APPROXIMATING METRICS WITH PLANAR BOUNDARY-LABELED PHYLOGENETIC NETWORKS

The evolutionary history of a set of species is usually described by a rooted phylogenetic tree. Although it is generally undisputed that bifurcating speciation events and descent with modifications are major forces of evolution, there is a growing belief that reticulate events also have a role to play. Phylogenetic networks provide an alternative to phylogenetic trees and may be more suitable for data sets where evolution involves significant amounts of reticulate events, such as hybridization, horizontal gene transfer, or recombination. In this article, we give an introduction to the topic of phylogenetic networks, very briefly describing the fundamental concepts and summarizing some of the most important combinatorial methods that are available for their computation.

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.

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.

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.

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BackgroundPhylogenetic reconstruction is one of the paramount challenges of contemporary bioinformatics. A subtask of existing tree reconstruction algorithms is modeled by the Small Parsimony problem: given a tree T and an assignment of character-states to its leaves, assign states to the internal nodes of T such as to minimize the parsimony score, that is, the number of edges of T connecting nodes with different states. While this problem is polynomial-time solvable on trees, the matter is more complicated if T contains reticulate events such as hybridizations or recombinations, i.e. when T is a network. Indeed, three different versions of the parsimony score on networks have been proposed and each of them is NP-hard to decide. Existing parameterized algorithms focus on combining the number c of possible character-states with the number of reticulate events (per biconnected component).ResultsWe consider the parameter treewidth t of the underlying undirected graph of the input network, presenting dynamic programming algorithms for (slight generalizations of) all three versions of the parsimony problem on size-n networks running in times c^t {n^{O(1)}}, (3c)^t {n^{O(1)}}, and 6^{tc}n^{O(1)}, respectively. Our algorithms use a formulation of the treewidth that may facilitate formalizing treewidth-based dynamic programming algorithms on phylogenetic networks for other problems.ConclusionsOur algorithms allow the computation of the three popular parsimony scores, modeling the evolutionary development of a (multistate) character on a given phylogenetic network of low treewidth. Our results subsume and improve previously known algorithm for all three variants. While our results rely on being given a “good” tree-decomposition of the input, encouraging theoretical results as well as practical implementations producing them are publicly available. We present a reformulation of tree decompositions in terms of “agreeing trees” on the same set of nodes. As this formulation may come more natural to researchers and engineers developing algorithms for phylogenetic networks, we hope to render exploiting the input network’s treewidth as parameter more accessible to this audience.

Phylogenetic networks are the generalization of the tree models used to represent evolutionary relationship between the species. Tree models of evolutionary process are not adequate to represent the evolutionary events such as, hybridization, lateral/ horizontal gene transfer and genetic recombination. A well-formulated problem in phylogenetic networks, due to recombination, is to derive a set of input sequences on a network with minimum number of recombinations. No efficient algorithm exists for this problem as it is known to be NP-hard. Efficient solutions exist for the constrained recombination networks, where the nodes on each recombination cycles are disjoint. These solutions are based on the assumption that the ancestral sequence is known in advance. On the other hand, the more biologically realistic case is that where the ancestor sequence is not known in advance. In this paper we propose an efficient classification based method for deriving a phylogenetic network under constrained recombination without knowing the ancestral sequence.

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.

In phylogenetics, tree-based networks are used to model and visualize the evolutionary history of species where reticulate events such as horizontal gene transfer have occurred. Formally, a tree-based network N consists of a phylogenetic tree T (a rooted, binary, leaf-labeled tree) and so-called reticulation edges that span between edges of T. The network N is typically visualized by drawing T downward and planar and reticulation edges with one of several different styles. One aesthetic criteria is to minimize the number of crossings between tree edges and reticulation edges. This optimization problem has not yet been researched. We show that, if reticulation edges are drawn x-monotone, the problem is NP-complete, but fixed-parameter tractable in the number of reticulation edges. If, on the other hand, reticulation edges are drawn like “ears”, the crossing minimization problem can be solved in quadratic time.

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Phylogenetic networks model the evolutionary history of sets of organisms when events such as hybrid speciation and horizontal gene transfer occur. In spite of their widely acknowledged importance in evolutionary biology, phylogenetic networks have so far been studied mostly for specific data sets. We present a general definition of phylogenetic networks in terms of directed acyclic graphs (DAGs) and a set of conditions. Further, we distinguish between model networks and reconstructible ones and characterize the effect of extinction and taxon sampling on the reconstructibility of the network. Simulation studies are a standard technique for assessing the performance of phylogenetic methods. A main step in such studies entails quantifying the topological error between the model and inferred phylogenies. While many measures of tree topological accuracy have been proposed, none exist for phylogenetic networks. Previously, we proposed the first such measure, which applied only to a restricted class of networks. In this paper, we extend that measure to apply to all networks, and prove that it is a metric on the space of phylogenetic networks. Our results allow for the systematic study of existing network methods, and for the design of new accurate ones.

Rooted phylogenetic networks, or more generally, directed acyclic graphs (DAGs), are widely used to model species or gene relationships that traditional rooted trees cannot fully capture, especially in the presence of reticulate processes or horizontal gene transfers. Such networks or DAGs are typically inferred from observable data (e.g., genomic sequences of extant species), providing only an estimate of the true evolutionary history. However, these inferred DAGs are often complex and difficult to interpret. In particular, many contain vertices that do not serve as least common ancestors (LCAs) for any subset of the underlying genes or species, thus may lack direct support from the observable data. In contrast, LCA vertices are witnessed by historical traces justifying their existence and thus represent ancestral states substantiated by the data. To reduce unnecessary complexity and eliminate unsupported vertices, we aim to simplify a DAG to retain only LCA vertices while preserving essential evolutionary information. In this paper, we characterize LCA-relevant and lca-relevant DAGs, defined as those in which every vertex serves as an LCA (or unique LCA) for some subset of taxa. We introduce methods to identify LCAs in DAGs and efficiently transform any DAG into an LCA-relevant or lca-relevant one while preserving key structural properties of the original DAG or network. This transformation is achieved using a simple operator “⊖” that mimics vertex suppression.

Phylogenetic networks are used to display the relationship among different species whose evolution is not treelike, which is the case, for instance, in the presence of hybridization events or horizontal gene transfers. Tree inference methods such as maximum parsimony need to be modified in order to be applicable to networks. In this paper, we discuss two different definitions of maximum parsimony on networks, “hardwired” and “softwired,” and examine the complexity of computing them given a network topology and a character. By exploiting a link with the problem Multiterminal Cut, we show that computing the hardwired parsimony score for 2-state characters is polynomial-time solvable, while for characters with more states this problem becomes NP-hard but is still approximable and fixed parameter tractable in the parsimony score. On the other hand we show that, for the softwired definition, obtaining even weak approximation guarantees is already difficult for binary characters and restricted network topologies, and fixed-parameter tractable algorithms in the parsimony score are unlikely. On the positive side we show that computing the softwired parsimony score is fixed-parameter tractable in the level of the network, a natural parameter describing how tangled reticulate activity is in the network. Finally, we show that both the hardwired and the softwired parsimony scores can be computed efficiently using integer linear programming. The software has been made freely available