On the Approximability of Comparing Genomes with Duplicates

International audienceA central problem in comparative genomics consists in computing a (dis-)simi- larity measure between two genomes, e.g. in order to construct a phylogenetic tree. A large number of such measures has been proposed in the recent past: number of reversals, number of breakpoints, number of common or conserved intervals, SAD etc. In their initial definitions, all these measures suppose that genomes contain no duplicates. However, we now know that genes can be duplicated within the same genome. One possible approach to overcome this difficulty is to establish a one-to-one correspondence (i.e. a matching) between genes of both genomes, where the correspondence is chosen in order to optimize the studied measure. Then, after a gene relabeling according to this matching and a deletion of the unmatched signed genes, two genomes without duplicates are obtained and the measure can be computed. In this paper, we are interested in three measures (number of breakpoints, number of common intervals and number of conserved intervals) and three models of matching (exemplar model, maximum matching model and non maximum matching model). We prove that, for each model and each measure, computing a matching between two genomes that optimizes the measure is APX-Hard. We show that this result remains true even for two genomes G1 and G2 such that G1 contains no duplicates and no gene of G2 appears more than twice. Therefore, our results extend those of [5–7]. Finally, we propose a 4-approximation algorithm for a measure closely related to the number of breakpoints, the number of adjacencies, under the maximum matching model, in the case where genomes contain the same number of duplications of each gene

Easy identification of generalized common and conserved nested intervals

In this paper we explain how to easily compute gene clusters, formalized by classical or generalized nested common or conserved intervals, between a set of K genomes represented as K permutations. A b-nested common (resp. conserved) interval I of size |I| is either an interval of size 1 or a common (resp. conserved) interval that contains another b-nested common (resp. conserved) interval of size at least |I|-b. When b=1, this corresponds to the classical notion of nested interval. We exhibit two simple algorithms to output all b-nested common or conserved intervals between K permutations in O(Kn+nocc) time, where nocc is the total number of such intervals. We also explain how to count all b-nested intervals in O(Kn) time. New properties of the family of conserved intervals are proposed to do so

Genomes containing Duplicates are Hard to compare

International audienceIn this paper, we are interested in the algorithmic complexity of computing (dis)similarity measures between two genomes when they contain duplicated genes. In that case, there are usually two main ways to compute a given (dis)similarity measure M between two genomes G1 and G2: the rst model, that we will call the matching model, consists in making a one-to-one correspondence between genes of G1 and genes of G2, in such a way that M is optimized. The second model, called the exemplar model, consists in keeping in G1 (resp. G2) exactly one copy of each gene, thus deleting all the other copies, in such a way that M is optimized. We present here dierent results concerning the algorithmic complexity of computing three dierent similarity measures (number of common intervals, MAD number and SAD number) in those two models, basically showing that the problem becomes NP-complete for each of them as soon as genomes contain duplicates. We show indeed that for common intervals, MAD and SAD, the problem is NP-complete when genes are duplicated in genomes, in both the exemplar and matching models. In the case of MAD and SAD, we actually prove that, under both models, both MAD and SAD problems are APX-har

MinMax-Profiles: A Unifying View of Common Intervals, Nested Common Intervals and Conserved Intervals of K Permutations

Common intervals of K permutations over the same set of n elements were firstly investigated by T. Uno and M.Yagiura (Algorithmica, 26:290:309, 2000), who proposed an efficient algorithm to find common intervals when K=2. Several particular classes of intervals have been defined since then, e.g. conserved intervals and nested common intervals, with applications mainly in genome comparison. Each such class, including common intervals, led to the development of a specific algorithmic approach for K=2, and - except for nested common intervals - for its extension to an arbitrary K. In this paper, we propose a common and efficient algorithmic framework for finding different types of common intervals in a set P of K permutations, with arbitrary K. Our generic algorithm is based on a global representation of the information stored in P, called the MinMax-profile of P, and an efficient data structure, called an LR-stack, that we introduce here. We show that common intervals (and their subclasses of irreducible common intervals and same-sign common intervals), nested common intervals (and their subclass of maximal nested common intervals) as well as conserved intervals (and their subclass of irreducible conserved intervals) may be obtained by appropriately setting the parameters of our algorithm in each case. All the resulting algorithms run in O(Kn+N)-time and need O(n) additional space, where N is the number of solutions. The algorithms for nested common intervals and maximal nested common intervals are new for K>2, in the sense that no other algorithm has been given so far to solve the problem with the same complexity, or better. The other algorithms are as efficient as the best known algorithms.Comment: 25 pages, 2 figure

Genomes are covered with ubiquitous 11 bp periodic patterns, the "class A flexible patterns"

BACKGROUND: The genomes of prokaryotes and lower eukaryotes display a very strong 11 bp periodic bias in the distribution of their nucleotides. This bias is present throughout a given genome, both in coding and non-coding sequences. Until now this bias remained of unknown origin. RESULTS: Using a technique for analysis of auto-correlations based on linear projection, we identified the sequences responsible for the bias. Prokaryotic and lower eukaryotic genomes are covered with ubiquitous patterns that we termed "class A flexible patterns". Each pattern is composed of up to ten conserved nucleotides or dinucleotides distributed into a discontinuous motif. Each occurrence spans a region up to 50 bp in length. They belong to what we named the "flexible pattern" type, in that there is some limited fluctuation in the distances between the nucleotides composing each occurrence of a given pattern. When taken together, these patterns cover up to half of the genome in the majority of prokaryotes. They generate the previously recognized 11 bp periodic bias. CONCLUSION: Judging from the structure of the patterns, we suggest that they may define a dense network of protein interaction sites in chromosomes

Finding approximate gene clusters with GECKO 3

Winter S, Jahn K, Wehner S, et al. Finding approximate gene clusters with GECKO 3. Nucleic Acids Research. 2016;44(20):9600-9610.Gene-order-based comparison of multiple genomes provides signals for functional analysis of genes and the evolutionary process of genome organization. Gene clusters are regions of co-localized genes on genomes of different species. The rapid increase in sequenced genomes necessitates bioinformatics tools for finding gene clusters in hundreds of genomes. Existing tools are often restricted to few (in many cases, only two) genomes, and often make restrictive assumptions such as short perfect conservation, conserved gene order or monophyletic gene clusters. We present Gecko 3, an open-source software for finding gene clusters in hundreds of bacterial genomes, that comes with an easy-to-use graphical user interface. The underlying gene cluster model is intuitive, can cope with low degrees of conservation as well as misannotations and is complemented by a sound statistical evaluation. To evaluate the biological benefit of Gecko 3 and to exemplify our method, we search for gene clusters in a dataset of 678 bacterial genomes using Synechocystis sp. PCC 6803 as a reference. We confirm detected gene clusters reviewing the literature and comparing them to a database of operons; we detect two novel clusters, which were confirmed by publicly available experimental RNA-Seq data. The computational analysis is carried out on a laptop computer in <40 min

Efficient algorithms for gene cluster detection in prokaryotic genomes

Schmidt T. Efficient algorithms for gene cluster detection in prokaryotic genomes. Bielefeld (Germany): Bielefeld University; 2005.The research in genomics science rapidly emerged in the last few years, and the availability of completely sequenced genomes continuously increases due to the use of semi-automatic sequencing machines. Also these sequences, mostly prokaryotic ones, are well annotated, which means that the positions of their genes and parts of their regulatory or metabolic pathways are known. A new task in the field of bioinformatics now is to gain gene or protein information from the comparison of genomes on a higher level. In the approach of "comparative genomics" researchers in bioinformatics are attempting to locate groups or clusters of orthologous genes that may have the same function in multiple genomes. These researches are often anchored on the simple, but biologically verified fact, that functionally related proteins are usually coded by genes placed in a region of close genomic neighborhood, in different species. From an algorithmic and combinatorial point of view, the first descriptions of the concept of "closely placed genes" were only fragmentary, and sometimes confusing. The given algorithms often lack the necessary grounds to prove their correctness, or assess their complexity. Within the first formal models of a conserved genomic neighborhood, genomes are often represented as permutations of their genes, and common intervals, i.e. intervals containing the same set of genes, are interpreted as gene clusters. But here the major disadvantage of representing genomes as permutations is the fact that paralogous copies of the same gene inside one genome can not be modelled. Since especially large genomes contain numerous paralogous genes, this model is insufficient to be used on real genomic data. In this work, we consider a modified model of gene clusters that allows paralogs, simply by representing genomes as sequences rather than permutations of genes. We define common intervals based on this model, and we present a simple algorithm that finds all common intervals of two sequences in [Theta](n2) time using [Theta](n2) space. Another, more complicated algorithm runs in [Omikron](n2) time and uses only linear space. We also show how to extend these algorithms to more than two genomes and present the implementation of the algorithms as well as the visualization of the located clusters in the tool Gecko. Since the creation of the string representation of a set of genomes is a non-trivial task, we also present the data preparation tool GhostFam that groups all genes from the given set of genomes to their families of homologs. In the evaluation on a set of 20 bacterial genomes, we show that with the presented approach it is possible to correctly locate gene clusters that are known from the literature, and to successfully predict new groups of functionally related genes