FlatNJ: A novel network-based approach to visualize evolutionary and biogeographical relationships

Split networks are a type of phylogenetic network that allow visualization of conflict in evolutionary data. We present a new method for constructing such networks called FlatNetJoining (FlatNJ). A key feature of FlatNJ is that it produces networks that can be drawn in the plane in which labels may appear inside of the network. For complex data sets that involve, for example, non-neutral molecular markers, this can allow additional detail to be visualized as compared to previous methods such as split decomposition and NeighborNet. We illustrate the application of FlatNJ by applying it to whole HIV genome sequences, where recombination has taken place, fluorescent proteins in corals, where ancestral sequences are present, and mitochondrial DNA sequences from gall wasps, where biogeographical relationships are of interest. We find that the networks generated by FlatNJ can facilitate the study of genetic variation in the underlying molecular sequence data and, in particular, may help to investigate processes such as intra-locus recombination. FlatNJ has been implemented in Java and is freely available at www.uea.ac.uk/computing/software/flatnj

Characterizing block graphs in terms of their vertex-induced partitions

Block graphs are a generalization of trees that arise in areas such as metric graph theory, molecular graphs, and phylogenetics. Given a finite connected simple graph $G=(V,E)$ with vertex set $V$ and edge set $E\subseteq \binom{V}{2}$, we will show that the (necessarily unique) smallest block graph with vertex set $V$ whose edge set contains $E$ is uniquely determined by the $V$-indexed family \Pp_G =\big(\pi_v)_{v \in V} of the partitions $\pi_v$ of the set $V$ into the set of connected components of the graph $(V,\{e\in E: v\notin e\})$. Moreover, we show that an arbitrary $V$-indexed family \Pp=(\p_v)_{v \in V} of partitions \p_v of the set $V$ is of the form \Pp=\Pp_G for some connected simple graph $G=(V,E)$ with vertex set $V$ as above if and only if, for any two distinct elements $u,v\in V$, the union of the set in \p_v that contains $u$ and the set in \p_u that contains $v$ coincides with the set $V$, and \{v\}\in \p_v holds for all $v \in V$. As well as being of inherent interest to the theory of block graphs,these facts are also useful in the analysis of compatible decompositions of finite metric spaces

Prioritizing Populations for Conservation Using Phylogenetic Networks

In the face of inevitable future losses to biodiversity, ranking species by conservation priority seems more than prudent. Setting conservation priorities within species (i.e., at the population level) may be critical as species ranges become fragmented and connectivity declines. However, existing approaches to prioritization (e.g., scoring organisms by their expected genetic contribution) are based on phylogenetic trees, which may be poor representations of differentiation below the species level. In this paper we extend evolutionary isolation indices used in conservation planning from phylogenetic trees to phylogenetic networks. Such networks better represent population differentiation, and our extension allows populations to be ranked in order of their expected contribution to the set. We illustrate the approach using data from two imperiled species: the spotted owl Strix occidentalis in North America and the mountain pygmy-possum Burramys parvus in Australia. Using previously published mitochondrial and microsatellite data, we construct phylogenetic networks and score each population by its relative genetic distinctiveness. In both cases, our phylogenetic networks capture the geographic structure of each species: geographically peripheral populations harbor less-redundant genetic information, increasing their conservation rankings. We note that our approach can be used with all conservation-relevant distances (e.g., those based on whole-genome, ecological, or adaptive variation) and suggest it be added to the assortment of tools available to wildlife managers for allocating effort among threatened populations

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(log2n) of the optimum

Configurations with few crossings in topological graphs

AbstractIn this paper we study the problem of computing subgraphs of a certain configuration in a given topological graph G such that the number of crossings in the subgraph is minimum. The configurations that we consider are spanning trees, s–t paths, cycles, matchings, and κ-factors for κ∈{1,2}. We show that it is NP-hard to approximate the minimum number of crossings for these configurations within a factor of k1−ε for any ε>0, where k is the number of crossings in G.We then give a simple fixed-parameter algorithm that tests in O⋆(2k) time whether G has a crossing-free configuration for any of the above, where the O⋆-notation neglects polynomial terms. For some configurations we have faster algorithms. The respective running times are O⋆(1.9999992k) for spanning trees and O⋆((3)k) for s-t paths and cycles. For spanning trees we also have an O⋆(1.968k)-time Monte-Carlo algorithm. Each O⋆(βk)-time decision algorithm can be turned into an O⋆((β+1)k)-time optimization algorithm that computes a configuration with the minimum number of crossings

The minimum evolution problem is hard:A link between tree inference and graph clustering problems

Motivation: Distance methods are well suited for constructing massive phylogenetic trees. However, the computational complexity for Rzhetsky and Nei’s minimum evolution (ME) approach, one of the earliest methods for constructing a phylogenetic tree from a distance matrix, remains open. Results: We show that Rzhetsky and Nei’s ME problem is NP-complete, and so probably computationally intractable. We do this by linking the ME problem to a graph clustering problem called the quasi-clique decomposition problem, which has recently also been shown to be NP-complete. We also discuss how this link could potentially open up some useful new connections between phylogenetics and graph clustering