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Extant timetrees are consistent with a myriad of diversification histories

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Stilianos Louca
Stilianos Louca
Stilianos Louca
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Matthew W. Pennell
Matthew W. Pennell
Matthew W. Pennell
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Abstract and Figures

Time-calibrated phylogenies of extant species (referred to here as ‘extant timetrees’) are widely used for estimating diversification dynamics¹. However, there has been considerable debate surrounding the reliability of these inferences2–5 and, to date, this critical question remains unresolved. Here we clarify the precise information that can be extracted from extant timetrees under the generalized birth–death model, which underlies most existing methods of estimation. We prove that, for any diversification scenario, there exists an infinite number of alternative diversification scenarios that are equally likely to have generated any given extant timetree. These ‘congruent’ scenarios cannot possibly be distinguished using extant timetrees alone, even in the presence of infinite data. Importantly, congruent diversification scenarios can exhibit markedly different and yet similarly plausible dynamics, which suggests that many previous studies may have over-interpreted phylogenetic evidence. We introduce identifiable and easily interpretable variables that contain all available information about past diversification dynamics, and demonstrate that these can be estimated from extant timetrees. We suggest that measuring and modelling these identifiable variables offers a more robust way to study historical diversification dynamics. Our findings also make it clear that palaeontological data will continue to be crucial for answering some macroevolutionary questions.
Pulled speciation and diversification rates a, b, Pulled speciation rate (a) and pulled diversification rate (b) of the four congruent models shown in Fig. 1.
Pulled speciation and diversification rates a, b, Pulled speciation rate (a) and pulled diversification rate (b) of the four congruent models shown in Fig. 1.
… 
Illustration of congruent birth–death processes (fossil data) a, Origination and extinction rates of marine invertebrate genera, estimated from fossil data. b, Congruent scenario to that in a, obtained by reversing the linear trend of μ (that is, fitting a linear curve to the original μ, and then subtracting that curve twice) and adjusting λ according to equation (2). c, Congruent scenario to that in a, assuming an extinction rate of zero. Further details are provided in Supplementary Information section S.10.
Illustration of congruent birth–death processes (fossil data) a, Origination and extinction rates of marine invertebrate genera, estimated from fossil data. b, Congruent scenario to that in a, obtained by reversing the linear trend of μ (that is, fitting a linear curve to the original μ, and then subtracting that curve twice) and adjusting λ according to equation (2). c, Congruent scenario to that in a, assuming an extinction rate of zero. Further details are provided in Supplementary Information section S.10.
… 
Previous studies are likely to have over-interpreted phylogenetic data Time-dependent birth–death model fitted to a nearly complete extant timetree of the Cetacea, under the assumption of extinction rates of zero (μ = 0), compared to a congruent model in which the extinction rate is close to the speciation rate (μ = 0.9λ). a, LTT of the tree, compared to the dLTT predicted by the two models. b, Speciation rates (λ) and extinction rates (μ) of the two models. c, Net diversification rates (r = λ − μ) of the two models. Further details are provided in Supplementary Information section S.4.
Previous studies are likely to have over-interpreted phylogenetic data Time-dependent birth–death model fitted to a nearly complete extant timetree of the Cetacea, under the assumption of extinction rates of zero (μ = 0), compared to a congruent model in which the extinction rate is close to the speciation rate (μ = 0.9λ). a, LTT of the tree, compared to the dLTT predicted by the two models. b, Speciation rates (λ) and extinction rates (μ) of the two models. c, Net diversification rates (r = λ − μ) of the two models. Further details are provided in Supplementary Information section S.4.
… 
Additional examples of congruent birth–death processes a–c, Example of two congruent—yet markedly different—birth–death models. Both models exhibit a temporary spike in the extinction rate and a temporary spike in the speciation rate; however, the timings of these events differ substantially between the two models. Both models exhibit the same dLTT and the same pulled diversification rate (rp) and would yield identical likelihoods for any given extant timetree. a, Speciation rates (λ and λ*) and extinction rates (μ and μ*) of the two models, plotted over time. Continuous curves correspond to the first model, and dashed curves correspond to the second model. b, Net diversification rates (r and r*) and pulled diversification rate (rp) of the two models. c, dLTT and deterministic total diversities (N and N*) predicted by the two models. d–f, Another example of two congruent models. In the first model, the speciation and extinction rates both decrease exponentially over time, whereas in the second model the extinction rate increases exponentially over time and the speciation rate exhibits variable directions of change over time. In all models, the sampling fraction is ρ = 0.5.
Additional examples of congruent birth–death processes a–c, Example of two congruent—yet markedly different—birth–death models. Both models exhibit a temporary spike in the extinction rate and a temporary spike in the speciation rate; however, the timings of these events differ substantially between the two models. Both models exhibit the same dLTT and the same pulled diversification rate (rp) and would yield identical likelihoods for any given extant timetree. a, Speciation rates (λ and λ*) and extinction rates (μ and μ*) of the two models, plotted over time. Continuous curves correspond to the first model, and dashed curves correspond to the second model. b, Net diversification rates (r and r*) and pulled diversification rate (rp) of the two models. c, dLTT and deterministic total diversities (N and N*) predicted by the two models. d–f, Another example of two congruent models. In the first model, the speciation and extinction rates both decrease exponentially over time, whereas in the second model the extinction rate increases exponentially over time and the speciation rate exhibits variable directions of change over time. In all models, the sampling fraction is ρ = 0.5.
… 
Identifiability issues persist in large trees a–c, Diversification analysis of a timetree (about 114,000 tips) simulated from a birth–death process that exhibits a mass extinction event at around 5 Myr before present. a, LTT of the generated tree (long-dashed curve), dLTT of the true model that generated the tree (continuous curve) and dLTT of a maximum-likelihood fitted model (short-dashed curve) are shown. The fitted dLTT is practically identical to the true dLTT and thus is covered by the latter. b, True speciation and extinction rates (continuous curves), compared to fitted speciation and extinction rates (dashed curves). There is considerable disagreement between the fitted and true λ and μ, despite the fact that the allowed model set could—in principle—approximate the true rates reasonably well. c, Pulled diversification rate (PDR) of the true model (continuous curve), compared to the pulled diversification rate of the fitted model (dashed curve). d–f, Diversification analysis of a timetree (about 785,000 tips) simulated from a birth–death process that exhibits a rapid radiation event at around 5 Myr before present and a mass extinction event at around 2 Myr before present. d–f are analogous to a–c. There is considerable disagreement between the fitted and true λ and μ, despite the fact that the allowed model set could—in principle—approximate the true rates reasonably well. Extended Data Figure 7 provides the fitting results when μ is fixed to its true value. g–i, Diversification analyses of an extant timetree of 79,874 seed plant species, performed either by fitting λ and μ on a grid of discrete time points or by fitting the parameters of generic polynomial or exponential functions for λ and μ. g, LTT of the tree, dLTT of the grid-fitted model and dLTT of the fitted parametric model. h, Speciation and extinction rates predicted by the grid-fitted model or the fitted parametric model. i, Pulled diversification rate predicted by the grid-fitted model and the fitted parametric model. Further details are provided in Supplementary Information sections S.10 and S.11.
+6
Identifiability issues persist in large trees a–c, Diversification analysis of a timetree (about 114,000 tips) simulated from a birth–death process that exhibits a mass extinction event at around 5 Myr before present. a, LTT of the generated tree (long-dashed curve), dLTT of the true model that generated the tree (continuous curve) and dLTT of a maximum-likelihood fitted model (short-dashed curve) are shown. The fitted dLTT is practically identical to the true dLTT and thus is covered by the latter. b, True speciation and extinction rates (continuous curves), compared to fitted speciation and extinction rates (dashed curves). There is considerable disagreement between the fitted and true λ and μ, despite the fact that the allowed model set could—in principle—approximate the true rates reasonably well. c, Pulled diversification rate (PDR) of the true model (continuous curve), compared to the pulled diversification rate of the fitted model (dashed curve). d–f, Diversification analysis of a timetree (about 785,000 tips) simulated from a birth–death process that exhibits a rapid radiation event at around 5 Myr before present and a mass extinction event at around 2 Myr before present. d–f are analogous to a–c. There is considerable disagreement between the fitted and true λ and μ, despite the fact that the allowed model set could—in principle—approximate the true rates reasonably well. Extended Data Figure 7 provides the fitting results when μ is fixed to its true value. g–i, Diversification analyses of an extant timetree of 79,874 seed plant species, performed either by fitting λ and μ on a grid of discrete time points or by fitting the parameters of generic polynomial or exponential functions for λ and μ. g, LTT of the tree, dLTT of the grid-fitted model and dLTT of the fitted parametric model. h, Speciation and extinction rates predicted by the grid-fitted model or the fitted parametric model. i, Pulled diversification rate predicted by the grid-fitted model and the fitted parametric model. Further details are provided in Supplementary Information sections S.10 and S.11.
… 
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502 | Nature | Vol 580 | 23 April 2020
Article
Extant timetrees are consistent with a
myriad of diversification histories
Stilianos Louca1,2 ✉ & Matthew W. Pennell3,4 ✉
Time-calibrated phylogenies of extant species (referred to here as ‘extant timetrees’)
are widely used for estimating diversication dynamics1. However, there has been
considerable debate surrounding the reliability of these inferences2–5 and, to date, this
critical question remains unresolved. Here we clarify the precise information that can
be extracted from extant timetrees under the generalized birth–death model, which
underlies most existing methods of estimation. We prove that, for any diversication
scenario, there exists an innite number of alternative diversication scenarios that
are equally likely to have generated any given extant timetree. These ‘congruent’
scenarios cannot possibly be distinguished using extant timetrees alone, even in the
presence of innite data. Importantly, congruent diversication scenarios can exhibit
markedly dierent and yet similarly plausible dynamics, which suggests that many
previous studies may have over-interpreted phylogenetic evidence. We introduce
identiable and easily interpretable variables that contain all available information
about past diversication dynamics, and demonstrate that these can be estimated
from extant timetrees. We suggest that measuring and modelling these identiable
variables oers a more robust way to study historical diversication dynamics. Our
ndings also make it clear that palaeontological data will continue to be crucial for
answering some macroevolutionary questions.
A central challenge in evolutionary biology is to reconstruct rates of
speciation and extinction over time5. Unfortunately, the majority of
taxa that have ever lived have not left much trace in the fossil record,
and the primary source of information on their past diversification
dynamics therefore comes from extant timetrees. Many methods
have been developed for extracting this information; most methods
fit variants of a birth–death process
1,6
. Despite the popularity of these
methods, which collectively have been used in thousands of studies
7–9
,
their reliability has been called into question by comparisons with
fossil-based estimates1,3,5,6,10. The reasoning behind these critiques
is that there may be insufficient information in extant timetrees to
fully reconstruct historical diversification dynamics. However, this
critical issue has remained unresolved; it is unknown precisely what
information on speciation and extinction rates is contained in extant
timetrees.
Here we present a definite answer to this question for the general
stochastic birth–death process with homogeneous (that is, lineage-
independent) rates, in which speciation (‘birth’) rates (λ) and extinc-
tion (‘death’) rates (μ) can vary over time, that underlies the majority
of existing methods for reconstructing diversification dynamics from
phylogenies
1
. We mathematically show that, for any givencandidate
birth–death model, there exists an infinite number of alternative birth–
death models that can explain any extant timetree equally as well as
can the candidate model. These alternative models may appear to be
similarly plausible and yet exhibit markedly different features, such as
different trends through time in both λ and μ. This severe ambiguity
persists for arbitrarily large trees and cannot be resolved even with an
infinite amount of data; it is thus impossible to design asymptotically
consistent estimators for λ and μ. Using simulated and real timetrees
as examples, we demonstrate how failing to recognize this issue can
seriously mislead our inferences about past diversification dynamics.
We present appropriately modified variables that are asymptotically
identifiable and that contain all available information on historical
diversification dynamics.
Lineages through time
An important feature of extant timetrees is the lineages-through-time
curve (LTT), which counts the number of lineages at each time in the
past that are represented by at least one sampled extant descending
species in the tree. The likelihood of a tree under a given birth–death
model, the LLT of the tree and the LTT that would be expected under
the model are linked as follows. Any given combination of (potentially
time-dependent) speciation and extinction rates (λ and μ, respectively)
and the probability that an extant species will be included in the tree
(‘sampling fraction’) (ρ) can be used to define a deterministic diver-
sification process, in which the number of lineages through time no
longer varies stochastically but instead according to a set of differ-
ential equations
6,11
(Supplementary Information section S.1). Given
a number of extant sampled species (M
o
), the LTT predicted by these
https://doi.org/10.1038/s41586-020-2176-1
Received: 14 September 2019
Accepted: 10 March 2020
Published online: 15 April 2020
Check for updates
1Department of Biology, University of Oregon, Eugene, OR, USA. 2Institute of Ecology and Evolution, University of Oregon, Eugene, OR, USA. 3Biodiversity Research Centre, University of British
Columbia, Vancouver, British Columbia, Canada. 4Department of Zoology, University of British Columbia, Vancouver, British Columbia, Canada. ✉e-mail: louca.research@gmail.com; pennell@
zoology.ubc.ca
Content courtesy of Springer Nature, terms of use apply. Rights reserved
... The potential nonidentifiability of evolutionary rates based on extant phylogenies has recently become a topic of fierce debate (Louca and Pennell 2020;Legried and Terhorst 2022;Morlon et al. 2022;Kopperud et al. 2023). Louca and Pennell (2020) describe how speciation and extinction functions are drawn from congruence classes, within which each pair of functions is equally likely to fit the evolutionary history of a given phylogeny. ...
... The potential nonidentifiability of evolutionary rates based on extant phylogenies has recently become a topic of fierce debate (Louca and Pennell 2020;Legried and Terhorst 2022;Morlon et al. 2022;Kopperud et al. 2023). Louca and Pennell (2020) describe how speciation and extinction functions are drawn from congruence classes, within which each pair of functions is equally likely to fit the evolutionary history of a given phylogeny. As a result, it may not be possible to discern which ...
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  • Sep 2019
  • ECOL LETT
Diversification rates vary over time, yet the factors driving these variations remain unclear. Temporal declines in speciation rates have often been interpreted as the effect of ecological limits, competition, and diversity dependence, emphasising the role of biotic factors. Abiotic factors, such as climate change, are also supposed to have affected diversification rates over geological time scales, yet direct tests of these presumed effects have mainly been limited to few clades well represented in the fossil record. If warmer climatic periods have sustained faster speciation, this could explain slowdowns in speciation during the Cenozoic climate cooling. Here, we apply state‐of‐the art diversity‐dependent and temperature‐dependent phylogenetic models of diversification to 218 tetrapod families, along with constant rate and time‐dependent models. We confirm the prevalence of diversification slowdowns, and find as much support for temperature‐dependent than diversity‐dependent models. These results call for a better integration of these two processes in studies of diversification dynamics.
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Swapping Birth and Death: Symmetries and Transformations in Phylodynamic Models
Article
Full-text available
  • May 2019
  • SYST BIOL
Stochastic birth-death models provide the foundation for studying and simulating evolutionary trees in phylodynamics. A curious feature of such models is that they exhibit fundamental symmetries when the birth and death rates are interchanged. In this paper, we first provide intuitive reasons for these known transformational symmetries. We then show that these transformational symmetries (encoded in algebraic identities) are preserved even when individuals at the present are sampled with some probability. However, these extended symmetries require the death rate parameter to sometimes take a negative value. In the last part of this paper, we describe the relevance of these transformations and their application to computational phylodynamics, particularly to maximum likelihood and Bayesian inference methods, as well as to model selection.
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Macroevolutionary diversification rates show time dependency
Article
Full-text available
  • Apr 2019
Significance Some branches of the tree of life are incredibly diverse, while others are represented by only a few living species. Ultimately, this difference reflects the balance of the formation and the extinction of species. Countless explanations have been proposed for why the rates of these two processes vary between lineages, including aspects of the organisms themselves and the environments they live in. Here we reveal that a substantial amount of variation in these rates is associated with a simple factor: time. Younger groups appear to accumulate diversity at much faster rates than older groups. This time scaling of macroevolutionary rates suggests that there may be hidden generalities governing the diversification of life on Earth.
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Closing the gap between palaeontological and neontological speciation and extinction rate estimates
Article
Full-text available
  • Dec 2018
Measuring the pace at which speciation and extinction occur is fundamental to understanding the origin and evolution of biodiversity. Both the fossil record and molecular phylogenies of living species can provide independent estimates of speciation and extinction rates, but often produce strikingly divergent results. Despite its implications, the theoretical reasons for this discrepancy remain unknown. Here, we reveal a conceptual and methodological basis able to reconcile palaeontological and molecular evidence: discrepancies are driven by different implicit assumptions about the processes of speciation and species evolution in palaeontological and neontological analyses. We present the “birth-death chronospecies” model that clarifies the definition of speciation and extinction processes allowing for a coherent joint analysis of fossil and phylogenetic data. Using simulations and empirical analyses we demonstrate not only that this model explains much of the apparent incongruence between fossils and phylogenies, but that differences in rate estimates are actually informative about the prevalence of different speciation modes.
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Bacterial diversification through geological time
Article
Full-text available
  • Sep 2018
  • Nat. Ecol. Evol.
Numerous studies have estimated plant and animal diversification dynamics; however, no comparable rigorous estimates exist for bacteria-the most ancient and widespread form of life on Earth. Here, we analyse phylogenies comprising up to 448,112 bacterial lineages to reconstruct global bacterial diversification dynamics. To handle such large phylogenies, we developed methods based on the statistical properties of infinitely large trees. We further analysed sequencing data from 60 environmental studies to determine the fraction of extant bacterial diversity missing from the phylogenies-a crucial parameter for estimating speciation and extinction rates. We estimate that there are about 1.4-1.9 million extant bacterial lineages when lineages are defined by 99% similarity in the 16S ribosomal RNA gene, and that bacterial diversity has been continuously increasing over the past 1 billion years (Gyr). Recent bacterial extinction rates are estimated at 0.03-0.05 per lineage per million years (lineage-1 Myr-1), and are only slightly below estimated recent bacterial speciation rates. Most bacterial lineages ever to have inhabited this planet are estimated to be extinct. Our findings disprove the notion that bacteria are unlikely to go extinct, and provide a valuable perspective on the evolutionary history of a domain of life with a sparse and cryptic fossil record.
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An inverse latitudinal gradient in speciation rate for marine fishes
Article
Full-text available
  • Jul 2018
  • NATURE
Far more species of organisms are found in the tropics than in temperate and polar regions, but the evolutionary and ecological causes of this pattern remain controversial1,2. Tropical marine fish communities are much more diverse than cold-water fish communities found at higher latitudes3,4, and several explanations for this latitudinal diversity gradient propose that warm reef environments serve as evolutionary ‘hotspots’ for species formation5,6,7,8. Here we test the relationship between latitude, species richness and speciation rate across marine fishes. We assembled a time-calibrated phylogeny of all ray-finned fishes (31,526 tips, of which 11,638 had genetic data) and used this framework to describe the spatial dynamics of speciation in the marine realm. We show that the fastest rates of speciation occur in species-poor regions outside the tropics, and that high-latitude fish lineages form new species at much faster rates than their tropical counterparts. High rates of speciation occur in geographical regions that are characterized by low surface temperatures and high endemism. Our results reject a broad class of mechanisms under which the tropics serve as an evolutionary cradle for marine fish diversity and raise new questions about why the coldest oceans on Earth are present-day hotspots of species formation.
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Constructing a broadly inclusive seed plant phylogeny
Article
Full-text available
  • Feb 2018
Premise of the Study: Large phylogenies can help shed light on macroevolutionary patterns that inform our understanding of fundamental processes that shape the tree of life. These phylogenies also serve as tools that facilitate other systematic, evolutionary, and ecological analyses. Here we combine genetic data from public repositories (GenBank) with phylogenetic data (Open Tree of Life project) to construct a dated phylogeny for seed plants. Methods: We conducted a hierarchical clustering analysis of publicly available molecular data for major clades within the Spermatophyta. We constructed phylogenies of major clades, estimated divergence times, and incorporated data from the Open Tree of Life project, resulting in a seed plant phylogeny. We estimated diversification rates, excluding those taxa without molecular data. We also summarized topological uncertainty and data overlap for each major clade. Key Results: The trees constructed for Spermatophyta consisted of 79,881 and 353,185 terminal taxa; the latter included the Open Tree of Life taxa for which we could not include molecular data from GenBank. The diversification analyses demonstrated nested patterns of rate shifts throughout the phylogeny. Data overlap and inference uncertainty show significant variation throughout and demonstrate the continued need for data collection across seed plants. Conclusions: This study demonstrates a means for combining available resources to construct a dated phylogeny for plants. However, this approach is an early step and more developments are needed to add data, better incorporating underlying uncertainty, and improve resolution. The methods discussed here can also be applied to other major clades in the tree of life.
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The fossilized birth-death model for the analysis of stratigraphic range data under different speciation concepts
Article
Full-text available
  • Jun 2017
A birth-death-sampling model gives rise to phylogenetic trees with samples from the past and the present. Interpreting “birth” as branching speciation, “death” as extinction, and “sampling” as fossil preservation and recovery, this model – also referred to as the fossilized birth-death (FBD) model – gives rise to phylogenetic trees on extant and fossil samples. The model has been mathematically analyzed and successfully applied to a range of datasets on different taxonomic levels, such as penguins, plants, and insects. However, the current mathematical treatment of this model does not allow for a group of temporally distinct fossil specimens to be assigned to the same species. In this paper, we provide a general mathematical FBD modeling framework that explicitly takes “stratigraphic ranges” into account, with a stratigraphic range being defined as the lineage interval associated with a single species, ranging through time from the first to the last fossil appearance of the species. To assign a sequence of fossil samples in the phylogenetic tree to the same species, i.e., to specify a stratigraphic range, we need to define the mode of speciation. We provide expressions to account for three common speciation modes: budding (or asymmetric) speciation, bifurcating (or symmetric) speciation, and anagenetic speciation. Our equations allow for flexible joint Bayesian analysis of paleontological and neontological data. Furthermore, our framework is directly applicable to epidemiology, where a stratigraphic range is the observed duration of infection of a single patient, “birth” via budding is transmission, “death” is recovery, and “sampling” is sequencing the pathogen of a patient. Thus, we present a model that allows for incorporation of multiple observations through time from a single patient.
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Simulating trees with millions of species
Article
  • Jan 2020
  • BIOINFORMATICS
Motivation: The birth-death model constitutes the theoretical backbone of most phylogenetic tools for reconstructing speciation/extinction dynamics over time. Performing simulations of reconstructed trees (linking extant taxa) under the birth-death model in backward time, conditioned on the number of species sampled at present day and, in some cases, a specific time interval since the most recent common ancestor (MRCA), is needed for assessing the performance of reconstruction tools, for parametric bootstrapping and for detecting data outliers. The few simulation tools that exist scale poorly to large modern phylogenies, which can comprise thousands or even millions of tips (and rising). Results: Here I present efficient software for simulating reconstructed phylogenies under time-dependent birth-death models in backward time, conditioned on the number of sampled species and (optionally) on the time since the MRCA. On large trees, my software is 1,000-10,000 times faster than existing tools. Availability: The presented software is incorporated into the R package "castor", which is available on The Comprehensive R Archive Network (CRAN).
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Efficient comparative phylogenetics on large trees
Article
  • Oct 2017
  • BIOINFORMATICS
Motivation: Biodiversity databases now comprise hundreds of thousands of sequences and trait records. For example, the Open Tree of Life includes over 1,491,000 metazoan and over 300,000 bacterial taxa. These data provide unique opportunities for analysis of phylogenetic trait distribution and reconstruction of ancestral biodiversity. However, existing tools for comparative phylogenetics scale poorly to such large trees, to the point of being almost unusable. Results: Here we present a new R package, named "castor", for comparative phylogenetics on large trees spanning millions of tips. On large trees castor is often 100-1000 times faster than existing tools. Availability: The castor source code, compiled binaries, documentation and usage examples are freely available at the Comprehensive R Archive Network (CRAN). Contact: louca.research@gmail.com.
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