^{*}

The author has declared that no competing interests exist.

Conceived and designed the experiments: DLR. Performed the experiments: DLR. Analyzed the data: DLR. Contributed reagents/materials/analysis tools: DLR. Wrote the paper: DLR.

A number of methods have been developed to infer differential rates of species diversification through time and among clades using time-calibrated phylogenetic trees. However, we lack a general framework that can delineate and quantify heterogeneous mixtures of dynamic processes within single phylogenies. I developed a method that can identify arbitrary numbers of time-varying diversification processes on phylogenies without specifying their locations in advance. The method uses reversible-jump Markov Chain Monte Carlo to move between model subspaces that vary in the number of distinct diversification regimes. The model assumes that changes in evolutionary regimes occur across the branches of phylogenetic trees under a compound Poisson process and explicitly accounts for rate variation through time and among lineages. Using simulated datasets, I demonstrate that the method can be used to quantify complex mixtures of time-dependent, diversity-dependent, and constant-rate diversification processes. I compared the performance of the method to the MEDUSA model of rate variation among lineages. As an empirical example, I analyzed the history of speciation and extinction during the radiation of modern whales. The method described here will greatly facilitate the exploration of macroevolutionary dynamics across large phylogenetic trees, which may have been shaped by heterogeneous mixtures of distinct evolutionary processes.

Perhaps the most general feature of biological diversity on Earth is the extent to which it varies - either through space, through time, or among different kinds of organisms. Biologists have long been fascinated by the observation that some groups of organisms contain far more species than other groups. For example, within vertebrates, lineages such as tetrapods (22000+ species), therian mammals (5000+ species), and teleosts (30000+ species) are several orders of magnitude more diverse than their respective sister clades (lungfishes, 6 species; monotremes, 5 species; holosteian fishes, <10 species). This phylogenetic variation in species richness is mirrored by analogous variation in diversity through time. Paleontological evidence indicates that species richness has undergone dramatic changes during the past 550 million years

The fossil record has provided insight into the temporal dynamics of species diversification

To date, few macroevolutionary studies have simultaneously accounted for rate variation through time and among lineages

In this article, I introduce a new framework for studying patterns of rate variation through time and among lineages using time-calibrated phylogenies of extant species. The approach is premised on the idea that phylogenetic trees are frequently shaped by heterogeneous mixtures of distinct processes. For example, some phylogenies may reflect mixtures of both diversity-dependent and constant-rate diversification processes (

(A) Clade diversification under constant-rate “background” diversification process with λ = 0.032 and μ = 0. (B) Shift to new adaptive zone with subsequent diversity-dependent regulation of speciation and diversity-independent extinction (blue branches; λ_{0} = 0.395; K = 66; μ = 0.041). (C) Another lineage shifts to diversity-dependent speciation regime (red branches; λ_{0} = 0.21; K = 97; μ = 0.012). Total tree depth is 100 time units. Despite undergoing two distinct diversity-dependent slowdowns in the rate of speciation, the overall gamma statistic

My general approach assumes that shifts between macroevolutionary regimes occur across the branches of a phylogenetic tree under a compound Poisson process. This framework has been used previously to model among-lineage variation in rates of molecular evolution

The method described here differs from previous methods in several key respects. First, the method does not assume that rates of speciation and extinction are constant through time within clades, thus relaxing the assumption of time-homogeneous diversification used in most previous multi-model approaches

The model assumes that phylogenetic trees are shaped by a countable set of distinct and potentially dynamic evolutionary processes of speciation and extinction. Transitions between processes, or “events”, are assumed to occur across the branches of the phylogeny under a compound Poisson process _{i}_{i}_{i}_{i}_{i}

Any tree is necessarily governed by at least one process that begins at the root node and the number of additional transitions is a Poisson-distributed random variable with rate parameter _{R}_{i}_{i} are governed by a new evolutionary process, _{i}_{k}. The addition of a process to a tree with _{k} to M_{k+1}. There is no upper bound on the number of transitions, as multiple transitions can occur on a given branch. The minimum model, with a single process, corresponds to

I assume that each process _{i} relative to the start of the process and where _{0,i}_{i}_{0,0}_{0}

I implemented the model in a Bayesian framework. Bayesian approaches have already been used effectively to model single processes on phylogenetic trees

The full model contains parameters for the overall rate at which transitions occur (_{k} → M_{k+1}_{k}_{k-1}_{1}_{2}_{N}_{i}_{0, i}_{i}_{i}

For within-model moves that do not involve changes in the dimensionality of the full model, acceptance probabilities follow the standard Metropolis-Hastings formulation

The acceptance probability for moves that transition between models requires a more general formulation _{k}_{k+1}_{k}_{k+1}_{k+1}_{k}_{k}_{k}

In the model described here, an increase from _{k}_{k+1}_{k+1}, _{0,k+1}, z_{k+1},_{k+1}

Variables _{1}, ν_{2}, ν_{2},_{3}_{0}, z,

Under the compound Poisson process, the overall (whole-tree) rate at which transitions occur under the model is _{k+1} to M_{k}, given Λ, is simply the ratio of Poisson densities with

To compute the proposal probability q(_{k}_{k+1}_{k+1}_{k}_{k}_{k}_{1}_{1}_{2}_{1}_{k-1}_{/}_{k}_{k+1}_{k}_{k}_{k+1}

Because elements of _{0}, z,_{k+1}_{k}

The acceptance probability for a move that deletes one of _{k-1}_{k}

The positions of transitions were updated using global and local moves. A global move entailed sampling a new map location

To update any of the zero-bounded rate parameters in the model (Λ, λ_{i}, µ_{i}), I used a proportional shrinking-expanding proposal

_{i}

I placed a uniform (0,

Likelihoods were computed on branches using a discretization of the constant-rate birth death model that enabled us to approximate time-dependent and diversity-dependent rate variation. Following the notation from Maddison et al. _{N}_{N}

_{0}_{0}_{0}_{0}_{0}_{0}

To discretize the rate calculations, I broke each branch of the phylogenetic tree into segments and computed the mean speciation rate under the exponential change model (_{0}_{0}

To compute the likelihood of the full tree under a given set of parameters, we perform the calculations described above on each terminal branch of the phylogeny. Initial values of _{0}_{0}_{0}_{0}_{L}_{R}_{R}(t)D_{L}(t)

I implemented the compound Poisson process model of rate variation described above in a C++ program, which I refer to as BAMM. BAMM (Bayesian Analysis of Macroevolutionary Mixtures) can estimate the number of distinct evolutionary regimes across phylogenetic trees and estimates marginal distributions of speciation and extinction rates for each branch in a phylogenetic tree. The model allows extinction rates to exceed speciation rates. BAMM and associated documentation is available from the BAMM project website (

To evaluate performance of the compound Poisson process model of diversification rate variation, I simulated phylogenetic trees under six general diversification models. I first considered a simple constant-rate birth death process (model CR; 1 process), to evaluate parameter bias and the frequency of overfitting when the generating model does not include a heterogeneous mixture of processes. Given the widespread interest in identifying well-supported rate shifts and key innovations on phylogenetic trees, we are particularly interested in the frequency with which the model described here will incorrectly identify a multi-process model as having the maximum

I also considered a model where a pure-birth diversification process shifts to an exponential change process at some point in time (model exp2; 2 processes). Finally, I considered four variants of diversity-dependent multi-process models. In each case, I assumed that a pure-birth process at the root of the tree underwent multiple (1, 2, 3, or 4) shifts to independent and decoupled diversity-dependent speciation-extinction processes (models DD2, DD3, DD4, DD5). I conducted 500 simulations per scenario.

Each multiprocess simulation was conducted by first simulating a pure-birth phylogeny for 100 time units with λ = 0.032. I then randomly chose a time T_{s} on the interval (40, 95) for the occurrence of a rate shift. A shift was then assigned randomly to one of the lineages that existed at time T_{s}. I then sampled parameters for the new process (see below). The tree was then broken at the shift point, and a new subtree was simulated forward in time from the shift point under the new process parameters. For trees with more than two processes, this procedure was repeated until the target number of processes had been added. For the exp2 model, this consisted of sampling

For the diversity-dependent models, diversification dynamics followed a linear diversity-dependent model _{t}_{0}

Each of the 500 simulations for each of 6 simulation scenarios was thus conducted under a potentially unique speciation-extinction parameterization. The number of taxa in each simulated tree also varied among datasets. I recorded the mean rate of speciation and extinction across each branch in each simulated tree. All simulations were conducted in C++; simulated trees are available through the Dryad data repository (doi:10.5061/dryad.hn1vn).

I analyzed each of the 3000 simulated datasets using BAMM with 3 million generations of MCMC sampling. I discarded the first half of samples from each simulation of the posterior as “burn-in” and estimated the overall “best model” as the model that was sampled most frequently by the Markov chain. I computed the mean of the posterior distribution of speciation and extinction rates on each branch for each tree. I then used OLS regression to assess the relationship between branch-specific rate estimates obtained using BAMM versus the true underlying evolutionary rates. As an additional estimate of bias, I computed the proportional error _{EST}_{TRUE}

I compared the performance of BAMM to that of MEDUSA

I analyzed each of the simulated datasets described above (500 datasets under each of 6 distinct models of diversification) using MEDUSA, using the implementation of MEDUSA available in the Geiger v1.99-3 package

Steeman et al.

_{1}_{0}

Example tree was simulated under three distinct processes (one constant rate and two diversity dependent processes; two transitions in total). The tree was analyzed under (i) the full multi-process BAMM model with time-variable speciation; (ii) a constrained multi-process BAMM with time-constant speciation; and (iii) a fully constrained 1-process constant-rate birth-death model. (A) Log-likelihoods for thinned MCMC chains for the constant rate birth-death process (bottom), the time-constant multi-process model (middle), and the full BAMM model with time-varying speciation (top). (B) Numbers of transitions during rjMCMC sampling when model is constrained to time-constant speciation rates; sidebar gives frequency distribution of sampled states. (C) Numbers of transitions under full BAMM model with time-variable speciation processes. Black sidebar denotes true number of transitions in generating model. The true number of transitions was estimated correctly only when the assumption of time-constant rates was relaxed.

Histograms in (A–C) display the frequency distribution of the estimated number of processes in the model with the maximum a posteriori (MAP) probability as a function of three different priors on the Poisson rate parameter _{0}). With a relatively flat prior on models (γ = 10), the MAP model is biased towards a model with 2 processes ( = 1 transition). However, the posterior probability of the true model M_{0} remains substantial (F), and M_{0} nonetheless had a posterior probability greater than 0.10 for the vast majority of simulations. Results are based on 500 simulated phylogenies per γ scenario.

For the five simulation scenarios with rate heterogeneity, the number of distinct processes estimated using BAMM was generally equal to the number of processes in the generating model (

For each simulation, the estimated number of processes was simply the model that was most frequently sampled during MCMC simulation of the posterior distribution. Black bars denote true number of processes in generating model. For example, 84% of trees simulated under a single-shift exponential change model (exp2; top panel; two processes in the generating model) were correctly inferred to have been generated under a two-process model. For a diversity-dependent model with five processes (DD5), power to detect the true number of processes is lower, although though most analyses (80.4%) recovered either 4 or 5 process models as the MAP model. Results for each model are based on 500 simulated phylogenies and used a conservative γ = 1 prior on the expected number of non-root processes (see

Estimates of speciation and extinction rates under the constant-rate model were highly correlated with rates in the generating model (

(A) Relationship between speciation rate in generating model and reconstructed mean rate across the tree under BAMM. Solid black line: identity line, expected if λ_{TRUE} = λ_{ESTIMATED}. Solid gray line: fitted OLS regression to estimates (black points) obtained using full BAMM model (multiple processes with time-variable speciation rates). (B) Corresponding extinction rate estimates for same set of trees.

Phylogenies were simulated under 5 distinct evolutionary scenarios. For each simulated phylogeny, I reconstructed branch-specific speciation rates using BAMM and modeled these as a function of the true branch rates from the generating model. Frequency distributions of the estimated slope of this relationship are shown in the left column for each simulation scenario. Center column denotes corresponding r^{2} values from the same OLS regressions. Right column is distribution of mean relative rate differences (RRD) for each scenario. A value of 1 implies that, on average, branch-specific speciation estimates are unbiased; a value of 0.5 would imply that branch-specific estimates are, on average, equal to 50% of the true value. Results for each simulation scenario are based on 500 simulated phylogenies (thus giving 500 slopes, r^{2} values, and RRD values for each simulation scenario).

A large fraction of the total variation in the underlying speciation rates is also explained by the estimated rates under the compound Poisson process model of diversification rate variation (

I performed a similar analysis for extinction rates, with one key difference. Each simulation scenario assumed constant extinction rates within each process; hence, the number of unique extinction rates in each simulation was equal to the number of processes in the generating model. I computed the mean branch-specific extinction rate across each subclade that was governed by a distinct evolutionary process in the simulation model; I then analyzed these extinction rates across all 500 trees in a given simulation scenario together. For example, consider the phylogeny shown in

Model | processes | slope |
r^{2} |
PE |

Exponential change (exp2) | 2 | 0.76 | 0.59 | 1.27 |

Diversity-dependent (DD2) | 2 | 0.81 | 0.16 | 1.85 |

Diversity-dependent (DD3) | 3 | 0.75 | 0.11 | 1.68 |

Diversity-dependent (DD4) | 4 | 0.81 | 0.13 | 1.67 |

Diversity-dependent (DD5) | 5 | 0.82 | 0.10 | 1.66 |

Slope and r^{2} denote the estimated slope and variance explained by the relationship between true and estimated extinction rates for 500 trees simulated under each model.

PE is the mean proportional error across all simulations under a given model.

I analyzed all 3000 simulated datasets using MEDUSA. The number of processes inferred for each simulation scenario are shown in

Phylogenies were simulated under 5 distinct evolutionary scenarios. For each simulation, the number of distinct rate partitions was estimated using the stepwise AICc algorithm as implemented in MEDUSA. Black bars denote true number of processes in generating model. MEDUSA consistently underestimates the true number of processes in simulated datasets when rates of speciation vary through time. Comparable results for BAMM using the same set of simulated datasets are shown in

Branch-specific estimates of speciation rates under MEDUSA were, in general, extremely poor (

For each simulated phylogeny, MEDUSA was used to estimate the number, location, and parameters of diversification rate shifts. The resulting branch-specific rates of speciation were compared with the true branch rates from the generating model. Results are based on the same simulated datasets analyzed with BAMM and can be directly compared to those shown in

Analysis of the time-calibrated cetacean phylogeny found strong support for a two-process model (

(A) Phylogeny of cetaceans

Using output from BAMM, I computed mean rates of speciation and extinction through time during the cetacean radiation. This was done by drawing an imaginary grid of vertical lines through the time-calibrated cetacean phylogeny at equally spaced points in time. Evolutionary rates were estimated as the mean branch-specific rates for all branches that intersected the line corresponding to a specific time point. This enabled estimation of the posterior density of speciation and extinction for any point in time. These results suggest an overall decline in the background rate of whale speciation, with a large spike during the Miocene driven by the radiation of the dolphin clade (Delphinidae). Extinction rates are inferred to be relatively low overall, with a mean per-branch relative extinction rate (

Finally, I assessed the sensitivity of the cetacean dataset to choice of prior on the number of processes in the phylogeny. I used BAMM to analyze the cetacean data under four additional values of γ (0.1, 0.5, 5, and 10). In each case, the single-process model had low posterior probability and was marginally worth considering (Pr (M_{1}) = 0.127) only under the strongest prior (γ = 0.1). For γ = 5 and γ = 10, the posterior probability of a model with a single process was approximately 0. The MAP model had two processes under γ = 0.5 and γ = 5. For γ = 10, the MAP model had four processes, but was not substantially more probable than models with two or three processes. The posterior odds ratio for M_{4} versus M_{2} was merely 1.63, and for M_{4} versus M_{3} it was only 1.26. I estimated marginal diversification rates for each branch in the cetacean phylogeny under these prior formulations; pairwise plots for speciation rate estimates under alternative priors suggest that these rates are robust to choice of prior (

Each panel shows a pairwise plot comparing branch-specific (marginal) diversification rate estimates for two values of γ for the Cetacean dataset, with results for speciation and extinction separated by the diagonal. Speciation rate estimates for the cetaceans are remarkably robust to choice of prior: even γ = 10 and γ = 0.1 yield strikingly similar marginal distributions for branch-specific speciation rates. This is generally not true for extinction, where mean marginal rates for each branch were more sensitive to prior formulation. However, extinction was nonetheless estimated to be low overall regardless of the prior γ.

Extracting information about the tempo and mode of species diversification remains a central methodological challenge in macroevolutionary studies. I developed a Poisson process model of diversification rate variation to address several limitations of current methodological approaches for studying evolutionary dynamics on phylogenetic trees. The approach described here views phylogenetic trees as the outcome of a complex mixture of potentially dynamic evolutionary processes and enables researchers to detect rate shifts, key innovations, time-dependent speciation, and diversity-dependence within single trees. Output from the BAMM implementation of the compound Poisson process model includes (i) estimates of the number of distinct process and posterior probabilities of each possible model; (ii) estimates of locations of those processes as well as associated parameter estimates; and (iii) estimates of branch-specific rates of speciation and extinction, which can further be used to infer temporal trends in evolutionary rates (

BAMM performed well throughout the parameter space explored here. For each of six distinct macroevolutionary scenarios, BAMM was usually able to identify the true number of processes in the generating model (

Extinction rate estimates from the model should be taken with caution. Branch specific estimates of extinction are potentially biased and, although these estimates are correlated with the true underlying rates, confidence in those estimates is low (

By implementing an exponential change function for speciation, I was able to accurately infer diversity-dependent dynamics across a range of simulation scenarios (

My results suggest that MEDUSA is not robust to violations of its assumption that diversification rates are constant through time. Whereas BAMM was often able to estimate the true number of distinct processes in the generating model (

To be clear, the model implementation in BAMM - in contrast to MEDUSA - was explicitly designed to account for variation in evolutionary rates both through time and among lineages. However, the MEDUSA method has been applied to many empirical datasets with little attention given to the potential consequences of violating the assumption of rate-constancy through time. Using

The analysis of the Cetacean phylogeny provides an important window into the history of cetacean diversification through time (

Many extensions are possible within the framework developed here. The computational machinery for adding, moving, and deleting processes from phylogenetic trees is flexible and can easily be extended to allow alternative functional models for speciation and/or extinction rate variation through time. Another obvious future extension is to explicitly account for phylogenetic uncertainty during simulation of the posterior. As currently implemented, BAMM simulates posterior distributions of models and parameters across a fixed topology. However, phylogenetic trees are rarely (if ever) known without error. Credible intervals on parameters inferred using BAMM (

I have described a methodological framework for inferring mixtures of processes that have influenced the structure of phylogenetic trees. By modeling phylogenies as collections of dynamic processes, the method greatly extends our ability to describe evolutionary dynamics. Most previous evolutionary studies using transdimensional MCMC on phylogenetic trees have assumed that dynamics within component processes are constant in time. By relaxing the assumption of time-homogeneous diversification, the model is better able to describe complex mixtures of both time-constant and time-varying processes. A number of recent studies have suggested that such complex dynamics might dominate speciation-extinction patterns in many empirical datasets. I suggest that the use of rjMCMC to fit time-inhomogeneous multiprocess models to phylogenetic data may have applications beyond those described here, including DNA sequence evolution, phenotypic evolution, and phylogeography.

I thank M. Alfaro. F. Bokma, R. FitzJohn, L. Harmon, J. Huelsenbeck, S. Kolokotronis, B. Moore, S. Otto, A. Pyron, J. Schraiber, J. Uyeda, and members of the Huelsenbeck lab for comments on the manuscript and/or discussion that improved the results presented here. I thank T. Nan for developing the PPSS code that facilitated parallel analysis of simulated datasets on multiprocessor computers.