Reconstructing phylogenetic level-1 networks from nondense binet and trinet sets

Binets and trinets are phylogenetic networks with two and three leaves, respectively. Here we consider the problem of deciding if there exists a binary level-1 phylogenetic network displaying a given set T of binary binets or trinets over a taxon set X, and constructing such a network whenever it exists. We show that this is NP-hard for trinets but polynomial-time solvable for binets. Moreover, we show that the problem is still polynomial-time solvable for inputs consisting of binets and trinets as long as the cycles in the trinets have size three. Finally, we present an O(3^{|X|} poly(|X|)) time algorithm for general sets of binets and trinets. The latter two algorithms generalise to instances containing level-1 networks with arbitrarily many leaves, and thus provide some of the first supernetwork algorithms for computing networks from a set of rooted 1 phylogenetic networks

Quarnet Inference Rules for Level-1 Networks

An important problem in phylogenetics is the construction of phylogenetic trees. One way to approach this problem, known as the supertree method, involves inferring a phylogenetic tree with leaves consisting of a set X of species from a collection of trees, each having leaf-set some subset of X. In the 1980s, Colonius and Schulze gave certain inference rules for deciding when a collection of 4-leaved trees, one for each 4-element subset of X, can be simultaneously displayed by a single supertree with leaf-set X. Recently, it has become of interest to extend this and related results to phylogenetic networks. These are a generalization of phylogenetic trees which can be used to represent reticulate evolution (where species can come together to form a new species). It has recently been shown that a certain type of phylogenetic network, called a (unrooted) level-1 network, can essentially be constructed from 4-leaved trees. However, the problem of providing appropriate inference rules for such networks remains unresolved. Here, we show that by considering 4-leaved networks, called quarnets, as opposed to 4-leaved trees, it is possible to provide such rules. In particular, we show that these rules can be used to characterize when a collection of quarnets, one for each 4-element subset of X, can all be simultaneously displayed by a level-1 network with leaf-set X. The rules are an intriguing mixture of tree inference rules, and an inference rule for building up a cyclic ordering of X from orderings on subsets of X of size 4. This opens up several new directions of research for inferring phylogenetic networks from smaller ones, which could yield new algorithms for solving the supernetwork problem in phylogenetics

Minimum triplet covers of binary phylogenetic X-trees

Trees with labelled leaves and with all other vertices of degree three play an important role in systematic biology and other areas of classification. A classical combinatorial result ensures that such trees can be uniquely reconstructed from the distances between the leaves (when the edges are given any strictly positive lengths). Moreover, a linear number of these pairwise distance values suffices to determine both the tree and its edge lengths. A natural set of pairs of leaves is provided by any `triplet cover' of the tree (based on the fact that each non-leaf vertex is the median vertex of three leaves). In this paper we describe a number of new results concerning triplet covers of minimum size. In particular, we characterize such covers in terms of an associated graph being a 2-tree. Also, we show that minimum triplet covers are `shellable' and thereby provide a set of pairs for which the inter-leaf distance values will uniquely determine the underlying tree and its associated branch lengths

Tree-Based Unrooted Phylogenetic Networks

Phylogenetic networks are a generalization of phylogenetic trees that are used to represent non-tree-like evolutionary histories that arise in organisms such as plants and bacteria, or uncertainty in evolutionary histories. An unrooted phylogenetic network on a non-empty, finite set X of taxa, or network, is a connected, simple graph in which every vertex has degree 1 or 3 and whose leaf set is X. It is called a phylogenetic tree if the underlying graph is a tree. In this paper we consider properties of tree-based networks, that is, networks that can be constructed by adding edges into a phylogenetic tree. We show that although they have some properties in common with their rooted analogues which have recently drawn much attention in the literature, they have some striking differences in terms of both their structural and computational properties. We expect that our results could eventually have applications to, for example, detecting horizontal gene transfer or hybridization which are important factors in the evolution of many organisms. Correction available at dx.doi.org/10.1007/s11538-018-0530-

Trinets encode tree-child and level-2 phylogenetic networks

Phylogenetic networks generalize evolutionary trees, and are commonly used to represent evolutionary histories of species that undergo reticulate evolutionary processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in trying to develop methods to construct rooted phylogenetic networks from triplets, that is rooted trees on three species. However, although triplets determine or encode rooted phylogenetic trees, they do not in general encode rooted phylogenetic networks, which is a potential issue for any such method. Motivated by this fact, Huber and Moulton recently introduced trinets as a natural extension of rooted triplets to networks. In particular, they showed that level-1 level-1 phylogenetic networks are encoded by their trinets, and also conjectured that all “recoverable” rooted phylogenetic networks are encoded by their trinets. Here we prove that recoverable binary level-2 networks and binary tree-child networks are also encoded by their trinets. To do this we prove two decomposition theorems based on trinets which hold for all recoverable binary rooted phylogenetic networks. Our results provide some additional evidence in support of the conjecture that trinets encode all recoverable rooted phylogenetic networks, and could also lead to new approaches to construct phylogenetic networks from trinets

Three-way symbolic tree-maps and ultrametrics

Three-way dissimilarities are a generalization of (two-way) dissimilarities which can be used to indicate the lack of homogeneity or resemblance between any three objects. Such maps have applications in cluster analysis and have been used in areas such as psychology and phylogenetics, where three-way data tables can arise. Special examples of such dissimilarities are three-way tree-metrics and ultrametrics, which arise from leaf-labelled trees with edges labelled by positive real numbers. Here we consider three-way maps which arise from leaf-labelled trees where instead the interior vertices are labelled by an arbitrary set of values. For unrooted trees, we call such maps three-way symbolic tree-maps; for rooted trees, we call them three-way symbolic ultrametrics since they can be considered as a generalization of the (two-way) symbolic ultrametrics of Bocker and Dress. We show that, as with two- and three-way tree-metrics and ultrametrics, three-way symbolic tree-maps and ultrametrics can be characterized via certain k-point conditions. In the unrooted case, our characterization is mathematically equivalent to one presented by Gurvich for a certain class of edge-labelled hypergraphs. We also show that it can be decided whether or not an arbitrary three-way symbolic map is a tree-map or a symbolic ultrametric using a triplet-based approach that relies on the so-called BUILD algorithm for deciding when a set of 3-leaved trees or triplets can be displayed by a single tree. We envisage that our results will be useful in developing new approaches and algorithms for understanding 3-way data, especially within the area of phylogenetics

The genetic architecture of the MHC class II region in British Texel sheep

Understanding the structure of the major histocompatibility complex, especially the number and frequency of alleles, loci and haplotypes, is crucial for efficient investigation of the way in which the MHC influences susceptibility to disease. Nematode infection is one of the most important diseases suffered by sheep, and the class II region has been repeatedly associated with differences in susceptibility and resistance to infection. Texel sheep are widely used in many different countries and are relatively resistant to infection. This study determined the number and frequency of MHC class II genes in a small flock of Texel sheep. There were 18 alleles at DRB1, 9 alleles at DQA1, 13 alleles at DQB1, 8 alleles at DQA2 and 16 alleles at DQB2. Several haplotypes had no detectable gene products at DQA1, DQB1 or DQB2, and these were defined as null alleles. Despite the large numbers of alleles, there were only 21 distinct haplotypes in the population. The relatively small number of observed haplotypes will simplify finding disease associations because common haplotypes provide more statistical power but complicate the discrimination of causative mutations from linked marker loci

Beyond representing orthology relations by trees

Reconstructing the evolutionary past of a family of genes is an important aspect of many genomic studies. To help with this, simple relations on a set of sequences called orthology relations may be employed. In addition to being interesting from a practical point of view they are also attractive from a theoretical perspective in that e.\,g.\,a characterization is known for when such a relation is representable by a certain type of phylogenetic tree. For an orthology relation inferred from real biological data it is however generally too much to hope for that it satisfies that characterization. Rather than trying to correct the data in some way or another which has its own drawbacks, as an alternative, we propose to represent an orthology relation $\delta$ in terms of a structure more general than a phylogenetic tree called a phylogenetic network. To compute such a network in the form of a level-1 representation for $\delta$, we formalize an orthology relation in terms of the novel concept of a symbolic 3- dissimilarity which is motivated by the biological concept of a ``cluster of orthologous groups'', or COG for short. For such maps which assign symbols rather that real values to elements, we introduce the novel {\sc Network-Popping} algorithm which has several attractive properties. In addition, we characterize an orthology relation $\delta$ on some set $X$ that has a level-1 representation in terms of eight natural properties for $\delta$ as well as in terms of level-1 representations of orthology relations on certain subsets of $X$

Seesaw Neutrino Signals at the Large Hadron Collider

We discuss the scenario with gauge singlet fermions (right-handed neutrinos) accessible at the energy of the Large Hadron Collider. The singlet fermions generate tiny neutrino masses via the seesaw mechanism and also have sizable couplings to the standard-model particles. We demonstrate that these two facts, which are naively not satisfied simultaneously, are reconciled in the five-dimensional framework in various fashions, which make the seesaw mechanism observable. The collider signal of tri-lepton final states with transverse missing energy is investigated for two explicit examples of the observable seesaw, taking account of three types of neutrino mass spectrum and the constraint from lepton flavor violation. We find by showing the significance of signal discovery that the collider experiment has a potential to find signals of extra dimensions and the origin of small neutrino masses.Comment: 27 pages, 4 figure

Evaluation of a novel nanocrystalline hydroxyapatite paste Ostim® in comparison to Alpha-BSM® - more bone ingrowth inside the implanted material with Ostim® compared to Alpha BSM®

<p>Abstract</p> <p>Background</p> <p>The purpose of this study was to evaluate the performance a newly developed nanocrystalline hydroxyapatite, OSTIM^®following functional implantation in femoral sites in thirty-eight sheep for 1, 2 or 3 months. Ostim^®35 was compared to an established calcium phosphate, Alpha BSM^®.</p> <p>Methods</p> <p>Biomechanical testing, μ-CT analysis, histological and histomorphological analyses were conducted to compare the treatments including evaluation of bone regeneration level, material degradation, implant biomechanical characteristics.</p> <p>Results</p> <p>The micro-computed tomography (μCT) analysis and macroscopic observations showed that Ostim^®seemed to diffuse easily particularly when the defects were created in a cancellous bone area. Alpha BSM^®remained in the defect.</p> <p>The performance of Ostim was good in terms of mechanical properties that were similar to Alpha BSM^®and the histological analysis showed that the bone regeneration was better with Ostim^®than with Alpha BSM^®. The histomorphometric analysis confirmed the qualitative analysis and showed more bone ingrowth inside the implanted material with Ostim^®when compared to Alpha BSM ^®at all time points.</p> <p>Conclusions</p> <p>The successful bone healing with osseous consolidation verifies the importance of the nanocrystalline hydroxyapatite in the treatment of metaphyseal osseous volume defects in the metaphyseal spongiosa.</p