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Abstract 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. 

Abstract We consider the evolution of phylogenetic gene trees along phylogenetic species networks, according to the network multispecies coalescent process, and introduce a new network coalescent model with correlated inheritance of gene flow. This model generalizes two traditional versions of the network coalescent: with independent or common inheritance. At each reticulation, multiple lineages of a given locus are inherited from parental populations chosen at random, either independently across lineages or with positive correlation according to a Dirichlet process. This process may account for locus-specific probabilities of inheritance, for example. We implemented the simulation of gene trees under these network coalescent models in the Julia package PhyloCoalSimulations, which depends on PhyloNetworks and its powerful network manipulation tools. Input species phylogenies can be read in extended Newick format, either in numbers of generations or in coalescent units. Simulated gene trees can be written in Newick format, and in a way that preserves information about their embedding within the species network. This embedding can be used for downstream purposes, such as to simulate species-specific processes like rate variation across species, or for other scenarios as illustrated in this note. This package should be useful for simulation studies and simulation-based inference methods. The software is available open source with documentation and a tutorial at https://github.com/cecileane/PhyloCoalSimulations.jl. 

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. 

Within-species trait variation may be the result of genetic variation, environmental variation, or measurement error, for example. In phylogenetic comparative studies, failing to account for within-species variation has many adverse effects, such as increased error in testing hypotheses about evolutionary correlations, biased estimates of evolutionary rates, and inaccurate inference of the mode of evolution. These adverse effects were demonstrated in studies that considered a tree-like underlying phylogeny. Comparative methods on phylogenetic networks are still in their infancy. The impact of within-species variation on network-based methods has not been studied. Here, we introduce a phylogenetic linear model in which the phylogeny can be a network to account for within-species variation in the continuous response trait assuming equal within-species variances across species. We show how inference based on the individual values can be reduced to a problem using species-level summaries, even when the within-species variance is estimated. Our method performs well under various simulation settings and is robust when within-species variances are unequal across species. When phenotypic (within-species) correlations differ from evolutionary (between-species) correlations, estimates of evolutionary coefficients are pulled towards the phenotypic coefficients for all methods we tested. Also, evolutionary rates are either underestimated or overestimated, depending on the mismatch between phenotypic and evolutionary relationships. We applied our method to morphological and geographical data from Polemonium. We find a strong negative correlation of leaflet size with elevation, despite a positive correlation within species. Our method can explore the role of gene flow in trait evolution by comparing the fit of a network to that of a tree. We find marginal evidence for leaflet size being affected by gene flow and support for previous observations on the challenges of using individual continuous traits to infer inheritance weights at reticulations. Our method is freely available in the Julia package PhyloNetworks. 

Rooted species trees are used in several downstream applications of phylogenetics. Most species tree estimation methods produce unrooted trees and additional methods are then used to root these unrooted trees. Recently, Quintet Rooting (QR) (Tabatabaee et al., ISMB and Bioinformatics 2022), a polynomial-time method for rooting an unrooted species tree given unrooted gene trees under the multispecies coalescent, was introduced. QR, which is based on a proof of identifiability of rooted 5-taxon trees in the presence of incomplete lineage sorting, was shown to have good accuracy, improving over other methods for rooting species trees when incomplete lineage sorting was the only cause of gene tree discordance, except when gene tree estimation error was very high. However, the statistical consistency of QR was left as an open question. Here, we present QR-STAR, a polynomial-time variant of QR that has an additional step for determining the rooted shape of each quintet tree. We prove that QR-STAR is statistically consistent under the multispecies coalescent model, and our simulation study shows that QR-STAR matches or improves on the accuracy of QR. QR-STAR is available in open source form at https://github.com/ytabatabaee/Quintet-Rooting.