A Survey of Combinatorial Methods for Phylogenetic Networks

Evolutionary trees have the assumption that evolution and phylogeny can be represented in a strictly bifurcating manner. Firmly speaking, from one ancestral taxon, two descendant taxa emerge. Nevertheless, hybridization, recombination and horizontal gene transfer is in conflict with this straightforward concept. In such cases, evolutionary lines do not only separate from each other, but have the possibility of melting again and are called reticulations. Consequently, networks can represent evolutionary events more realistically than phylogenetic trees. Networks can display alternative topologies and co-existence of ancestors and descendants, which are otherwise not obvious when a comparison is done on several single trees or a consensus tree. Therefore, networks have the ability to visualize the conflicting information in a given data set. Moreover, the distribution, frequencies and arrangement of haplotypes in populations can reveal the phylogenetic histories of the taxa, regarding predictions from the coalescent theory. This review aims to: (1) give a brief comparison between phylogenetic trees and networks, (2) provide the overall concept of the coalescent theory, (3) clarify how phylogenetic networks can be used to display conflict data and evaluate phylogenetic histories, and (4) offer a useful starting point and guide for sequence analysis, with the aim to discover population dynamics. Key words: Phylogenetic networks, reticulation, coalescent theory, population history, character conflict.

Motivation: Genetic material is transferred in a non-reproductive manner across species more frequently than commonly thought, particularly in the bacteria kingdom. On one hand, extant genomes are thus more properly considered as a fusion product of both reproductive and non-reproductive genetic transfers. This has motivated researchers to adopt phylogenetic networks to study genome evolution. On the other hand, a gene’s evolution is usually tree-like and has been studied for over half a century. Accordingly, the relationships between phylogenetic trees and networks are the basis for the reconstruction and verification of phylogenetic networks. One important problem in verifying a network model is determining whether or not certain existing phylogenetic trees are displayed in a phylogenetic network. This problem is formally called the tree containment problem. It is NP-complete even for binary phylogenetic networks. Results: We design an exponential time but efficient method for determining whether or not a phylogenetic tree is displayed in an arbitrary phylogenetic network. It is developed on the basis of the so-called reticulation-visible property of phylogenetic networks.

Phylogenetic networks are useful for visualizing evolutionary relationships between species with reticulate events such as hybridizations and horizontal gene transfers. In this paper, we consider the problem of constructing undirected phylogenetic networks that (1) are planar graphs and (2) admit embeddings in the plane where the vertices labeling all taxa are on the boundary of the network. We develop a new algorithm for constructing phylogenetic networks satisfying these constraints. First, we show that only approximate networks can be constructed for some distance matrices with at least five taxa. Then we prove that any five-point metric can be represented approximately by a planar boundary-labeled network with guaranteed fit value of 94.79. We extend the networks constructed in the proof to design an algorithm for computing planar boundary-labeled networks for any number of taxa.

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BackgroundPhylogenetic trees resulting from molecular phylogenetic analysis are available in Newick format from specialized databases but when it comes to phylogenetic networks, which provide an explicit representation of reticulate evolutionary events such as recombination, hybridization or lateral gene transfer, the lack of a standard format for their representation has hindered the publication of explicit phylogenetic networks in the specialized literature and their incorporation in specialized databases. Two different proposals to represent phylogenetic networks exist: as a single Newick string (where each hybrid node is splitted once for each parent) or as a set of Newick strings (one for each hybrid node plus another one for the phylogenetic network).ResultsThe standard we advocate as extended Newick format describes a whole phylogenetic network with k hybrid nodes as a single Newick string with k repeated nodes, and this representation is unique once the phylogenetic network is drawn or the ordering among children nodes is fixed. The extended Newick format facilitates phylogenetic data sharing and exchange, and also allows for the practical use of phylogenetic networks in computer programs and scripts. This standard has been recently agreed upon by a number of computational biologists, is already supported by several phylogenetic tools, and avoids the different drawbacks of using an a priori unknown number of Newick strings without any additional mark-up to represent a phylogenetic network.ConclusionThe adoption of the extended Newick format as a standard for the representation of phylogenetic network is an important step towards the publication of explicit phylogenetic networks in peer-reviewed journals and their incorporation in a future database of published phylogenetic networks.

Phylogenetic networks are powerful models for representing evolutionary history in studies of genome evolution involving reticulation events. The complex structure of phylogenetic networks makes their inference challenging, especially for large numbers of taxa. To better understand the full space of phylogenetic networks, we derive closed-form formulas for counting binary phylogenetic networks with two and three reticulation events. In addition, we also use our approach to give a simple proof of the known asymptotic result for a fixed number of reticulations.

Reconstructing unobserved ancestral states of a phylogenetic tree provides insight into the history of evolving systems and is one of the fundamental problems in phylogenetics. For a fixed phylogenetic tree, the most parsimonious ancestral reconstruction - a solution to the small parsimony problem - can be efficiently found using the dynamic programming algorithms of Fitch-Hartigan and Sankoff. Ancestral reconstruction is important in many applications including inferring the routes of metastases in cancer, deriving the transmission history of viruses, determining the direction of cellular differentiation in organismal development, and detecting recombination and horizontal gene transfer in phylogenetic networks. However, most of these applications impose additional global constraints on the reconstructed ancestral states, which break the local structure required in the recurrences of Fitch-Hartigan and Sankoff. We introduce an alternative, polyhedral approach to ancestral reconstruction problems using the tree labeling polytope, a geometric object whose vertices represent the feasible ancestral labelings of a tree. This framework yields a polynomial-time linear programming algorithm for the small parsimony problem. More importantly, the tree labeling polytope facilitates the incorporation of additional constraints that arise in modern ancestral reconstruction problems. We demonstrate the utility of our approach by deriving mixed-integer programming algorithms with a small number of integer variables and strong linear relaxations for three such problems: the parsimonious migration history problem, the softwired small parsimony problem on phylogenetic networks, and the convex recoloring problem on trees. Our algorithms outperform existing state-of-the-art methods on both simulated and real datasets. For instance, our algorithm scales to trace routes of cancer metastases in trees with thousands of leaves, enabling the analysis of large trees generated by recent single-cell sequencing technologies. On a mouse model of metastatic lung adenocarcinoma, the tree labeling polytope allows us to infer simpler migration histories compared to previous results. Python implementations of the algorithms provided in this work are available at: github.com/raphael-group/tree-labeling-polytope.

Phylogenetic networks can represent evolutionary events that cannot be described by phylogenetic trees. These networks are able to incorporate reticulate evolutionary events such as hybridization, introgression, and lateral gene transfer. Recently, network-based Markov models of DNA sequence evolution have been introduced along with model-based methods for reconstructing phylogenetic networks. For these methods to be consistent, the network parameter needs to be identifiable from data generated under the model. Here, we show that the semi-directed network parameter of a triangle-free, level-1 network model with any fixed number of reticulation vertices is generically identifiable under the Jukes–Cantor, Kimura 2-parameter, or Kimura 3-parameter constraints.

Dendroscope 3 is a new program for working with rooted phylogenetic trees and networks. It provides a number of methods for drawing and comparing rooted phylogenetic networks, and for computing them from rooted trees. The program can be used interactively or in command-line mode. The program is written in Java, use of the software is free, and installers for all 3 major operating systems can be downloaded from www.dendroscope.org. [Phylogenetic trees; phylogenetic networks; software.].

BackgroundBranching events in phylogenetic trees reflect bifurcating and/or multifurcating speciation and splitting events. In the presence of gene flow, a phylogeny cannot be described by a tree but is instead a directed acyclic graph known as a phylogenetic network. Both phylogenetic trees and networks are typically reconstructed using computational analysis of multi-locus sequence data. The advent of high-throughput sequencing technologies has brought about two main scalability challenges: (1) dataset size in terms of the number of taxa and (2) the evolutionary divergence of the taxa in a study. The impact of both dimensions of scale on phylogenetic tree inference has been well characterized by recent studies; in contrast, the scalability limits of phylogenetic network inference methods are largely unknown.ResultsIn this study, we quantify the performance of state-of-the-art phylogenetic network inference methods on large-scale datasets using empirical data sampled from natural mouse populations and a range of simulations using model phylogenies with a single reticulation. We find that, as in the case of phylogenetic tree inference, the performance of leading network inference methods is negatively impacted by both dimensions of dataset scale. In general, we found that topological accuracy degrades as the number of taxa increases; a similar effect was observed with increased sequence mutation rate. The most accurate methods were probabilistic inference methods which maximize either likelihood under coalescent-based models or pseudo-likelihood approximations to the model likelihood. The improved accuracy obtained with probabilistic inference methods comes at a computational cost in terms of runtime and main memory usage, which become prohibitive as dataset size grows past twenty-five taxa. None of the probabilistic methods completed analyses of datasets with 30 taxa or more after many weeks of CPU runtime.ConclusionsWe conclude that the state of the art of phylogenetic network inference lags well behind the scope of current phylogenomic studies. New algorithmic development is critically needed to address this methodological gap.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-016-1277-1) contains supplementary material, which is available to authorized users.

Phylogeny and evolutionary histories of Pyrus L. revealed by phylogenetic trees and networks based on data from multiple DNA sequences

The multispecies coalescent (MSC) is a statistical framework that models how gene genealogies grow within the branches of a species tree. The field of computational phylogenetics has witnessed an explosion in the development of methods for species tree inference under MSC, owing mainly to the accumulating evidence of incomplete lineage sorting in phylogenomic analyses. However, the evolutionary history of a set of genomes, or species, could be reticulate due to the occurrence of evolutionary processes such as hybridization or horizontal gene transfer. We report on a novel method for Bayesian inference of genome and species phylogenies under the multispecies network coalescent (MSNC). This framework models gene evolution within the branches of a phylogenetic network, thus incorporating reticulate evolutionary processes, such as hybridization, in addition to incomplete lineage sorting. As phylogenetic networks with different numbers of reticulation events correspond to points of different dimensions in the space of models, we devise a reversible-jump Markov chain Monte Carlo (RJMCMC) technique for sampling the posterior distribution of phylogenetic networks under MSNC. We implemented the methods in the publicly available, open-source software package PhyloNet and studied their performance on simulated and biological data. The work extends the reach of Bayesian inference to phylogenetic networks and enables new evolutionary analyses that account for reticulation.

Phylogenetic networks provide a mathematical model to represent the evolution of a set of species where, apart from speciation, reticulate evolutionary events have to be taken into account. Among these events, lateral gene transfers need special consideration due to the asymmetry in the roles of the species involved in such an event. To take into account this asymmetry, LGT networks were introduced. Contrarily to the case of phylogenetic trees, the combinatorial structure of phylogenetic networks is much less known and difficult to describe. One of the approaches in the literature is to classify them according to their level and find generators of the given level that can be used to recursively generate all networks. In this paper, we adapt the concept of generators to the case of LGT networks. We show how these generators, classified by their level, give rise to simple LGT networks of the specified level, and how any LGT network can be obtained from these simple networks, that act as building blocks of the generic structure. The stochastic models of evolution of phylogenetic networks are also much less studied than those for phylogenetic trees. In this setting, we introduce a novel two-parameter model that generates LGT networks. Finally, we present some computer simulations using this model in order to investigate the complexity of the generated networks, depending on the parameters of the model.

Co-evolution has an important function in the evolution of species and it is clearly manifested in certain scenarios such as host–parasite and predator–prey interactions, symbiosis and mutualism. The extrapolation of the concepts and methodologies developed for the study of species co-evolution at the molecular level has prompted the development of a variety of computational methods able to predict protein interactions through the characteristics of co-evolution. Particularly successful have been those methods that predict interactions at the genomic level based on the detection of pairs of protein families with similar evolutionary histories (similarity of phylogenetic trees: mirrortree). Future advances in this field will require a better understanding of the molecular basis of the co-evolution of protein families. Thus, it will be important to decipher the molecular mechanisms underlying the similarity observed in phylogenetic trees of interacting proteins, distinguishing direct specific molecular interactions from other general functional constraints. In particular, it will be important to separate the effects of physical interactions within protein complexes (‘co-adaptation') from other forces that, in a less specific way, can also create general patterns of co-evolution.

Motivation: The presence of reticulate evolutionary events in phylogenies turn phylogenetic trees into phylogenetic networks. These events imply in particular that there may exist multiple evolutionary paths from a non-extant species to an extant one, and this multiplicity makes the comparison of phylogenetic networks much more difficult than the comparison of phylogenetic trees. In fact, all attempts to define a sound distance measure on the class of all phylogenetic networks have failed so far. Thus, the only practical solutions have been either the use of rough estimates of similarity (based on comparison of the trees embedded in the networks), or narrowing the class of phylogenetic networks to a certain class where such a distance is known and can be efficiently computed. The first approach has the problem that one may identify two networks as equivalent, when they are not; the second one has the drawback that there may not exist algorithms to reconstruct such networks from biological sequences.Results: We present in this article a distance measure on the class of semi-binary tree-sibling time consistent phylogenetic networks, which generalize tree-child time consistent phylogenetic networks, and thus also galled-trees. The practical interest of this distance measure is 2-fold: it can be computed in polynomial time by means of simple algorithms, and there also exist polynomial-time algorithms for reconstructing networks of this class from DNA sequence data.Availability: The Perl package Bio::PhyloNetwork, included in the BioPerl bundle, implements many algorithms on phylogenetic networks, including the computation of the distance presented in this article.Contact: gabriel.cardona@uib.esSupplementary information: Some counterexamples, proofs of the results not included in this article, and some computational experiments are available at Bioinformatics online.