments and compared PMAG with InferCarsPro, GASTS, GapAdj and SCJ. According to the results, PMAG demonstrated great performance in terms of the true positive rate of gene adjacency. PMAG also achieved comparable running time to the other methods, even when the traveling sales man problem (TSP) were exactly solved. Although PMAG can give good performance, it is strongly restricted from analyzing datasets underwent only rearrangements. To infer ancestral genomes under a more general model of evolution with an arbitrary rate of indels , we proposed an enhanced method PMAG+ based on PMAG. PMAG+ includes a novel approach to infer ancestral gene contents and a detail description to reduce the adjacency assembly problem to an instance of TSP. We designed a series of experiments to validate PMAG+ and compared the results with the most recent and comparable method GapAdj. According to the results, ancestral gene contents predicted by PMAG+ coincided highly with the actual contents with error rates less than 1%. Under various degrees of indels, PMAG+ consistently achieved more accurate prediction of ancestral gene orders and at the same time, produced contigs very close to the actual chromosomes

Assessing the robustness of parsimonious predictions for gene neighborhoods from reconciled phylogenies

The availability of a large number of assembled genomes opens the way to study the evolution of syntenic character within a phylogenetic context. The DeCo algorithm, recently introduced by B{\'e}rard et al. allows the computation of parsimonious evolutionary scenarios for gene adjacencies, from pairs of reconciled gene trees. Following the approach pioneered by Sturmfels and Pachter, we describe how to modify the DeCo dynamic programming algorithm to identify classes of cost schemes that generates similar parsimonious evolutionary scenarios for gene adjacencies, as well as the robustness to changes to the cost scheme of evolutionary events of the presence or absence of specific ancestral gene adjacencies. We apply our method to six thousands mammalian gene families, and show that computing the robustness to changes to cost schemes provides new and interesting insights on the evolution of gene adjacencies and the DeCo model.Comment: Accepted, to appear in ISBRA - 11th International Symposium on Bioinformatics Research and Applications - 2015, Jun 2015, Norfolk, Virginia, United State

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

Robustness Evaluation for Phylogenetic Reconstruction Methods and Evolutionary Models Reconstruction of Tumor Progression

During evolutionary history, genomes evolve by DNA mutation, genome rearrangement, duplication and gene loss events. There has been endless effort to the phylogenetic and ancestral genome inference study. Due to the great development of various technology, the information about genomes is exponentially increasing, which make it possible figure the problem out. The problem has been shown so interesting that a great number of algorithms have been developed rigorously over the past decades in attempts to tackle these problems following different kind of principles. However, difficulties and limits in performance and capacity, and also low consistency largely prevent us from confidently statement that the problem is solved. To know the detailed evolutionary history, we need to infer the phylogeny of the evolutionary history (Big Phylogeny Problem) and also infer the internal nodes information (Small Phylogeny Problem). The work presented in this thesis focuses on assessing methods designed for attacking Small Phylogeny Problem and algorithms and models design for genome evolution history inference from FISH data for cancer data. During the recent decades, a number of evolutionary models and related algorithms have been designed to infer ancestral genome sequences or gene orders. Due to the difficulty of knowing the true scenario of the ancestral genomes, there must be some tools used to test the robustness of the adjacencies found by various methods. When it comes to methods for Big Phylogeny Problem, to test the confidence rate of the inferred branches, previous work has tested bootstrapping, jackknifing, and isolating and found them good resampling tools to corresponding phylogenetic inference methods. However, till now there is still no system work done to try and tackle this problem for small phylogeny. We tested the earlier resampling schemes and a new method inversion on different ancestral genome reconstruction methods and showed different resampling methods are appropriate for their corresponding methods. Cancer is famous for its heterogeneity, which is developed by an evolutionary process driven by mutations in tumor cells. Rapid, simultaneous linear and branching evolution has been observed and analyzed by earlier research. Such process can be modeled by a phylogenetic tree using different methods. Previous phylogenetic research used various kinds of dataset, such as FISH data, genome sequence, and gene order. FISH data is quite clean for the reason that it comes form single cells and shown to be enough to infer evolutionary process for cancer development. RSMT was shown to be a good model for phylogenetic analysis by using FISH cell count pattern data, but it need efficient heuristics because it is a NP-hard problem. To attack this problem, we proposed an iterative approach to approximate solutions to the steiner tree in the small phylogeny tree. It is shown to give better results comparing to earlier method on both real and simulation data. In this thesis, we continued the investigation on designing new method to better approximate evolutionary process of tumor and applying our method to other kinds of data such as information using high-throughput technology. Our thesis work can be divided into two parts. First, we designed new algorithms which can give the same parsimony tree as exact method in most situation and modified it to be a general phylogeny building tool. Second, we applied our methods to different kinds data such as copy number variation information inferred form next generation sequencing technology and predict key changes during evolution