Arboreal networks are multi-rooted phylogenetic networks whose underlying graph is a tree. We give an encoding of stack-free arboreal networks in terms of triplets and the novel concept of a duet. This yields a polynomial time algorithm to construct these networks from complete triplet and duet systems. The classification results show correctness and lead to a natural metric on these multi-rooted networks.

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.

Horizontal gene transfer (HGT) is an important process in bacterial evolution. Current phylogeny-based approaches to capture it cannot however appropriately account for the fact that HGT can occur between bacteria living in different ecological niches. Due to the fact that arboreal networks are a type of multiple-rooted phylogenetic network that can be thought of as a forest of rooted phylogenetic trees along with a set of additional arcs each joining two different trees in the forest, understanding the combinatorial structure of such networks might therefore pave the way to extending current phylogeny-based HGT-inference methods in this direction. A central question in this context is, how can we construct an arboreal network? Answering this question is strongly informed by finding ways to encode an arboreal network, that is, breaking up the network into simpler combinatorial structures that, in a well defined sense uniquely determine the network. In the form of triplets, trinets and quarnets such encodings are known for certain types of single-rooted phylogenetic networks. By studying the underlying tree of an arboreal network, we complement them here with an answer for arboreal networks.

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.

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.

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/).

In evolutionary biology, phylogenetic networks are graphs that provide a flexible framework for representing complex evolutionary histories that involve reticulate evolutionary events. Recently phylogenetic studies have started to focus on a special class of such networks called semi-directed networks. These graphs are defined as mixed graphs that can be obtained by de-orienting some of the arcs in some rooted phylogenetic network, that is, a directed acyclic graph whose leaves correspond to a collection of species and that has a single source or root vertex. However, this definition of semi-directed networks is implicit in nature since it is not clear when a mixed-graph enjoys this property or not. In this paper, we introduce novel, explicit mathematical characterizations of semi-directed networks, and also multi-semi-directed networks, that is, mixed graphs that can be obtained from directed phylogenetic networks that may have more than one root. In addition, through extending foundational tools from the theory of rooted networks into the semi-directed setting - such as cherry picking sequences, omnians, and path partitions - we characterize when a (multi-)semi-directed network can be obtained by de-orienting some rooted network that is contained in one of the well-known classes of tree-child, orchard, tree-based or forest-based networks. These results address structural aspects of (multi-)semi-directed networks and pave the way to improved theoretical and computational analyses of such networks, for example, within the development of algebraic evolutionary models that are based on such networks.

We characterize when a set of distances d(x, y) between elements in a set X have a subtree representation, a real tree T and a collection {Sx}x is an element of X of subtrees of T such that d(x, y) equals the length of the shortest path in T from a point in Sx to a point in Sy for all x, y is an element of X. The characterization was first established for finite X by Hirai (2006) using a tight span construction defined for distance spaces, metric spaces without the triangle inequality. To extend Hirai's result beyond finite X we establish fundamental results of tight span theory for general distance spaces, including the surprising observation that the tight span of a distance space is hyperconvex. We apply the results to obtain the first characterization of when a diversity-a generalization of a metric space which assigns values to all finite subsets of X, not just to pairs-has a tight span which is tree-like. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

An equidistant X-cactus is a type of rooted, arc-weighted, directed acyclic graph with leaf set X, that is used in biology to represent the evolutionary history of a set X of species. In this paper, we introduce and investigate the space of equidistant X-cactuses. This space contains, as a subset, the space of ultrametric trees on X that was introduced by Gavryushkin and Drummond. We show that equidistant-cactus space is a CAT(0)-metric space which implies, for example, that there are unique geodesic paths between points. As a key step to proving this, we present a combinatorial result concerning ranked rooted X-cactuses. In particular, we show that such graphs can be encoded in terms of a pairwise compatibility condition arising from a poset of collections of pairs of subsets of X that satisfy certain set-theoretic properties. As a corollary, we also obtain an encoding of ranked, rooted X-trees in terms of partitions of X, which provides an alternative proof that the space of ultrametric trees on X is CAT(0). We expect that our results will provide the basis for novel ways to perform statistical analyses on collections of equidistant X-cactuses, as well as new directions for defining and understanding spaces of more general, arc-weighted phylogenetic networks.

Bacterivorous nematodes represent numerically abundant bacterial grazers in the soil micro-food web. Their trophic regulation shapes the soil microbiome, but the underlying population dynamics of bacteria and archaea are poorly understood. Here, we followed bulk soil respiration and time-resolved population dynamics (32 days) of bacterial and archaeal species in response to top-down control by a common bacterivorous soil nematode, Acrobeloides buetschlii , bottom-up control by resource amendment via maize litter as well as the combination of both. Addition of maize litter significantly increased soil respiration rates, while bacterivorous nematodes shifted the maximum rate of soil respiration from day 12 to day 6. Underlying bacterial and archaeal abundance changes could be separated into five major response types, dominating in different top-down and bottom-up control scenarios. Individual microbial species switched between response types depending on the different scenarios. In-depth analysis of these differential abundance patterns revealed a broad feeding behavior for A. buetschlii on dominating populations of gram-negative bacteria ( Acidobacteriota, Bacteroidota, Gemmatimonatoda, Pseudomonadota ) and ammonia-oxidizing archaea ( Nitrososphaerota ), while discriminating against dominant populations of gram-positive bacteria ( Actinobacteriota, Bacillota ). Combined bottom-up control by maize litter and top-down control by nematode grazing caused a succession of soil microbiota, which was driven by population changes first in the Bacteroidota , then in the Pseudomonadota , and last in the Acidobacteriota and Nitrososphaerota . This mechanistic understanding of nematode grazing on soil microbiota population dynamics is essential to inform predictive models of the soil food web.### Competing Interest StatementThe authors have declared no competing interest.

A comprehensive polyphasic taxonomic investigation integrating taxongenomic criteria was conducted on strain IRAMC:0171T isolated from the root nodules of Retama raetam in Tunisia. This Gram-stain-negative and aerobic bacterium thrived within a temperature range of 5–45 °C, optimal at 28 °C, and tolerated salt concentrations from 0–6 % NaCl, with an optimal range of 0–3 %. It displayed pH tolerance from pH 4 to 10, thriving best at pH 6.8–7.5. Chemotaxonomically, strain IRAMC:0171T was characterized by diphosphatidylglycerol, phosphatidylglycerol, phosphatidylcholine, and phosphatidylethanolamine as polar lipids. Its predominant fatty acid composition was C18 : 1 ω7c (61.2 %), and the primary ubiquinone was Q10 (97 %). Analysis of the 16S rRNA gene of strain IRAMC:0171T showed 99.08 % similarity to Mesorhizobium waimense ICMP 19557T, Mesorhizobium amorphae ACCC 19665T, and Mesorhizobium huakuii IAM 14158. However, digital DNA–DNA hybridization and average nucleotide identity analyses revealed values ranging from 21.1 to 25.2 % and 77.05 to 82.24 %, respectively, signifying significant deviation from established species demarcation thresholds. Phylogenetic studies, encompassing 16S rRNA, whole-genome-based tree reconstruction, and core protein analysis, positioned strain IRAMC:0171T closest to Mesorhizobium terrae KCTC 72278T and ‘Mesorhizobium hungaricum’ UASWS1009T, forming together a distinct branch within the genus Mesorhizobium. In consideration of this comprehensive data, we propose strain IRAMC:0171T (=DSM 112841T=CECT 30767T) as the type strain of a new species named Mesorhizobium retamae sp. nov.

Two new strains JP48T and JP55 affiliated with the acidobacterial class Terriglobia have been isolated from fen soil sampled in the Fichtelgebirge Mountains near Bayreuth, Germany. Both strains were Gram-stain-negative, non-motile, non-spore-forming rods that divide by binary fission, segregate exopolysaccharide-like material and form capsules. Strains JP48T and JP55 grew at 4–36 °C (optimum at 27 °C), pH 3.6–7.3 (optimum at pH 4.6–5.5) and with NaCl concentrations of 0–3% (optimum at 1.0%; w/v). Strains JP48T and JP55 grew aerobically on a wide range of organic substrates including mono- and oligosaccharides, amino acids and short-chained fatty acids. MK-8 was identified as the major respiratory quinone. The major fatty acids for strains JP48T and JP55 were iso-C15 : 0, C16 : 1 ω7c, C16 : 0 and iso-diabolic acid. Phosphatidylglycerol, phosphatidylethanolamine, diphosphatidylglycerol, lysophophatidylethanolamine, phosphatidylcholine, unidentified glyco- and glycophospholipids, and unidentified high mass lipid species were the major polar membrane lipids. The G+C content of strains JP48T and JP55 was 57.4 and 57.2 mol%, respectively. The genomes of strains JP48T and JP55 contained nine potential secondary metabolite regions encoding for the compound classes NRPS(-like), T3PKS, terpene, or lanthipeptide class IV. Phylogenetic reconstruction and 16S rRNA gene sequence similarities of 98.3 and 96.9% identified Edaphobacter dinghuensis DHF9T and Edaphobacter lichenicola DSM 104462T as the most closely related type strains to strains JP48T and JP55. Based on their phenotype, phylogeny and chemotaxonomy, we propose the novel species Edaphobacter paludis sp. nov. (type strain JP48T=DSM 109919T=CECT 30269T; additional strain JP55=DSM 109920=CECT 30268) within the class Terriglobia of the phylum Acidobacteriota.

Bacterivorous nematodes are important grazers in the soil micro-food web. Their trophic regulation shapes the composition and ecosystem services of the soil microbiome, but the underlying population dynamics of bacteria and archaea are poorly understood. We followed soil respiration and 221 dominant bacterial and archaeal 16S rRNA gene amplicon sequencing variants (ASVs) in response to top-down control by a common bacterivorous soil nematode, Acrobeloides buetschlii, bottom-up control by maize litter amendment and their combination over 32 days. Maize litter amendment significantly increased soil respiration, while A. buetschlii addition caused an earlier peak in soil respiration. Underlying bacterial and archaeal population dynamics separated into five major response types, differentiating in their temporal abundance maxima and minima. In-depth analysis of these population dynamics identified a broad imprint of A. buetschlii grazing on dominant bacterial (Acidobacteriota, Bacteroidota, Gemmatimonadota, Pseudomonadota) and archaeal (Nitrososphaerota) ASVs. Combined bottom-up control by maize litter and top-down control by A. buetschlii grazing caused a succession of soil microbiota, driven by population changes first in the Bacteroidota, then in the Pseudomonadota and finally in the Acidobacteriota and Nitrososphaerota. Our results are an essential step forward in understanding trophic modulation of soil microbiota and its feedback on soil respiration.

Convergent evolution is an important process in which independent species evolve similar features usually over a long period of time. It occurs with many different species across the tree of life, and is often caused by the fact that species have to adapt to similar environmental niches. In this paper, we introduce and study properties of a distance-based model for convergent evolution in which we assume that two ancestral species converge for a certain period of time within a collection of species that have otherwise evolved according to an evolutionary clock. Under these assumptions it follows that we obtain a distance on the collection that is a modification of an ultrametric distance arising from an equidistant phylogenetic tree. As well as characterising when this modified distance is a tree metric, we give conditions in terms of the model's parameters for when it is still possible to recover the underlying tree and also its height, even in case the modified distance is not a tree metric.

Phylogenetic networks model the evolutionary history of taxa while allowing for reticulate events such as hybridization and horizontal gene transfer. As is the case for phylogenetic trees, it is often not possible to infer the root location of such a network directly from biological data for several evolutionary models. Hence, 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. To build a network from quarnets, Squirrel first builds a tree, after which it repeatedly solves the Travelling Salesman Problem (TSP) to replace each high-degree vertex by a cycle. We demonstrate Squirrel's performance on randomly generated networks and on real sequence data sets, the largest of which contains 29 aligned sequences close to 1.7 Mpb 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. ### Competing Interest Statement The authors have declared no competing interest.

A network $N$ on a finite set $X$, $|X|\geq 2$, is a connected directed acyclic graph with leaf set $X$ in which every root in $N$ has outdegree at least 2 and no vertex in $N$ has indegree and outdegree equal to 1; $N$ is arboreal if the underlying unrooted, undirected graph of $N$ is a tree. Networks are of interest in evolutionary biology since they are used, for example, to represent the evolutionary history of a set $X$ of species whose ancestors have exchanged genes in the past. For $M$ some arbitrary set of symbols, $d:{X \choose 2} \to M \cup \{\odot\}$ is a symbolic arboreal map if there exists some arboreal network $N$ whose vertices with outdegree two or more are labelled by elements in $M$ and so that $d(\{x,y\})$, $\{x,y\} \in {X \choose 2}$, is equal to the label of the least common ancestor of $x$ and $y$ in $N$ if this exists and $\odot$ else. Important examples of symbolic arboreal maps include the symbolic ultrametrics, which arise in areas such as game theory, phylogenetics and cograph theory. In this paper we show that a map $d:{X \choose 2} \to M \cup \{\odot\}$ is a symbolic arboreal map if and only if $d$ satisfies certain 3- and 4-point conditions and the graph with vertex set $X$ and edge set consisting of those pairs $\{x,y\} \in {X \choose 2}$ with $d(\{x,y\}) \neq \odot$ is Ptolemaic. To do this, we introduce and prove a key theorem concerning the shared ancestry graph for a network $N$ on $X$, where this is the graph with vertex set $X$ and edge set consisting of those $\{x,y\} \in {X \choose 2}$ such that $x$ and $y$ share a common ancestor in $N$. In particular, we show that for any connected graph $G$ with vertex set $X$ and edge clique cover $K$ in which there are no two distinct sets in $K$ with one a subset of the other, there is some network with $|K|$ roots and leaf set $X$ whose shared ancestry graph is $G$.

This paper studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks. We describe a polynomial-time algorithm for deciding whether an undirected nonbinary phylogenetic network, given the locations of the root and reticulation vertices, can be oriented as a directed nonbinary phylogenetic network. Moreover, we characterize when this is possible and show that, in such instances, the resulting directed nonbinary phylogenetic network is unique. In addition, without being given the location of the root and the reticulation vertices, we describe an algorithm for deciding whether an undirected binary phylogenetic network N can be oriented as a directed binary phylogenetic network of a certain class. The algorithm is fixed-parameter tractable (FPT) when the parameter is the level of N and is applicable to classes of directed phylogenetic networks that satisfy certain conditions. As an example, we show that the well-studied class of binary tree-child networks satisfies these conditions.