A maximum pseudo-likelihood approach for estimating species trees under the coalescent model

Background  

Several phylogenetic approaches have been developed to estimate species trees from collections of gene trees. However, maximum likelihood approaches for estimating species trees under the coalescent model are limited. Although the likelihood of a species tree under the multispecies coalescent model has already been derived by Rannala and Yang, it can be shown that the maximum likelihood estimate (MLE) of the species tree (topology, branch lengths, and population sizes) from gene trees under this formula does not exist. In this paper, we develop a pseudo-likelihood function of the species tree to obtain maximum pseudo-likelihood estimates (MPE) of species trees, with branch lengths of the species tree in coalescent units.     

Discordance of species trees with their most likely gene trees

Because of the stochastic way in which lineages sort during speciation, gene trees may differ in topology from each other and from species trees. Surprisingly, assuming that genetic lineages follow a coalescent model of within-species evolution, we find that for any species tree topology with five or more species, there exist branch lengths for which gene tree discordance is so common that the most likely gene tree topology to evolve along the branches of a species tree differs from the species phylogeny. This counterintuitive result implies that in combining data on multiple loci, the straightforward procedure of using the most frequently observed gene tree topology as an estimate of the species tree topology can be asymptotically guaranteed to produce an incorrect estimate. We conclude with suggestions that can aid in overcoming this new obstacle to accurate genomic inference of species phylogenies.     

Identifying the rooted species tree from the distribution of unrooted gene trees under the coalescent

Gene trees are evolutionary trees representing the ancestry of genes sampled from multiple populations. Species trees represent populations of individuals—each with many genes—splitting into new populations or species. The coalescent process, which models ancestry of gene copies within populations, is often used to model the probability distribution of gene trees given a fixed species tree. This multispecies coalescent model provides a framework for phylogeneticists to infer species trees from gene trees using maximum likelihood or Bayesian approaches. Because the coalescent models a branching process over time, all trees are typically assumed to be rooted in this setting. Often, however, gene trees inferred by traditional phylogenetic methods are unrooted. We investigate probabilities of unrooted gene trees under the multispecies coalescent model. We show that when there are four species with one gene sampled per species, the distribution of unrooted gene tree topologies identifies the unrooted species tree topology and some, but not all, information in the species tree edges (branch lengths). The location of the root on the species tree is not identifiable in this situation. However, for 5 or more species with one gene sampled per species, we show that the distribution of unrooted gene tree topologies identifies the rooted species tree topology and all its internal branch lengths. The length of any pendant branch leading to a leaf of the species tree is also identifiable for any species from which more than one gene is sampled.     

Bayes estimation of species divergence times and ancestral population sizes using DNA sequences from multiple loci

The effective population sizes of ancestral as well as modern species are important parameters in models of population genetics and human evolution. The commonly used method for estimating ancestral population sizes, based on counting mismatches between the species tree and the inferred gene trees, is highly biased as it ignores uncertainties in gene tree reconstruction. In this article, we develop a Bayes method for simultaneous estimation of the species divergence times and current and ancestral population sizes. The method uses DNA sequence data from multiple loci and extracts information about conflicts among gene tree topologies and coalescent times to estimate ancestral population sizes. The topology of the species tree is assumed known. A Markov chain Monte Carlo algorithm is implemented to integrate over uncertain gene trees and branch lengths (or coalescence times) at each locus as well as species divergence times. The method can handle any species tree and allows different numbers of sequences at different loci. We apply the method to published noncoding DNA sequences from the human and the great apes. There are strong correlations between posterior estimates of speciation times and ancestral population sizes. With the use of an informative prior for the human-chimpanzee divergence date, the population size of the common ancestor of the two species is estimated to be approximately 20,000, with a 95% credibility interval (8000, 40,000). Our estimates, however, are affected by model assumptions as well as data quality. We suggest that reliable estimates have yet to await more data and more realistic models.     

Maximum tree: a consistent estimator of the species tree

We propose a model based approach to use multiple gene trees to estimate the species tree. The coalescent process requires that gene divergences occur earlier than species divergences when there is any polymorphism in the ancestral species. Under this scenario, speciation times are restricted to be smaller than the corresponding gene split times. The maximum tree (MT) is the tree with the largest possible speciation times in the space of species trees restricted by available gene trees. If all populations have the same population size, the MT is the maximum likelihood estimate of the species tree. It can be shown the MT is a consistent estimator of the species tree even when the MT is built upon the estimates of the true gene trees if the gene tree estimates are statistically consistent. The MT converges in probability to the true species tree at an exponential rate.     

Coalescent methods for estimating species trees from phylogenomic data

Genome-scale sequence data have become increasingly available in the phylogenetic studies for understanding the evolutionary histories of species. However, it is challenging to develop probabilistic models to account for heterogeneity of phylogenomic data. The multispecies coalescent model describes gene trees as independent random variables generated from a coalescence process occurring along the lineages of the species tree. Since the multispecies coalescent model allows gene trees to vary across genes, coalescent-based methods have been popularly used to account for heterogeneous gene trees in phylogenomic data analysis. In this paper, we summarize and evaluate the performance of coalescent-based methods for estimating species trees from genome-scale sequence data. We investigate the effects of deep coalescence and mutation on the performance of species tree estimation methods. We found that the coalescent-based methods perform well in estimating species trees for a large number of genes, regardless of the degree of deep coalescence and mutation. The performance of the coalescent methods is negatively correlated with the lengths of internal branches of the species tree.     

Discordance of species trees with their most likely gene trees: the case of five taxa

Under a coalescent model for within-species evolution, gene trees may differ from species trees to such an extent that the gene tree topology most likely to evolve along the branches of a species tree can disagree with the species tree topology. Gene tree topologies that are more likely to be produced than the topology that matches that of the species tree are termed anomalous, and the region of branch-length space that gives rise to anomalous gene trees (AGTs) is the anomaly zone. We examine the occurrence of anomalous gene trees for the case of five taxa, the smallest number of taxa for which every species tree topology has a nonempty anomaly zone. Considering all sets of branch lengths that give rise to anomalous gene trees, the largest value possible for the smallest branch length in the species tree is greater in the five-taxon case (0.1934 coalescent time units) than in the previously studied case of four taxa (0.1568). The five-taxon case demonstrates the existence of three phenomena that do not occur in the four-taxon case. First, anomalous gene trees can have the same unlabeled topology as the species tree. Second, the anomaly zone does not necessarily enclose a ball centered at the origin in branch-length space, in which all branches are short. Third, as a branch length increases, it is possible for the number of AGTs to increase rather than decrease or remain constant. These results, which help to describe how the properties of anomalous gene trees increase in complexity as the number of taxa increases, will be useful in formulating strategies for evading the problem of anomalous gene trees during species tree inference from multilocus data.     

Gene tree quality affects empirical coalescent branch length estimation

Assessing effects of gene tree error in coalescent analyses have widely ignored coalescent branch lengths (CBLs) despite their potential utility in estimating ancestral population demographics and detecting species tree anomaly zones. However, the ability of coalescent methods to obtain accurate estimates remains largely unexplored. Errors in gene trees should lead to underestimates of the true CBL, and for a given set of comparisons, longer CBLs should be more accurate. Here, we furthered our empirical understanding of how error in gene tree quality (i.e., locus informativeness and gene tree resolution) affect CBLs using four datasets comprised of ultraconserved elements (UCE) or exons for clades that exhibit wide ranges of branch lengths. For each dataset, we compared the impact of locus informativeness (assessed using number of parsimony-informative sites) and gene tree resolution on CBL estimates. Our results, in general, showed that CBLs were drastically shorter when estimates included low informative loci. Gene tree resolution also had an impact on UCE datasets, with polytomous gene trees producing longer branches than randomly resolved gene trees. However, resolution did not appear to affect CBL estimates from the more informative exon datasets. Thus, as expected, gene tree quality affects CBL estimates, though this can generally be minimized by using moderate filtering to select more informative loci and/or by allowing polytomies in gene trees. These approaches, as well as additional contributions to improve CBL estimation, should lead to CBLs that are useful for addressing evolutionary and biological questions.     

Gene tree distributions under the coalescent process

Under the coalescent model for population divergence, lineage sorting can cause considerable variability in gene trees generated from any given species tree. In this paper, we derive a method for computing the distribution of gene tree topologies given a bifurcating species tree for trees with an arbitrary number of taxa in the case that there is one gene sampled per species. Applications for gene tree distributions include determining exact probabilities of topological equivalence between gene trees and species trees and inferring species trees from multiple datasets. In addition, we examine the shapes of gene tree distributions and their sensitivity to changes in branch lengths, species tree shape, and tree size. The method for computing gene tree distributions is implemented in the computer program COAL.     

Bayesian Phylogenetic Inference using Relaxed-clocks and the Multispecies Coalescent

The multispecies coalescent (MSC) model accommodates both species divergences and within-species coalescent and provides a natural framework for phylogenetic analysis of genomic data when the gene trees vary across the genome. The MSC model implemented in the program bpp assumes a molecular clock and the Jukes–Cantor model, and is suitable for analyzing genomic data from closely related species. Here we extend our implementation to more general substitution models and relaxed clocks to allow the rate to vary among species. The MSC-with-relaxed-clock model allows the estimation of species divergence times and ancestral population sizes using genomic sequences sampled from contemporary species when the strict clock assumption is violated, and provides a simulation framework for evaluating species tree estimation methods. We conducted simulations and analyzed two real datasets to evaluate the utility of the new models. We confirm that the clock-JC model is adequate for inference of shallow trees with closely related species, but it is important to account for clock violation for distant species. Our simulation suggests that there is valuable phylogenetic information in the gene-tree branch lengths even if the molecular clock assumption is seriously violated, and the relaxed-clock models implemented in bpp are able to extract such information. Our Markov chain Monte Carlo algorithms suffer from mixing problems when used for species tree estimation under the relaxed clock and we discuss possible improvements. We conclude that the new models are currently most effective for estimating population parameters such as species divergence times when the species tree is fixed.     

Maximum likelihood models and algorithms for gene tree evolution with duplications and losses

Background

The abundance of new genomic data provides the opportunity to map the location of gene duplication and loss events on a species phylogeny. The first methods for mapping gene duplications and losses were based on a parsimony criterion, finding the mapping that minimizes the number of duplication and loss events. Probabilistic modeling of gene duplication and loss is relatively new and has largely focused on birth-death processes.

Results

We introduce a new maximum likelihood model that estimates the speciation and gene duplication and loss events in a gene tree within a species tree with branch lengths. We also provide an, in practice, efficient algorithm that computes optimal evolutionary scenarios for this model. We implemented the algorithm in the program DrML and verified its performance with empirical and simulated data.

Conclusions

In test data sets, DrML finds optimal gene duplication and loss scenarios within minutes, even when the gene trees contain sequences from several hundred species. In many cases, these optimal scenarios differ from the lca-mapping that results from a parsimony gene tree reconciliation. Thus, DrML provides a new, practical statistical framework on which to study gene duplication.

     

iGLASS: an improvement to the GLASS method for estimating species trees from gene trees

Several methods have been designed to infer species trees from gene trees while taking into account gene tree/species tree discordance. Although some of these methods provide consistent species tree topology estimates under a standard model, most either do not estimate branch lengths or are computationally slow. An exception, the GLASS method of Mossel and Roch, is consistent for the species tree topology, estimates branch lengths, and is computationally fast. However, GLASS systematically overestimates divergence times, leading to biased estimates of species tree branch lengths. By assuming a multispecies coalescent model in which multiple lineages are sampled from each of two taxa at L independent loci, we derive the distribution of the waiting time until the first interspecific coalescence occurs between the two taxa, considering all loci and measuring from the divergence time. We then use the mean of this distribution to derive a correction to the GLASS estimator of pairwise divergence times. We show that our improved estimator, which we call iGLASS, consistently estimates the divergence time between a pair of taxa as the number of loci approaches infinity, and that it is an unbiased estimator of divergence times when one lineage is sampled per taxon. We also show that many commonly used clustering methods can be combined with the iGLASS estimator of pairwise divergence times to produce a consistent estimator of the species tree topology. Through simulations, we show that iGLASS can greatly reduce the bias and mean squared error in obtaining estimates of divergence times in a species tree.     

Estimating species trees using approximate Bayesian computation

Development of methods for estimating species trees from multilocus data is a current challenge in evolutionary biology. We propose a method for estimating the species tree topology and branch lengths using approximate Bayesian computation (ABC). The method takes as data a sample of observed rooted gene tree topologies, and then iterates through the following sequence of steps: First, a randomly selected species tree is used to compute the distribution of rooted gene tree topologies. This distribution is then compared to the observed gene topology frequencies, and if the fit between the observed and the predicted distributions is close enough, the proposed species tree is retained. Repeating this many times leads to a collection of retained species trees that are then used to form the estimate of the overall species tree. We test the performance of the method, which we call ST-ABC, using both simulated and empirical data. The simulation study examines both symmetric and asymmetric species trees over a range of branch lengths and sample sizes. The results from the simulation study show that the model performs very well, giving accurate estimates for both the topology and the branch lengths across the conditions studied, and that a sample size of 25 loci appears to be adequate for the method. Further, we apply the method to two empirical cases: a 4-taxon data set for primates and a 7-taxon data set for yeast. In both cases, we find that estimates obtained with ST-ABC agree with previous studies. The method provides efficient estimation of the species tree, and does not require sequence data, but rather the observed distribution of rooted gene topologies without branch lengths. Therefore, this method is a useful alternative to other currently available methods for species tree estimation.     

The concordance of gene trees and species trees at two linked loci

The gene genealogies of two linked loci in three species are analyzed using a series of Markov chain models. We calculate the probability that the gene tree of one locus is concordant with the species tree, given that the gene tree of the other locus is concordant. We define a threshold value of the recombination rate, r*, to be the rate for which the difference between the conditional probability of concordance and its asymptotic value is reduced to 5% of the initial difference. We find that, although r* depends in a complicated way on the times of speciation and effective population sizes, it is always relatively small, <10/N4, where N4 is the effective size of the species represented by the internal branch of the species tree. Consequently, the concordance of gene trees of neutral loci with the species tree is expected to be on roughly the same length scale on the chromosome as the extent of significant linkage disequilibrium within species unless the effective size of contemporary populations is very different from the effective sizes of their ancestral populations. Both balancing selection and selective sweeps can result in much longer genomic regions having concordant gene trees.     

Accuracy and precision of species trees: effects of locus, individual, and base pair sampling on inference of species trees in lizards of the Liolaemus darwinii group (Squamata, Liolaemidae)

Molecular phylogenetics has entered a new era in which species trees are estimated from a collection of gene trees using methods that accommodate their heterogeneity and discordance with the species tree. Empirical evaluation of species trees is necessary to assess the performance (i.e., accuracy and precision) of these methods with real data, which consists of gene genealogies likely shaped by different historical and demographic processes. We analyzed 20 loci for 16 species of the South American lizards of the Liolaemus darwinii species group and reconstructed a species tree with *BEAST, then compared the performance of this method under different sampling strategies of loci, individuals, and sequence lengths. We found an increase in the accuracy and precision of species trees with the number of loci, but for any number of loci, accuracy substantially decreased only when using only one individual per species or 25% of the full sequence length (～ 147 bp). In addition, locus "informativeness" was an important factor in the accuracy/precision of species trees when using a few loci, but it became increasingly irrelevant with additional loci. Our empirical results combined with the previous simulation studies suggest that there is an optimal range of sampling effort of loci, individuals, and sequence lengths for a given speciation history and information content of the data. Future studies should be directed toward further assessment of other factors that can impact performance of species trees, including gene flow, locus "informativeness," tree shape, missing data, and errors in species delimitation.     

Long branch effects distort maximum likelihood phylogenies in simulations despite selection of the correct model

The aim of our study was to test the robustness and efficiency of maximum likelihood with respect to different long branch effects on multiple-taxon trees. We simulated data of different alignment lengths under two different 11-taxon trees and a broad range of different branch length conditions. The data were analyzed with the true model parameters as well as with estimated and incorrect assumptions about among-site rate variation. If length differences between connected branches strongly increase, tree inference with the correct likelihood model assumptions can fail. We found that incorporating invariant sites together with Γ distributed site rates in the tree reconstruction (Γ+I) increases the robustness of maximum likelihood in comparison with models using only Γ. The results show that for some topologies and branch lengths the reconstruction success of maximum likelihood under the correct model is still low for alignments with a length of 100,000 base positions. Altogether, the high confidence that is put in maximum likelihood trees is not always justified under certain tree shapes even if alignment lengths reach 100,000 base positions.     

Coalescent time distributions in trees of arbitrary size

The relationship between speciation times and the corresponding times of gene divergence is of interest in phylogenetic inference as a means of understanding the past evolutionary dynamics of populations and of estimating the timing of speciation events. It has long been recognized that gene divergence times might substantially pre-date speciation events. Although the distribution of the difference between these has previously been studied for the case of two populations, this distribution has not been explicitly computed for larger species phylogenies. Here we derive a simple method for computing this distribution for trees of arbitrary size. A two-stage procedure is proposed which (i) considers the probability distribution of the time from the speciation event at the root of the species tree to the gene coalescent time conditionally on the number of gene lineages available at the root; and (ii) calculates the probability mass function for the number of gene lineages at the root. This two-stage approach dramatically simplifies numerical analysis, because in the first step the conditional distribution does not depend on an underlying species tree, while in the second step the pattern of gene coalescence prior to the species tree root is irrelevant. In addition, the algorithm provides intuition concerning the properties of the distribution with respect to the various features of the underlying species tree. The methodology is complemented by developing probabilistic formulae and software, written in R. The method and software are tested on five-taxon species trees with varying levels of symmetry. The examples demonstrate that more symmetric species trees tend to have larger mean coalescent times and are more likely to have a unimodal gamma-like distribution with a long right tail, while asymmetric trees tend to have smaller mean coalescent times with an exponential-like distribution. In addition, species trees with longer branches generally have shorter mean coalescent times, with branches closest to the root of the tree being most influential.     

JML: testing hybridization from species trees

I introduce the software JML that tests for the presence of hybridization in multispecies sequence data sets by posterior predictive checking following Joly, McLenachan and Lockhart (2009, American Naturalist 174, e54). Although their method could potentially be applied on any data set, the lack of appropriate software made its application difficult. The software JML thus fills a need for an easy application of the method but also includes improvements such as the possibility to incorporate uncertainty in the species tree topology. The JML software uses a posterior distribution of species trees, population sizes and branch lengths to simulate replicate sequence data sets using the coalescent with no migration. A test quantity, defined as the minimum pairwise sequence distance between sequences of two species, is then evaluated on the simulated data sets and compared to the one estimated from the original data. Because the test quantity is a good predictor of hybridization events, departure from the bifurcating species tree model could be interpreted as evidence of hybridization. Software performance in terms of computing time is evaluated for several parameters. I also show an application example of the software for detecting hybridization among native diploid North American roses.     

Divergence and support among slightly suboptimal likelihood gene trees

Contemporary phylogenomic studies frequently incorporate two-step coalescent analyses wherein the first step is to infer individual-gene trees, generally using maximum-likelihood implemented in the popular programs PhyML or RAxML . Four concerns with this approach are that these programs only present a single fully resolved gene tree to the user despite potential for ambiguous support, insufficient phylogenetic signal to fully resolve each gene tree, inexact computer arithmetic affecting the reported likelihood of gene trees, and an exclusive focus on the most likely tree while ignoring trees that are only slightly suboptimal or within the error tolerance. Taken together, these four concerns are sufficient for RAxML and Phy ML users to be suspicious of the resulting (perhaps over-resolved) gene-tree topologies and (perhaps unjustifiably high) bootstrap support for individual clades. In this study, we sought to determine how frequently these concerns apply in practice to contemporary phylogenomic studies that use RAxML for gene-tree inference. We did so by re-analyzing 100 genes from each of ten studies that, taken together, are representative of many empirical phylogenomic studies. Our seven findings are as follows. First, the few search replicates that are frequently applied in phylogenomic studies are generally insufficient to find the optimal gene-tree topology. Second, there is often more topological variation among slightly suboptimal gene trees relative to the best-reported tree than can be safely ignored. Third, the Shimodaira–Hasegawa-like approximate likelihood ratio test is highly effective at identifying dubiously supported clades and outperforms the alternative approaches of relying on bootstrap support or collapsing minimum-length branches. Fourth, the bootstrap can, but rarely does, indicate high support for clades that are not supported amongst slightly suboptimal trees. Fifth, increasing the accuracy by which RA xML optimizes model-parameter values generally has a nominal effect on selection of optimal trees. Sixth, tree searches using the GTRCAT model were generally less effective at finding optimal known trees than those using the GTRGAMMA model. Seventh, choice of gene-tree sampling strategy can affect inferred coalescent branch lengths, species-tree topology and branch support.     

CALIBRATING DIVERGENCE TIMES ON SPECIES TREES VERSUS GENE TREES: IMPLICATIONS FOR SPECIATION HISTORY OF APHELOCOMA JAYS

Estimates of the timing of divergence are central to testing the underlying causes of speciation. Relaxed molecular clocks and fossil calibration have improved these estimates; however, these advances are implemented in the context of gene trees, which can overestimate divergence times. Here we couple recent innovations for dating speciation events with the analytical power of species trees, where multilocus data are considered in a coalescent context. Divergence times are estimated in the bird genus Aphelocoma to test whether speciation in these jays coincided with mountain uplift or glacial cycles. Gene trees and species trees show general agreement that diversification began in the Miocene amid mountain uplift. However, dates from the multilocus species tree are more recent, occurring predominately in the Pleistocene, consistent with theory that divergence times can be significantly overestimated with gene‐tree based approaches that do not correct for genetic divergence that predates speciation. In addition to coalescent stochasticity, Haldane's rule could account for some differences in timing estimates between mitochondrial DNA and nuclear genes. By incorporating a fossil calibration applied to the species tree, in addition to the process of gene lineage coalescence, the present approach provides a more biologically realistic framework for dating speciation events, and hence for testing the links between diversification and specific biogeographic and geologic events.