We construct a new family of permutationally in-variant codes that correct$t$Pauli errors for any$t\geqslant 1$. We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. This is an extended abstract of the preprint [1].

MOTIVATION: One of the key computational problems in comparative genomics is the reconstruction of genomes of ancestral species based on genomes of extant species. Since most dramatic changes in genomic architectures are caused by genome rearrangements, this problem is often posed as minimization of the number of genome rearrangements between extant and ancestral genomes. The basic case of three given genomes is known as the genome median problem. Whole-genome duplications (WGDs) represent yet another type of dramatic evolutionary events and inspire the reconstruction of preduplicated ancestral genomes, referred to as the genome halving problem. Generalization of WGDs to whole-genome multiplication events leads to the genome aliquoting problem. RESULTS: In this study, we propose polynomial-size integer linear programming (ILP) formulations for the aforementioned problems. We further obtain such formulations for the restricted and conserved versions of the median and halving problems, which have been recently introduced to improve biological relevance of the solutions. Extensive evaluation of solutions to the different ILP problems demonstrates their good accuracy. Furthermore, since the ILP formulations for the conserved versions have linear size, they provide a novel practical approach to ancestral genome reconstruction, which combines the advantages of homology- and rearrangements-based methods. AVAILABILITY AND IMPLEMENTATION: Code and data are available in https://github.com/AvdeevPavel/ILP-WGD-reconstructor. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

BACKGROUND: Despite the recent progress in genome sequencing and assembly, many of the currently available assembled genomes come in a draft form. Such draft genomes consist of a large number of genomic fragments (scaffolds), whose positions and orientations along the genome are unknown. While there exists a number of methods for reconstruction of the genome from its scaffolds, utilizing various computational and wet-lab techniques, they often can produce only partial error-prone scaffold assemblies. It therefore becomes important to compare and merge scaffold assemblies produced by different methods, thus combining their advantages and highlighting present conflicts for further investigation. These tasks may be labor intensive if performed manually. RESULTS: We present CAMSA-a tool for comparative analysis and merging of two or more given scaffold assemblies. The tool (i) creates an extensive report with several comparative quality metrics; (ii) constructs the most confident merged scaffold assembly; and (iii) provides an interactive framework for a visual comparative analysis of the given assemblies. Among the CAMSA features, only scaffold merging can be evaluated in comparison to existing methods. Namely, it resembles the functionality of assembly reconciliation tools, although their primary targets are somewhat different. Our evaluations show that CAMSA produces merged assemblies of comparable or better quality than existing assembly reconciliation tools while being the fastest in terms of the total running time. CONCLUSIONS: CAMSA addresses the current deficiency of tools for automated comparison and analysis of multiple assemblies of the same set scaffolds. Since there exist numerous methods and techniques for scaffold assembly, identifying similarities and dissimilarities across assemblies produced by different methods is beneficial both for the developers of scaffold assembly algorithms and for the researchers focused on improving draft assemblies of specific organisms.

BACKGROUND: Genome median and genome halving are combinatorial optimization problems that aim at reconstruction of ancestral genomes by minimizing the number of evolutionary events between them and genomes of the extant species. While these problems have been widely studied in past decades, their solutions are often either not efficient or not biologically adequate. These shortcomings have been recently addressed by restricting the problems solution space. RESULTS: We show that the restricted variants of genome median and halving problems are, in fact, closely related. We demonstrate that these problems have a neat topological interpretation in terms of embedded graphs and polygon gluings. We illustrate how such interpretation can lead to solutions to these problems in particular cases. CONCLUSIONS: This study provides an unexpected link between comparative genomics and topology, and demonstrates advantages of solving genome median and halving problems within the topological framework.

It is known that a minimal teaching set of any threshold function on the two-dimensional rectangular grid consists of 3 or 4 points. We derive exact formulae for the numbers of functions corresponding to these values and further refine them in the case of a minimal teaching set of size 3. We also prove that the average cardinality of the minimal teaching sets of threshold functions is asymptotically $\nicefrac{7}{2}$. We further present corollaries of these results concerning some special arrangements of lines in the plane.

In comparative genomics, the rearrangement distance between two genomes (equal the minimal number of genome rearrangements required to transform them into a single genome) is often used for measuring their evolutionary remoteness. Generalization of this measure to three genomes is known as the median score (while a resulting genome is called median genome). In contrast to the rearrangement distance between two genomes which can be computed in linear time, computing the median score for three genomes is NP-hard. This inspires a quest for simpler and faster approximations for the median score, the most natural of which appears to be the halved sum of pairwise distances which in fact represents a lower bound for the median score.In this work, we study relationship and interplay of pairwise distances between three genomes and their median score under the model of Double-Cut-and-Join (DCJ) rearrangements. Most remarkably we show that while a rearrangement may change the sum of pairwise distances by at most 2 (and thus change the lower bound by at most 1), even the most "powerful" rearrangements in this respect that increase the lower bound by 1 (by moving one genome farther away from each of the other two genomes), which we call strong, do not necessarily affect the median score. This observation implies that the two measures are not as well-correlated as one's intuition may suggest.We further prove that the median score attains the lower bound exactly on the triples of genomes that can be obtained from a single genome with strong rearrangements. While the sum of pairwise distances with the factor 2/3 represents an upper bound for the median score, its tightness remains unclear. Nonetheless, we show that the difference of the median score and its lower bound is not bounded by a constant.

A two-dimensional threshold function of k-valued logic can be viewed as a coloring of the points of a $k\times k$ square lattice into two colors such that there exists a straight line separating points of different colors. For the number of such functions only asymptotic bounds are known. We give an exact formula for the number of two-dimensional threshold functions and derive more accurate asymptotics.

In a landmark paper, Nadeau and Taylor [18] formulated the random breakage model (RBM) of chromosome evolution that postulates that there are no rearrangement hotspots in the human genome. In the next two decades, numerous studies with progressively increasing levels of resolution made RBM the de facto theory of chromosome evolution. Despite the fact that RBM had prophetic prediction power, it was recently refuted by Pevzner and Tesler [4], who introduced the fragile breakage model (FBM), postulating that the human genome is a mosaic of solid regions (with low propensity for rearrangements) and fragile regions (rearrangement hotspots). However, the rebuttal of RBM caused a controversy and led to a split among researchers studying genome evolution. In particular, it remains unclear whether some complex rearrangements (e.g., transpositions) can create an appearance of rearrangement hotspots. We contribute to the ongoing debate by analyzing multi-break rearrangements that break a genome into multiple fragments and further glue them together in a new order. In particular, we demonstrate that (1) even if transpositions were a dominant force in mammalian evolution, the arguments in favor of FBM still stand, and (2) the "gene deletion" argument against FBM is flawed.

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.

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.

MOTIVATION: The traditional approach to annotate alternative splicing is to investigate every splicing variant of the gene in a case-by-case fashion. This approach, while useful, has some serious shortcomings. Recent studies indicate that alternative splicing is more frequent than previously thought and some genes may produce tens of thousands of different transcripts. A list of alternatively spliced variants for such genes would be difficult to build and hard to analyse. Moreover, such a list does not show the relationships between different transcripts and does not show the overall structure of all transcripts. A better approach would be to represent all splicing variants for a given gene in a way that captures the relationships between different splicing variants. RESULTS: We introduce the notion of the splicing graph that is a natural and convenient representation of all splicing variants. The key difference with the existing approaches is that we abandon the linear (sequence) representation of each transcript and replace it with a graph representation where each transcript corresponds to a path in the graph. We further design an algorithm to assemble EST reads into the splicing graph rather than assembling them into each splicing variant in a case-by-case fashion.