Calibrated tree priors for relaxed phylogenetics and divergence time estimation

The use of fossil evidence to calibrate divergence time estimation has a long history. More recently, Bayesian Markov chain Monte Carlo has become the dominant method of divergence time estimation, and fossil evidence has been reinterpreted as the specification of prior distributions on the divergence times of calibration nodes. These so-called "soft calibrations" have become widely used but the statistical properties of calibrated tree priors in a Bayesian setting hashave not been carefully investigated. Here, we clarify that calibration densities, such as those defined in BEAST 1.5, do not represent the marginal prior distribution of the calibration node. We illustrate this with a number of analytical results on small trees. We also describe an alternative construction for a calibrated Yule prior on trees that allows direct specification of the marginal prior distribution of the calibrated divergence time, with or without the restriction of monophyly. This method requires the computation of the Yule prior conditional on the height of the divergence being calibrated. Unfortunately, a practical solution for multiple calibrations remains elusive. Our results suggest that direct estimation of the prior induced by specifying multiple calibration densities should be a prerequisite of any divergence time dating analysis.     

Inferring speciation times under an episodic molecular clock

We extend our recently developed Markov chain Monte Carlo algorithm for Bayesian estimation of species divergence times to allow variable evolutionary rates among lineages. The method can use heterogeneous data from multiple gene loci and accommodate multiple fossil calibrations. Uncertainties in fossil calibrations are described using flexible statistical distributions. The prior for divergence times for nodes lacking fossil calibrations is specified by use of a birth-death process with species sampling. The prior for lineage-specific substitution rates is specified using either a model with autocorrelated rates among adjacent lineages (based on a geometric Brownian motion model of rate drift) or a model with independent rates among lineages specified by a log-normal probability distribution. We develop an infinite-sites theory, which predicts that when the amount of sequence data approaches infinity, the width of the posterior credibility interval and the posterior mean of divergence times form a perfect linear relationship, with the slope indicating uncertainties in time estimates that cannot be reduced by sequence data alone. Simulations are used to study the influence of among-lineage rate variation and the number of loci sampled on the uncertainty of divergence time estimates. The analysis suggests that posterior time estimates typically involve considerable uncertainties even with an infinite amount of sequence data, and that the reliability and precision of fossil calibrations are critically important to divergence time estimation. We apply our new algorithms to two empirical data sets and compare the results with those obtained in previous Bayesian and likelihood analyses. The results demonstrate the utility of our new algorithms.     

A hierarchical Bayesian model for calibrating estimates of species divergence times

In Bayesian divergence time estimation methods, incorporating calibrating information from the fossil record is commonly done by assigning prior densities to ancestral nodes in the tree. Calibration prior densities are typically parametric distributions offset by minimum age estimates provided by the fossil record. Specification of the parameters of calibration densities requires the user to quantify his or her prior knowledge of the age of the ancestral node relative to the age of its calibrating fossil. The values of these parameters can, potentially, result in biased estimates of node ages if they lead to overly informative prior distributions. Accordingly, determining parameter values that lead to adequate prior densities is not straightforward. In this study, I present a hierarchical Bayesian model for calibrating divergence time analyses with multiple fossil age constraints. This approach applies a Dirichlet process prior as a hyperprior on the parameters of calibration prior densities. Specifically, this model assumes that the rate parameters of exponential prior distributions on calibrated nodes are distributed according to a Dirichlet process, whereby the rate parameters are clustered into distinct parameter categories. Both simulated and biological data are analyzed to evaluate the performance of the Dirichlet process hyperprior. Compared with fixed exponential prior densities, the hierarchical Bayesian approach results in more accurate and precise estimates of internal node ages. When this hyperprior is applied using Markov chain Monte Carlo methods, the ages of calibrated nodes are sampled from mixtures of exponential distributions and uncertainty in the values of calibration density parameters is taken into account.     

A Simple Method for Estimating Informative Node Age Priors for the Fossil Calibration of Molecular Divergence Time Analyses

Molecular divergence time analyses often rely on the age of fossil lineages to calibrate node age estimates. Most divergence time analyses are now performed in a Bayesian framework, where fossil calibrations are incorporated as parametric prior probabilities on node ages. It is widely accepted that an ideal parameterization of such node age prior probabilities should be based on a comprehensive analysis of the fossil record of the clade of interest, but there is currently no generally applicable approach for calculating such informative priors. We provide here a simple and easily implemented method that employs fossil data to estimate the likely amount of missing history prior to the oldest fossil occurrence of a clade, which can be used to fit an informative parametric prior probability distribution on a node age. Specifically, our method uses the extant diversity and the stratigraphic distribution of fossil lineages confidently assigned to a clade to fit a branching model of lineage diversification. Conditioning this on a simple model of fossil preservation, we estimate the likely amount of missing history prior to the oldest fossil occurrence of a clade. The likelihood surface of missing history can then be translated into a parametric prior probability distribution on the age of the clade of interest. We show that the method performs well with simulated fossil distribution data, but that the likelihood surface of missing history can at times be too complex for the distribution-fitting algorithm employed by our software tool. An empirical example of the application of our method is performed to estimate echinoid node ages. A simulation-based sensitivity analysis using the echinoid data set shows that node age prior distributions estimated under poor preservation rates are significantly less informative than those estimated under high preservation rates.     

Divergence time uncertainty and historical biogeography reconstruction – an example from Urophylleae (Rubiaceae)

Aim When hypotheses of historical biogeography are evaluated, age estimates of individual nodes in a phylogeny often have a direct impact on what explanation is concluded to be most likely. Confidence intervals of estimated divergence times obtained in molecular dating analyses are usually very large, but the uncertainty is rarely incorporated in biogeographical analyses. The aim of this study is to use the group Urophylleae, which has a disjunct pantropical distribution, to explore how the uncertainty in estimated divergence times affects conclusions in biogeographical analysis. Two hypotheses are evaluated: (1) long‐distance dispersal from Africa to Asia and the Neotropics, and (2) a continuous distribution in the boreotropics, probably involving migration across the North Atlantic Land Bridge, followed by isolation in equatorial refugia. Location Tropical and subtropical Asia, tropical Africa, and central and southern tropical America. Methods This study uses parsimony and Bayesian phylogenetic analyses of chloroplast DNA and nuclear ribosomal DNA data from 56 ingroup species, beast molecular dating and a Bayesian approach to dispersal–vicariance analysis (Bayes‐DIVA) to reconstruct the ancestral area of the group, and the dispersal–extinction–cladogenesis method to test biogeographical hypotheses. Results When the two models of geographic range evolution were compared using the maximum likelihood (ML) tree with mean estimates of divergence times, boreotropical migration was indicated to be much more likely than long‐distance dispersal. Analyses of a large sample of dated phylogenies did, however, show that this result was not consistent. The age estimate of one specific node had a major impact on likelihood values and on which model performed best. The results show that boreotropical migration provides a slightly better explanation of the geographical distribution patterns of extant Urophylleae than long‐distance dispersal. Main conclusions This study shows that results from biogeographical analyses based on single phylogenetic trees, such as a ML or consensus tree, can be misleading, and that it may be very important to take the uncertainty in age estimates into account. Methods that account for the uncertainty in topology, branch lengths and estimated divergence times are not commonly used in biogeographical inference today but should definitely be preferred in order to avoid unwarranted conclusions.     

Testing the impact of calibration on molecular divergence times using a fossil-rich group: the case of Nothofagus (Fagales)

Although temporal calibration is widely recognized as critical for obtaining accurate divergence-time estimates using molecular dating methods, few studies have evaluated the variation resulting from different calibration strategies. Depending on the information available, researchers have often used primary calibrations from the fossil record or secondary calibrations from previous molecular dating studies. In analyses of flowering plants, primary calibration data can be obtained from macro- and mesofossils (e.g., leaves, flowers, and fruits) or microfossils (e.g., pollen). Fossil data can vary substantially in accuracy and precision, presenting a difficult choice when selecting appropriate calibrations. Here, we test the impact of eight plausible calibration scenarios for Nothofagus (Nothofagaceae, Fagales), a plant genus with a particularly rich and well-studied fossil record. To do so, we reviewed the phylogenetic placement and geochronology of 38 fossil taxa of Nothofagus and other Fagales, and we identified minimum age constraints for up to 18 nodes of the phylogeny of Fagales. Molecular dating analyses were conducted for each scenario using maximum likelihood (RAxML + r8s) and Bayesian (BEAST) approaches on sequence data from six regions of the chloroplast and nuclear genomes. Using either ingroup or outgroup constraints, or both, led to similar age estimates, except near strongly influential calibration nodes. Using "early but risky" fossil constraints in addition to "safe but late" constraints, or using assumptions of vicariance instead of fossil constraints, led to older age estimates. In contrast, using secondary calibration points yielded drastically younger age estimates. This empirical study highlights the critical influence of calibration on molecular dating analyses. Even in a best-case situation, with many thoroughly vetted fossils available, substantial uncertainties can remain in the estimates of divergence times. For example, our estimates for the crown group age of Nothofagus varied from 13 to 113 Ma across our full range of calibration scenarios. We suggest that increased background research should be made at all stages of the calibration process to reduce errors wherever possible, from verifying the geochronological data on the fossils to critical reassessment of their phylogenetic position.     

Change in community phylogenetic structure during tropical forest succession: evidence from New Guinea

Ecologists have recently interpreted patterns of phylogenetic distance among coexisting species as indicative of processes affecting community assembly during forest succession. We investigated plant community phylogenetic structure along a successional gradient in New Guinean lowland rain forest. We surveyed all trees with diameter at breast height ≥ 5 cm in nineteen 0.25 ha plots representing younger secondary, older secondary, and primary forest. We estimated plant community phylogeny from rbcL gene sequences to quantify change in phylogenetic structure during succession. Mean phylogenetic distance among co‐occurring trees increased with total basal area per plot, a proxy for forest age. Significant phylogenetic clustering was detected in secondary forest whereas primary forest was significantly over‐dispersed relative to null expectations. We examined the sensitivity of these patterns to various methods of branch length estimation and phylogenetic uncertainty. Power to detect community phylogenetic patterns when equal branch lengths were assumed was weak in comparison to direct molecular and time‐calibrated measures of divergence. Inferred change during forest succession was also robust to phylogenetic uncertainty so long as temporal information was incorporated in estimates of divergence. The observed patterns are consistent with processes of environmental filtering during tropical forest succession giving way to other processes in primary forests including density‐dependence.     

The impact of fossils and taxon sampling on ancient molecular dating analyses

The number and complexity of molecular dating studies has increased over the past decade. Along with a broadening acceptance of their utility has come significant controversy over the methods and models that are appropriate, as well as the accuracy of the estimates yielded by molecular clock analyses. Radically different age estimates have been published for the same divergences from analyses of different datasets with different fossil constraints obtained with different methods, and the underlying explanation for these differences is often unclear. Here we utilize two previously published datasets to examine the effect of fossil calibrations and taxon sampling on the age estimates for two deep eukaryote divergences in an attempt to discern the relative impact of these factors. Penalized likelihood, non-parametric rate smoothing, and Bayesian methods were utilized to generate age estimates for the origin of the Metazoa from a 7-gene dataset and for the divergence of Eukaryotes from a 129-gene dataset. From these analyses, it is clear that the fossil calibrations chosen and the method for applying constraints to these nodes have a large impact on age estimates, while the degree of taxon sampling within a dataset is less important in terms of the resulting age estimates. Concerns and recommendations for addressing these two factors when initiating a dating analysis are discussed.     

Integrating fossil preservation biases in the selection of calibrations for molecular divergence time estimation

The selection of fossil data to use as calibration age priors in molecular divergence time estimates inherently links neontological methods with paleontological theory. However, few neontological studies have taken into account the possibility of a taphonomic bias in the fossil record when developing approaches to fossil calibration selection. The Sppil-Rongis effect may bias the first appearance of a lineage toward the recent causing most objective calibration selection approaches to erroneously exclude appropriate calibrations or to incorporate multiple calibrations that are too young to accurately represent the divergence times of target lineages. Using turtles as a case study, we develop a Bayesian extension to the fossil selection approach developed by Marshall (2008. A simple method for bracketing absolute divergence times on molecular phylogenies using multiple fossil calibrations points. Am. Nat. 171:726-742) that takes into account this taphonomic bias. Our method has the advantage of identifying calibrations that may bias age estimates to be too recent while incorporating uncertainty in phylogenetic parameter estimates such as tree topology and branch lengths. Additionally, this method is easily adapted to assess the consistency of potential calibrations to any one calibration in the candidate pool.     

Taking into account phylogenetic and divergence‐time uncertainty in a parametric biogeographical analysis of the Northern Hemisphere plant clade Caprifolieae

Aim To examine the biogeographical history of the angiosperm clade Caprifolieae (Caprifoliaceae) using parametric biogeographical reconstruction methods. The existing parametric method was extended to evaluate biogeographical reconstructions over the distribution of dated phylogenies. This method provides a framework for reconstructing large‐scale biogeography with parametric methods, while accounting for uncertainty in the phylogenetic relationships and divergence time. Location Asia, Europe and North America. Methods The biogeographical history of the major lineages of Caprifolieae was reconstructed over the posterior distribution of dated trees generated from Bayesian divergence‐time analyses. Results from a model with no geological constraints were compared with those from one where movement is disallowed across the North Atlantic after the Eocene. The most plausible scenarios were segregated at each node to test whether particular scenarios were reconstructed for particular divergence times. The parametric biogeographical method was also extended to estimate connectivity between areas so that the probability of dispersal between the major areas of the Northern Hemisphere could be explored. Results Phylogenetic results for Caprifolieae agreed with previous estimates using smaller sampling, but uncertainty remained despite efforts to resolve the relationships of the four genera within this clade using multiple markers. In addition to topological uncertainty, there were few fossils available for calibrations, resulting in large confidence intervals for divergence times. Divergence‐time analyses put the diversification of Caprifolieae at between 36 and 51 Ma and showed that Caprifolieae probably originated in Asia, with multiple movements into Europe and western and eastern North America. Main conclusions Newly developed parametric methods for biogeographical reconstruction incorporate more data and better models. Here, the parametric biogeographical reconstruction method has been extended to allow for topological and divergence‐time uncertainty. The analyses of Caprifolieae demonstrated that biogeographical hypotheses can be explored even where there are large confidence intervals on divergence times and uncertainty in topology. These results add to the growing evidence that Asia was an important source of Northern Hemisphere diversity throughout the Cenozoic.     

Bayesian estimation of species divergence times under a molecular clock using multiple fossil calibrations with soft bounds

We implement a Bayesian Markov chain Monte Carlo algorithm for estimating species divergence times that uses heterogeneous data from multiple gene loci and accommodates multiple fossil calibration nodes. A birth-death process with species sampling is used to specify a prior for divergence times, which allows easy assessment of the effects of that prior on posterior time estimates. We propose a new approach for specifying calibration points on the phylogeny, which allows the use of arbitrary and flexible statistical distributions to describe uncertainties in fossil dates. In particular, we use soft bounds, so that the probability that the true divergence time is outside the bounds is small but nonzero. A strict molecular clock is assumed in the current implementation, although this assumption may be relaxed. We apply our new algorithm to two data sets concerning divergences of several primate species, to examine the effects of the substitution model and of the prior for divergence times on Bayesian time estimation. We also conduct computer simulation to examine the differences between soft and hard bounds. We demonstrate that divergence time estimation is intrinsically hampered by uncertainties in fossil calibrations, and the error in Bayesian time estimates will not go to zero with increased amounts of sequence data. Our analyses of both real and simulated data demonstrate potentially large differences between divergence time estimates obtained using soft versus hard bounds and a general superiority of soft bounds. Our main findings are as follows. (1) When the fossils are consistent with each other and with the molecular data, and the posterior time estimates are well within the prior bounds, soft and hard bounds produce similar results. (2) When the fossils are in conflict with each other or with the molecules, soft and hard bounds behave very differently; soft bounds allow sequence data to correct poor calibrations, while poor hard bounds are impossible to overcome by any amount of data. (3) Soft bounds eliminate the need for "safe" but unrealistically high upper bounds, which may bias posterior time estimates. (4) Soft bounds allow more reliable assessment of estimation errors, while hard bounds generate misleadingly high precisions when fossils and molecules are in conflict.     

Dating primate divergences through an integrated analysis of palaeontological and molecular data

Estimation of divergence times is usually done using either the fossil record or sequence data from modern species. We provide an integrated analysis of palaeontological and molecular data to give estimates of primate divergence times that utilize both sources of information. The number of preserved primate species discovered in the fossil record, along with their geological age distribution, is combined with the number of extant primate species to provide initial estimates of the primate and anthropoid divergence times. This is done by using a stochastic forwards-modeling approach where speciation and fossil preservation and discovery are simulated forward in time. We use the posterior distribution from the fossil analysis as a prior distribution on node ages in a molecular analysis. Sequence data from two genomic regions (CFTR on human chromosome 7 and the CYP7A1 region on chromosome 8) from 15 primate species are used with the birth-death model implemented in mcmctree in PAML to infer the posterior distribution of the ages of 14 nodes in the primate tree. We find that these age estimates are older than previously reported dates for all but one of these nodes. To perform the inference, a new approximate Bayesian computation (ABC) algorithm is introduced, where the structure of the model can be exploited in an ABC-within-Gibbs algorithm to provide a more efficient analysis.     

Relaxed clocks and inferences of heterogeneous patterns of nucleotide substitution and divergence time estimates across whales and dolphins (Mammalia: Cetacea)

Various nucleotide substitution models have been developed to accommodate among lineage rate heterogeneity, thereby relaxing the assumptions of the strict molecular clock. Recently developed "uncorrelated relaxed clock" and "random local clock" (RLC) models allow decoupling of nucleotide substitution rates between descendant lineages and are thus predicted to perform better in the presence of lineage-specific rate heterogeneity. However, it is uncertain how these models perform in the presence of punctuated shifts in substitution rate, especially between closely related clades. Using cetaceans (whales and dolphins) as a case study, we test the performance of these two substitution models in estimating both molecular rates and divergence times in the presence of substantial lineage-specific rate heterogeneity. Our RLC analyses of whole mitochondrial genome alignments find evidence for up to ten clade-specific nucleotide substitution rate shifts in cetaceans. We provide evidence that in the uncorrelated relaxed clock framework, a punctuated shift in the rate of molecular evolution within a subclade results in posterior rate estimates that are either misled or intermediate between the disparate rate classes present in baleen and toothed whales. Using simulations, we demonstrate abrupt changes in rate isolated to one or a few lineages in the phylogeny can mislead rate and age estimation, even when the node of interest is calibrated. We further demonstrate how increasing prior age uncertainty can bias rate and age estimates, even while the 95% highest posterior density around age estimates decreases; in other words, increased precision for an inaccurate estimate. We interpret the use of external calibrations in divergence time studies in light of these results, suggesting that rate shifts at deep time scales may mislead inferences of absolute molecular rates and ages.     

To what extent do new fossil discoveries change our understanding of clade evolution? A cautionary tale from burying beetles (Coleoptera: Nicrophorus)

Divergence time estimates derived from phylogenies are crucial to infer historical biogeography and diversification dynamics. Yet, the impact of fossil record incompleteness on macroevolutionary reconstructions remains equivocal. Here, we investigate to what extent gaps in the fossil record can impinge downstream evolutionary inferences in the beetle family Silphidae. Recent discoveries have pushed back the fossil record of this group from the Eocene into the Jurassic. We estimated the divergence times of the family using both its currently understood fossil record and the fossil record known prior to these recent discoveries. All fossil calibrations were informed with different parametric distributions to investigate the weight of priors on posterior age estimates. Based on time‐calibrated trees, we assessed the impact of fossil calibrations on the inference of ancestral ranges and diversification rate dynamics in the genus Nicrophorus. Depending upon the selected sets of fossil constraints, the age discrepancies had a major impact on the macroevolutionary inferences: the biogeographic extrapolations relative to paleogeography are markedly contrasting, and the calculated rates at which species form or go extinct (and when they varied) are strikingly different. We show that soft prior distributions do not necessarily alleviate such shortcomings therefore preventing the inference of reliable macroevolutionary patterns in groups presenting a taphonomic bias in their fossil record.     

ESTIMATING DIVERSIFICATION RATES: HOW USEFUL ARE DIVERGENCE TIMES?

The dynamics of species diversification rates are a key component of macroevolutionary patterns. Although not absolutely necessary, the use of divergence times inferred from sequence data has led to development of more powerful methods for inferring diversification rates. However, it is unclear what impact uncertainty in age estimates have on diversification rate inferences. Here, we quantify these effects using both Bayesian and frequentist methodology. Through simulation, we demonstrate that adding sequence data results in more precise estimates of internal node ages, but a reasonable approximation of these node ages is often sufficient to approach the theoretical minimum variance in speciation rate estimates. We also find that even crude estimates of divergence times increase the power of tests of diversification rate differences between sister clades. Finally, because Bayesian and frequentist methods provided similar assessments of error, novel Bayesian approaches may provide a useful framework for tests of diversification rates in more complex contexts than are addressed here.     

Exploring uncertainty in the calibration of the molecular clock

Calibration is a critical step in every molecular clock analysis but it has been the least considered. Bayesian approaches to divergence time estimation make it possible to incorporate the uncertainty in the degree to which fossil evidence approximates the true time of divergence. We explored the impact of different approaches in expressing this relationship, using arthropod phylogeny as an example for which we established novel calibrations. We demonstrate that the parameters distinguishing calibration densities have a major impact upon the prior and posterior of the divergence times, and it is critically important that users evaluate the joint prior distribution of divergence times used by their dating programmes. We illustrate a procedure for deriving calibration densities in Bayesian divergence dating through the use of soft maximum constraints.     

Calibration age and quartet divergence date estimation

The date of a single divergence point--between living alligators and crocodiles--was estimated with quartet dating using calibrations of widely divergent ages. For five mitochondrial sequence datasets, there is a clear relationship between calibration age and quartet estimate--quartets based on two relatively recent calibrations support younger divergence estimates than do quartets based on two older calibrations. Some of the estimates supported by young quartets are impossibly young and exclude the first appearance of the group in the fossil record as too old. The older estimates--those based on two relatively old calibrations--may be overestimates, and those based on one old and one recent calibration support divergence estimates very close to fossil data. This suggests that quartet dating methods may be most effective when calibrations are applied from different parts of a clade's history.     

Bayesian estimation of ancestral character states on phylogenies

Biologists frequently attempt to infer the character states at ancestral nodes of a phylogeny from the distribution of traits observed in contemporary organisms. Because phylogenies are normally inferences from data, it is desirable to account for the uncertainty in estimates of the tree and its branch lengths when making inferences about ancestral states or other comparative parameters. Here we present a general Bayesian approach for testing comparative hypotheses across statistically justified samples of phylogenies, focusing on the specific issue of reconstructing ancestral states. The method uses Markov chain Monte Carlo techniques for sampling phylogenetic trees and for investigating the parameters of a statistical model of trait evolution. We describe how to combine information about the uncertainty of the phylogeny with uncertainty in the estimate of the ancestral state. Our approach does not constrain the sample of trees only to those that contain the ancestral node or nodes of interest, and we show how to reconstruct ancestral states of uncertain nodes using a most-recent-common-ancestor approach. We illustrate the methods with data on ribonuclease evolution in the Artiodactyla. Software implementing the methods (BayesMultiState) is available from the authors.     

Assessing the sensitivity of divergence time estimates to locus sampling,calibration points,and model priors in a RAD-seq phylogeny of Carex section Schoenoxiphium

Restriction site-associated DNA sequencing (RAD-seq) and related methods have become relatively common approaches to resolve species-level phylogeny. It is not clear, however, whether RAD-seq data matrices are well suited to relaxed clock inference of divergence times, given the size of the matrices and the abundance of missing data. We investigated the sensitivity of Bayesian relaxed clock estimates of divergence times to alternative analytical decisions on an empirical RAD-seq phylogenetic matrix. We explored the relative contribution of secondary calibration strategies, amount of missing data, and the data partition analyzed to overall variance in divergence times inferred using BEAST MCMC analyses of Carex section Schoenoxiphium (Cyperaceae)—a recent radiation for which we have nearly complete species sampling of RAD-seq data. The crown node for Schoenoxiphium was estimated to be 15.22 (9.56–21.18) Ma using a single calibration point and low missing data, 11.93 (8.07–16.03) Ma using multiple calibration points and low missing data, and 8.34 (5.41–11.22) using multiple calibrations but high missing data. We found that using matrices with more than half of the individuals with missing data inferred younger mean ages for all nodes. Moreover, we have found that our molecular clock estimates are sensitive to the positions of the calibration(s) in our phylogenetic tree (using matrices with low missing data), especially when only a single calibration was applied to estimate divergence times. These results argue for sensitivity analyses and caution in interpreting divergence time estimates from RAD-seq data.