Identifying Species Network Features from Gene Tree Quartets Under the Coalescent Model

Squirrel: Reconstructing Semi-directed Phylogenetic Level-1 Networks from Four-Leaved Networks or Sequence Alignments

With the increasing availability of genomic data, biologists aim to find more accurate descriptions of evolutionary histories influenced by secondary contact, where diverging lineages reconnect before diverging again. Such reticulate evolutionary events can be more accurately represented in phylogenetic networks than in phylogenetic trees. Since the root location of phylogenetic networks cannot be inferred from biological data under several evolutionary models, we consider semi-directed (phylogenetic) networks: partially directed graphs without a root in which the directed edges represent reticulate evolutionary events. By specifying a known outgroup, the rooted topology can be recovered from such networks. We introduce the algorithm Squirrel (Semi-directed Quarnet-based Inference to Reconstruct Level-1 Networks) which constructs a semi-directed level-1 network from a full set of quarnets (four-leaf semi-directed networks). Our method also includes a heuristic to construct such a quarnet set directly from sequence alignments. We demonstrate Squirrel's performance through simulations and on real sequence data sets, the largest of which contains 29 aligned sequences close to 1.7 Mb long. The resulting networks are obtained on a standard laptop within a few minutes. Lastly, we prove that Squirrel is combinatorially consistent: given a full set of quarnets coming from a triangle-free semi-directed level-1 network, it is guaranteed to reconstruct the original network. Squirrel is implemented in Python, has an easy-to-use graphical user interface that takes sequence alignments or quarnets as input, and is freely available at https://github.com/nholtgrefe/squirrel.

Misspecification Strikes: ASTRAL can Mislead in the Presence of Hybridization, even for Nonanomalous Scenarios

ASTRAL is a powerful and widely used tool for species tree inference, known for its computational speed and robustness under incomplete lineage sorting. The method has often been used as an initial step in species network inference to provide a backbone tree structure upon which hybridization events are later added to such a tree via other methods. However, we show empirically and theoretically, that this methodology can yield flawed results. Specifically, we demonstrate that under the network multispecies coalescent model-including nonanomalous scenarios-ASTRAL can produce a tree that does not correspond to any topology displayed by the true underlying network. This finding highlights the need for caution when using ASTRAL-based inferences in suspected hybridization cases.

Leveraging graphical model techniques to study evolution on phylogenetic networks

The evolution of molecular and phenotypic traits is commonly modelled using Markov processes along a phylogeny. This phylogeny can be a tree, or a network if it includes reticulations, representing events such as hybridization or admixture. Computing the likelihood of data observed at the leaves is costly as the size and complexity of the phylogeny grows. Efficient algorithms exist for trees, but cannot be applied to networks. We show that a vast array of models for trait evolution along phylogenetic networks can be reformulated as graphical models, for which efficient belief propagation algorithms exist. We provide a brief review of belief propagation on general graphical models, then focus on linear Gaussian models for continuous traits. We show how belief propagation techniques can be applied for exact or approximate (but more scalable) likelihood and gradient calculations, and prove novel results for efficient parameter inference of some models. We highlight the possible fruitful interactions between graphical models and phylogenetic methods. For example, approximate likelihood approaches have the potential to greatly reduce computational costs for phylogenies with reticulations.This article is part of the theme issue '"A mathematical theory of evolution": phylogenetic models dating back 100 years'.

Complex Hybridization in a Clade of Polytypic Salamanders (Plethodontidae: Desmognathus) Uncovered by Estimating Higher-Level Phylogenetic Networks

Reticulation between radiating lineages is a common feature of diversification. We examine these phenomena in the Pisgah clade of Desmognathus salamanders from the southern Appalachian Mountains of the eastern United States. The group contains 4-7 species exhibiting 2 discrete phenotypes, aquatic "shovel-nosed" and semi-aquatic "black-bellied" forms. These ecomorphologies are ancient and have apparently been transmitted repeatedly between lineages through introgression. Geographically proximate populations of both phenotypes exhibit admixture, and at least 2 black-bellied lineages have been produced via reticulations between shovel-nosed parentals, suggesting potential hybrid speciation dynamics. However, computational constraints currently limit our ability to reconstruct network radiations from gene-tree data. Available methods are limited to level-1 networks wherein reticulations do not share edges, and higher-level networks may be non-identifiable in many cases. We present a heuristic approach to recover information from higher-level networks across a range of potentially identifiable empirical scenarios, supported by theory and simulation. When extrinsic information indicates the location and direction of reticulations, our method can successfully estimate a reduced possible set of nonlevel-1 networks. Phylogenomic data support a single backbone topology with up to 5 overlapping hybrid edges in the Pisgah clade. These results suggest an unusual mechanism of ecomorphological hybrid speciation, wherein a binary threshold trait causes some hybrid populations to shift between microhabitat niches, promoting ecological divergence between sympatric hybrids and parentals. This contrasts with other well-known systems in which hybrids exhibit intermediate, novel, or transgressive phenotypes. The genetic basis of these phenotypes is unclear and further data are needed to clarify the evolutionary basis of morphological changes with ecological consequences.

Advances in Estimating Level-1 Phylogenetic Networks from Unrooted SNPs

We address the problem of how to estimate a phylogenetic network when given single-nucleotide polymorphisms (i.e., SNPs, or bi-allelic markers that have evolved under the infinite sites assumption). We focus on level-1 phylogenetic networks (i.e., networks where the cycles are node-disjoint), since more complex networks are unidentifiable. We provide a polynomial time quartet-based method that we prove correct for reconstructing the semi-directed level-1 phylogenetic network N, if we are given a set of SNPs that covers all the bipartitions of N, even if the ancestral state is not known, provided that the cycles are of length at least 5; we also prove that an algorithm developed by Dan Gusfield in the Journal of Computer and System Sciences in 2005 correctly recovers semi-directed level-1 phylogenetic networks in polynomial time in this case. We present a stochastic model for DNA evolution, and we prove that the two methods (our quartet-based method and Gusfield's method) are statistically consistent estimators of the semi-directed level-1 phylogenetic network. For the case of multi-state homoplasy-free characters, we prove that our quartet-based method correctly constructs semi-directed level-1 networks under the required conditions (all cycles of length at least five), while Gusfield's algorithm cannot be used in that case. These results assume that we have access to an oracle for indicating which sites in the DNA alignment are homoplasy-free, and we show that the methods are robust, under some conditions, to oracle errors.

TINNiK: inference of the tree of blobs of a species network under the coalescent model

The tree of blobs of a species network shows only the tree-like aspects of relationships of taxa on a network, omitting information on network substructures where hybridization or other types of lateral transfer of genetic information occur. By isolating such regions of a network, inference of the tree of blobs can serve as a starting point for a more detailed investigation, or indicate the limit of what may be inferrable without additional assumptions. Building on our theoretical work on the identifiability of the tree of blobs from gene quartet distributions under the Network Multispecies Coalescent model, we develop an algorithm, TINNiK, for statistically consistent tree of blobs inference. We provide examples of its application to both simulated and empirical datasets, utilizing an implementation in the MSCquartets 2.0 R package.

Identifiability of Level-1 Species Networks from Gene Tree Quartets

When hybridization or other forms of lateral gene transfer have occurred, evolutionary relationships of species are better represented by phylogenetic networks than by trees. While inference of such networks remains challenging, several recently proposed methods are based on quartet concordance factors-the probabilities that a tree relating a gene sampled from the species displays the possible 4-taxon relationships. Building on earlier results, we investigate what level-1 network features are identifiable from concordance factors under the network multispecies coalescent model. We obtain results on both topological features of the network, and numerical parameters, uncovering a number of failures of identifiability related to 3-cycles in the network. Addressing these identifiability issues is essential for designing statistically consistent inference methods.

TINNiK: Inference of the Tree of Blobs of a Species Network Under the Coalescent

The tree of blobs of a species network shows only the tree-like aspects of relationships of taxa on a network, omitting information on network substructures where hybridization or other types of lateral transfer of genetic information occur. By isolating such regions of a network, inference of the tree of blobs can serve as a starting point for a more detailed investigation, or indicate the limit of what may be inferrable without additional assumptions. Building on our theoretical work on the identifiability of the tree of blobs from gene quartet distributions under the Network Multispecies Coalescent model, we develop an algorithm, TINNiK, for statistically consistent tree of blobs inference. We provide examples of its application to both simulated and empirical datasets, utilizing an implementation in the MSCquartets 2.0 R package.

Anomalous networks under the multispecies coalescent: theory and prevalence

Reticulations in a phylogenetic network represent processes such as gene flow, admixture, recombination and hybrid speciation. Extending definitions from the tree setting, an anomalous network is one in which some unrooted tree topology displayed in the network appears in gene trees with a lower frequency than a tree not displayed in the network. We investigate anomalous networks under the Network Multispecies Coalescent Model with possible correlated inheritance at reticulations. Focusing on subsets of 4 taxa, we describe a new algorithm to calculate quartet concordance factors on networks of any level, faster than previous algorithms because of its focus on 4 taxa. We then study topological properties required for a 4-taxon network to be anomalous, uncovering the key role of [Formula: see text]-cycles: cycles of 3 edges parent to a sister group of 2 taxa. Under the model of common inheritance, that is, when each gene tree coalesces within a species tree displayed in the network, we prove that 4-taxon networks are never anomalous. Under independent and various levels of correlated inheritance, we use simulations under realistic parameters to quantify the prevalence of anomalous 4-taxon networks, finding that truly anomalous networks are rare. At the same time, however, we find a significant fraction of networks close enough to the anomaly zone to appear anomalous, when considering the quartet concordance factors observed from a few hundred genes. These apparent anomalies may challenge network inference methods.

IDENTIFIABILITY OF LEVEL-1 SPECIES NETWORKS FROM GENE TREE QUARTETS

When hybridization or other forms of lateral gene transfer have occurred, evolutionary relationships of species are better represented by phylogenetic networks than by trees. While inference of such networks remains challenging, several recently proposed methods are based on quartet concordance factors - the probabilities that a tree relating a gene sampled from the species displays the possible 4-taxon relationships. Building on earlier results, we investigate what level-1 network features are identifiable from concordance factors under the network multispecies coalescent model. We obtain results on both topological features of the network, and numerical parameters, uncovering a number of failures of identifiability related to 3-cycles in the network.

Summary Tests of Introgression Are Highly Sensitive to Rate Variation Across Lineages

The evolutionary implications and frequency of hybridization and introgression are increasingly being recognized across the tree of life. To detect hybridization from multi-locus and genome-wide sequence data, a popular class of methods are based on summary statistics from subsets of 3 or 4 taxa. However, these methods often carry the assumption of a constant substitution rate across lineages and genes, which is commonly violated in many groups. In this work, we quantify the effects of rate variation on the D test (also known as ABBA-BABA test), the D3 test, and HyDe. All 3 tests are used widely across a range of taxonomic groups, in part because they are very fast to compute. We consider rate variation across species lineages, across genes, their lineage-by-gene interaction, and rate variation across gene-tree edges. We simulated species networks according to a birth-death-hybridization process, so as to capture a range of realistic species phylogenies. For all 3 methods tested, we found a marked increase in the false discovery of reticulation (type-1 error rate) when there is rate variation across species lineages. The D3 test was the most sensitive, with around 80% type-1 error, such that D3 appears to more sensitive to a departure from the clock than to the presence of reticulation. For all 3 tests, the power to detect hybridization events decreased as the number of hybridization events increased, indicating that multiple hybridization events can obscure one another if they occur within a small subset of taxa. Our study highlights the need to consider rate variation when using site-based summary statistics, and points to the advantages of methods that do not require assumptions on evolutionary rates across lineages or across genes.

Exploring spaces of semi-directed level-1 networks

Semi-directed phylogenetic networks have recently emerged as a class of phylogenetic networks sitting between rooted (directed) and unrooted (undirected) phylogenetic networks as they contain both directed as well as undirected edges. While various spaces of rooted phylogenetic networks and unrooted phylogenetic networks have been analyzed in recent years and several rearrangement moves to traverse these spaces have been introduced, little is known about spaces of semi-directed phylogenetic networks. Here, we propose a simple rearrangement move for semi-directed phylogenetic networks, called cut edge transfer (CET), and show that the space of semi-directed level-1 networks with precisely k reticulations is connected under CET. This level-1 space is currently the predominantly used search space for most algorithms that reconstruct semi-directed phylogenetic networks. Our results imply that every semi-directed level-1 network with a fixed number of reticulations and leaf set can be reached from any other such network by a sequence of CETs. By introducing two additional moves, R[Formula: see text] and R[Formula: see text], that allow for the addition and deletion, respectively, of a reticulation, we then establish connectedness for the space of all semi-directed level-1 networks on a fixed leaf set. As a byproduct of our results for semi-directed phylogenetic networks, we also show that the space of rooted level-1 networks with a fixed number of reticulations and leaf set is connected under CET, when translated into the rooted setting.

ANOMALOUS NETWORKS UNDER THE MULTISPECIES COALESCENT: THEORY AND PREVALENCE

Reticulations in a phylogenetic network represent processes such as gene flow, admixture, recombination and hybrid speciation. Extending definitions from the tree setting, an anomalous network is one in which some unrooted tree topology displayed in the network appears in gene trees with a lower frequency than a tree not displayed in the network. We investigate anomalous networks under the Network Multispecies Coalescent Model with possible correlated inheritance at reticulations. Focusing on subsets of 4 taxa, we describe a new algorithm to calculate quartet concordance factors on networks of any level, faster than previous algorithms because of its focus on 4 taxa. We then study topological properties required for a 4-taxon network to be anomalous, uncovering the key role of 32-cycles: cycles of 3 edges parent to a sister group of 2 taxa. Under the model of common inheritance, that is, when each gene tree coalesces within a species tree displayed in the network, we prove that 4-taxon networks are never anomalous. Under independent and various levels of correlated inheritance, we use simulations under realistic parameters to quantify the prevalence of anomalous 4-taxon networks, finding that truly anomalous networks are rare. At the same time, however, we find a significant fraction of networks close enough to the anomaly zone to appear anomalous, when considering the quartet concordance factors observed from a few hundred genes. These apparent anomalies may challenge network inference methods.

Identifiability of local and global features of phylogenetic networks from average distances

Phylogenetic networks extend phylogenetic trees to model non-vertical inheritance, by which a lineage inherits material from multiple parents. The computational complexity of estimating phylogenetic networks from genome-wide data with likelihood-based methods limits the size of networks that can be handled. Methods based on pairwise distances could offer faster alternatives. We study here the information that average pairwise distances contain on the underlying phylogenetic network, by characterizing local and global features that can or cannot be identified. For general networks, we clarify that the root and edge lengths adjacent to reticulations are not identifiable, and then focus on the class of zipped-up semidirected networks. We provide a criterion to swap subgraphs locally, such as 3-cycles, resulting in indistinguishable networks. We propose the "distance split tree", which can be constructed from pairwise distances, and prove that it is a refinement of the network's tree of blobs, capturing the tree-like features of the network. For level-1 networks, this distance split tree is equal to the tree of blobs refined to separate polytomies from blobs, and we prove that the mixed representation of the network is identifiable. The information loss is localized around 4-cycles, for which the placement of the reticulation is unidentifiable. The mixed representation combines split edges for 4-cycles, regular tree and hybrid edges from the semidirected network, and edge parameters that encode all information identifiable from average pairwise distances.

The tree of blobs of a species network: identifiability under the coalescent

Inference of species networks from genomic data under the Network Multispecies Coalescent Model is currently severely limited by heavy computational demands. It also remains unclear how complicated networks can be for consistent inference to be possible. As a step toward inferring a general species network, this work considers its tree of blobs, in which non-cut edges are contracted to nodes, so only tree-like relationships between the taxa are shown. An identifiability theorem, that most features of the unrooted tree of blobs can be determined from the distribution of gene quartet topologies, is established. This depends upon an analysis of gene quartet concordance factors under the model, together with a new combinatorial inference rule. The arguments for this theoretical result suggest a practical algorithm for tree of blobs inference, to be fully developed in a subsequent work.

Classes of explicit phylogenetic networks and their biological and mathematical significance

The evolutionary relationships among organisms have traditionally been represented using rooted phylogenetic trees. However, due to reticulate processes such as hybridization or lateral gene transfer, evolution cannot always be adequately represented by a phylogenetic tree, and rooted phylogenetic networks that describe such complex processes have been introduced as a generalization of rooted phylogenetic trees. In fact, estimating rooted phylogenetic networks from genomic sequence data and analyzing their structural properties is one of the most important tasks in contemporary phylogenetics. Over the last two decades, several subclasses of rooted phylogenetic networks (characterized by certain structural constraints) have been introduced in the literature, either to model specific biological phenomena or to enable tractable mathematical and computational analyses. In the present manuscript, we provide a thorough review of these network classes, as well as provide a biological interpretation of the structural constraints underlying these networks where possible. In addition, we discuss how imposing structural constraints on the network topology can be used to address the scalability and identifiability challenges faced in the estimation of phylogenetic networks from empirical data.

Identifiability of species network topologies from genomic sequences using the logDet distance

Inference of network-like evolutionary relationships between species from genomic data must address the interwoven signals from both gene flow and incomplete lineage sorting. The heavy computational demands of standard approaches to this problem severely limit the size of datasets that may be analyzed, in both the number of species and the number of genetic loci. Here we provide a theoretical pointer to more efficient methods, by showing that logDet distances computed from genomic-scale sequences retain sufficient information to recover network relationships in the level-1 ultrametric case. This result is obtained under the Network Multispecies Coalescent model combined with a mixture of General Time-Reversible sequence evolution models across individual gene trees. It applies to both unlinked site data, such as for SNPs, and to sequence data in which many contiguous sites may have evolved on a common tree, such as concatenated gene sequences. Thus under standard stochastic models statistically justifiable inference of network relationships from sequences can be accomplished without consideration of individual genes or gene trees.

Distinguishing level-1 phylogenetic networks on the basis of data generated by Markov processes

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.

Gene tree discord, simplex plots, and statistical tests under the coalescent

A simple graphical device, the simplex plot of quartet concordance factors, is introduced to aid in the exploration of a collection of gene trees on a common set of taxa. A single plot summarizes all gene tree discord, and allows for visual comparison to the expected discord from the multispecies coalescent model (MSC) of incomplete lineage sorting on a species tree. A formal statistical procedure is described that can quantify the deviation from expectation for each subset of four taxa, suggesting when the data is not in accord with the MSC, and thus that either gene tree inference error is substantial or a more complex model such as that on a network may be required. If the collection of gene trees is in accord with the MSC, the plots reveal when substantial incomplete lineage sorting is present. Applications to both simulated and empirical multilocus data sets illustrate the insights provided.

NANUQ: a method for inferring species networks from gene trees under the coalescent model

Species networks generalize the notion of species trees to allow for hybridization or other lateral gene transfer. Under the network multispecies coalescent model, individual gene trees arising from a network can have any topology, but arise with frequencies dependent on the network structure and numerical parameters. We propose a new algorithm for statistical inference of a level-1 species network under this model, from data consisting of gene tree topologies, and provide the theoretical justification for it. The algorithm is based on an analysis of quartets displayed on gene trees, combining several statistical hypothesis tests with combinatorial ideas such as a quartet-based intertaxon distance appropriate to networks, the NeighborNet algorithm for circular split systems, and the Circular Network algorithm for constructing a splits graph.

Modeling Hybridization Under the Network Multispecies Coalescent

Simultaneously modeling hybridization and the multispecies coalescent is becoming increasingly common, and inference of species networks in this context is now implemented in several software packages. This article addresses some of the conceptual issues and decisions to be made in this modeling, including whether or not to use branch lengths and issues with model identifiability. This article is based on a talk given at a Spotlight Session at Evolution 2017 meeting in Portland, Oregon. This session included several talks about modeling hybridization and gene flow in the presence of incomplete lineage sorting. Other talks given at this meeting are also included in this special issue of Systematic Biology.