The tree-child network inference problem for line trees and the shortest common supersequence problem for permutation strings

An evolutionary algorithm based on parsimony for the multiobjective phylogenetic network inference problem

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.

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.

The combinatorial study of phylogenetic networks has attracted much attention in recent times. In particular, one class of them, the so-called tree-child networks, are becoming the most prominent ones. However, their combinatorial properties are largely unknown. In this paper we address the problem of exactly counting them. We conjecture a relationship with the cardinality of a certain class of words. By solving the counting problem for the words, and on the basis of the conjecture, several simple recurrence formulas for general cases arise. Moreover, a precise asymptotic analysis is provided. Our results coincide with all current formulas in the literature for particular subclasses of tree-child networks, as well as with numerical results obtained for small networks. We expect that the study of the relationship between the newly defined words and the networks will lead to further combinatoric characterizations of this class of phylogenetic networks.

PhyloNet was released in 2008 as a software package for representing and analyzing phylogenetic networks. At the time of its release, the main functionalities in PhyloNet consisted of measures for comparing network topologies and a single heuristic for reconciling gene trees with a species tree. Since then, PhyloNet has grown significantly. The software package now includes a wide array of methods for inferring phylogenetic networks from data sets of unlinked loci while accounting for both reticulation (e.g., hybridization) and incomplete lineage sorting. In particular, PhyloNet now allows for maximum parsimony, maximum likelihood, and Bayesian inference of phylogenetic networks from gene tree estimates. Furthermore, Bayesian inference directly from sequence data (sequence alignments or biallelic markers) is implemented. Maximum parsimony is based on an extension of the "minimizing deep coalescences" criterion to phylogenetic networks, whereas maximum likelihood and Bayesian inference are based on the multispecies network coalescent. All methods allow for multiple individuals per species. As computing the likelihood of a phylogenetic network is computationally hard, PhyloNet allows for evaluation and inference of networks using a pseudolikelihood measure. PhyloNet summarizes the results of the various analyzes and generates phylogenetic networks in the extended Newick format that is readily viewable by existing visualization software.

Phylogenetic networks extend the phylogenetic tree structure and allow for modeling vertical and evolution in a single framework. Statistical inference of phylogenetic networks is prohibitive and currently limited to small networks. An approach that could significantly improve phylogenetic network space exploration is based on first inferring an evolutionary tree of the species under consideration, and then augmenting the tree into a network by adding a set of horizontal edges to better fit the data. In this paper, we study the performance of such an approach on networks generated under a birth-hybridization model and explore its feasibility as an alternative to approaches that search the phylogenetic network space directly (without relying on a fixed underlying tree). We find that the concatenation method does poorly at obtaining a tree that could be augmented into the correct network, whereas the popular species tree inference method ASTRAL does significantly better at such a task. We then evaluated the tree-to-network augmentation phase under the minimizing deep coalescence and pseudo-likelihood criteria. We find that even though this is a much faster approach than the direct search of the network space, the accuracy is much poorer, even when the backbone tree is a good starting tree. Our results show that tree-based inference of phylogenetic networks could yield very poor results. As exploration of the network space directly in search of maximum likelihood estimates or a representative sample of the posterior is very expensive, significant improvements to the computational complexity of phylogenetic network inference are imperative if analyses of large data sets are to be performed. We show that a recently developed divide-and-conquer approach significantly outperforms tree-based inference in terms of accuracy, albeit still at a higher computational cost.

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.

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.].

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.

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

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.

Inference of phylogenetic networks is of increasing interest in the genomic era. However, the extent to which phylogenetic networks are identifiable from various types of data remains poorly understood, despite its crucial role in justifying methods. This work obtains strong identifiability results for large sub-classes of galled tree-child semidirected networks. Some of the conditions our proofs require, such as the identifiability of a network’s tree of blobs or the circular order of 4 taxa around a cycle in a level-1 network, are already known to hold for many data types. We show that all these conditions hold for quartet concordance factor data under various gene tree models, yielding the strongest results from 2 or more samples per taxon. Although the network classes we consider have topological restrictions, they include non-planar networks of any level and are substantially more general than level-1 networks — the only class previously known to enjoy identifiability from many data types. Our work establishes a route for proving future identifiability results for tree-child galled networks from data types other than quartet concordance factors, by checking that explicit conditions are met.

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.

Attaching leaves and picking cherries to characterise the hybridisation number for a set of phylogenies

Summary The fields of phylogenetic tree and network inference have dramatically advanced in the past decade, but independently with few attempts to bridge them. Here we provide a framework, implemented in the phangorn library in R, to transfer information between trees and networks. This includes: (i) identifying and labelling equivalent tree branches and network edges, (ii) transferring tree branch support to network edges, and (iii) mapping bipartition support from a sample of trees (e.g. from bootstrapping or Bayesian inference) onto network edges. The ability to readily combine tree and network information should lead to more comprehensive evolutionary comparisons and inferences.