arXiv subjects

Vincent Moulton

Publications and source records attributed to Vincent Moulton.

At least 55 records · Page 3Linked to original sources

arXiv2019

Order distances and split systems

Given a distance $D$ on a finite set $X$ with $n$ elements, it is interesting to understand how the ranking $R_x = z_1,z_2,\dots,z_n$ obtained by ordering the elements in $X$ according to increasing distance $D(x,z_i)$ from $x$, varies with different choices of $x \in X$. The order distance $O_{p,q}(D)$ is a distance on $X$ associated to $D$ which quantifies these variations, where $q \geq \frac{p}{2} > 0$ are parameters that control how ties in the rankings are handled. The order distance $O_{p,q}(D)$ of a distance $D$ has been intensively studied in case $D$ is a treelike distance (that is, $D$ arises as the shortest path distances in an edge-weighted tree with leaves labeled by $X$), but relatively little is known about properties of $O_{p,q}(D)$ for general $D$. In this paper we study the order distance for various types of distances that naturally generalize treelike distances in that they can be generated by split systems, i.e. they are examples of so-called $l_1$-distances. In particular we show how and to what extent properties of the split systems associated to the distances $D$ that we study can be used to infer properties of $O_{p,q}(D)$.

cs.DM↗

arXiv2019

Reconstructing Tree-Child Networks from Reticulate-Edge-Deleted Subnetworks

Network reconstruction lies at the heart of phylogenetic research. Two well studied classes of phylogenetic networks include tree-child networks and level-$k$ networks. In a tree-child network, every non-leaf node has a child that is a tree node or a leaf. In a level-$k$ network, the maximum number of reticulations contained in a biconnected component is $k$. Here, we show that level-$k$ tree-child networks are encoded by their reticulate-edge-deleted subnetworks, which are subnetworks obtained by deleting a single reticulation edge, if $k\geq 2$. Following this, we provide a polynomial-time algorithm for uniquely reconstructing such networks from their reticulate-edge-deleted subnetworks. Moreover, we show that this can even be done when considering subnetworks obtained by deleting one reticulation edge from each biconnected component with $k$ reticulations.

math.CO↗

arXiv2019

Reconciling Event-Labeled Gene Trees with MUL-trees and Species Networks

Phylogenomics commonly aims to construct evolutionary trees from genomic sequence information. One way to approach this problem is to first estimate event-labeled gene trees (i.e., rooted trees whose non-leaf vertices are labeled by speciation or gene duplication events), and to then look for a species tree which can be reconciled with this tree through a \emph{reconciliation map} between the trees. In practice, however, it can happen that there is no such map from a given event-labeled tree to \emph{any} species tree. An important situation where this might arise is where the species evolution is better represented by a \emph{network} instead of a tree. In this paper, we therefore consider the problem of reconciling event-labeled trees with species networks. In particular, we prove that any event-labeled gene tree can be reconciled with some network and that, under certain mild assumptions on the gene tree, the network can even be assumed to be multi-arc free. To prove this result, we show that we can always reconcile the gene tree with some multi-labeled (MUL-)tree, which can then be "folded up" to produce the desired reconciliation and network. In addition, we study the interplay between reconciliation maps from event-labeled gene trees to MUL-trees and networks. Our results could be useful for understanding how genomes have evolved after undergoing complex evolutionary events such as polyploidy.

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arXiv2018

The complexity of comparing multiply-labelled trees by extending phylogenetic-tree metrics

A multilabeled tree (or MUL-tree) is a rooted tree in which every leaf is labelled by an element from some set, but in which more than one leaf may be labelled by the same element of that set. In phylogenetics, such trees are used in biogeographical studies, to study the evolution of gene families, and also within approaches to construct phylogenetic networks. A multilabelled tree in which no leaf-labels are repeated is called a phylogenetic tree, and one in which every label is the same is also known as a tree-shape. In this paper, we consider the complexity of computing metrics on MUL-trees that are obtained by extending metrics on phylogenetic trees. In particular, by restricting our attention to tree shapes, we show that computing the metric extension on MUL-trees is NP complete for two well-known metrics on phylogenetic trees, namely, the path-difference and Robinson Foulds distances. We also show that the extension of the Robinson Foulds distance is fixed parameter tractable with respect to the distance parameter. The path distance complexity result allows us to also answer an open problem concerning the complexity of solving the quadratic assignment problem for two matrices that are a Robinson similarity and a Robinson dissimilarity, which we show to be NP-complete. We conclude by considering the maximum agreement subtree (MAST) distance on phylogenetic trees to MUL-trees. Although its extension to MUL-trees can be computed in polynomial time, we show that computing its natural generalization to more than two MUL-trees is NP-complete, although fixed-parameter tractable in the maximum degree when the number of given trees is bounded.

q-bio.PE↗

arXiv2018

Phylogenetic flexibility via Hall-type inequalities and submodularity

Given a collection $τ$ of subsets of a finite set $X$, we say that $τ$ is {\em phylogenetically flexible} if, for any collection $R$ of rooted phylogenetic trees whose leaf sets comprise the collection $τ$, $R$ is compatible (i.e. there is a rooted phylogenetic $X$--tree that displays each tree in $R$). We show that $τ$ is phylogenetically flexible if and only if it satisfies a Hall-type inequality condition of being `slim'. Using submodularity arguments, we show that there is a polynomial-time algorithm for determining whether or not $τ$ is slim. This `slim' condition reduces to a simpler inequality in the case where all of the sets in $τ$ have size 3, a property we call `thin'. Thin sets were recently shown to be equivalent to the existence of an (unrooted) tree for which the median function provides an injective mapping to its vertex set; we show here that the unrooted tree in this representation can always be chosen to be a caterpillar tree. We also characterise when a collection $τ$ of subsets of size 2 is thin (in terms of the flexibility of total orders rather than phylogenies) and show that this holds if and only if an associated bipartite graph is a forest. The significance of our results for phylogenetics is in providing precise and efficiently verifiable conditions under which supertree methods that require consistent inputs of trees, can be applied to any input trees on given subsets of species.

math.CO↗

arXiv2017

Identifiability of tree-child phylogenetic networks under a probabilistic recombination-mutation model of evolution

Phylogenetic networks are an extension of phylogenetic trees which are used to represent evolutionary histories in which reticulation events (such as recombination and hybridization) have occurred. A central question for such networks is that of identifiability, which essentially asks under what circumstances can we reliably identify the phylogenetic network that gave rise to the observed data? Recently, identifiability results have appeared for networks relative to a model of sequence evolution that generalizes the standard Markov models used for phylogenetic trees. However, these results are quite limited in terms of the complexity of the networks that are considered. In this paper, by introducing an alternative probabilistic model for evolution along a network that is based on some ground-breaking work by Thatte for pedigrees, we are able to obtain an identifiability result for a much larger class of phylogenetic networks (essentially the class of so-called tree-child networks). To prove our main theorem, we derive some new results for identifying tree-child networks combinatorially, and then adapt some techniques developed by Thatte for pedigrees to show that our combinatorial results imply identifiability in the probabilistic setting. We hope that the introduction of our new model for networks could lead to new approaches to reliably construct phylogenetic networks.

q-bio.PE↗

arXiv2017

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 \emph{unrooted} phylogenetic network on a nonempty, finite set $X$ of taxa, or \emph{network}, is a connected graph in which every vertex has degree 1 or 3 and whose leaf-set is $X$. It is called a \emph{phylogenetic tree} if the underlying graph is a tree. In this paper we consider properties of \emph{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 hyrbridization which are important factors in the evolution of many organisms.

q-bio.PE↗

arXiv2017

Recovering tree-child 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 ignore redundant edges, 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.

math.CO↗

arXiv2017

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 1980's characterizations, certain inference rules were given for when a collection of 4-leaved trees, one for each 4-element subset of $X$, can all be simultaneously displayed by a single supertree with leaf-set $X$. Recently, it has become of interest to extend such 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 been shown that a certain type of phylogenetic network, called a 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.

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arXiv2017

Three-way symbolic tree-maps and ultrametrics

Three-way dissimilarities are a generalization of (two-way) dissimilarities which can be used to indicate the lack of homogeneity or resemblance between any three objects. Such maps have applications in cluster analysis, and have been used in areas such as psychology and phylogenetics, where three-way data tables can arise. Special examples of such dissimilarities are three-way tree-metrics and ultrametrics, which arise from leaf-labelled trees with edges labelled by positive real numbers. Here we consider three-way maps which arise from leaf-labelled trees where instead the interior vertices are labelled by an arbitrary set of values. For unrooted trees we call such maps three-way symbolic tree-maps; for rooted trees we call them three-way symbolic ultrametrics since they can be considered as a generalization of the (two-way) symbolic ultrametrics of Böcker and Dress. We show that, as with two- and three-way tree-metrics and ultrametrics, three-way symbolic tree-maps and ultrametrics can be characterized via certain $k$-point conditions. In the unrooted case, our characterization is mathematically equivalent to one presented by Gurvich for a certain class of edge-labelled hypergraphs. We also show that it can be decided whether or not an arbitrary three-way symbolic map is a tree-map or a symbolic ultrametric using a triplet-based approach that relies on the so-called BUILD algorithm for deciding when a set of 3-leaved trees or triplets can be displayed by a single tree. We envisage that our results will be useful in developing new approaches and algorithms for understanding 3-way data, especially within the area of phylogenetics.

math.CO↗

arXiv2017

Combinatorial properties of triplet covers for binary trees

It is a classical result that an unrooted tree $T$ having positive real-valued edge lengths and no vertices of degree two can be reconstructed from the induced distance between each pair of leaves. Moreover, if each non-leaf vertex of $T$ has degree 3 then the number of distance values required is linear in the number of leaves. A canonical candidate for such a set of pairs of leaves in $T$ is the following: for each non-leaf vertex $v$, choose a leaf in each of the three components of $T-v$, group these three leaves into three pairs, and take the union of this set over all choices of $v$. This forms a so-called 'triplet cover' for $T$. In the first part of this paper we answer an open question (from 2012) by showing that the induced leaf-to-leaf distances for any triplet cover for $T$ uniquely determine $T$ and its edge lengths. We then investigate the finer combinatorial properties of triplet covers. In particular, we describe the structure of triplet covers that satisfy one or more of the following properties of being minimal, 'sparse', and 'shellable'.

math.CO↗

arXiv2017

Geometric medians in reconciliation spaces

In evolutionary biology, it is common to study how various entities evolve together, for example, how parasites coevolve with their host, or genes with their species. Coevolution is commonly modelled by considering certain maps or reconciliations from one evolutionary tree $P$ to another $H$, all of which induce the same map $ϕ$ between the leaf-sets of $P$ and $H$ (corresponding to present-day associations). Recently, there has been much interest in studying spaces of reconciliations, which arise by defining some metric $d$ on the set $Rec(P,H,ϕ)$ of all possible reconciliations between $P$ and $H$. In this paper, we study the following question: How do we compute a geometric median for a given subset $Ψ$ of $Rec(P,H,ϕ)$ relative to $d$, i.e. an element $ψ_{med} \in Rec(P,H,ϕ)$ such that $$ \sum_{ψ' \in Ψ} d(ψ_{med},ψ') \le \sum_{ψ' \in Ψ} d(ψ,ψ') $$ holds for all $ψ\in Rec(P,H,ϕ)$? For a model where so-called host-switches or transfers are not allowed, and for a commonly used metric $d$ called the edit-distance, we show that although the cardinality of $Rec(P,H,ϕ)$ can be super-exponential, it is still possible to compute a geometric median for a set $Ψ$ in $Rec(P,H,ϕ)$ in polynomial time. We expect that this result could be useful for computing a summary or consensus for a set of reconciliations (e.g. for a set of suboptimal reconciliations).

q-bio.PE↗

arXiv2017

From event labeled gene trees to species trees

Background: Tree reconciliation problems have long been studied in phylogenetics. A particular variant of the reconciliation problem for a gene tree T and a species tree S assumes that for each interior vertex x of T it is known whether x represents a speciation or a duplication. This problem appears in the context of analyzing orthology data. Results: We show that S is a species tree for T if and only if S displays all rooted triples of T that have three distinct species as their leaves and are rooted in a speciation vertex. A valid reconciliation map can then be found in polynomial time. Simulated data shows that the event-labeled gene trees convey a large amount of information on underlying species trees, even for a large percentage of losses. Conclusions: The knowledge of event labels in a gene tree strongly constrains the possible species tree and, for a given species tree, also the possible reconciliation maps. Nevertheless, many degrees of freedom remain in the space of feasible solutions. In order to disambiguate the alternative solutions additional external constraints as well as optimization criteria could be employed.

cs.DM↗

arXiv2017

UPGMA and the normalized equidistant minimum evolution problem

UPGMA (Unweighted Pair Group Method with Arithmetic Mean) is a widely used clustering method. Here we show that UPGMA is a greedy heuristic for the normalized equidistant minimum evolution (NEME) problem, that is, finding a rooted tree that minimizes the minimum evolution score relative to the dissimilarity matrix among all rooted trees with the same leaf-set in which all leaves have the same distance to the root. We prove that the NEME problem is NP-hard. In addition, we present some heuristic and approximation algorithms for solving the NEME problem, including a polynomial time algorithm that yields a binary, rooted tree whose NEME score is within O(log^2 n) of the optimum. We expect that these results to eventually provide further insights into the behavior of the UPGMA algorithm.

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arXiv2017

A cubic-time algorithm for computing the trinet distance between level-1 networks

In evolutionary biology, phylogenetic networks are constructed to represent the evolution of species in which reticulate events are thought to have occurred, such as recombination and hybridization. It is therefore useful to have efficiently computable metrics with which to systematically compare such networks. Through developing an optimal algorithm to enumerate all trinets displayed by a level-1 network (a type of network that is slightly more general than an evolutionary tree), here we propose a cubic-time algorithm to compute the trinet distance between two level-1 networks. Employing simulations, we also present a comparison between the trinet metric and the so-called Robinson-Foulds phylogenetic network metric restricted to level-1 networks. The algorithms described in this paper have been implemented in JAVA and are freely available at https://www.uea.ac.uk/computing/TriLoNet.

q-bio.PE↗

arXiv2017

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.

q-bio.PE↗

arXiv2017

Minimum triplet covers of binary phylogenetic $X$-trees

Trees with labelled leaves and with all other vertices of degree three play an important role in systematic biology and other areas of classification. A classical combinatorial result ensures that such trees can be uniquely reconstructed from the distances between the leaves (when the edges are given any strictly positive lengths). Moreover, a linear number of these pairwise distance values suffices to determine both the tree and its edge lengths. A natural set of pairs of leaves is provided by any `triplet cover' of the tree (based on the fact that each non-leaf vertex is the median vertex of three leaves). In this paper we describe a number of new results concerning triplet covers of minimum size. In particular, we characterize such covers in terms of an associated graph being a 2-tree. Also, we show that minimum triplet covers are `shellable' and thereby provide a set of pairs for which the inter-leaf distance values will uniquely determine the underlying tree and its associated branch lengths.

math.CO↗

arXiv2017

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 non-trivial 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.

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