PyRate: a new program to estimate speciation and extinction rates ...

Methods in Ecology and Evolution 2014, 5, 1126–1131

doi: 10.1111/2041-210X.12263

APPLICATION

PyRate: a new program to estimate speciation and extinction rates from incomplete fossil data Daniele Silvestro1*, Nicolas Salamin2,3 and Jan Schnitzler4,5 1

Department of Plant and Environmental Sciences, University of Gothenburg, Carl Skottsbergs gata 22B, 413 19 Gothenburg, Sweden; 2Department of Ecology and Evolution, University of Lausanne, 1015 Lausanne, Switzerland; 3Swiss Institute of Bioinformatics, Quartier Sorge, 1015 Lausanne, Switzerland; 4Biodiversity and Climate Research Centre, Senckenberg Research Institute, Senckenberganlage 25, 60325 Frankfurt am Main, Germany; and 5Department of Biological Sciences, Goethe University, Max-von-Laue-Str. 13, 60438 Frankfurt am Main, Germany

Summary 1. Despite the advancement of phylogenetic methods to estimate speciation and extinction rates, their power can be limited under variable rates, in particular for clades with high extinction rates and small number of extant species. Fossil data can provide a powerful alternative source of information to investigate diversification processes. 2. Here, we present PyRate, a computer program to estimate speciation and extinction rates and their temporal dynamics from fossil occurrence data. The rates are inferred in a Bayesian framework and are comparable to those estimated from phylogenetic trees. 3. We describe how PyRate can be used to explore different models of diversification. In addition to the diversification rates, it provides estimates of the parameters of the preservation process (fossilization and sampling) and the times of speciation and extinction of each species in the data set. Moreover, we develop a new birth–death model to correlate the variation of speciation/extinction rates with changes of a continuous trait. 4. Finally, we demonstrate the use of Bayes factors for model selection and show how the posterior estimates of a PyRate analysis can be used to generate calibration densities for Bayesian molecular clock analysis. PyRate is an open-source command-line Python program available at http://sourceforge.net/projects/pyrate/.

Key-words: diversification rates, birth–death process, macroevolution, diversity dynamics, Bayesian, BDMCMC Introduction The estimation of speciation and extinction rates forms an essential part of macroevolutionary analyses and can contribute substantially to understanding the processes that generate and maintain diversity in time and space (Ricklefs 2007; Fritz et al. 2013; Quental & Marshall 2013). This topic has received increased attention in comparative phylogenetics where significant methodological progress has been made over the past few years with the development of robust probabilistic frameworks (reviewed in Stadler 2013; Morlon 2014). Despite these advances, the estimation of complex diversification dynamics from phylogenies is still difficult, due to the lack of direct information about extinct taxa, and the accuracy of the estimates might be severely affected by high extinction and small number of extant species (Rabosky 2010). Thus, the use of fossil data to estimate speciation and extinction rates is preferable, when available, over phylogenetic data (Quental & Marshall 2010; Silvestro et al. 2014), while the integration of both types of data would be ideal (Fritz et al. 2013). The estimation of speciation and extinction rates using the fossil record has a long

*Correspondence author. E-mail: [email protected]

tradition in paleobiology (Raup et al. 1973; Sepkoski et al. 1981; Alroy 1996; Foote 2003) and provided significant progress towards understanding past evolutionary dynamics (Bapst et al. 2012; Quental & Marshall 2013). The main limitations of these methods are that they usually (i) use discrete time bins, (ii) analyse only first and last appearances of a taxon, thus ignoring other occurrences if available, (iii) lack the ability to perform model testing against over-parameterization, and (iv) do not incorporate extant species in the analyses (Foote 2000). These features make the estimated speciation and extinction rates difficult to compare to those estimated from molecular phylogenies. In this application note, we present PyRate, a computer program to perform macroevolutionary analyses based on paleontological data using a Bayesian framework. PyRate allows researchers to utilize the growing information stored in large fossil data bases (e.g. NOW, Paleobiology Data base, MioMap) to estimate speciation and extinction rates and their temporal dynamics within a hierarchical Bayesian framework. The program performs a joint estimation of the preservation and the diversification processes in continuous time, using all fossil occurrences for a taxon (not only first and last appearances), and provides a robust probabilistic framework to assess the best fitting birth–death

© 2014 The Authors. Methods in Ecology and Evolution © 2014 British Ecological Society

Speciation and extinction rates from fossils model. Extant species and species known only from one occurrence (singletons) can be included in the analysis. Furthermore, the number of extant species (N) can be used to construct an informative hyperprior for the birth–death parameters as the probability that a clade originating at some time in the past results in N extant species after diversifying under a birth–death process. The method also provides estimates of the parameters of the preservation process (rate of fossilization and sampling) and the times of speciation and extinction of each species in the data set. The method was described by Silvestro et al. (2014) and tested via simulations with a particular focus on the properties and power of the model. Here, we present the application implementing the model and develop user-friendly utilities that help setting up the analyses and generating graphical representations of the results. Additionally, we implement a new birth–death model to test for trait-correlated diversification. We use an empirical data set of the mammal family Ursidae to provide a step-by-step tutorial of an analysis while demonstrating the main features implemented in PyRate.

Description The main parameters estimated in a PyRate analysis are (i) the parameters of the preservation process (fossilization and sampling), (ii) the times of speciation and extinction of each species in the data set, and (iii) the speciation and extinction rates and their variation through time. All parameters are estimated jointly in a single analysis. Preservation is mod-

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elled on a lineage-specific basis as a non-homogeneous Poisson process in which the preservation rate follows a ‘hat’shaped trajectory (Liow et al. 2010) and is estimated from the data as the expected number of occurrences/lineage*Myr. Based on the preservation process, the individual speciation and extinction times (s and e respectively) are estimated for each lineage. This procedure is important because the first and last appearances of a species are likely to underestimate the true extent of its life span (Liow & Stenseth 2007). To avoid over-parameterization, the mean preservation rate q is assumed to be constant across species. However, the gamma model can relax this assumption to account for rate heterogeneity across species using a gamma distribution with shape parameter a estimated from the data. Large a values thus indicate a low variability of the preservation rate, whereas an estimated a 20 is considered as evidence for significant rate variation (Silvestro et al. 2014). The diversification process is modelled by a birth–death model estimated from the temporal distribution of lineages (s, e) and in the simplest case includes a speciation rate (k) and extinction rate (l), expressed in terms of expected number of speciation/extinction events/lineage*Myr. Rate variation can be introduced in two ways: Through time, with rates changing for all lineages at estimated times of rate shift, and across taxa, based on an estimated correlation with a continuous lineagespecific trait. The support for alternative birth–death models (e.g. with different number of rate shifts) can be estimated by their respective marginal likelihoods and compared using Bayes factors (Kass & Raftery 1995). The major functions of the package are summarized and explained in Table 1.

Table 1. Major functions of the PyRate program (detailed list is available in the PyRate manual) MCMC Algorithm -A 0 -A 1 -A 2 (default) Main model specifications -mG -mHPP

-mL -mM -mCov

Utility functions -plot

-mProb

Uses Markov chain Monte Carlo methods to estimate the parameter values under a specific model of preservation and diversification (see Model specifications). Estimates the marginal likelihood of a birth–death model. This value can be used to compare the fit of alternative models using Bayes factors. Jointly estimates the number of rate shifts in the birth–death rates, the temporal placement of the shifts, and the marginal speciation and extinction rates through time using the BDMCMC algorithm. Allows heterogeneity of the preservation rate across lineages in the data set. The rate variation is assumed to be gamma distributed, with a shape parameter estimated from the data. Use homogeneous Poisson process for preservation rates instead of non-homogeneous Poisson process (NHPP) with ‘hat-shaped’ (PERT) distributed preservation rates (Liow et al. 2010). Note that the NHPP model is used unless differently specified. Sets the number of rates for a birth–death model with rate shifts (e.g. ‘-mL 2’ sets one shift in the speciation rate and ‘-mM 3’ two shifts in the extinction rate). The rates and times of shifts are estimated from the data. Sets the Covar model, in which speciation and extinction rates vary across lineages as the result of a correlation with a continuous trait, provided as an observed variable in the input data. Usage: -mCov 1 correlated speciation rate -mCov 2 correlated extinction rate -mCov 3 correlated speciation and extinction rates Takes the marginal speciation and extinction rates logged in a PyRate analysis (named ‘*_marginal_ rates.log’) and generates a rates-through-time plot (RTT) using the scripting language R. Mean speciation, extinction, and net diversification rates through time are plotted in 1 Myr time bins with the respective 95% HPD. Several log files (e.g. from different replicates) can be loaded at once and combined in a single plot. Takes the posterior samples logged in a BDMCMC analysis to a file (named ‘*_mcmc.log’) to calculate the sampling frequencies of birth–death models with different number of rate shifts after a BDMCMC analysis.

© 2014 The Authors. Methods in Ecology and Evolution © 2014 British Ecological Society, Methods in Ecology and Evolution, 5, 1126–1131

1128 D. Silvestro et al. DATA PREPARATION

As an example, we analyse the Neogene and Quarternary diversification of bears (Ursidae). Fossil data were downloaded from two online data bases (Paleobiology data base and NOW data base; accessed 13/10/2013). The final data set (available from the Supplementary Data on Sourceforge), after excluding all occurrences that were not identified to species level, comprised 965 records for 75 species, 69 of which are extinct. To prepare the input file for analysis, all occurrences need to be entered in a table (tab-delimited text file) with species names, their status (‘extant’ or ‘extinct’), and minimum and maximum ages. These ages correspond to the upper and lower temporal boundaries of the stage a particular fossil specimen is assigned to and are generally available from the data bases. Additionally, one column can be added providing a trait value, if available, which can be used in the birth–death analysis (see paragraphs below). A typical input file may look like this: Species

Status

MinT MaxT

Trait

Ursus etruscus Ursus etruscus Ursus etruscus Agriotherium africanum Ailuropoda melanoleuca Ursavus brevirhinus Ursavus brevirhinus

extinct 1.95

2.6

90

extinct 1.2

1.8

90

extinct 2.6

3.4

90

extinct 3.6

5.3

NA

extant

0.6

1.3

118

extinct 8.2

9.0

80

extinct 11.2 15.2

80

These data can be processed using the R function extract.ages from the R utilities script provided with PyRate package. To load and call the extract.ages function, open a R console and type:

the model (in this case the number of rate shifts) and the parameter values (i.e. birth–death rates, times of shift, . . .). We launch this analysis using: python PyRate.py /path_to_file/Ursidae_PyRate.py -N 8 -mG

where ‘-N 8’ specifies the number of extant species (including those which lack a fossil record and thus do not appear in the input data set), and ‘-mG’ sets the gamma model for the preservation rates. By default, 10 million BDMCMC iterations are carried out and logged to files. Three output files are generated by the analysis: (i) a ‘*sum.txt’ file with the complete list of parameters and settings used in the analysis, (ii) a ‘*mcmc.log’ file containing the posterior sample of the model parameters, such as the preservation rate and the speciation and extinction times of each species (Fig. 1), and (iii) a ‘*marginal_rates.log’ file where birth–death marginal rates are logged within 1 Myr time bins. The BDMCMC sample can be inspected by loading the ‘*mcmc.log’ and ‘*marginal_rates.log’ files into the program Tracer (Rambaut & Drummond 2007), to determine the extent of the burn-in phase, check the effective sample sizes (ESS) and obtain posterior estimates of the parameters. For instance, after removing the first 10% of samples as burn in, the estimated mean preservation rate for the Ursidae data set was q = 1393 (95% HPD: 1157–1597) and the shape parameter of the gamma distribution had a mean value of a = 061 (95% HPD: 055–068), indicating a very strong heterogeneity across lineages. In the BDMCMC, the proportion of time any particular birth–death model is visited during the run is proportional to its posterior probability. This sampling frequency can be obtained using the following command: python PyRate.py -mProb /path_to_file/ Ursidae_1_G_mcmc.log -b 200

where with ‘-b returns:

200’

we exclude the burn-in samples. This

> source(file = ’/path_to_file/pyrate_utilities.r’) > extract.ages(file = ’/path_to_file/Ursidae.txt’)

ESTIMATION OF SPECIATION AND EXTINCTION RATES THROUGH TIME

A first analysis is aimed at evaluating the temporal dynamics of the speciation and extinction rates, using a birth–death MCMC algorithm (Silvestro et al. 2014), which, similarly to the reversible-jump MCMC (Green 1995), can jointly estimate the number of parameters in

s

Sampling frequency

This function resamples the age of fossil occurrences randomly within the respective temporal ranges and generates a Python file (here called ‘Ursidae_PyRate.py’) that can now be imported in PyRate for diversification rate analyses.

7

6

e

5

4

3

2

1

0

Time [Ma] Fig. 1. Fossil record and inferred times of speciation and extinction (s and e) and for Ursus minimus. Circles represent the sampled fossil occurrences (from uniform distributions between their minimum and maximum ages), the frequency distributions show the posterior samples of speciation and extinction times (dark grey 95% HPD).


Speciation and extinction rates from fossils

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Probability

python PyRate.py -plot /path_to_file/ Ursidae_marginal_rates.log -b 200

The command will generate (and execute) an R script and save a plot for speciation (k), extinction (l) and net diversification (k-l) rates through time in pdf format, showing in this case a decrease in speciation rate around 20 Ma, and an increase in extinction around 8 Ma (Fig. 2).

2·5 2·0 1·5 1·0

1·0

0·0

0·5

Speciation rate

0·5

20

15

10

5

Time [Ma]

0

(b)

1·0

Thus, the birth–death model with the highest posterior probability is a model with 2 speciation rates (0785) and two extinction rates (0581). We can observe, however, some uncertainty, which indicates that three speciation rates might be also plausible (0177) and that extinction rates might not vary significantly through time (1-rate model: 029; for extensive model testing see below). The output of a BDMCMC analysis can be used to plot marginal speciation and extinction rates through time. This is done with the following command:

0

0·8

0.2901 0.581 0.1121 0.0153 0.0014

–0·5

0·6

0.0095 0.7853 0.1773 0.0258 0.002

(a)

0

0·5

1

0·4

1-rate 2-rate 3-rate 4-rate 5-rate

Extinction rate

Extinction

0·2

Speciation

0

Model

20

15

20

15

10

5

0

10

5

0

Time [Ma]

kðbi Þ ¼ expðlogðk0 Þ þ ak logðbi ÞÞ where k0 is a baseline speciation rate and ak is an estimated correlation parameter. Similarly, the extinction rate can be transformed using an analogous equation with a baseline extinction rate l0 and the correlation parameter al. Given these definitions, positive correlations are represented by a > 0, and negative correlations are expressed by a < 0, whereas a = 0 indicates that the birth–death rates do not covary with the trait value. Assuming that s and e represent the times of speciation and extinction estimated from the preservation process, the likelihood of a birth–death process with rates varying across lineages is calculated, based on Keiding (1975) and Silvestro et al. (2014), as: Pðs; e; bjk; l; ak ; al Þ ¼

N h i Y kðbi Þlðbi Þe½kðbi Þþlðbi Þðsi ei Þ i

We used this function to investigate the presence of correlations between speciation and extinction rates and the

0 –0·5

We here introduce an alternative birth–death model (hereafter the ‘Covar model’) in which the speciation and extinction rates can vary on a lineage-specific basis as the result of a correlation with a continuous trait. Let us assume that for a set of N species we have the corresponding values of a quantitative trait, which we indicate with the vector B = [b1, . . ., bN]. The speciation rate is modelled on a species-specific basis using the following transformation:

0·5 1·0 1·5 2·0 2·5

TRAIT-CORRELATED DIVERSIFICATION

Time [Ma]

Fig. 2. Rate-through-time plots for the marginal rates of speciation (green), extinction (blue) and net diversification (grey) through time for the Ursidae obtained through BDMCMC. Solid lines show the mean rate estimates, shaded areas display the associated 95% credibility intervals. Insets show the posterior estimates of the parameters correlating speciation (a) and extinction (b) rates with changes in body mass.

log-transformed body mass within the Ursidae. The correlation parameters (ak, al) were assigned a standard normal prior distribution and estimated from the data. Body mass (BM) data for the extinct and extant species were obtained from the NOW (Fortelius 2013) and PanTHERIA (Jones et al. 2009) data bases, respectively. Since BM data were available only for 38% of the extinct species, the missing


1130 D. Silvestro et al. data were iteratively sampled during the MCMC. The analysis was set up by using: python Pyrate.py Ursidae_PyRate.py -N 8 -mG -mCov 1 -A

Table 2. Comparison of different birth–death models by marginal likelihoods. Bayes factors are used to quantify the support of the best model (highlighted in bold) against all alternative models (see text for detailed model specifications)

0

where ‘-mCov 1’ sets the Covar model and ‘-A 0’ specifies that the MCMC should sample under a fixed number of birth–death rates (by default constant through time) that is without sampling the number of rate shifts by BDMCMC. The resulting posterior distributions of the correlation parameters were found to be positive, but not significantly different from zero (ak = 027 HPD: 008–060, al = 030 HPD: 090–072). The correlation between the diversification rates and the trait might be confounded by the existence of temporal changes in the baseline rates. Therefore, a second analysis was run to jointly estimate variation of the baseline rate through time by BDMCMC and rate changes across clades according to the Covar model. This was done using the command: python Pyrate.py Ursidae_PyRate.py -N 8 -mG -mCov 3

As in the previous BDMCMC analysis, we found one shift in both speciation and extinction (baseline) rates. The correlation parameters, however, differed from the previous estimates indicating a strong positive correlation between speciation rate and log BM (ak = 052 HPD: 015–090; Fig. 2a), whereas no significant correlation was detected with the extinction rates (al = 022 HPD: 023–065; Fig. 2b). HYPOTHESIS TESTING USING BAYES FACTORS

An important component of PyRate is the ability to compare the fit of alternative diversification scenarios, for example the different birth–death models described above. In Bayesian statistics, we can rank competing hypotheses by their relative probability and use Bayes factors (BF) to compare their fit (Kass & Raftery 1995). This is done in PyRate by computing the marginal likelihoods of the different models using thermodynamic integration (TI; Lartillot & Philippe 2006). The estimation of the marginal likelihood is performed iteratively under each model using the command ‘-A 1’ to select the TI algorithm. For instance with:

Model

Number of parameters

Marginal likelihood

BF

BD11 BD21 BD11Cov BD22 BD22Covk BD22Cov

2 4 4 6 7 8

39079 38332 37967 36821 36587 36699

4984 (very strong) 3490 (very strong) 276 (very strong) 468 (positive) 0 224 (positive)

rates and Covar models (Table 2). As indicated in the BDMCMC analysis, the birth–death model with a rate shift in both speciation and extinction (BD22) obtained a better fit than a model with a shift in speciation and constant extinction rate (BD21). The BD22 also received strong support against a model with constant baseline rates and birth–death rates correlated with BM (BD11Covar). An improvement of the model fit was obtained after combining the Covar model with shifts in the baseline rates of speciation and extinction (BD22Covar). The previous analysis for parameter estimation under this model suggested that a significant correlation with the BM is only detected for speciation rates, while extinction appears to be independent of it (Fig. 1 insets A,B). This observation is confirmed by the model comparisons, in which indeed a model featuring only a correlation with speciation rate (BD22Covk), that is with constrained al = 0 (specified by ‘-mCov 1’), yielded the best fit among all tested models (Table 2).

System requirements and resources PyRate is written in Python 2.7 (using the Numpy and Scipy libraries; Python Software Foundation. Python Language Reference, version 2.7. Available at http://www.python.org) and has been tested on different UNIX operating systems. The program supports multithread likelihood computation and parallelization of the MCMC runs. The pyrate_utilities.r script is written in R and runs cross-platform using the package ‘fitdistrplus’ (Delignette-Muller et al. 2013).

python PyRate.py Ursidae_PyRate.py -N 8 -mG -A 1 -mL 2 -mM 1 -b 500000

we define a birth–death model with two speciation rates and one extinction rate (specified by ‘-mL 2 -mM 1’) and set the burn-in to exclude the first 500,000 iterations. This analysis generates a ‘*_marginal_likelihood.txt’ file where the model’s marginal likelihood is logged (note that posterior parameter estimates cannot be obtained from TI runs). Bayes factors can be computed using the function -BF. To demonstrate the use of BF for model selection, we calculated the marginal likelihoods under a range of different models including birth–death with constant and variable

Acknowledgements Analyses were run on the VITAL-IT cluster of the Swiss Institute for Bioinformatics. We thank Liam J. Revell, David W. Bapst and Rampal S. Etienne for helpful comments on this paper and John Alroy and Alan Turner for fossil data. DS was funded by Wenner-Gren foundation; JS was supported by the funding €koprogram LOEWE – ‘Landes-Offensive zur Entwicklung Wissenschaftlich- o nomischer Exzellenz’ of Hesse’s Ministry of Higher Education, Research, and the Arts.

Data accessibility The PyRate package, with detailed manual and template input files (including the Ursidae data set used in this paper), is accessible at http://sourceforge.net/projects/pyrate/.


Speciation and extinction rates from fossils

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Supporting Information Additional Supporting Information may be found in the online version of this article. Appendix S1. Ursidae fossil occurrence data. Appendix S2. PyRate input file for diversification analysis.
