Abstract
Divergence time estimation — the calibration of a phylogeny to geological time — is a integral first step in modelling the tempo of biological evolution (traits and lineages). However, despite increasingly sophisticated methods to infer divergence times from molecular genetic sequences, the estimated age of many nodes across the tree of life contrast significantly and consistently with timeframes conveyed by the fossil record. This is perhaps best exemplified by crown angiosperms, where molecular clock (Triassic) estimates predate the oldest (Early Cretaceous) undisputed angiosperm fossils by tens of millions of years or more. While the incompleteness of the fossil record is a common concern, issues of data limitation and model inadequacy are viable (if underexplored) alternative explanations. In this vein, Beaulieu et al. (2015) convincingly demonstrated how methods of divergence time inference can be misled by both (i) extreme state-dependent molecular substitution rate heterogeneity and (ii) biased sampling of representative major lineages. While these (essentially model-violation) results are robust (and probably common in empirical data sets), we note a further alternative: that the configuration of the statistical inference problem alone precluded the reconstruction of the paleontological timeframe for the crown age of angiosperms. We demonstrate, through sampling from the joint prior (formed by combining the tree (diversification) prior with the various calibration densities specified for fossil-calibrated nodes), that with no data present at all, an Early Cretaceous crown angiosperms is rejected (i.e., has essentially zero probability). More worrisome, however, is that for the 24. nodes calibrated by fossils, almost all have indistinguishable marginal prior and posterior age distributions, indicating an absence of relevant information in the data. Given that these calibrated nodes are strategically placed in disparate regions of the tree, they essentially anchor the tree scaffold, and so the posterior inference for the tree as a whole is largely determined by the pseudo-data present in the (often arbitrary) calibration densities. We recommend, as for any Bayesian analysis, that marginal prior and posterior distributions be carefully compared, especially for parameters of direct interest. Finally, we note that the results presented here do not refute the biological modelling concerns identified by Beaulieu et al.(2015). Both sets of issues remain apposite to the goals of accurate divergence time estimation, and only by considering them in tandem can we move forward more confidently. [Angiosperms; divergence time estimation; fossil record; marginal priors; information content; diptych; wild speculation.]
“Molecular clocks are not up to the job, but neither isthe fossil record.” Donoghue and Benton (2007)
Divergence time estimation from molecular geneticsequences is fraught with uncertainty. The errorsinvolved in routine phylogenetic reconstruction (suboptimal alignments, inadequate substitutionmodels, insufficient taxon/gene sampling, real gene e tree/species tree conflict, etc.) are compounded by assumptions required to transform a phylogram (inunits of expected number of substitutions per site)into a chronogram (in units of geological time):1) an appropriate model of substitution rate heterogeneityamong lineages and across time, and 2) temporal calibrations, generated from the fossil (or biogeographic) record, used to inform and constrain the extent of rate variation.
It is therefore not surprising that there are discrepancies between inferred molecular genetic and paleontological timescales. However, while many disagreements are minor and may innocuously be attributed to insufficient sampling (genes, taxa, orfossils), others are so severe and consistent that theycast serious doubt on the appropriateness of molecularclock models, the fossil record, or both. One prominent example concerns placental mammals, where molecular estimates (e.g.,Meredith et al.2011) of the crownage almost double those from the fossil record (e.g.,O’Leary et al.2013), obscuring the role of the K-Pgmass extinction on the evolutionary trajectory of thisgroup. Another conspicuous example, also spanning the K-Pg boundary, is crown birds (Neornithes;Ksepka et al. 2014), where (re)analyses have repeatedly lead to incongruous inferred evolutionary timeframes (e.g., Ericson et al. 2006vs.Brown et al. 2007; Jarvis et al. 2014vs.Mitchell et al. 2015; Prurn et al. 2015).
However, perhaps the best exemplified recalcitrant node in terms of absolute age is that of crown angiosperms, where molecular clocks pervasively infer a Triassic age (e.g.,Bell et al. 2010;Smith et al. 2010;Zeng et al. 2014; Beaulieu et al. 2015;Foster et al. 2016; see a comprehensive review of estimates in Magallón et al. 2015) while the oldest undisputed fossil remains are restricted to the Early Cretaceous (136 Ma;Brenner 1996). Moreover, using a model of uniform random fossilization (Marshall 2008) applied to 137 fossils lead Magallón et al. (2015)to infer an upper bound on the origin of crown angiosperms of just ~140 Ma. The age of this one node, more than any other, has seriously called into question the utility of both molecular clock models and the fossil record.
Numerous reasons have been put forth to explain the disparity of molecular and paleontological timescales. On the one hand we have concerns with the fossil record. By their nature, fossils must postdate the origin of taxa, meaning that molecular estimates should predate those from the fossil record. Furthermore, it is clear that the fossil record is imperfect (more so for some groups than others), such that in some instances it may prove impossible to ever have a tight correlation with molecular clock estimates. However, ‘absence of evidence’ vs. ‘evidence of absence’ is a complex matter, so a more productive avenue to pursue may be that of considering molecular approaches. While it has been speculated that molecular clocks might ‘run fast’ during radiations (thereby misleading clocks into inferring a long period oftime has occurred;Benton 1999), this has no empirical support. However, it is known that substitution model mis-specification can mislead divergence time estimation(Phillips 2009; Schenk and Hufford 2010) and ultimately downstream analyses (Revell et al.,2005). Likewise,mis-specification of relaxed clock models may also leadto inaccurate results (Dornburg et al. 2012;Worobey et al. 2014). Being only semi-identifiable, molecularclock methods require calibration from the fossil record,so appropriate calibration use is critically important (Inoue et al. 2010;Sauquet et al. 2011;Warnock et al.2012;Magallón et al. 2013;Zhu et al. 2015). Finally,Beaulieu et al.(2015) recently demonstrated through simulation two further ways where molecular clocksmight be mislead: (i) through extreme state-dependent molecular substitution rate heterogeneity, and (ii) biased sampling of representative major lineages (both of whichare essentially instances of model violation). While theseare all valid concerns that should always be kept in mind,we explore below a further non-biological possibility thatmay unknowingly be at play in many data analyses.
Diptych: A Metaphor for Data Analysis?
A diptych is a device commonly used in westernart and literature. It consists of paired, complementary works, in the artistic tradition typically two imagesjoined at a hinge (e.g., Fig. 1). The function of a diptychis to reciprocally illuminate the component images,ideally revealing some more holistic concept. It is thisfeature that suggests an association with Bayesian data analysis. It is de rigueur in any Bayesian analysis tocarefully compare paired prior and posterior parameter distributions to gauge how information content (via the likelihood) drives the results, as well as to establishthe sensitivity of inferences to prior specifications.A diptych is, we argue, therefore a useful metaphorfor describing the process of changes in belief in parameter values from the prior (before data havebeen observed) to the posterior (after data have been observed). We note that the phylogenetic systematics community has been generally lax in this respect,despite available Bayesian software packages making such reflections straight forward. We argue that thisis especially important in divergence time estimation analyses, as it is generally unappreciated that there are three sets of distributions to consider. In addition to the temporal fossil calibration specified by the investigator(the ‘user prior’) and the resulting marginal posterior distribution, there exists an intermediate distribution,the marginal prior (also called the ‘effective’ or ‘joint’prior by some authors), which is formed by the interaction among user priors and the underlying ‘tree prior’ (for nodes not directly calibrated by a fossil; typically a birth-death prior). Here we turn our attention to these distributions to see what, if anything, we can glean about the age of crown angiosperms.
A Re-reanalysis of the Age of Angiosperms
We reanalyzed the data set provided byBeaulieu et al.(2015). The molecular alignment consists of four genes (chloroplast: atpB, psbB, and rbcL; nuclear: 18S) for 124 taxa including 91 angiosperms representing all extant orders (data file provided in the Supplementary Material; see also data from the original paper available on Dryad at http://dx.doi.org/10.5061/dryad.629sc). Sampling was originally designed specifically for dating the origin of angiosperms by allowing the placement of 24 (15 angiosperm) fossil calibrations from across landplants. All dating analyses reported here, like those in the original paper, were performed using the uncorrelated lognormal (UCLN) clock model and birth-death tree prior in BEAST v1.8.2 (Drummond et al. 2006;Drummond and Rambaut 2007) using the CPU implementation of the BEAGLE v2.1.2 library (Ayres et al. 2012).
We regenerated the posterior results of Beaulieu et al. (2015)by employing the original lognormal user priors specified in Table 1 and the following analysis settings: 3 replicate analyses of 50 million generations, sampling every 1000 generations. As in the original Beaulieu et al. (2015)paper we fixed the tree topology (their Figure 1). These analyses were re-run without any data (i.e., sampling from the marginal prior). Because these latter analyses were not as computationally demanding, as welas to perform a thorough exploration of parameter space analyses were run in duplicate for 1 billion generations each, sampling every 10 thousand generations. Finally we replicated all analyses but replaced the log normaluser prior calibrations from Table 1 with ‘extreme exponential’ distributions with a mean of 1.0 (offset byfossil ages); such distributions lend the utmost credenceto the fossil record, as 95% of the prior masses liewithin 3 Ma of the relevant fossil ages. Importantly all analyses (posterior and prior-only) were initializer with chronograms where in crown angiosperms originated~140 Ma (i.e., consistent with the prescription of Magallón et al. 2015).
We focus here on the data (taxon, gene, and fossil sampling) and settings (model and priors) of Beaulieu et al.(2015) because it is representative of a is standard dating analysis. However, to demonstrate thatour own results are not restricted to this particular data set, we also reanalyze the data set of Magallóiet al.(2015), albeit to a more limited degree because computational requirements. This data set consists of 5 genes (chloroplast: atpB, rbcL and matK; nuclear: 18Sand 26S) for 799 taxa (792 angiosperms) and 121 fossil calibrations (note that the original analysis included 137 fossil calibrations, but 16 non-essential calibrations were subsequently found to be missing from the file shared bythe original authors). All analyses employed the UCLN model as above and a fixed tree topology (their Figure 3)
The posterior results of Magallón et al.(2015)were regenerated by running 3 replicate analyses of 100 million generations, sampling every 5000 generations. These analyses included the uniform calibration prior (139.5–136 Ma) on the age of crown angiosperms. To assess the influence of this single prior, 4 replicate analyses of 50 million generations were performed without the prior. Finally, as above, analyses sampling from the marginal prior (i.e., without any data) were performed for both sets of fossil calibrations (each with 4 replicates of 50 million generations).
Prior to summarization, the MCMC log files from each set of replicated analyses were concatenated while removing a conservative 10% sample burnin using the pxlog program from the phyx package (Brown et al. 2017). All results were processed in R v3.3.2 (R Core Team, 2016) and were visualized using ggplot2 v2.2.1 (Wickham, 2009) and code adapted from phyloch v1.5-3 (Heibl, 2008).
The Inaccessibility of an Early Cretaceous Crown Angiosperms
As in the original Beaulieu et al.(2015)paper we were unable to recover a posterior age estimate for crown angiosperms that corresponded with the prevailing paleontological time frame, even when employing overly precise exponential fossil calibration user priors. However, when considering the diptych interpretation by examining the joint marginal prior, it is clear that we need not invoke modelling complications (e.g., due to biased lineage sampling or structured excessive rate heterogeneity) to explain the results. Rather, when running the analysis without any data, we see that an Early Cretaceous crown angiosperms is precluded based on the configuration of the statistical problem alone (Fig. 2). From the trace plots (Fig. 3) we see that the parameter regarding the age of crown angiosperms departs immediately from ~140 Ma to >200 Ma. In no instance did the MCMC samplers ever return to a ‘young’ age of angiosperms. The minimum post-burnin value for the prior and posterior analyses for the original lognormal calibration priors were 185.9 Ma and 192.0 Ma, respectively (181.1 Ma and 176.0 Ma for the exponential calibration priors). We note that these findings do not have to do with any peculiarity of the Beaulieu et al.(2015)data set as the results generated from the Magallón et al.(2015)data set without the uniform prior on crown angiosperms confirms the findings (minimum post-burnin ages for the prior and posterior analyses are 226.1 Ma and 197.2 Ma, respectively). In fact, the Magallónet al.(2015)data set, containing far more data (genes, taxa, and fossils) generated the oldest mean posterior estimate (249.7 Ma vs. 233.0 Ma for the original Beaulieu et al.(2015)priors vs. 213.3 Ma for the same data set using exponential calibration priors).
While rejecting an Early Cretaceous origin based on the marginal prior alone, both data sets do seem to contain signal relevant to the age of crown angiosperms,as the marginal posterior estimates are shifted significantly younger than the marginal prior (Fig. 2).This raises the question: if true, what kind/amount ofdata would be required to recover an Early Cretaceous age for crown angiosperms? Ultimately this comesdown to quantifying phylogenetic ‘information content’.Intriguing possibilities to addressing this sort of question therefore lie in Shannon information theory (Shannon 1948), a field which the systematics community has largely ignored. Recently Lewis et al.(2016) made substantial strides forward by applying this theory (inthe discrete case) towards assessing the information content of prior and posterior tree topology distributions.Applications to the continuous case, however, are muchmore difficult, and no theory (let alone software)currently exists to address the present problem (PaulLewis, personal communication).
Priors and Posteriors: a Diptych in Three Parts
Above we introduced ‘diptych’ as a useful metaphor for interpreting Bayesian analytical results. The paired nature of a diptych mirrors the before (prior) and after 2 (posterior) reflection on what has been learned about probable parameter values.
The metaphor is slightly more complicated for someparameters involved in divergence time analyses. Nodes not explicitly calibrated by fossildata (hence forth, ‘uncalibrated’ nodes) still require an age prior, which is provided by a ‘tree prior’, typically a birth-death,Yule, or coalescent prior. Such nodes thus have marginal prior and posterior distributions, and are conducive to 3 the diptych metaphor. For those nodes that are directly calibrated using fossil information, the interpretation of 3 the results of inference are more complicated. Thesenodes have a ‘user prior’, a distribution constructed in some way using information from the fossil record.
However, these nodes are also involved in the tree prior. The resulting ‘marginal prior’ is a multiplicative combination of the user and tree priors, and may also be influenced by interactions with adjacent user-calibrated nodes (e.g., ancestor and descendant nodes which have overlapping user priors). The resulting marginal prior thus does not necessarily reflect theoriginal user prior.Thisis, in our judgement, fartoo rarely assessed. Finally, these nodes have marginal posterior distributions. However, unlike the uncalibrated nodes (which involve only two distributions, and thus a simple interpretation), calibrated nodes involve three distributions (two priorsandone posterior), which complicates interpretation. [We prefer to continue with the diptych metaphor for these nodes, rather than the obvious ‘triptych’, as the focus lies still on the change in belief on parameter values before (prior) and after(posterior) observing the data, even if the prior involves two components.] The difference between the marginalprior and marginal posterior, like the uncalibrated nodes, reflects information in the data (that is, thelikelihood). However, the difference between the user andmarginal priors, if present, may be better described as demonstrating the influence of ‘pseudo-data’ present inthe various user priors.
We now turn our attention to the fossil-calibrated nodes. Ideally we would find that the user and marginal priors are identical (that is, that the marginal priors reflect the intentions of the researcher), and the marginal priors and posteriors to differ (indicating information present in the data relevant to the parameter of interest).We plot in Fig. 4 the three sets of distributions for the24 fossil-calibrated nodes for the Beaulieu et al. 2015 data set. In general, user and marginal priors match quite closely. However, there is a stark exception involving the Tricolpites constraint: while the user prior specifies that 95% of the prior mass should lie between 135.2–125 Ma,the marginal prior has a 95% highest posterior density(HPD) range of 226.1–128.7 Ma. The marginal posterior of this node age has a 95% HPD range of 170.7–146.6 Ma, which already surpasses the angiosperm paleontological estimate of ~140 Ma despite being well nested within the clade. We note that Tricolpites is the oldest constraint within angiosperms used by Beaulieu et al. 2015(see Table 1 and Fig. 5). However, it is clear that it is not this particular calibration which is forcing angiosperms to be ‘too old’. Reanalysis without this specific constraint yielded even older posterior estimates for both this node (95% HPD: 181.2–154.3 Ma) and crown angiosperms (mean 241.0 Ma vs. 233.0 Ma with the constraint; data not shown). We cannot currently identify the cause of the disruption of the Tricolpites user prior.
Finally, we consider differences between the marginal prior and posterior distributions of age estimates forthese same fossil-calibrated nodes. As discussed above,shifts in these paired distributions (i.e., following the diptych metaphor) would indicate the presence/degree of relevant phylogenetic ‘information’ (although theory has not yet been worked out on how to quantify this). However, from Fig. 4 we note that, for the mostpart, these distributions are nearly indistinguishable.This pattern is even more clear in Fig. 5 where,while uncalibrated nodes show significant shifts between prior and posterior analyses, calibrated nodes show no movement. This lack of updating in belief from prior to posterior for the calibrated nodes suggests two non-mutually exclusive interpretations: (i) that the UCLN model, through allowing independent molecular substitution rates on all edges, may be over fitting the data, essentially allowing all calibrations to be mutually consistent (we do not expect, for example, that all ofour calibration densities are equally ‘good’ - that is, that some constraints should conflict); (ii) that the data set considered here lacks relevant phylogenetic ‘information’,or at least insufficient information to overrule the pseudo-data present in the fossil calibration densities.
The finding of equivalent prior and posterior distributions may come as a surprise to some users,as unbounded (e.g., lognormal) temporal priors are typically used to ‘let the data speak for themselves’.Certainly, divergence time inference is only semi-identifiable (dos Reis and Yang 2013), so we do expect some level of association. It is the degree of association that is worrying. We are unaware of any other type of Bayesian analysis in evolutionary biology where identical prior and posterior distributions would not cause concern. The present results go a long way to explaining why divergence time estimation shows such a strong sensitivity to the fossil calibrations used (Inoue et al.2010;Sauquet et al. 2011;Warnock et al. 2012). The interpretation of fossil calibrations contributing pseudo-data (rather than, say, fossils setting simple minimumage constraints as they have been traditionally) suggests that we might benefit from rethinking lessons that have been learned in the early days of phylogenetic divergence time estimation. We briefly consider one now.
Are More Fossils Really Better in Node Dating Analyses?
There is an adage in the divergence time estimation literature that as many fossils as possible should beused to calibrate nodes (Benton and Donoghue 2007).This makes sense, as of course we would like to includeas much information as possible into a reconstruction.However, this advice largely came about when dating methods (e.g.,r8s,Sanderson 2003; multidivtime,Thorne and Kishino 2002) employed constraints (e.g.,boolean minimum ages for the age of the fossil) rather than probabilistic distributions. As long as fossils were correctly placed within the phylogeny, the inclusion of more fossils should not produce misleading results.For instance, fossils that are ‘too young’ (that is, do not closely approximate in age the node they are calibrating) are either simply uninformative, or appropriately represent limitations of the fossil record.
As an extreme example, a chicken bone found in a back alley gutter is a valid (if imprecise) minimum age constraint for Gallus gallus (recently estimated at 2.9 Ma; Stein et al. 2015).
However, from the results reported above (Figs. 4,5) we find several concerns with including as many fossils as possible in a probabilistic node-dating analysis. [We note that these concerns do not apply to the fossilized birth-death model (Heath et al. 2014) or tip-dating (Ronquist et al. 2012) approaches to divergence time estimation, which do not involve such calibrations.] First, as with the Tricolpites example above, calibrations can interact with each other and the tree prior in unpredictable ways to produce marginal priors that do not represent the originally intended user priors. While this is a recognized (though under-appreciated) issue, the available solutions work only for a small number of calibrations (Heled and Drummond 2012;Heled and Drummond 2015). Second, given that the marginal prior and posterior calibrated node ages are often indistinguishable (suggesting little relevant phylogenetic information content), it is worrisome that the act of employing temporal calibration priors can directly determine the resulting posterior patterns of rate heterogeneity across a tree. It is not inconceivable, for example, that the parametric use of the best available fossils from an incomplete fossil record could turn a clock-like data set unclock-like, needlessly increasing the model complexity (and therefore uncertainty) involved.
Our final concern with unrestrained parametric calibration use is the form of the calibrations themselves.A flexible assortment of distribution families are available (Ho and Phillips 2009; see also discussion in Brown and van Tuinen 2011), allowing essentially any prior belief to be employed. In addition, researchers can make use of the fossil calibration database (Ksepka et al. 2015), and prescribed ‘best practices’ (Parham et al. 2012) can help avoid naive errors when dealing with the fossil record. Nevertheless, the vast majority of user calibration priors employed in the literature are wholly idiosyncratic and arbitrary (we include ourselves here). This is not necessarily a result of molecular phylogeneticists lacking the appropriate paleontological expertise (and isn’t that what collaboration is for?), but rather a property of data involved.
While methods exist to generate a distribution from a set of fossils (Marshall 2008; Nowak et al. 2013Claramunt and Cracraft 2015), these require well sampled data. Scant data is an entirely different problem.How does one fit a distribution to a single (exceptionally old, and therefore exceptionally informative) fossil? Minimum bounds are simple (the age of the oldest i fossil), but as Parham et al.(2012)note, there exists no standard protocol for formulating maximum ages (let alone the shape of the distribution spanning the upper and lower bounds). Indeed, the process of constructing temporal priors is so nebulous that Lee and Skinner (2011)likened it to “educated guesswork”. However, it is not the arbitrariness of the calibrations per se that is of concern, but rather that they act as a strong source of pseudo-data. Taken to a hyberbolic extreme, if calibration priors were applied to every node in a tree, then the results above would suggest that there would be no use in running the analysis at all. In other types of Bayesian analysis we expect that an increase in data can overrule poorly constructed priors and converge on an answer. It is presently unclear whether this is true when performing node-dating using the UCLN model, and if so, how much data (or ‘information’, whatever that turns out to be) would be required. Certainly it is known that the uncertainty in divergence time estimates cannot be reduced arbitrarily, even with infinite amounts of data (Rannala and Yang 2007;Yang and Rannala 2006; dos Reis and Yang 2013), but it is unclear how data can override the pseudo-data present in the node calibration priors.
Where to Go from Here?
The results presented here highlight several issues that should be considered as the field moves forward. In regard to angiosperms, is the amount of temporal ‘information’ present in empirical data (or, on the other hand, the adequacy of current relaxed clock models) insufficient to reconstruct such recalcitrant nodes as the age of crown angiosperms? If this is the case, the only way forward, given the methods of inference, may be to apply user priors that are intended to constitute pseudo-data. However, this fact needs to be more widely recognized. In some respects this is a defensible position as, if fossil calibrations are constructed with significant information about the fossil record, estimations will be constrained to existing fossil information. In this vein, the results of Magallón et al.(2015), which estimate nested angiosperm divergence times within a strict paleontologically-imposed age of the ancestral crown node, are reasonable in the context of the data available (Sanderson, 2015). Nevertheless, it should be clear when the molecular data, in this context, do not significantly alter the posterior distribution. If this is indeed the way forward, then care should be taken to assess both the validity of the fossils being used (Sanders and Lee 2007;Brown and Sorhannus 2010) and the form of the calibration priors (Inoue et al. 2010; Brown and Sorhannus 2010;Sauquet et al. 2011; Warnock et al. 2012; Foster et al. 2016). In addition, we strongly advocate the regular use of the diptych approach to data analysis by habitually comparing prior and posterior distributions: it is imperative to understand which parameters in our models are informed by the data present, and which simply recapitulate the prior. When hypothesis testing it is even more critically important to determine whether a hypothesis is rejected by the data or, as with the crown angiosperm age results above, are effectively precluded by the joint prior.
However, new methods of divergence time inference are emerging that largely bypass the concerns associated with node-dating (Heath and Moore, 2014). The fossilized birth-death model of Heath et al.(2014) incorporates extant and extinct (i.e., sampled fossils) lineages as evolving according to the same underlying diversification model. Alternatively, when morphological data are available for both extinct and extant taxa, divergence times can be estimated using the tipdating approach of Ronquist et al.(2012). Both of these methods can take advantage of an arbitrary number of fossils within a lineage (rather than being reduced to a single distribution as in node-dating) and incorporate fossil temporal information directly without extrapolation. The excitement surrounding these methods might lead us to think it not unreasonable to suppose that in the near future node-dating will be regarded as a useful tool that was ultimately replaced by methods that more directly make use of the available data. However, both of these methods are relatively new, and it is unclear whether they will overthrow node-dating results for the most recalcitrant nodes (i.e., placental mammals, crown birds, crown angiosperms, etc.). These methods also raise new questions in regard to model adequacy, implied and explicit assumptions regarding both diversification and morphology models, and data availability and quality for extinct and extant lineages. Furthermore, the resulting divergence time estimates from these new methods may not differ as much as expected. For example, Eguchi and Tamura (2016)employed the fossilized birth-death model and found monocots arose 174.26–134.14 Ma, which does not conflict strongly with previous node-dating results. It thus appears far too premature to consider the tempo of angiosperm diversification solved.
Final Thoughts on the Age of Angiosperms
Finally, we note that the results presented above do not refute the concerns identified and demonstrated through simulation by Beaulieu et al.(2015)regarding violations of biological modelling. While among-lineage molecular substitution rate heterogeneity is regarded as ubiquitous, clade-and trait-specific correlations of rate variation explored by Beaulieu et al.(2015)are becoming increasingly recognized as important biological patterns of molecular evolution (Smith et al. 2010;Dornburg et al. 2012;Lartillot and Delsuc 2012;Worobey et al. 2014). Such processes must be correctly modelled if our divergence time estimates are to be accurate. In this vein, we note that the fit (Lepage et al.,2007) and adequacy (Duchênene et al.,2015) of alternative clock models are far too rarely assessed. In addition, the artefacts of lineage sampling identified by Beaulieu et al. (2015)casts doubt on the suitability of a homogeneous birth-death model as a prior on node ages. This doubt is especially manifested with respect to dating the evolution of angiosperms, where it is known a priori that lineages exhibit an incredible breadth of diversification rates (Tank et al.,2015), not to mention that the extant angiosperm diversity dwarfs other embryophyte clades. We thus regard our results above as complementary to those ofBeaulieu et al.(2015), and only with both in mind can we confidently move forward.
Funding
J.W.B. and S.A.S. were both supported by the NSF AVATOL Grant 1207915.
Acknowledgements
We thank Jeremy Beaulieu and Brian O’Meara for sharing their thoughts on these issues, and Susana Magallón for graciously sharing her BEAST xml input file. We would also like to thank the following for helpful discussions: Ben Redelings, Joseph Walker, Ning Wang, Greg Stull, Aaron King, Michael Landis, Alex Taylor, Paul Lewis, and members of the Smith laboratory. J.W.B. thanks Don Van Vliet for help with the title, Emo Philips for help with broader dissemination, Mark Kozelek for encouragement, and Linda Slote for tracking down an obscure Testudines reference (that was unfortunately dropped from an earlier version of this paper).