Recently the poset of ranked cactuses (𝔓(X),≼) was introduced. For a finite set X, this poset consists of a set 𝔓(X) of certain collections of ordered pairs of subsets of X together with an ordering ≼ that is similar to the refinement ordering of partitions of a finite set. In addition, the maximal chains in this poset correspond to binary ranked cactuses, a fact which can be used to construct the so-called space of equidistant cactuses. In this paper, we show that the poset of ranked cactuses is EL-shellable. As a consequence we also show that the proper part of the link of the origin of the space of equidistant cactuses has the homotopy type of a wedge of spheres.

A core goal of phylogenomics is to determine the evolutionary history of a set of species from biological sequence data. Phylogenetic networks are able to describe more complex evolutionary phenomena than phylogenetic trees but are more difficult to accurately reconstruct. Recently, there has been growing interest in developing methods to infer semi-directed phylogenetic networks. As computing such networks can be computationally intensive, one approach to building such networks is to puzzle together smaller networks. Thus, it is essential to have robust methods for inferring semi-directed phylogenetic networks on small numbers of taxa. In this paper, we investigate an algebraic method for performing phylogenetic network inference from nucleotide sequence data on 4-leaf semi-directed phylogenetic networks by analyzing the distribution of leaf-pattern probabilities. On simulated data, we found that we can correctly identify with high accuracy the undirected phylogenetic network for sequences of length at least 10 kbp. We found that identifying the semi-directed network is more challenging and requires sequences of length approaching 10 Mbp. We are also able to use our approach to identify treelike evolution and determine the underlying tree. Finally, we employ our method on a real data set from Xiphophorus species and use the results to build a phylogenetic network.

Ultametrics are an important class of distances used in applications such as phylogenetics, clustering and classification theory. Ultrametrics are essentially distances that can be represented by an edge-weighted rooted tree so that all of the distances in the tree from the root to any leaf of the tree are equal. In this paper, we introduce a generalization of ultrametrics called arboreal ultrametrics which have applications in phylogenetics and also arise in the theory of distance-hereditary graphs. These are partial distances, that is distances that are not necessarily defined for every pair of elements in the groundset, that can be represented by an ultrametric arboreal network, that is, an edge-weighted rooted network whose underlying graph is a tree. As with ultrametrics all of the distances in the ultrametric arboreal network from any root to any leaf below it are are equal but, in contrast, the network may have more than one root. In our two main results we characterize when a partial distance is an arboreal ultrametric as well as proving that, somewhat surprisingly, given any unrooted edge-weighted phylogenetic tree there is a necessarily unique way to insert roots into this tree so as to obtain an arboreal ultrametric.

One hundred diatom species have been selected for genome and transcriptome sequencing. The 100 Diatom Genomes Project aims to provide a scalable framework for understanding diatom biodiversity, ecology and evolution, and for investigating their use in biotechnology.

2. Abstract Staphylococcus aureus produces a broad range of enterotoxins that act as superantigens, disrupting host immune responses and resulting in a myriad of clinical symptoms. However, large-scale analyses determining enterotoxin gene diversity, lineage structure and isolate metadata remain scarce. We analysed 15,887 S. aureus RefSeq genomes using a machine learning pipeline combining profile Hidden Markov Model-based enterotoxin gene identification, lineage typing, gene profile-based strain clustering and association rule mining using a broad range of gene and metadata features. This approach identified 35 distinct enterotoxin genes and five variant forms, including two putative novel enterotoxin genes, sel34 and sel35 . HDBSCAN clustering distinguished 45 enterotoxin gene profile groups, revealing strong associations between the two major egc enterotoxin gene cluster variants (OMIWNG and OMIUNG) and Clonal Complex membership: CC5, CC22 and CC45 with OMIWNG; CC30 and CC121 with OMIUNG. Integration of isolate metadata exposed distinct geographic and temporal trends, including a recent rise in non-egc lineages derived from Asia and animal sources. These findings show that S. aureus enterotoxin diversity is structured by lineage, mobile genetic element composition and Clonal Complex association. The discovery of sel34 and sel35 , together with the comprehensive overview of lineage-specific enterotoxin profiles, expands current understanding of S. aureus virulence evolution and provides a scalable analytical framework for monitoring toxin gene dynamics in clinical and environmental populations. 3. Impact Statement Understanding how virulence genes evolve and spread in Staphylococcus aureus is vital for predicting pathological potential and managing infection risk for this species. By analysis of over 15,000 publicly available S. aureus genomes, this study provides the most comprehensive overview to date of enterotoxin gene diversity and lineage structure. Using machine learning and large-scale genomic mining, we reveal clear evolutionary and epidemiological patterns linking enterotoxin gene clusters to specific Clonal Complexes and identify two previously unknown enterotoxin genes. These findings highlight how recombination and horizontal gene transfer shape S. aureus toxins across hosts, continents and time. The resulting analytical framework offers a scalable foundation for future genomic surveillance of virulence evolution in both clinical and environmental settings. 4. Data Summary Genome assemblies and associated metadata for the 15,887 Staphylococcus aureus strains analysed within this study were generated elsewhere prior to this study and were here downloaded from the NCBI RefSeq database. Secondary datasets and descriptions of algorithmic approaches used to develop them are presented in the main text and within the Supplementary Files 1 and 2.

Phylogenetic networks and, more generally, directed acyclic graphs (DAGs) represent hierarchical structure beyond trees, for instance in the presence of reticulate evolutionary events such as hybridization or horizontal gene transfer. A central question is which parts of such graphs are essential with respect to leaf-observable information, and which parts can be removed without changing this information. Resolving this question can lead to principled simplification methods for phylogenetic networks, such as the recent normalization approach of Francis et al. In this paper, we study this question from three related perspectives: clusters displayed by a DAG G, least common ancestors (LCAs) of subsets of its leaf set, and visibility, a path-based property of vertices. We first introduce an LCA-based simplification procedure called i-regularization. For a DAG G and i≥ 1, the DAG _i(G) retains precisely those vertices that occur as unique LCAs of leaf subsets of size at most i, removes the remaining non-leaf vertices by a graph-editing operation ⊖, and then deletes shortcuts. We show that _i(G) preserves all such LCAs, is i-lca-relevant, and admits a cluster-level description: it is regular, i.e., isomorphic to the Hasse diagram of the corresponding lca-clusters. We then compare LCA-based regularization with normalization. Using the same ⊖-operator, we describe the cover construction underlying normalization, identify visible vertices that are nevertheless removed, and characterize when regularization and normalization coincide. Together, these results provide a unified framework for cluster-based, LCA-based, and visibility-based simplifications of DAGs and phylogenetic networks.

In phylogenetics, a key problem is to construct evolutionary trees from collections of characters where, for a set X of species, a character is simply a function from X onto a set of states. In this context, a key concept is convexity, where a character is convex on a tree with leaf set X if the collection of subtrees spanned by the leaves of the tree that have the same state are pairwise disjoint. Although collections of convex characters on a single tree have been extensively studied over the past few decades, very little is known about coconvex characters, that is, characters that are simultaneously convex on a collection of trees. As a starting point to better understand coconvexity, in this paper we prove a number of extremal results for the following question: What is the minimal number of coconvex characters on a collection of n-leaved trees taken over all collections of size t >= 2, also if we restrict to coconvex characters which map to k states? As an application of coconvexity, we introduce a new one-parameter family of tree metrics, which range between the coarse Robinson-Foulds distance and the much finer quartet distance. We show that bounds on the quantities in the above question translate into bounds for the diameter of the tree space for the new distances. Our results open up several new interesting directions and questions which have potential applications to, for example, tree spaces and phylogenomics.

A least common ancestor (LCA) of two leaves in a directed acyclic graph (DAG) is a vertex that is an ancestor of both of the leaves, and for which no proper descendant of this vertex is also an ancestor of the two leaves. LCAs play a central role in representing hierarchical relationships in rooted trees and, more generally, in DAGs. In 1981, Aho et al. introduced the problem of determining whether a collection of pairwise LCA constraints on a set X, of the form (i, j) < (k, l) with i, j, k, l is an element of X, can be realized by a rooted tree with leaf set X such that, whenever (i, j) < (k, l) holds, the LCA of i and j in the tree is a descendant of the LCA of k and l. In particular, they presented a polynomial-time algorithm, called Build, to solve this problem. In many cases, however, such constraints cannot be realized by any tree. In these situations, it is natural to ask whether they can still be realized by a more general directed acyclic graph (DAG). In this paper, we extend Aho et al.'s problem from trees to DAGs, providing a theoretical and algorithmic framework for reasoning about LCA constraints in this more general setting. More specifically, given a collection R of LCA constraints, we introduce the notion of the +-closure R+ which captures additional LCA relations implied by R. Using this closure, we then associate canonical DAG G(R) to R and show that a collection of constraints R can be realized by a DAG in terms of LCAs if and only if R is realized in this way by G(R). In addition, we adapt this construction to obtain phylogenetic networks, a special class of DAGs widely used in phylogenetics. In particular, we define a canonical phylogenetic network N-R for realizing R, and prove that N-R is regular, that is, it coincides with the Hasse diagram of the set system underlying N-R. Finally, we show that, for any DAG-realizable collection R, the classical closure of R, that is the collection of all LCA constraints that must hold in every DAG realizing R, coincides with its +-closure. All constructions introduced in this paper can be computed in polynomial time, and we provide explicit algorithms for each. All algorithms developed in this paper are implemented in the freely available Python package RealLCA.

Polyploidy occurs in plants and animals, and is an important force in speciation and genome evolution. The main focus of this paper is the following fundamental question that was recently posed by Huber and Maher: Given the ploidy numbers of a collection of extant species, or their ploidy profile, what is the smallest number of hybridizations needed in any evolutionary history for these species to completely represent these numbers? In this paper, we shall show that this question can be rephrased in terms of addition chains and the closely related addition sequences, which have been studied for over a century in mathematics and computer science. These are sequences of natural numbers that start with 1, so that each number in the sequence larger than 1 is the sum of two other numbers arising earlier in the sequence. In our first main result, we show that finding the smallest number of hybridization events to explain a ploidy profile, or the hybrid number, is equivalent to solving the so-called addition sequence problem. This immediately implies that computing the hybridization number is computationally intractable. Even so, it also leads to new connections to representing polyploid evolution using networks. More specifically, in our second main result we show that ploidy profiles representable by tree-child networks are exactly the addition chains, implying a polynomial-time algorithm for identifying these profiles. We then consider beaded tree-child networks, which permit the representation of autopolyploidy events, and in our third main result we provide a greedy polynomial-time algorithm to decide whether a given profile can be realized by such a network. We expect that our results can be leveraged in future work through, for example, making use of known algorithms for computing short addition sequences to give bounds for the hybrid number, and in guiding network reconstruction for polyploid species.

Encoding phylogenetic networks by suitable substructures is a central problem in phylogenetic combinatorics. We study encodings based on least common ancestor (LCA) constraints. For a directed acyclic graph (DAG) $G$ with leaf set $X$, we consider the relation on pairs of leaves in which $(ab,xy)$ records that the LCAs of $a,b$ and $x,y$ are well-defined and that the former is a descendant of the latter. We first identify precisely which part of $G$ is determined by this relation. To this end, we compare the canonical DAG constructed from the LCA relation with the 2-regularization of $G$, obtained by removing all vertices that are not LCAs of one or two leaves and then deleting shortcut edges. We prove that these two DAGs are isomorphic. Hence the obstruction to encoding a graph by its LCA relation is exactly the information lost under 2-regularization. This yields a general reconstruction principle, which we apply to several natural classes of phylogenetic networks. In particular, we show that shortcut-free 2-LCA-relevant DAGs, phylogenetic trees, regular level-1 networks, regular networks with binary clustering systems, regular networks whose clustering systems are closed weak hierarchies, strong-phylogenetic normal networks, separated phylogenetic normal networks, and binary normal networks are encoded by their LCA relations. We also introduce a sparse triple-like restriction consisting only of comparisons of the form $(ab,ac)$, where $a,b,c\in X$ are pairwise distinct. For graphs with the 2-LCA property, we show that this sparse relation, together with the leaf set, determines the full LCA relation after a natural closure operation. Consequently, several of the above classes can be reconstructed, up to isomorphism, from the sparse relation in polynomial time.

Phylogenetic networks are graphs that are used to represent evolutionary relationships between different taxa. They generalize phylogenetic trees since for example, unlike trees, they permit lineages to combine. Recently, there has been rising interest in semi-directed phylogenetic networks, which are mixed graphs in which certain lineage combination events are represented by directed edges coming together, whereas the remaining edges are left undirected. One reason to consider such networks is that it can be difficult to root a network using real data. In this paper, we consider the problem of when a semi-directed phylogenetic network is defined or encoded by the smaller networks that it induces on the 4-leaf subsets of its leaf set. These smaller networks are called quarnets. We prove that semi-directed binary level-2 phylogenetic networks are encoded by their quarnets, but that this is not the case for level-3. In addition, we prove that the so-called blob tree of a semi-directed binary network, a tree that gives the coarse-grained structure of the network, is always encoded by the quarnets of the network. These results are relevant for proving the statistical consistency of programs that are currently being developed for reconstructing phylogenetic networks from practical data, such as the recently developed Squirrel software tool.

Phylogenetic networks are a special type of graph which generalize phylogenetic trees and that are used to model non-treelike evolutionary processes such as recombination and hybridization. In this paper, we consider {\em unrooted} phylogenetic networks, i.e. simple, connected graphs $\mathcal{N}=(V,E)$ with leaf set $X$, for $X$ some set of species, in which every internal vertex in $\mathcal{N}$ has degree three. One approach used to construct such phylogenetic networks is to take as input a collection $\mathcal{P}$ of phylogenetic trees and to look for a network $\mathcal{N}$ that contains each tree in $\mathcal{P}$ and that minimizes the quantity $r(\mathcal{N}) = |E|-(|V|-1)$ over all such networks. Such a network always exists, and the quantity $r(\mathcal{N})$ for an optimal network $\mathcal{N}$ is called the hybrid number of $\mathcal{P}$. In this paper, we give a new characterization for the hybrid number in case $\mathcal{P}$ consists of two trees. This characterization is given in terms of a cherry picking sequence for the two trees, although to prove that our characterization holds we need to define the sequence more generally for two forests. Cherry picking sequences have been intensively studied for collections of rooted phylogenetic trees, but our new sequences are the first variant of this concept that can be applied in the unrooted setting. Since the hybrid number of two trees is equal to the well-known tree bisection and reconnection distance between the two trees, our new characterization also provides an alternative way to understand this important tree distance.

In 1989 Erdős and Székely showed that there is a bijection between (i) the set of rooted trees with n+1 vertices whose leaves are bijectively labeled with the elements of [ℓ]={1,2,…,ℓ} for some ℓ≤ n, and (ii) the set of partitions of [n]={1,2,…,n}. They established this via a labeling algorithm based on the anti-lexicographic ordering of non-empty subsets of [n] which extends the labeling of the leaves of a given tree to a labeling of all of the vertices of that tree. In this paper, we generalize their approach by developing a labeling algorithm for multi-labeled trees, that is, rooted trees whose leaves are labeled by positive integers but in which distinct leaves may have the same label. In particular, we show that certain orderings of the set of all finite, non-empty multisets of positive integers can be used to characterize partitions of a multiset that arise from labelings of multi-labeled trees. As an application, we show that the recently introduced class of labelable phylogenetic networks is precisely the class of phylogenetic networks that are stable relative to the so-called folding process on multi-labeled trees. We also give a bijection between the labelable phylogenetic networks with leaf-set [n] and certain partitions of multisets.

Ranked tree-child networks are a recently introduced class of rooted phylogenetic networks in which the evolutionary events represented by the network are ordered so as to respect the flow of time. This class includes the well-studied ranked phylogenetic trees (also known as ranked genealogies). An important problem in phylogenetic analysis is to define distances between phylogenetic trees and networks in order to systematically compare them. Various distances have been defined on ranked binary phylogenetic trees, but very little is known about comparing ranked tree-child networks. In this paper, we introduce an approach to compare binary ranked tree-child networks on the same leaf set that is based on a new encoding of such networks that is given in terms of a certain partially ordered set. This allows us to define two new spaces of ranked binary tree-child networks. The first space can be considered as a generalization of the recently introduced space of ranked binary phylogenetic trees whose distance is defined in terms of ranked nearest neighbor interchange moves. The second space is a continuous space that captures all equidistant tree-child networks and generalizes the space of ultrametric trees. In particular, we show that this continuous space is a so-called CAT(0)-orthant space which, for example, implies that the distance between two equidistant tree-child networks can be efficiently computed.

The inference of phylogenetic networks, which model complex evolutionary processes including hybridization and gene flow, remains a central challenge in evolutionary biology. Until now, statistically consistent inference methods have been limited to phylogenetic level-1 networks, which allow no interdependence between reticulate events. In this work, we establish the theoretical foundations for a statistically consistent inference method for a much broader class: semi-directed level-2 networks that are outer-labeled planar and galled. We precisely characterize the features of these networks that are distinguishable from the topologies of their displayed quartet trees. Moreover, we prove that an inter-taxon distance derived from these quartets is circular decomposable, enabling future robust inference of these networks from quartet data, such as concordance factors obtained from gene tree distributions under the Network Multispecies Coalescent model. Our results also have novel identifiability implications across different data types and evolutionary models, applying to any setting in which displayed quartets can be distinguished.

The Multidisciplinary Observatory for Study of the Arctic Climate (MOSAiC) expedition consisted of a year-long drifting survey of the Central Arctic Ocean. The ecosystems component of MOSAiC included the sampling of molecular data, with metagenomes collected from a diverse range of environments. The generation of metagenome-assembled-genomes (MAGs) from metagenomes are a starting point for genome-resolved analyses. This dataset presents a catalogue of MAGs recovered from a set of 73 samples from MOSAiC, including 2407 prokaryotic and 56 eukaryotic MAGs, as well as annotations of a near complete eukaryotic MAG using the Joint Genome Institute (JGI) annotation pipeline. The metagenomic samples are from the surface ocean, chlorophyll maximum, mesopelagic and bathypelagic, within leads and under-ice ocean, as well as melt ponds, ice ridges, and first- and second-year sea ice. This set of MAGs can be used to benchmark microbial biodiversity in the Central Arctic Ocean, compare individual strains across space and time, and to study changes in Arctic microbial communities from the winter to summer, at a genomic level.

With the increasing availability of genomic data, biologists aim to find more accurate descriptions of evolutionary histories influenced by secondary contact, where diverging lineages reconnect before diverging again. Such reticulate evolutionary events can be more accurately represented in phylogenetic networks than in phylogenetic trees. Since the root location of phylogenetic networks cannot be inferred from biological data under several evolutionary models, we consider semi-directed (phylogenetic) networks: partially directed graphs without a root in which the directed edges represent reticulate evolutionary events. By specifying a known outgroup, the rooted topology can be recovered from such networks. We introduce the algorithm Squirrel (Semi-directed Quarnet-based Inference to Reconstruct Level-1 Networks) which constructs a semi-directed level-1 network from a full set of quarnets (four-leaf semi-directed networks). Our method also includes a heuristic to construct such a quarnet set directly from sequence alignments. We demonstrate Squirrel's performance through simulations and on real sequence data sets, the largest of which contains 29 aligned sequences close to 1.7 Mb long. The resulting networks are obtained on a standard laptop within a few minutes. Lastly, we prove that Squirrel is combinatorially consistent: given a full set of quarnets coming from a triangle-free semi-directed level-1 network, it is guaranteed to reconstruct the original network. Squirrel is implemented in Python, has an easy-to-use graphical user interface that takes sequence alignments or quarnets as input, and is freely available at https://github.com/nholtgrefe/squirrel.

In evolutionary biology, networks are becoming increasingly used to represent evolutionary histories for species that have undergone non-treelike or reticulate evolution. Such networks are essentially directed acyclic graphs with a leaf set that corresponds to a collection of species, and in which non-leaf vertices with indegree 1 correspond to speciation events and vertices with indegree greater than 1 correspond to reticulate events such as gene transfer. Recently forest-based networks have been introduced, which are essentially (multi-rooted) networks that can be formed by adding some arcs to a collection of phylogenetic trees (or phylogenetic forest), where each arc is added in such a way that its ends always lie in two different trees in the forest. In this paper, we consider the complexity of deciding whether a given network is proper forest-based, that is, whether it can be formed by adding arcs to some underlying phylogenetic forest which contains the same number of trees as there are roots in the network. More specifically, we show that it is NP-complete to decide whether a tree-child network with m roots is proper forest-based, for each m≥2. Moreover, for binary networks the problem remains NP-complete when m≥3 but becomes polynomial-time solvable for m=2. We also give a fixed parameter tractable (FPT) algorithm, with parameters the maximum outdegree of a vertex, the number of roots, and the number of indegree 2 vertices, for deciding if a semi-binary network is proper forest-based. A key element in proving our results is a new characterization for when a network with m roots is proper forest-based in terms of certain m-colorings.

In phylogenetics and other areas of classification, the Buneman graph is commonly used to represent a collection of bipartitions or splits of a (finite) set X in order to display evolutionary relationships. The set X usually corresponds to a set of taxa (or species), and the splits are usually derived from molecular sequence data associated to the taxa. One issue with this approach is that missing molecular data can lead to bipartitions of subsets of X or partial splits, instead of splits of the full set X. In this paper, we show that the definition of the Buneman graph can be naturally extended to collections of partial splits of a set X. Just as with splits, we show that the graph so obtained is an X-labeled median graph but, in contrast to the usual Buneman graph, the elements in X are represented by convex subsets of the vertex set of the graph instead of single vertices. We also show that the Buneman graph for a collection of partial splits is closely related to subtree distances. In particular, for a collection S of weighted partial splits that satisfies a certain pairwise compatibility condition, we show that the corresponding edge-weighted Buneman graph is the unique minimal tree that represents the subtree distanced corresponding to S. Moreover, we show that in this special situation the Buneman graph can also be considered as a type of configuration space for the set of all tree-metrics that minimally extend the subtree distanced. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).