Quantifying the difference between phylogenetic diversity and diversity indices.

Phylogenetic diversity is a popular measure for quantifying the biodiversity of a collection Y of species, while phylogenetic diversity indices provide a way to apportion phylogenetic diversity to individual species. Typically, for some specific diversity index, the phylogenetic diversity of Y is not equal to the sum of the diversity indices of the species in Y. In this paper, we investigate the extent of this difference for two commonly-used indices: Fair Proportion and Equal Splits. In particular, we determine the maximum value of this difference under various instances including when the associated rooted phylogenetic tree is allowed to vary across all rooted phylogenetic trees with the same leaf set and whose edge lengths are constrained by either their total sum or their maximum value.

Counting and optimising maximum phylogenetic diversity sets.

In conservation biology, phylogenetic diversity (PD) provides a way to quantify the impact of the current rapid extinction of species on the evolutionary 'Tree of Life'. This approach recognises that extinction not only removes species but also the branches of the tree on which unique features shared by the extinct species arose. In this paper, we investigate three questions that are relevant to PD. The first asks how many sets of species of given size k preserve the maximum possible amount of PD in a given tree. The number of such maximum PD sets can be very large, even for moderate-sized phylogenies. We provide a combinatorial characterisation of maximum PD sets, focusing on the setting where the branch lengths are ultrametric (e.g. proportional to time). This leads to a polynomial-time algorithm for calculating the number of maximum PD sets of size k by applying a generating function; we also investigate the types of tree shapes that harbour the most (or fewest) maximum PD sets of size k. Our second question concerns optimising a linear function on the species (regarded as leaves of the phylogenetic tree) across all the maximum PD sets of a given size. Using the characterisation result from the first question, we show how this optimisation problem can be solved in polynomial time, even though the number of maximum PD sets can grow exponentially. Our third question considers a dual problem: If k species were to become extinct, then what is the largest possible loss of PD in the resulting tree? For this question, we describe a polynomial-time solution based on dynamical programming.

Trinets encode orchard phylogenetic networks.

Rooted triples, rooted binary phylogenetic trees on three leaves, are sufficient to encode rooted binary phylogenetic trees. That is, if [Formula: see text] and [Formula: see text] are rooted binary phylogenetic X-trees that infer the same set of rooted triples, then [Formula: see text] and [Formula: see text] are isomorphic. However, in general, this sufficiency does not extend to rooted binary phylogenetic networks. In this paper, we show that trinets, phylogenetic network analogues of rooted triples, are sufficient to encode rooted binary orchard networks. Rooted binary orchard networks naturally generalise rooted binary tree-child networks. Moreover, we present a polynomial-time algorithm for building a rooted binary orchard network from its set of trinets. As a consequence, this algorithm affirmatively answers a previously-posed question of whether there is a polynomial-time algorithm for building a rooted binary tree-child network from the set of trinets it infers.

Caterpillars on three and four leaves are sufficient to reconstruct binary normal networks.

While every rooted binary phylogenetic tree is determined by its set of displayed rooted triples, such a result does not hold for an arbitrary rooted binary phylogenetic network. In particular, there exist two non-isomorphic rooted binary temporal normal networks that display the same set of rooted triples. Moreover, without any structural constraint on the rooted phylogenetic networks under consideration, similarly negative results have also been established for binets and trinets which are rooted subnetworks on two and three leaves, respectively. Hence, in general, piecing together a rooted phylogenetic network from such a set of small building blocks appears insurmountable. In contrast to these results, in this paper, we show that a rooted binary normal network is determined by its sets of displayed caterpillars (particular type of subtrees) on three and four leaves. The proof is constructive and realises a polynomial-time algorithm that takes the sets of caterpillars on three and four leaves displayed by a rooted binary normal network and, up to isomorphism, reconstructs this network.

Tree-based networks: characterisations, metrics, and support trees.

Phylogenetic networks generalise phylogenetic trees and allow for the accurate representation of the evolutionary history of a set of present-day species whose past includes reticulate events such as hybridisation and lateral gene transfer. One way to obtain such a network is by starting with a (rooted) phylogenetic tree T, called a base tree, and adding arcs between arcs of T. The class of phylogenetic networks that can be obtained in this way is called tree-based networks and includes the prominent classes of tree-child and reticulation-visible networks. Initially defined for binary phylogenetic networks, tree-based networks naturally extend to arbitrary phylogenetic networks. In this paper, we generalise recent tree-based characterisations and associated proximity measures for binary phylogenetic networks to arbitrary phylogenetic networks. These characterisations are in terms of matchings in bipartite graphs, path partitions, and antichains. Some of the generalisations are straightforward to establish using the original approach, while others require a very different approach. Furthermore, for an arbitrary tree-based network N, we characterise the support trees of N, that is, the tree-based embeddings of N. We use this characterisation to give an explicit formula for the number of support trees of N when N is binary. This formula is written in terms of the components of a bipartite graph.

When is a Phylogenetic Network Simply an Amalgamation of Two Trees?

Phylogenetic networks generalise phylogenetic (evolutionary) trees by allowing for the representation of reticulation (non-treelike) events. The structure of such networks is often viewed by the phylogenetic trees they embed. In this paper, we determine when a phylogenetic network [Formula: see text] has two phylogenetic tree embeddings which collectively contain all of the edges of [Formula: see text]. This determination leads to a polynomial-time algorithm for recognising such networks and an unexpected characterisation of the class of reticulation-visible networks.

Quarnet Inference Rules for Level-1 Networks.

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

Recovering normal networks from shortest inter-taxa distance information.

Phylogenetic networks are a type of leaf-labelled, acyclic, directed graph used by biologists to represent the evolutionary history of species whose past includes reticulation events. A phylogenetic network is tree-child if each non-leaf vertex is the parent of a tree vertex or a leaf. Up to a certain equivalence, it has been recently shown that, under two different types of weightings, edge-weighted tree-child networks are determined by their collection of distances between each pair of taxa. However, the size of these collections can be exponential in the size of the taxa set. In this paper, we show that, if we have no "shortcuts", that is, the networks are normal, the same results are obtained with only a quadratic number of inter-taxa distances by using the shortest distance between each pair of taxa. The proofs are constructive and give cubic-time algorithms in the size of the taxa sets for building such weighted networks.

On the information content of discrete phylogenetic characters.

Phylogenetic inference aims to reconstruct the evolutionary relationships of different species based on genetic (or other) data. Discrete characters are a particular type of data, which contain information on how the species should be grouped together. However, it has long been known that some characters contain more information than others. For instance, a character that assigns the same state to each species groups all of them together and so provides no insight into the relationships of the species considered. At the other extreme, a character that assigns a different state to each species also conveys no phylogenetic signal. In this manuscript, we study a natural combinatorial measure of the information content of an individual character and analyse properties of characters that provide the maximum phylogenetic information, particularly, the number of states such a character uses and how the different states have to be distributed among the species or taxa of the phylogenetic tree.

Lost in space? Generalising subtree prune and regraft to spaces of phylogenetic networks.

Over the last fifteen years, phylogenetic networks have become a popular tool to analyse relationships between species whose past includes reticulation events such as hybridisation or horizontal gene transfer. However, the space of phylogenetic networks is significantly larger than that of phylogenetic trees, and how to analyse and search this enlarged space remains a poorly understood problem. Inspired by the widely-used rooted subtree prune and regraft (rSPR) operation on rooted phylogenetic trees, we propose a new operation-called subnet prune and regraft (SNPR)-that induces a metric on the space of all rooted phylogenetic networks on a fixed set of leaves. We show that the spaces of several popular classes of rooted phylogenetic networks (e.g. tree child, reticulation visible, and tree based) are connected under SNPR and that connectedness remains for the subclasses of these networks with a fixed number of reticulations. Lastly, we bound the distance between two rooted phylogenetic networks under the SNPR operation, show that it is computationally hard to compute this distance exactly, and analyse how the SNPR-distance between two such networks relates to the rSPR-distance between rooted phylogenetic trees that are embedded in these networks.

On the quirks of maximum parsimony and likelihood on phylogenetic networks.

Maximum parsimony is one of the most frequently-discussed tree reconstruction methods in phylogenetic estimation. However, in recent years it has become more and more apparent that phylogenetic trees are often not sufficient to describe evolution accurately. For instance, processes like hybridization or lateral gene transfer that are commonplace in many groups of organisms and result in mosaic patterns of relationships cannot be represented by a single phylogenetic tree. This is why phylogenetic networks, which can display such events, are becoming of more and more interest in phylogenetic research. It is therefore necessary to extend concepts like maximum parsimony from phylogenetic trees to networks. Several suggestions for possible extensions can be found in recent literature, for instance the softwired and the hardwired parsimony concepts. In this paper, we analyze the so-called big parsimony problem under these two concepts, i.e. we investigate maximum parsimonious networks and analyze their properties. In particular, we show that finding a softwired maximum parsimony network is possible in polynomial time. We also show that the set of maximum parsimony networks for the hardwired definition always contains at least one phylogenetic tree. Lastly, we investigate some parallels of parsimony to different likelihood concepts on phylogenetic networks.

Phylogenetic Networks with Every Embedded Phylogenetic Tree a Base Tree.

We show that the class of tree-child networks is precisely the class of tree-based networks with the property that every embedded phylogenetic tree is a base tree.

Determining phylogenetic networks from inter-taxa distances.

We consider the problem of determining the topological structure of a phylogenetic network given only information about the path-length distances between taxa. In particular, one of the main results of the paper shows that binary tree-child networks are essentially determined by such information.

Amalgamating source trees with different taxonomic levels.

Supertree methods combine a collection of source trees into a single parent tree or supertree. For almost all such methods, the terminal taxa across the source trees have to be non-nested for the output supertree to make sense. Motivated by Page, the first supertree method for combining rooted source trees where the taxa can be hierarchically nested is called AncestralBuild. In addition to taxa labeling the leaves, this method allows the rooted source trees to have taxa labeling some of the interior nodes at a higher taxonomic level than their descendants (e.g., genera vs. species). However, the utility of AncestralBuild is somewhat restricted as it is mostly intended to decide if a collection of rooted source trees is compatible. If the initial collection is not compatible, then no tree is returned. To overcome this restriction, we introduce here the MultiLevelSupertree (MLS) supertree method whose input is the same as that for AncestralBuild, but which accommodates incompatibilities among rooted source trees using a MinCut-like procedure. We show that MLS has several desirable properties including the preservation of common subtrees among the source trees, the preservation of ancestral relationships whenever they are compatible, as well as running in polynomial time. Furthermore, application to a small test data set (the mammalian carnivore family Phocidae) indicates that the method correctly places nested taxa at different taxonomic levels (reflecting vertical signal), even in cases where the input trees harbor a significant level of conflict between their clades (i.e., in their horizontal signal).

Phylogenetic networks that display a tree twice.

In the last decade, the use of phylogenetic networks to analyze the evolution of species whose past is likely to include reticulation events, such as horizontal gene transfer or hybridization, has gained popularity among evolutionary biologists. Nevertheless, the evolution of a particular gene can generally be described without reticulation events and therefore be represented by a phylogenetic tree. While this is not in contrast to each other, it places emphasis on the necessity of algorithms that analyze and summarize the tree-like information that is contained in a phylogenetic network. We contribute to the toolbox of such algorithms by investigating the question of whether or not a phylogenetic network embeds a tree twice and give a quadratic-time algorithm to solve this problem for a class of networks that is more general than tree-child networks.

Cherry picking: a characterization of the temporal hybridization number for a set of phylogenies.

Recently, we have shown that calculating the minimum-temporal-hybridization number for a set [Formula: see text] of rooted binary phylogenetic trees is NP-hard and have characterized this minimum number when [Formula: see text] consists of exactly two trees. In this paper, we give the first characterization of the problem for [Formula: see text] being arbitrarily large. The characterization is in terms of cherries and the existence of a particular type of sequence. Furthermore, in an online appendix to the paper, we show that this new characterization can be used to show that computing the minimum-temporal hybridization number for two trees is fixed-parameter tractable.

Budgeted Nature Reserve Selection with diversity feature loss and arbitrary split systems.

Arising in the context of biodiversity conservation, the Budgeted Nature Reserve Selection (BNRS) problem is to select, subject to budgetary constraints, a set of regions to conserve so that the phylogenetic diversity (PD) of the set of species contained within those regions is maximized. Here PD is measured across either a single rooted tree or a single unrooted tree. Nevertheless, in both settings, this problem is NP-hard. However, it was recently shown that, for each setting, there is a polynomial-time [Formula: see text] -approximation algorithm for it and that this algorithm is tight. In the first part of the paper, we consider two extensions of BNRS. In the rooted setting we additionally allow for the disappearance of features, for varying survival probabilities across species, and for PD to be measured across multiple trees. In the unrooted setting, we extend to arbitrary split systems. We show that, despite these additional allowances, there remains a polynomial-time (1 - 1/e)-approximation algorithm for each extension. In the second part of the paper, we resolve a complexity problem on computing PD across an arbitrary split system left open by Spillner et al.

Defining phylogenetic networks using ancestral profiles.

Rooted phylogenetic networks provide a more complete representation of the ancestral relationship between species than phylogenetic trees when reticulate evolutionary processes are at play. One way to reconstruct a phylogenetic network is to consider its 'ancestral profile' (the number of paths from each ancestral vertex to each leaf). In general, this information does not uniquely determine the underlying phylogenetic network. A recent paper considered a new class of phylogenetic networks called 'orchard networks' where this uniqueness was claimed to hold. Here we show that an additional restriction on the network, that of being 'stack-free', is required in order for the original uniqueness claim to hold. On the other hand, if the additional stack-free restriction is lifted, we establish an alternative result; namely, there is uniqueness within the class of orchard networks up to the resolution of vertices of high in-degree.

Quantifying the extent of lateral gene transfer required to Avert a 'genome of Eden'.

The complex pattern of presence and absence of many genes across different species provides tantalising clues as to how genes evolved through the processes of gene genesis, gene loss, and lateral gene transfer (LGT). The extent of LGT, particularly in prokaryotes, and its implications for creating a 'network of life' rather than a 'tree of life' is controversial. In this paper, we formally model the problem of quantifying LGT, and provide exact mathematical bounds, and new computational results. In particular, we investigate the computational complexity of quantifying the extent of LGT under the simple models of gene genesis, loss, and transfer on which a recent heuristic analysis of biological data relied. Our approach takes advantage of a relationship between LGT optimization and graph-theoretical concepts such as tree width and network flow.

Analyzing and reconstructing reticulation networks under timing constraints.

Reticulation networks are now frequently used to model the history of life for various groups of species whose evolutionary past is likely to include reticulation events such as horizontal gene transfer or hybridization. However, the reconstructed networks are rarely guaranteed to be temporal. If a reticulation network is temporal, then it satisfies the two biologically motivated timing constraints of instantaneously occurring reticulation events and successively occurring speciation events. On the other hand, if a reticulation network is not temporal, it is always possible to make it temporal by adding a number of additional unsampled or extinct taxa. In the first half of the paper, we show that deciding whether a given number of additional taxa is sufficient to transform a non-temporal reticulation network into a temporal one is an NP-complete problem. As one is often given a set of gene trees instead of a network in the context of hybridization, this motivates the second half of the paper which provides an algorithm, called TemporalHybrid, for reconstructing a temporal hybridization network that simultaneously explains the ancestral history of two trees or indicates that no such network exists. We further derive two methods to decide whether or not a temporal hybridization network exists for two given trees and illustrate one of the methods on a grass data set.