GTRpmix: A Linked General Time-Reversible Model for Profile Mixture Models.

Profile mixture models capture distinct biochemical constraints on the amino acid substitution process at different sites in proteins. These models feature a mixture of time-reversible models with a common matrix of exchangeabilities and distinct sets of equilibrium amino acid frequencies known as profiles. Combining the exchangeability matrix with each profile generates the matrix of instantaneous rates of amino acid exchange for that profile. Currently, empirically estimated exchangeability matrices (e.g. the LG matrix) are widely used for phylogenetic inference under profile mixture models. However, these were estimated using a single profile and are unlikely optimal for profile mixture models. Here, we describe the GTRpmix model that allows maximum likelihood estimation of a common exchangeability matrix under any profile mixture model. We show that exchangeability matrices estimated under profile mixture models differ from the LG matrix, dramatically improving model fit and topological estimation accuracy for empirical test cases. Because the GTRpmix model is computationally expensive, we provide two exchangeability matrices estimated from large concatenated phylogenomic-supermatrices to be used for phylogenetic analyses. One, called Eukaryotic Linked Mixture (ELM), is designed for phylogenetic analysis of proteins encoded by nuclear genomes of eukaryotes, and the other, Eukaryotic and Archaeal Linked mixture (EAL), for reconstructing relationships between eukaryotes and Archaea. These matrices, combined with profile mixture models, fit data better and have improved topology estimation relative to the LG matrix combined with the same mixture models. Starting with version 2.3.1, IQ-TREE2 allows users to estimate linked exchangeabilities (i.e. amino acid exchange rates) under profile mixture models.

Is Over-parameterization a Problem for Profile Mixture Models?

Biochemical constraints on the admissible amino acids at specific sites in proteins lead to heterogeneity of the amino acid substitution process over sites in alignments. It is well known that phylogenetic models of protein sequence evolution that do not account for site heterogeneity are prone to long-branch attraction (LBA) artifacts. Profile mixture models were developed to model heterogeneity of preferred amino acids at sites via a finite distribution of site classes each with a distinct set of equilibrium amino acid frequencies. However, it is unknown whether the large number of parameters in such models associated with the many amino acid frequency vectors can adversely affect tree topology estimates because of over-parameterization. Here we demonstrate theoretically that for long sequences, over-parameterization does not create problems for estimation with profile mixture models. Under mild conditions, tree, amino acid frequencies, and other model parameters converge to true values as sequence length increases, even when there are large numbers of components in the frequency profile distributions. Because large sample theory does not necessarily imply good behavior for shorter alignments we explore the performance of these models with short alignments simulated with tree topologies that are prone to LBA artifacts. We find that over-parameterization is not a problem for complex profile mixture models even when there are many amino acid frequency vectors. In fact, simple models with few site classes behave poorly. Interestingly, we also found that misspecification of the amino acid frequency vectors does not lead to increased LBA artifacts as long as the estimated cumulative distribution function of the amino acid frequencies at sites adequately approximates the true one. In contrast, misspecification of the amino acid exchangeability rates can severely negatively affect parameter estimation. Finally, we explore the effects of including in the profile mixture model an additional 'F-class' representing the overall frequencies of amino acids in the data set. Surprisingly, the F-class does not help parameter estimation significantly and can decrease the probability of correct tree estimation, depending on the scenario, even though it tends to improve likelihood scores.

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.

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.

MSCquartets 1.0: quartet methods for species trees and networks under the multispecies coalescent model in R.

SUMMARY: MSCquartets is an R package for species tree hypothesis testing, inference of species trees and inference of species networks under the Multispecies Coalescent model of incomplete lineage sorting and its network analog. Input for these analyses are collections of metric or topological locus trees which are then summarized by the quartets displayed on them. Results of hypothesis tests at user-supplied levels are displayed in a simplex plot by color-coded points. The package implements the QDC and WQDC algorithms for topological and metric species tree inference, and the NANUQ algorithm for level-1 topological species network inference, all of which give statistically consistent estimators under the model. AVAILABILITY AND IMPLEMENTATION: MSCquartets is available through the Comprehensive R Archive Network: https://CRAN.R-project.org/package=MSCquartets.

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.

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.

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

We show that many topological features of level-1 species networks are identifiable from the distribution of the gene tree quartets under the network multi-species coalescent model. In particular, every cycle of size at least 4 and every hybrid node in a cycle of size at least 5 are identifiable. This is a step toward justifying the inference of such networks which was recently implemented by Solís-Lemus and Ané. We show additionally how to compute quartet concordance factors for a network in terms of simpler networks, and explore some circumstances in which cycles of size 3 and hybrid nodes in 4-cycles can be detected.

Dimensions of Group-Based Phylogenetic Mixtures.

Mixtures of group-based Markov models of evolution correspond to joins of toric varieties. In this paper, we establish a large number of cases for which these phylogenetic join varieties realize their expected dimension, meaning that they are nondefective. Nondefectiveness is not only interesting from a geometric point-of-view, but has been used to establish combinatorial identifiability for several classes of phylogenetic mixture models. Our focus is on group-based models where the equivalence classes of identified parameters are orbits of a subgroup of the automorphism group of the abelian group defining the model. In particular, we show that for these group-based models, the variety corresponding to the mixture of r trees with n leaves is nondefective when [Formula: see text]. We also give improved bounds for claw trees and give computational evidence that 2-tree and 3-tree mixtures are nondefective for small n.

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.

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.

Testing Multispecies Coalescent Simulators Using Summary Statistics.

As genomic-scale datasets motivate research on species tree inference, simulators of the multispecies coalescent (MSC) process have become essential for the testing and evaluation of new inference methods. However, the simulators themselves must be tested to ensure that they give valid samples. This work develops methods for checking whether a collection of gene trees is in accord with the MSC model on a given species tree. When applied to well-known simulators, we find that several give flawed samples. The tests presented are capable of validating both topological and metric properties of gene tree samples, and are implemented in a freely available R package MSCsimtester so that developers and users may easily apply them.

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.

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.