On the Comparison of Incompatibility of Split Systems Across Different Numbers of Taxa.

We consider the problem of the minimum number of phylogenetic trees it would take to display all splits in a given set, a problem related to k-compatibility. A set of trees that displays every single possible split is termed a universal tree set. In this note, we find the universal incompatibility U(n), the minimal size of a universal tree set for n taxa. By normalising incompatibility using U(n), one can then compare incompatibility of split systems across different numbers of taxa. We demonstrate this application by comparing two SplitsTree networks derived from archaeal genomes, with different numbers of taxa.

Phylosymmetric Algebras: Mathematical Properties of a New Tool in Phylogenetics.

In phylogenetics, it is of interest for rate matrix sets to satisfy closure under matrix multiplication as this makes finding the set of corresponding transition matrices possible without having to compute matrix exponentials. It is also advantageous to have a small number of free parameters as this, in applications, will result in a reduction in computation time. We explore a method of building a rate matrix set from a rooted tree structure by assigning rates to internal tree nodes and states to the leaves, then defining the rate of change between two states as the rate assigned to the most recent common ancestor of those two states. We investigate the properties of these matrix sets from both a linear algebra and a graph theory perspective and show that any rate matrix set generated this way is closed under matrix multiplication. The consequences of setting two rates assigned to internal tree nodes to be equal are then considered. This methodology could be used to develop parameterised models of amino acid substitution which have a small number of parameters but convey biological meaning.

A partial order and cluster-similarity metric on rooted phylogenetic trees.

Metrics on rooted phylogenetic trees are integral to a number of areas of phylogenetic analysis. Cluster-similarity metrics have recently been introduced in order to limit skew in the distribution of distances, and to ensure that trees in the neighbourhood of each other have similar hierarchies. In the present paper we introduce a new cluster-similarity metric on rooted phylogenetic tree space that has an associated local operation, allowing for easy calculation of neighbourhoods, a trait that is desirable for MCMC calculations. The metric is defined by the distance on the Hasse diagram induced by a partial order on the set of rooted phylogenetic trees, itself based on the notion of a hierarchy-preserving map between trees. The partial order we introduce is a refinement of the well-known refinement order on hierarchies. Both the partial order and the hierarchy-preserving maps may also be of independent interest.

A polynomial invariant for a new class of phylogenetic networks.

Invariants for complicated objects such as those arising in phylogenetics, whether they are invariants as matrices, polynomials, or other mathematical structures, are important tools for distinguishing and working with such objects. In this paper, we generalize a complete polynomial invariant on trees to a class of phylogenetic networks called separable networks, which will include orchard networks. Networks are becoming increasingly important for their ability to represent reticulation events, such as hybridization, in evolutionary history. We provide a function from the space of internally multi-labelled phylogenetic networks, a more generic graph structure than phylogenetic networks where the reticulations are also labelled, to a polynomial ring. We prove that the separability condition allows us to characterize, via the polynomial, the phylogenetic networks with the same number of leaves and same number of reticulations by considering their internally labelled versions. While the invariant for trees is a polynomial in [Formula: see text] where n is the number of leaves, the invariant for internally multi-labelled phylogenetic networks is an element of [Formula: see text], where r is the number of reticulations in the network. When the networks are considered without leaf labels the number of variables reduces to r + 2.

Evidence for a Syncytial Origin of Eukaryotes from Ancestral State Reconstruction.

Modern accounts of eukaryogenesis entail an endosymbiotic encounter between an archaeal host and a proteobacterial endosymbiont, with subsequent evolution giving rise to a unicell possessing a single nucleus and mitochondria. The mononucleate state of the last eukaryotic common ancestor (LECA) is seldom, if ever, questioned, even though cells harboring multiple (syncytia, coenocytes, and polykaryons) are surprisingly common across eukaryotic supergroups. Here, we present a survey of multinucleated forms. Ancestral character state reconstruction for representatives of 106 eukaryotic taxa using 16 different possible roots and supergroup sister relationships, indicate that LECA, in addition to being mitochondriate, sexual, and meiotic, was multinucleate. LECA exhibited closed mitosis, which is the rule for modern syncytial forms, shedding light on the mechanics of its chromosome segregation. A simple mathematical model shows that within LECA's multinucleate cytosol, relationships among mitochondria and nuclei were neither one-to-one, nor one-to-many, but many-to-many, placing mitonuclear interactions and cytonuclear compatibility at the evolutionary base of eukaryotic cell origin. Within a syncytium, individual nuclei and individual mitochondria function as the initial lower-level evolutionary units of selection, as opposed to individual cells, during eukaryogenesis. Nuclei within a syncytium rescue each other's lethal mutations, thereby postponing selection for viable nuclei and cytonuclear compatibility to the generation of spores, buffering transitional bottlenecks at eukaryogenesis. The prokaryote-to-eukaryote transition is traditionally thought to have left no intermediates, yet if eukaryogenesis proceeded via a syncytial common ancestor, intermediate forms have persisted to the present throughout the eukaryotic tree as syncytia but have so far gone unrecognized.

Online social media tells a story of Anaselina, Paraselina, and Selivinga (Orthoptera, Tetrigidae), rare Australian pygmy grasshoppers.

Knowledge on the pygmy grasshoppers of Australia is, despite the numerous endemics being described from this unique continent, still scarce. Of interest is the Vingselina genus group, including genera Anaselina Storozhenko, 2019, Paraselina Storozhenko, 2019, Selivinga Storozhenko, 2019 and Vingselina Sjöstedt, 1921. The systematic position of this group, currently assigned to Batrachideinae (Bufonidini), is probably not correct. In this study new records are presented of Anaselina minor (Sjöstedt, 1921), Paraselina brunneri (Bolívar, 1887), P. trituberculata (Sjöstedt, 1932), and Selivinga tribulata Storozhenko, 2019, all except A. minor the first records of the species since their original descriptions. The first photographs of living specimens of A. minor, P. brunneri, P. trituberculata and S. tribulata are provided and their habitats described. All the records were compiled by citizen scientists who use online social media, such as iNaturalist. Lastly, P. multifora (Rehn, 1952) syn. nov. represents a junior synonym of P. brunneri.

Tree-metrizable HGT networks.

Phylogenetic trees are often constructed by using a metric on the set of taxa that label the leaves of the tree. While there are a number of methods for constructing a tree using a given metric, such trees will only display the metric if it satisfies the so-called "four point condition", established by Buneman in 1971. While this condition guarantees that a unique tree will display the metric, meaning that the distance between any two leaves can be found by adding the distances on arcs in the path between the leaves, it doesn't exclude the possibility that a phylogenetic network might also display the metric. This possibility was recently pointed out and "tree-metrized" networks - that display a tree metric - with a single reticulation were characterized. In this paper, we show that in the case of HGT (horizontal gene transfer) networks, in fact there are tree-metrized networks containing many reticulations.

Tree-based unrooted nonbinary phylogenetic networks.

Phylogenetic networks are a generalisation of phylogenetic trees that allow for more complex evolutionary histories that include hybridisation-like processes. It is of considerable interest whether a network can be considered 'tree-like' or not, which leads to the introduction of tree-based networks in the rooted, binary context. Tree-based networks are those networks which can be constructed by adding additional edges into a phylogenetic tree, called the base tree. Previous extensions have considered extending to the binary, unrooted case and the nonbinary, rooted case. In this paper, we extend tree-based networks to the context of unrooted, nonbinary networks in three ways, depending on the types of additional edges that are permitted. Further, we study fully tree-based networks which are phylogenetic networks in which every embedded tree is a base tree. We also extend this concept to unrooted, nonbinary, phylogenetic networks and classify the resulting networks. Finally, we derive some results on the colourability of tree-based networks, which can be useful to determine whether a network is tree-based.