Bayesian Molecular Clock Dating Using Genome-Scale Datasets

Abstract

Bayesian methods for molecular clock dating of species divergences have been greatly developed during the past decade. Advantages of the methods include the use of relaxed-clock models to describe evolutionary rate variation in the branches of a phylogenetic tree and the use of flexible fossil calibration densities to describe the uncertainty in node ages. The advent of next-generation sequencing technologies has led to a flood of genome-scale datasets for organisms belonging to all domains in the tree of life. Thus, a new era has begun where dating the tree of life using genome-scale data is now within reach. In this protocol, we explain how to use the computer program MCMCTree to perform Bayesian inference of divergence times using genome-scale datasets. We use a ten-species primate phylogeny, with a molecular alignment of over three million base pairs, as an exemplar on how to carry out the analysis. We pay particular attention to how to set up the analysis and the priors and how to diagnose the MCMC algorithm used to obtain the posterior estimates of divergence times and evolutionary rates.

1 Introduction

The molecular clock hypothesis, which states that the rate of molecular evolution is approximately constant with time, provides a powerful way to estimate the times of divergence of species in a phylogeny. Since its proposal over 50 years ago [1], the molecular clock hypothesis has been used countless times to calibrate molecular phylogenies to geological time, with the ultimate aim of dating the tree of life [2, 3]. Several statistical inference methodologies have been developed for molecular clock dating analyses; however, during the past decade, the Bayesian method has emerged as the method of choice [4, 5], and several Bayesian inference software packages now exist to carry out this type of analysis [6,7,8,9,10].

In this protocol, we will explain how to use the computer program MCMCTree to estimate times of species divergences using genome-scale datasets within the Bayesian inference framework. Bayesian inference is well suited for divergence time estimation because it allows the natural integration of information from the fossil record (in the form of prior statistical distributions describing the ages of nodes in a phylogeny) with information from molecular sequences to estimate node ages, or geological times of divergence, of a species phylogeny [6, 11]. Another advantage of the Bayesian clock dating method is that relaxed-clock models, which allow for violations of the molecular clock, can be easily implemented as the prior on the evolutionary rates for the branches in the phylogeny [6]. MCMCTree allows analyses to be carried out using two popular relaxed-clock models (the autocorrelated and independent log-normally distributed rates models [12, 13]), as well as under the strict molecular clock. Furthermore, MCMCTree allows the user to build flexible fossil calibrations based on various statistical distributions (such as the uniform, truncated-Cauchy, and skew-t, and skew-normal distributions [12, 14, 15]). But perhaps the main advantage of MCMCTree is the implementation of an approximate algorithm to calculate the likelihood [6, 16], which allows the computer analysis of genome-scale datasets to be completed in reasonable amounts of time. The disadvantage of the algorithm is that it only works on fixed tree topologies. Several software packages that perform co-estimation of times and tree topology, but which do not use the approximation, are available [8, 9, 17, 18].

In this protocol, we focus on how to carry out a clock dating analysis with MCMCTree, paying particular attention to diagnosing the MCMC algorithm (the workhorse algorithm within the Bayesian method). Theoretical details of the Bayesian clock dating methods implemented in the program MCMCTree are described in [12,13,14,15,16, 19]. For general introductions to Bayesian statistics and Bayesian molecular clock dating, the reader may consult [20, 21].

2 Software and Data Files

To run the protocol, you will need the MCMCTree and BASEML programs, which are part of the PAML software package for phylogenetic analysis [22]. The source code and compiled versions of the code are freely available from bit.ly/ziheng-paml. All the data files necessary to run the protocol can be obtained from github.com/mariodosreis/divtime. Please create a directory called divtime in your computer and download all the data files from the GitHub repository. This protocol was tested with PAML version 4.9e.

You are assumed to have basic knowledge of the command line in Unix or Windows (also known as command prompt, shell, or terminal). Simple tutorials for users of Windows, Mac OS, and Linux are posted at bit.ly/ziheng-software. Install MCMCTree and BASEML in your computer system, and make sure you have the mcmctree and baseml executables in your system’s path (see bit.ly/ziheng-paml for details on how to do this). Finally, it is helpful (but not indispensable) to have knowledge of the R statistical environment (www.r-project.org). R is quite useful to analyze the output of the program, perform convergence diagnostics, and create nice-looking plots. File R/analysis.R contains some examples for this tutorial.

In this protocol, we will estimate the divergence times of nine primates and one scandentian (an out-group), using a very long alignment (over three million nucleotides long). This dataset was chosen because it can be analyzed very quickly with MCMCTree and it is thus suitable to illustrate the method. We also provide a dataset of 330 species (276 primates and 4 out-groups) with a shorter alignment, to illustrate time estimation in a taxon-rich dataset (see Sect. 5.5 for details).

2.1 Tree and Fossil Calibrations

The phylogenetic tree of the ten species is shown in Fig. 1. The tree encompasses members of all the main primate lineages. The ten species were chosen because they have had their complete genomes sequenced. They are a subset of the 36 mammal species analyzed in [23]. File data/10s.tree contains the tree with fossil calibrations in Newick format, which is the format required by MCMCTree. The eight fossil calibrations are shown in Table 1. The calibrations are the same used to estimate primate divergence times in [24]. We discuss fossil calibrations in detail in the “Sampling from the Prior” section. The time unit in the analysis is 100 million years (My). Thus, the calibration B(0.075, 0.10) means the node age is constrained to be between 7.5 and 10 million years ago (Ma).

Fig. 1

The tree of ten species. Nodes with fossil calibrations are indicated with black dots (see Table 1 for calibration densities). Internal nodes are numbered from 11 to 19 according to the nomenclature used by MCMCTree

Table 1 List of fossil calibrations used in this tutorial

2.2 Molecular Sequence Data

The molecular data are an alignment of 5614 protein-coding genes from the ten species. All ambiguous codon sites were removed, and thus the alignment contains no missing data. The alignment was separated into two partitions: A partition consisting of all the first and second codon positions (2,253,316 nucleotides long) and a partition of third codon positions (1,126,658 nucleotides long). The alignment is a subset of the larger 36-mammal-species alignment in [23]. See also ref. 24. File 10s.phys in the data directory contains the alignment. The alignment is compressed into site patterns (a site pattern is a unique combination of character states in an alignment column) to save disk space.

3 Tutorial

We seek to obtain the posterior distribution (i.e., the estimates) of the divergence times (t) and the molecular evolutionary rates (r, μ, σ ²) for the species in the phylogeny of Fig. 1. Here t = (t ₁₁, …, t ₁₉) are the nine species divergence times; r = (r _1,12, …, r _1,19, r _2,12, …, r _2,19) are the 2 × 8 = 16 molecular rates, one per branch and partition (i.e., there are eight branches in the tree and two partitions in the molecular data); and μ = (μ ₁, μ ₂) and σ ² = (${\sigma}_1^2,{\sigma}_2^2$) are the mean rates and the log-variance of the rates, for each partition. The posterior distribution is

$$ f\left(\mathbf{t},\mathbf{r},\mu, {\sigma}^2|D\right)\propto f\left(\mathbf{t}\right)f\left(\mathbf{r}|\mathbf{t},\mu, {\sigma}^2\right)f\left(\mu \right)f\left({\sigma}^2\right)f\left(D|\mathbf{r},\mathbf{t}\right), $$

where f(t) is the prior on times; f(r|t, μ, σ ²)f(μ)f(σ ²) is the prior on the branch rates, mean rates, and variances of the log-rates; and f(D|t, r) is the molecular sequence likelihood. The prior on the times is constructed by combining the birth-death process with the fossil calibration densities (see ref. 13 for details). The prior on the rates is constructed under a model of rate evolution, assuming, in this tutorial, that the branch rates are independent draws from a log-normal distribution with mean μ _i and log-variance ${\sigma}_i^2$ [13].

Bayesian phylogenetic inference using MCMC is computationally expensive because of the repeated calculation of the likelihood on a sequence alignment. The time it takes to compute the likelihood is proportional to the number of site patterns in the alignment. Thus, longer alignments take longer to compute. For genome-scale alignments, the computation time is prohibitive.

MCMCTree implements an approximation to the likelihood that speeds computation time substantially, making analysis of genome-scale data feasible. The approximate likelihood method for clock dating was proposed by Thorne et al. [6] and extended within MCMCTree [16]. The method relies on approximating the log-likelihood surface on the branch lengths by its Taylor expansion. Write ℓ(b _j) =  log f(D| b _j) for the log-likelihood as a function of the branch lengths b _j = (b _j,i = r _j,i t _i) for the alignment partition j. The Taylor approximation is

$$ \mathrm{\ell}\left({\mathbf{b}}_j\right)\approx \mathrm{\ell}\left({\widehat{\mathbf{b}}}_j\right)+{\left({\mathbf{b}}_j-{\widehat{\mathbf{b}}}_j\right)}^{\mathrm{T}}{\mathbf{g}}_j+\frac{1}{2}{\left({\mathbf{b}}_j-{\widehat{\mathbf{b}}}_j\right)}^{\mathrm{T}}{\mathbf{H}}_j\left({\mathbf{b}}_j-{\widehat{\mathbf{b}}}_j\right), $$

where ${\widehat{\mathbf{b}}}_j$ are the maximum likelihood estimates (MLEs) of the branch lengths and g _j and H _j are the gradient (vector of first derivatives) and Hessian (matrix of second derivatives) of the log-likelihood surface evaluated at the MLEs for the partition. The approximation can be improved by applying transformations to the branch lengths (see ref. 16 for details).

To use the approximation, one first fixes the topology of the phylogeny, and then estimates the branch lengths for each alignment partition on the fixed tree by maximum likelihood. The gradient and Hessian of the log-likelihood are obtained for each partition at the same time as the MLEs of the branch lengths. Note that parameters of the substitution model—such as the transition/transversion ratio, κ, in the HKY model or the α parameter in the discrete gamma model of rate variation among sites—are estimated at this step. Thus, different substitution models will generate different approximations, because they will have different MLEs for the branch lengths, gradient, and Hessian. Note that the time it takes to compute the approximate likelihood depends only on the number of species (which determines the size of b and H) and not on the alignment length, that is, once g and H have been calculated, MCMC sampling on the approximation takes the same time regardless of the length of the original alignment.

3.1 Overview

We will use the approximate likelihood method to speed up the computation of the likelihood on the large genome alignment. The general strategy for the analysis is as follows:

• 1. Approximate likelihood calculation: First, we will calculate the gradient (g) and Hessian (H) matrix of the branch lengths on the unrooted tree. For this step, we will need to use the MCMCTree and BASEML programs (BASEML will carry out the actual computation of g and H). The substitution model is chosen at this step.

• 2. MCMC sampling from the posterior: Once g and H have been calculated and we have decided on our priors, we can use MCMCTree to perform MCMC sampling from the posterior distribution of times and rates. We will then look at the summaries of the posterior (such as posterior mean times and rates and 95% credibility intervals).

• 3. Convergence diagnostics: The MCMC algorithm is a stochastic algorithm that visits regions of the parameter space in proportion to the posterior distribution. Due to its very nature, it is possible that sometimes the MCMC chain is terminated before it has had a chance to explore the parameter space appropriately. The way to guard against this is to run the analysis two or more times and compare the summary statistics from the two (or more) MCMC chains. If the results from different runs are very similar, then convergence to the posterior distribution can be reasonably assumed.

• 4. MCMC sampling from the prior: Finally, we will sample directly from the prior of times and rates. This is particularly important in Bayesian molecular clock dating because in most cases the prior on times may look quite different from the fossil calibration densities specified by the user. Thus, sampling from the prior allows the user to check the soundness of the prior actually used.

Note that in this protocol we assume the user has chosen a suitable sequence alignment and a phylogenetic tree to carry out the analysis. For genome-scale alignments, it is important that the genes chosen among the various species are orthologous and that the alignment has been checked for accuracy. Several chapters in this volume can guide the user in this purpose.

3.2 Calculation of the Gradient and Hessian to Approximate the Likelihood

Go into the gH directory, and open the mcmctree-outBV.ctl file using your favorite text editor. This control file contains the set of parameters necessary for MCMCTree to carry out the calculations of the gradient and Hessian needed for the approximate likelihood method. Figure 2 shows the contents of the mcmctree-outBV.ctl file.

Fig. 2

The gH/mcmctree-outBV.ctl file, with appropriate options to set up calculation of the gradient and Hessian matrix for the approximate likelihood method

The first two items, seqfile and treefile, indicate the alignment and tree files to be used. The third item, ndata, indicates the number of partitions in the sequence file, in this case, two partitions. The fifth item, usedata, is very important, as it tells MCMCTree the type of analysis being carried out. The options are 0, to sample from the prior; 1, to sample from the posterior using exact likelihood; 2, to sample from the posterior using approximate likelihood; and 3, to prepare the data for calculation of g and H. The last is the option we will be using in this step. The next three items, model, alpha, and ncatG, set up the nucleotide substitution model, in this case the HKY + Gamma model [25]. Finally, the cleandata option tells MCMCTree whether to remove ambiguous data. Our alignment has no ambiguous sites, so this option has no effect in this case.

Using a terminal, go to the gH directory and type

$ mcmctree mcmctree-outBV.ctl

(Don’t type in the $ as this represents the command prompt!) This will start the MCMCTree program. MCMCTree will prepare several tmp????.* files and will then call the BASEML program to estimate g and H. For this step to work correctly, the baseml executable must be in your system’s path. Once BASEML and MCMCTree have finished, you will notice a file called out.BV has been created. Figure 3 shows part of the contents of this file. The first line indicates the number of species (10), followed by the tree with branch lengths estimated under maximum likelihood for the first partition (first and second codon sites). Next, we have the MLEs of the 17 branch lengths (these are the same as in the tree but printed in a different order). Then we have the gradient, g ₁, the vector of 17 first derivatives of the likelihood at the branch length MLEs for partition 1. For small datasets, the gradient is usually zero. For large datasets, the likelihood surface is too sharp (i.e., bends downward sharply and it is very narrow at the MLEs), and the gradient is not zero for numerical issues. But this is fine. Next, we have the 17 × 17 Hessian matrix, H ₁, the matrix of second derivatives of the likelihood at the branch length MLEs for partition 1. If you scroll down the file, you will find the second block, with the tree, branch length MLEs, g ₂, and H ₂ for partition 2 (third codon positions).

Fig. 3

The gH/out.BV file produced by BASEML. The first line has the number of species (10), the second line has the tree topology with MLEs of branch lengths, and the MLEs of branch lengths are given again in the third line. The fourth line contains the gradient, g, followed by the Hessian, H, for partition 1. This file will be renamed in.BV and placed into the mcmc/ directory to carry out MCMC sampling using the approximate likelihood method

3.3 Calculation of the Posterior of Times and Rates

3.3.1 Control File and Priors

Now that we have calculated g and H, we can proceed to MCMC sampling of the posterior distribution using the approximate likelihood method. Copy the gH/out.BV file into the mcmc directory, and rename it as in.BV. Now go into the mcmc directory. There you will find mcmctree.ctl, the necessary MCMCTree control file to carry out MCMC sampling from the posterior. Figure 4 shows the contents of the file. The first item, seed, is the seed for the random number generator used by the MCMC algorithm. Here it is set to −1, which tells MCMCTree to use the system’s clock time as the seed. This is useful, as running the program multiple times will generate different outputs.

Fig. 4

The mcmc/mcmctree.ctl file necessary to sample from the posterior distribution using the approximate likelihood method

The mcmcfile option tells MCMCTree where to save the parameters sampled (divergence times and rates) during the MCMC iterations. Here we will save them to a file named mcmc.txt. Once the MCMC sampling has completed, MCMCTree will read the sample from the mcmc.txt file and generate a summary of the MCMC output. This summary will be saved to a file called out.txt (outfile option).

The option usedata is set to 2 here, which tells MCMCTree to calculate the likelihood approximately by using the g and H values saved in the in.BV file. Option clock sets the clock model. Here we use clock = 2, which assumes rates are identical, independent realizations from a log-normal distribution [7, 26]. Option RootAge sets the calibration on the root node of the phylogeny, if none are present in the tree file. In our case, we already have a calibration on the root, so this option has no effect. The next three options, model, alpha, and ncatG, have no effect as the substitution model was chosen during estimation of g and H.

The following options are very important as they determine the prior used in the analysis. BDparams sets the prior on node ages for those nodes without fossil calibrations by using the birth-death process [12]. Here we use 1 1 0, which means node ages are uniformly distributed between present time and the age of the root. Options kappa_gamma and alpha_gamma set gamma priors for the κ and α parameters in the substitution model. These have no effect as we are using the likelihood approximation. Options rgene_gamma and sigma2_gamma set the gamma-Dirichlet prior on the mean substitution rate for partitions and for the rate variance parameter, σ ² [19]. The prior on the mean rate is Gamma(2, 40), which has mean 0.05 substitutions per time 100 My. A symmetric Dirichlet distribution with concentration parameter equal to 1 is used to spread the rate prior across partitions (thus rgene_gamma = 2 40 1). See ref. 19 for details. The prior on σ ² is Gamma(1, 10) which has mean 0.1. A Dirichlet is also used to spread the prior across partitions.

The final block of options, print, burnin, sampfreq, and nsample, control the length and sampling frequency of the MCMC. We will discard the first 20,000 iterations as the burn-in and then print parameter values to the mcmc.txt file every 100 iterations, to a maximum of 20,000 + 1 samples. Thus, our MCMC chain will run for a total of 20,000 + 20,000 × 100 = 2,020,000 iterations.

3.3.2 Running and Summarizing the MCMC

Go into the mcmc directory and type

$ mcmctree mcmctree.ctl

This will start the MCMC sampling. First, MCMCTree will iterate the chain for a set number of iterations, known as the burn-in. During this period, the program will fine-tune the step sizes for proposing parameters in the chain. Once the burn-in is finished, sampling from the posterior will start. Figure 5 shows a screenshot of MCMCTree in action. The leftmost column indicates the progress of the sampling as a percentage of the total (5%, 10% of total iterations, and so on). The next numbers represent the acceptance proportions, which are close to 30% (this is the result of fine-tuning by the program). After the five acceptance proportions, the programs prints a few parameters to the screen and in the last columns the log-likelihood and the time taken.

Fig. 5

Screenshot of MCMCTree’s output during MCMC sampling of the posterior. Different runs of the program will give slightly different output values

The above analysis takes about 2 min and 30 s to complete on a 2.2 GHz Intel Core i7 Processor. Once the analysis has finished, you will see that MCMCTree has created several new files in the mcmc directory. Rename mcmc.txt to mcmc1.txt and out.txt to out1.txt. Now, on the command line, type again

$ mcmctree mcmctree.ctl

This will run the analysis a second time. The results should be slightly different to the previous run due to the stochastic nature of the algorithm. Once the second run has finished, rename mcmc.txt to mcmc2.txt and out.txt to out2.txt. If you want to conduct two runs simultaneously, you can create two directories (say r1/ and r2/) and copy the necessary files into them. Then open two terminal windows to start the runs from within each directory.

Using your favorite text editor, open file out1.txt, which contains the summary of the first MCMC run. Scroll to the end of the file (see screenshot, Fig. 6). You will see the time used by the program (in my case 2:32), the posterior means of the parameters sampled, and three phylogenetic trees in Newick format. The first tree simply has internal nodes labelled with a number. This is useful to compare the tree with the posterior means of times at the end of the file. The second tree is the tree with branch lengths in absolute time units. The third tree is like the second by including the 95% credibility intervals (CIs) of the node ages. At the bottom of the file, you have a table with all the divergence times (from t_n11 to t_n19), the mean substitution rates for the two partitions (mu1 and mu2), the rate variation coefficients (sigma2_1 and sigma2_2), and finally the log-likelihood (lnL). The table gives the posterior means, equal-tail CIs, and high-posterior-density CIs. For example, the posterior age of the root (node 11, Fig. 1) is 116.8 Ma (95% CI, 144.2–92.4 Ma) while for the divergence between human and chimp (node 19, Fig. 1) is 8.52 Ma (95% CI, 7.58–9.81 Ma).

Fig. 6

The end of the mcmc/out.txt file produced by MCMCTree at the end of the MCMC sampling of the posterior

You will also notice that MCMCTree created a file called FigTree.tre. This contains the posterior tree in Nexus format, suitable for plotting in the program FigTree (tree.bio.ed.ac.uk/software/figtree/). Figure 7 shows the posterior tree plotted in FigTree, with the time unit set to 1 My.

Fig. 7

The dated primate phylogeny with error bars (representing 95% CIs of node ages), drawn with FigTree. The time unit is 1 My

3.4 Convergence Diagnostics of the MCMC

Diagnosing convergence of the MCMC chains is extremely important. Several software tools have been written for this purpose. For example, the user-friendly Tracer program (beast.bio.ed.ac.uk/tracer) can be used to read in the mcmc1.txt and mcmc2.txt files and calculate several convergence statistics. Here we will use R to perform basic convergence tests (check out file R/analysis.R).

The first step to assess convergence is to compare the posterior means among the different runs. You can visually inspect the posterior means reported in the out1.txt and out2.txt files (Fig. 8), although this may be cumbersome. Figure 8a shows a plot, made with R, of posterior times for run 1 vs. those from run 2. You can see that the points fall almost perfectly on the y = x line, indicating that both runs have converged to the same distribution (hopefully the posterior!).

Fig. 8

Convergence diagnostic plots of the MCMC drawn with R (see R/analysis.R)

Another useful statistic to be calculated is the effective sample size (ESS). This gives the user an idea about whether an MCMC chain has been run long enough. Tracer calculates ESS automatically for all parameters. Function coda::effectiveSize in R will do the same. Figure 9 shows the posterior mean, ESS, posterior variance, and standard error of posterior means calculated with R for run 1 of the MCMC. The longer the ESS, the better. As a rule of thumb, one should seek ESS larger than 1000, although this may not always be practical in phylogenetic analysis. Note in Fig. 9 that some estimates have very low ESSs, while others have substantially higher ESSs. For example, t_n11 has ESS = 76.1, while t_n19 has ESS = 1261. Running the analysis again and increasing the total number of iterations (e.g., by increasing samplefreq or nsample) will lead to higher ESS values for all parameters.

Fig. 9

Calculations of posterior mean, ESS, posterior variance, and standard error of the posterior mean in R (see R/analysis.R)

Let v be the posterior variance of a parameter. The standard error of the posterior mean of the parameter is S.E. = √(v/ESS). This is why having large ESS is important: Large ESS leads to small S.E. and better estimates of the posterior mean. For example, for t_n11, the posterior mean is 116.8 Ma, with standard error 1.53 My (Fig. 9). That is, we have estimated the mean accurately to within 2 × 1.53 My = 3.06 My. To reduce the S.E. by half, you need to increase the ESS four times. Note that independent MCMC runs can be combined into a single run. Thus, you may save time by running several MCMC chains in parallel for computationally expensive analyses, although care must be taken to ensure each chain has run long enough to exit the burn-in phase and explore the posterior appropriately.

Trace plots and histograms are useful to spot problems and check convergence. Figure 8b, c shows trace plots for t_n19 and t_n11, respectively. The trace of t_n19, which has high ESS, looks like a “hairy caterpillar.” Compare it to the trace of t_n11, which has low ESS. Visual inspection of a trace plot usually gives a sense of whether the parameter has an adequate ESS without calculating it. Note that both traces are trendless, that is, the traces oscillate around a mean value (the posterior mean). If you see a persistent trend in the trace (such as an increase or a decrease), that most likely means the MCMC did not converge to the posterior and needs a longer burn-in period.

Figure 8d shows the smoothed histograms (calculated using density in R) for t_n11 for the two runs. Notice that the two histograms are slightly different. As the ESS becomes larger, histograms for different runs will converge in shape until becoming indistinguishable. If you see large discrepancies between histograms, that may indicate serious problems with the MCMC, such as lack of convergence due to short burn-in or the MCMC getting stuck in different modes of a multimodal posterior.

3.5 MCMC Sampling from the Prior

Note that fossil calibrations (such as those of Table 1) are represented as statistical distributions of node ages. MCMCTree uses these distributions to construct the prior on times. However, the resulting time prior used by the program may be substantially different from the original fossil calibrations, because the program applies a truncation so that daughter nodes are younger than their ancestors [14, 27]. Thus, it is advisable to calculate the time prior explicitly by running the MCMC with no data so that it can be examined and compared with the fossil calibrations and the posterior.

Go to the prior directory and type

$ mcmctree mcmctree-pr.ctl

This will start the MCMC sampling from the prior. File mcmctree-pr.ctl is identical to mcmc/mcmctree.ctl except that option usedata has been set to 0. Sampling from the prior is much quicker because the likelihood does not need to be calculated. It takes about 1 min on the Intel Core i7 for MCMCTree to complete the analysis. Rename files mcmc.txt and out.txt to mcmc1.txt and out1.txt, and run the analysis again. Rename the new files as appropriate. Check for convergence by calculating the ESS and plotting the traces and histograms.

Figure 10 shows the prior densities of node ages obtained by MCMC sampling (shown in gray) vs. the posterior densities (shown in black). Notice that for four nodes t_n19, t_n18, t_n17, and t_n16, the posterior times “agree” with the prior, that is, the posterior density is contained within the prior density. For nodes t_n15, t_n13, and t_n11, there is some conflict between the prior and posterior densities. However, for nodes t_n14 and t_n12, there is substantial conflict between the prior and the posterior. In both cases the molecular data (together with the clock model) suggest the node age is much older than that implied by the calibrations. This highlights the problems in construction of fossil calibrations.

Fig. 10

Prior (gray) and posterior (black) density plots of node ages plotted with R (see R/analysis.R)

Each fossil calibration represents the paleontologist’s best guess about the age of a node. For example, the calibration for the human-chimp ancestor is B(0.075, 0.10, 0.01, 0.20); thus, the calibration is a uniform distribution between 7.5 and 10 million years ago (Ma). The bounds of the calibration are soft, that is, there is a set probability that the bound is violated. In this case the probabilities are 1% for the minimum bound and 20% for the maximum bound. The bound probabilities are asymmetrical because they reflect the nature of the fossil information. Minimum bounds are usually set with confidence because they are based on the age of the oldest fossil member of a clade. For example, the minimum of 7.5 Ma is based on the age of †Sahelanthropus tchadensis, recognized as the oldest fossil within the human lineage [28]. On the other hand, establishing maximum bounds is difficult, as absence of fossils for certain clades cannot be interpreted as evidence that the clade in question did not exist during a particular geological time [29]. Our maximum here of 10 Ma represents the paleontologist’s informed guess about the likely oldest age of the clade; however, a large probability of 20% is given to allow for the fact that the node age could be older. The conflict between the prior and posterior seen in Fig. 10 evidences this.

Note that when constructing the time prior, the Bayesian dating software must respect the constraints whereby daughter nodes must be younger than their parents. This means that calibration densities are truncated to accommodate the constraint, with the result that the actual prior used on node ages can be substantially different to the calibration density used (see Sect. 5.4). Detailed analyses of the interactions between fossil calibrations and the time prior and the effect of truncation are given in [14, 27].

4 General Recommendations for Bayesian Clock Dating

Extensive reviews of best practice in Bayesian clock dating are given elsewhere [4, 20, 21, 30, 31]. Here we give a few brief recommendations.

4.1 Taxon Sampling, Data Partitioning, and Estimation of Tree Topology

In this tutorial we used a small phylogeny to illustrate Bayesian time estimation using approximate likelihood calculation. In practical data analysis, it may be desirable to analyze much larger phylogenies (see Sect. 5.5). In large phylogenies, there may be uncertainties in the relationships of some groups. The approximate method discussed here can only be applied to a fixed (known) tree topology. If the uncertainties in the tree are few so that just a handful of tree topologies appear reasonable, the approximate method can be used by analyzing each topology separately [23, 32]. This involves estimating g and H for each topology and then running separate MCMC chains on each topology to estimate the times. Several methods to co-estimate divergence times and tree topology are available [8, 9, 17, 18], although they do not implement the approximate likelihood method and are thus unsuitable for the analysis of genome-scale datasets.

We note that partitioning of sites in genomic datasets may have important effects on divergence time estimation. The infinite-sites theory [13, 33] studies the asymptotic behavior of the posterior distribution of times when the amount of molecular data (measured by the number of partitions and the number of sites per partition) increases in a relaxed-clock dating analysis. This theory shows that increasing the number of sites per partition will have minimal effects on time estimation when the sequences per partition are moderately long (>1000 sites, say), but the precision improves when the number of partitions increases, eventually approximating a limit when the number of partitions is infinite. The theory also predicts that very different time estimates may be obtained if the same genomic sequence alignment is analyzed as one partition or as multiple partitions [34]. Furthermore, while more partitions tend to produce more precise time estimates, with narrow CIs, they may not necessarily be more reliable, depending on the correctness of the fossil calibrations and the appropriateness of the partitioning strategies. Unfortunately it is hard to decide on a good partitioning strategy given the genome-scale sequence data, despite efforts to design automatic partitioning strategies for phylogenetic analysis and divergence time estimation [34,35,36]. Commonly used approaches partition sites in the alignment by codon position or by protein-coding genes of different relative rates [23]. We recommend the use of the infinite-sites plot [14], in which uncertainty in divergence time estimates (measured as the CI width) is plotted against the posterior mean of times. If the scatter points fall on a straight line, information due to the molecular sequence data has reached saturation, and uncertainty in time estimate is predominantly due to uncertainties in fossil calibrations.

4.2 Selection of Fossil Calibrations

Fossil calibrations are one of the most important pieces of information needed to perform divergence time estimation and thus should be chosen after careful consideration of the fossil record, although this may involve some subjectivity [29]. Parham et al. [30] discuss best practice for construction of fossil calibrations. For example, minimum bounds on node ages are normally set to be the age of the oldest fossil member of the crown group. A small probability (say 2.5%) should be set for the probability that the node age violates the minimum bound (e.g., to guard against misidentified or incorrectly dated fossils). Specifying maximum bounds is more difficult, as absence of fossils for a given geological period is not evidence that the clade in question was absent during the period [31]. Current practice is to set the maximum bound to a reasonable value according to the expertise of the paleontologist (see ref. 29 for examples), although a large probability (say 10% or even 20%) may be required to guard against badly specified maximum bounds. Calibration densities based on statistical modeling of species diversification, fossil preservation, and discovery are also possible [15]. In so-called tip-dating approaches, fossil species are included as taxa in the analysis (which may or may not include morphological information for the fossil and extant taxa) [37,38,39]. Thus, in tip-dating, explicit specification of a fossil calibration density for a node age is not necessary.

4.3 Construction of the Time Prior

The birth-death process with species sampling was used here to construct the time prior for nodes in the phylogeny for which fossil calibrations are not available. Varying the birth (μ), death (λ), and sampling (ρ), parameters can result in substantially different time priors. For example, using μ = λ = 1 and ρ = 0 leads to a uniform distribution prior on node ages. This diffuse prior appears appropriate for most analyses. Varying the values of μ, λ, and ρ is useful to assess whether the time estimates are robust to the time prior. Parameter configurations can be set up to generate time densities that result in young node ages or in very old node ages (see p. 381 in [20] for examples).

4.4 Selection of the Clock Model

In analysis of closely related species (such as the apes), the clock assumption appears to be appropriate for time estimation. A likelihood ratio test can be used to determine whether the strict clock is appropriate for a given dataset [40]. If the clock is rejected, then Bayesian molecular clock dating should proceed using one of the various relaxed-clock models available [7, 13]. In this case, Bayesian model selection may be used to choose the most appropriate relaxed-clock model [41], although the method is computationally expensive and thus only applicable to small datasets. The use of different relaxed-clock models (such as the autocorrelated vs. the independent log-normally distributed rates) may result in substantially different time estimates (see ref. 32 for an example). In such cases, repeating the analysis under the different clock models may be desirable.

5 Exercises

5.1 Autocorrelated Rate Model

Modify file mcmc/mcmctree.ctl and set clock = 3. This activates the autocorrelated log-normal rates model, also known as the geometric Brownian motion rates model [6, 13]. Run the MCMC twice and check for convergence. Compare the posterior times obtained with those obtained under the independent log-normal model (clock = 2). Are there any systematic differences in node age estimates between the two analyses? Which clock model produces the most precise (i.e., narrower CIs) divergence time estimates?

5.2 MCMC Sampling with Exact Likelihood Calculation

Modify file mcmc/mcmctree.ctl and set clock = 2 (independent rates), usedata = 1 (exact likelihood), burnin = 200, sampfreq = 2, and nsample = 500. These last three options will lead to a much shorter MCMC chain, with a total of 1200 iterations. Run the MCMC sampling twice, and check for convergence using the ESS, histograms, and trace plots. How long does it take for the sampling to complete? Can you estimate how long it would take to run the analysis using 2,020,000 iterations, as long as for the approximate method of Sect. 3.3.2? Did the two chains converge despite the low number of iterations?

5.3 Change of Fossil Calibrations

There is some controversy over whether †Sahelanthropus, used to set the minimum bound for the human-chimp divergence, is indeed part of the human lineage. The next (younger) fossil in the human lineage is †Orrorin which dates to around 6 Ma. Modify file data/10s.tree and change the calibration in the human-chimp node to B(0.057, 0.10, 0.01, 0.2). Also change the calibration on the root node to B(0.615, 1.315, 0.01, 0.05). Run the MCMC analysis with the approximate method and again sampling from the prior. Are there any substantial differences in the posterior distributions of times under the new fossil calibrations? Which nodes are affected? How bad is the truncation effect among the calibration densities and the prior?

5.4 Comparing Calibration Densities and Prior Densities

This is a difficult exercise. Use R to plot the prior densities of times sampled using MCMC (the same as in Fig. 10). Now try to work out how to overlay the calibration densities onto the plots. For example, see Fig. 3 in [23] for an idea. First, write functions that calculate the calibration densities. The dunif function in R is useful to plot uniform calibrations. Functions sn::dsn and sn::dst (in the SN package) are useful to plot the skew-t (ST) and skew-normal (SN) distributions. Calibration type S2N (Table 1) is a mixture of two skew-normal distributions [15]. How do the sampled priors compare to the calibration densities? Are there any substantial truncation effects?

5.5 Time Estimation in a Supermatrix of 330 Species

Good taxon sampling is critical to obtaining robust estimates of divergence times for clades. In the data/ directory, an alignment of the first and second codon positions from mitochondrial protein-coding genes from 330 species (326 primate and 4 out-group species) is provided, 330s.phys, with corresponding tree topology, 330s.tree. First, place the fossil calibrations of Table 1 on the appropriate nodes of the species tree. Then obtain the gradient and Hessian matrix for the 330-species alignment using the HKY + G model. Finally, estimate the divergence times on the 330-species phylogeny by using the approximate likelihood method. How does taxon sampling affect node age estimates when comparing the 10-species and 330-species trees? How does uncertainty in node ages in the large tree, which was estimated on a short alignment, compare with the estimates on the small tree, but with a large alignment?