Unrooted Phylogenetic Networks Research Articles

Exploring spaces of semi-directed level-1 networks

Semi-directed phylogenetic networks have recently emerged as a class of phylogenetic networks sitting between rooted (directed) and unrooted (undirected) phylogenetic networks as they contain both directed as well as undirected edges. While various spaces of rooted phylogenetic networks and unrooted phylogenetic networks have been analyzed in recent years and several rearrangement moves to traverse these spaces have been introduced, little is known about spaces of semi-directed phylogenetic networks. Here, we propose a simple rearrangement move for semi-directed phylogenetic networks, called cut edge transfer (CET), and show that the space of semi-directed level-1 networks with precisely k reticulations is connected under CET. This level-1 space is currently the predominantly used search space for most algorithms that reconstruct semi-directed phylogenetic networks. Our results imply that every semi-directed level-1 network with a fixed number of reticulations and leaf set can be reached from any other such network by a sequence of CETs. By introducing two additional moves, R^+ and R^-, that allow for the addition and deletion, respectively, of a reticulation, we then establish connectedness for the space of all semi-directed level-1 networks on a fixed leaf set. As a byproduct of our results for semi-directed phylogenetic networks, we also show that the space of rooted level-1 networks with a fixed number of reticulations and leaf set is connected under CET, when translated into the rooted setting.

How far is my network from being edge-based? Proximity measures for edge-basedness of unrooted phylogenetic networks

Phylogenetic networks which are, as opposed to trees, suitable to describe processes like hybridization and horizontal gene transfer, play a substantial role in evolutionary research. However, while non-treelike events need to be taken into account, they are relatively rare, which implies that biologically relevant networks are often assumed to be similar to trees in the sense that they can be obtained by taking a tree and adding some additional edges. This observation led to the concept of so-called tree-based networks, which recently gained substantial interest in the literature. Unfortunately, though, identifying such networks in the unrooted case is an NP-complete problem. Therefore, classes of networks for which tree-basedness can be guaranteed are of the utmost interest. The most prominent such class is formed by so-called edge-based networks, which have a close relationship to generalized series–parallel graphs known from graph theory. They can be identified in linear time and are in some regards biologically more plausible than general tree-based networks. While concerning the latter proximity measures for general networks have already been introduced, such measures are not yet available for edge-basedness. This means that for an arbitrary unrooted network, the “distance” to the nearest edge-based network could so far not be determined. The present manuscript fills this gap by introducing two classes of proximity measures for edge-basedness, one based on the given network itself and one based on its so-called leaf shrink graph (LS graph). Both classes contain four different proximity measures, whose similarities and differences we study subsequently.

English

Phoenix dactylifera L. (Date palm) is one of the leading perennial fruit crops regarding its nutritional profile, medicinal significance, excellent yields, long life, and high adaptability to diverse climatic conditions of semi-arid and arid regions globally. Eleven major palm-growing districts of Punjab, Sindh, and Khyber Pakhtunkhwa were surveyed for wilt disease assessment on date palms. The date palm plantation was highly affected with Fusarium wilt diseases with 55%, the incidence in Khairpur (Sindh province), followed by Bahawalpur 47.4% (Punjab Province) and 45% in Dera Ismail Khan (KPK province). Disease severity was recorded as 65% in Khairpur, 55% in Bahawalpur, 50% in Dera Ismail Khan (Sindh, Punjab, and KPK, respectively). Sixty-three Fusarium isolates were obtained from three hundred and eighty (380) diseased samples, and thirty isolates randomly selected out of 63 were studied for morpho-cultural and microscopic characterization. Morpho-cultural characters i.e colony pigmentation, growth pattern, conidia (macro and microconidia), shape, apical cell shape, and phalidies studied were coinciding with typical characters of Fusarium oxysporum. Twenty isolates of Fusarium population was subjected for the ITS region sequence data-based phylogenetic network. The Neighbor net algorithm computed an unrooted phylogenetic network displayed as four branched genetic-network that depicted the division of twenty fungal isolates into four groups. F. oxysporum (FMB-FO-PD-011) inoculated onto healthy symptomless detached date palm leaves, seedlings and suckers under controlled as well as field conditions showed typical wilt disease symptoms. Hence this study proved that F. oxysporum is the potential cause of date palm wilt in Pakistan.

Improved Fixed-Parameter Algorithm for the Tree Containment Problem on Unrooted Phylogenetic Network.

Phylogenetic trees are unable to represent the evolutionary process for a collection of species if reticulation events happened, and a generalized model named phylogenetic network was introduced consequently. However, the representation of the evolutionary process for one gene is actually a phylogenetic tree that is "contained" in the phylogenetic network for the considered species containing the gene. Thus a fundamental computational problem named Tree Containment problem arises, which asks whether a phylogenetic tree is contained in a phylogenetic network. The previous research on the problem mainly focused on its rooted version of which the considered tree and network are rooted, and several algorithms were proposed when the considered network is binary or structure-restricted. There is almost no algorithm for its unrooted version except the recent fixed-parameter algorithm with runtime O(4kn2), where k and n are the reticulation number and size of the considered unrooted binary phylogenetic network N, respectively. As the runtime is a little expensive when considering big values of k, we aim to improve it and successfully propose a fixed-parameter algorithm with runtime O(2.594kn2) in the paper. Additionally, we experimentally show its effectiveness on biological data and simulated data.

Nuclear DNA reveals multiple waves of colonisation, reticulate evolution and a large impact of stocking on trout in north-west Africa

Using nuclear DNA sequences we studied the phylogeny and genetic structure of trout in Morocco, and estimated the impact of stocking upon natural populations. We supplemented the sample set from a previous study with new sample localities and with specimens from the main hatchery in Morocco. Phylogeny was performed using Bayesian inference, identification of genetic clusters and the genetic structure of populations were assessed from microsatellite polymorphism using discriminant analysis of principal components, and genealogical relationships among mitochondrial haplotypes were re-estimated with an unrooted phylogenetic network using statistical parsimony. Nuclear DNA sequences demonstrated the existence of three major clades in north-west Africa: (1) an Afro-Atlantic clade in Moroccan Mediterranean coastal rivers and the Sous drainage, stemming from a colonisation wave of the Atlantic lineage from Iberia, (2) a relict and native Draa clade (Dades trout), which has maintained an incipient nuclear DNA and mitochondrial DNA signature, and (3) a native Atlas clade, sister to Draa, in which only Atlantic mitochondrial DNA was found, suggesting invasion by Atlantic trout that introgressed into local populations and resulted in a reticulate evolutionary event. Microsatellite analysis and the distribution of the haplotypes revealed a large impact of hatchery trout stocking on wild populations.

Counting phylogenetic networks of level 1 and 2.

Phylogenetic networks generalize phylogenetic trees, and have been introduced in order to describe evolution in the case of transfer of genetic material between coexisting species. There are many classes of phylogenetic networks, which can all be modeled as families of graphs with labeled leaves. In this paper, we focus on rooted and unrooted level-k networks and provide enumeration formulas (exact and asymptotic) for rooted and unrooted level-1 and level-2 phylogenetic networks with a given number of leaves. We also prove that the distribution of some parameters of these networks (such as their number of cycles) are asymptotically normally distributed. These results are obtained by first providing a recursive description (also called combinatorial specification) of our networks, and by next applying classical methods of enumerative, symbolic and analytic combinatorics.

Reconstructibility of unrooted level-k phylogenetic networks from distances

A phylogenetic network is a graph-theoretical tool that is used by biologists to represent the evolutionary history of a collection of species. One potential way of constructing such networks is via a distance-based approach, where one is asked to find a phylogenetic network that in some way represents a given distance matrix, which gives information on the evolutionary distances between present-day taxa. Here, we consider the following question. For which k are unrooted level-k networks uniquely determined by their distance matrices? We consider this question for shortest distances as well as for the case that the multisets of all distances is given. We prove that level-1 networks and level-2 networks are reconstructible from their shortest distances and multisets of distances, respectively. Furthermore we show that, in general, networks of level higher than 1 are not reconstructible from shortest distances and that networks of level higher than 2 are not reconstructible from their multisets of distances.

The Space of Tree-Based Phylogenetic Networks.

Phylogenetic networks are generalizations of phylogenetic trees that allow the representation of reticulation events such as horizontal gene transfer or hybridization, and can also represent uncertainty in inference. A subclass of these, tree-based phylogenetic networks, have been introduced to capture the extent to which reticulate evolution nevertheless broadly follows tree-like patterns. Several important operations that change a general phylogenetic network have been developed in recent years and are important for allowing algorithms to move around spaces of networks; a vital ingredient in finding an optimal network given some biological data. A key such operation is the nearest neighbour interchange, or NNI. While it is already known that the space of unrooted phylogenetic networks is connected under NNI, it has been unclear whether this also holds for the subspace of tree-based networks. In this paper, we show that the space of unrooted tree-based phylogenetic networks is indeed connected under the NNI operation. We do so by explicitly showing how to get from one such network to another one without losing tree-basedness along the way. Moreover, we introduce some new concepts, for instance "shoat networks", and derive some interesting aspects concerning tree-basedness. Last, we use our results to derive an upper bound on the size of the space of tree-based networks.

How tree-based is my network? Proximity measures for unrooted phylogenetic networks

Tree-based networks are a class of phylogenetic networks that attempt to formally capture what is meant by “tree-like” evolution. A given non-tree-based phylogenetic network, however, might appear to be very close to being tree-based, or very far. In this paper, we formalize the notion of proximity to tree-based for unrooted phylogenetic networks, with a range of proximity measures. These measures also provide characterizations of tree-based networks. One measure in particular, related to the nearest neighbour interchange operation, allows us to define the notion of “tree-based rank”. This provides a subclassification within the tree-based networks themselves, identifying those networks that are “very” tree-based. Finally, we prove results relating tree-based networks in the settings of rooted and unrooted phylogenetic networks, showing effectively that an unrooted network is tree-based if and only if it can be made a rooted tree-based network by rooting it and orienting the edges appropriately. This leads to a clarification of the contrasting decision problems for tree-based networks, which are polynomial in the rooted case but NP complete in the unrooted.

Not all phylogenetic networks are leaf-reconstructible

Unrooted phylogenetic networks are graphs used to represent reticulate evolutionary relationships. Accurately reconstructing such networks is of great relevance for evolutionary biology. It has recently been conjectured that all unrooted phylogenetic networks for at least five taxa can be uniquely reconstructed from their subnetworks obtained by deleting a single taxon. Here, we show that this conjecture is false, by presenting a counter-example for each possible number of taxa that is at least 4. Moreover, we show that the conjecture is still false when restricted to binary networks. This means that, even if we are able to reconstruct the unrooted evolutionary history of each proper subset of some taxon set, this still does not give us enough information to reconstruct their full unrooted evolutionary history.

Rearrangement operations on unrooted phylogenetic networks

Rearrangement operations transform a phylogenetic tree into another one and hence induce a metric on the space of phylogenetic trees. Popular operations for unrooted phylogenetic trees are NNI (nearest neighbour interchange), SPR (subtree prune and regraft), and TBR (tree bisection and reconnection). Recently, these operations have been extended to unrooted phylogenetic networks, which are generalisations of phylogenetic trees that can model reticulated evolutionary relationships. Here, we study global and local properties of spaces of phylogenetic networks under these three operations. In particular, we prove connectedness and asymptotic bounds on the diameters of spaces of different classes of phylogenetic networks, including tree-based and level-k networks. We also examine the behaviour of shortest TBR-sequence between two phylogenetic networks in a class, and whether the TBR-distance changes if intermediate networks from other classes are allowed: for example, the space of phylogenetic trees is an isometric subgraph of the space of phylogenetic networks under TBR. Lastly, we show that computing the TBR-distance and the PR-distance of two phylogenetic networks is NP-hard.

Mitochondrial sequences retrieve an ancient lineage of Bicolored shrew in the Hyrcanian refugium

The bicolored shrew Crocidura leucodon consists of two phylogeographic lineages which diverged during the Middle Pleistocene and are roughly separated by the Bosporus Strait at the contact point of Europe and Asia Minor. In this study we analysed mitochondrial cytochrome b genes (1137 bp) of nine shrews from the Caspian region in Iran. Phylogenetic trees obtained in Maximum Likelihood and Bayesian analyses retrieved a sister position of our Iranian haplotypes (Iranian lineage) against all the remaining C. leucodon samples from Europe, Turkey and Georgia (Euro-Asian lineage). Identical topology was also evident from the unrooted phylogenetic network. The Euro-Asian and the Iranian lineages were separated by a K2P genetic distance of 7.5 ± 0.9 and diverged 1.14 Mya (95% CI: 0.841–1.616). The geographic range of the Iranian lineage is evidently restricted to the Hyrcanian region south of the Caspian Sea which abounds with small range mammalian endemics.

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.

Leaf-Reconstructibility of Phylogenetic Networks

An important problem in evolutionary biology is to reconstruct the evolutionary history of a set $X$ of species. This history is often represented as a phylogenetic network, that is, a connected graph with leaves labelled by elements in $X$ (for example, an evolutionary tree), which is usually also binary, i.e., all vertices have degree 1 or 3. A common approach used in phylogenetics to build a phylogenetic network on $X$ involves constructing it from networks on subsets of $X$. Here we consider the question of which (unrooted) phylogenetic networks are leaf-reconstructible, i.e., which networks can be uniquely reconstructed from the set of networks obtained from it by deleting a single leaf (its $X$-deck). This problem is closely related to the (in)famous reconstruction conjecture in graph theory but, as we shall show, presents distinct challenges. We show that some large classes of phylogenetic networks are reconstructible from their $X$-deck. This includes phylogenetic trees, binary networks containing at least one nontrivial cut-edge, and binary level-4 networks. (The level of a network measures how far it is from being a tree.) We also show that for fixed $k$, almost all binary level-$k$ phylogenetic networks are leaf-reconstructible. As an application of our results, we show that a level-3 network $N$ can be reconstructed from its quarnets, that is, 4-leaved networks that are induced by $N$ in a certain recursive fashion. Our results lead to several interesting open problems which we discuss, including the conjecture that all phylogenetic networks with at least five leaves are leaf-reconstructible.

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.

Bounds for phylogenetic network space metrics

Phylogenetic networks are a generalization of phylogenetic trees that allow for representation of reticulate evolution. Recently, a space of unrooted phylogenetic networks was introduced, where such a network is a connected graph in which every vertex has degree 1 or 3 and whose leaf-set is a fixed set X of taxa. This space, denoted mathcal {N}(X), is defined in terms of two operations on networks—the nearest neighbor interchange and triangle operations—which can be used to transform any network with leaf set X into any other network with that leaf set. In particular, it gives rise to a metric d on {mathcal {N}}(X) which is given by the smallest number of operations required to transform one network in {mathcal {N}}(X) into another in {mathcal {N}}(X). The metric generalizes the well-known NNI-metric on phylogenetic trees which has been intensively studied in the literature. In this paper, we derive a bound for the metric d as well as a related metric d_{N!N!I} which arises when restricting d to the subset of mathcal {N}(X) consisting of all networks with 2(|X|-1+i) vertices, i ge 1. We also introduce two new metrics on networks—the SPR and TBR metrics—which generalize the metrics on phylogenetic trees with the same name and give bounds for these new metrics. We expect our results to eventually have applications to the development and understanding of network search algorithms.

On Unrooted and Root-Uncertain Variants of Several Well-Known Phylogenetic Network Problems

The hybridization number problem requires us to embed a set of binary rooted phylogenetic trees into a binary rooted phylogenetic network such that the number of nodes with indegree two is minimized. However, from a biological point of view accurately inferring the root location in a phylogenetic tree is notoriously difficult and poor root placement can artificially inflate the hybridization number. To this end we study a number of relaxed variants of this problem. We start by showing that the fundamental problem of determining whether an unrooted phylogenetic network displays (i.e. embeds) an unrooted phylogenetic tree, is NP-hard. On the positive side we show that this problem is FPT in reticulation number. In the rooted case the corresponding FPT result is trivial, but here we require more subtle argumentation. Next we show that the hybridization number problem for unrooted networks (when given two unrooted trees) is equivalent to the problem of computing the tree bisection and reconnect distance of the two unrooted trees. In the third part of the paper we consider the “root uncertain” variant of hybridization number. Here we are free to choose the root location in each of a set of unrooted input trees such that the hybridization number of the resulting rooted trees is minimized. On the negative side we show that this problem is APX-hard. On the positive side, we show that the problem is FPT in the hybridization number, via kernelization, for any number of input trees.

Uprooted Phylogenetic Networks

The need for structures capable of accommodating complex evolutionary signals such as those found in, for example, wheat has fueled research into phylogenetic networks. Such structures generalize the standard model of a phylogenetic tree by also allowing for cycles and have been introduced in rooted and unrooted form. In contrast to phylogenetic trees or their unrooted versions, rooted phylogenetic networks are notoriously difficult to understand. To help alleviate this, recent work on them has also centered on their “uprooted” versions. By focusing on such graphs and the combinatorial concept of a split system which underpins an unrooted phylogenetic network, we show that not only can a so-called (uprooted) 1-nested network N be obtained from the Buneman graph (sometimes also called a median network) associated with the split system Sigma (N) induced on the set of leaves of N but also that that graph is, in a well-defined sense, optimal. Along the way, we establish the 1-nested analogue of the fundamental “splits equivalence theorem” for phylogenetic trees and characterize maximal circular split systems.

Transforming phylogenetic networks: Moving beyond tree space

Phylogenetic networks are a generalization of phylogenetic trees that are used to represent reticulate evolution. Unrooted phylogenetic networks form a special class of such networks, which naturally generalize unrooted phylogenetic trees. In this paper we define two operations on unrooted phylogenetic networks, one of which is a generalization of the well-known nearest-neighbor interchange (NNI) operation on phylogenetic trees. We show that any unrooted phylogenetic network can be transformed into any other such network using only these operations. This generalizes the well-known fact that any phylogenetic tree can be transformed into any other such tree using only NNI operations. It also allows us to define a generalization of tree space and to define some new metrics on unrooted phylogenetic networks. To prove our main results, we employ some fascinating new connections between phylogenetic networks and cubic graphs that we have recently discovered. Our results should be useful in developing new strategies to search for optimal phylogenetic networks, a topic that has recently generated some interest in the literature, as well as for providing new ways to compare networks.

QUARTETS AND UNROOTED PHYLOGENETIC NETWORKS

Phylogenetic networks were introduced to describe evolution in the presence of exchanges of genetic material between coexisting species or individuals. Split networks in particular were introduced as a special kind of abstract network to visualize conflicts between phylogenetic trees which may correspond to such exchanges. More recently, methods were designed to reconstruct explicit phylogenetic networks (whose vertices can be interpreted as biological events) from triplet data. In this article, we link abstract and explicit networks through their combinatorial properties, by introducing the unrooted analog of level-k networks. In particular, we give an equivalence theorem between circular split systems and unrooted level-1 networks. We also show how to adapt to quartets some existing results on triplets, in order to reconstruct unrooted level-k phylogenetic networks. These results give an interesting perspective on the combinatorics of phylogenetic networks and also raise algorithmic and combinatorial questions.