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Keywords:

  • extinction;
  • inference;
  • macroevolution;
  • phylogenetics;
  • speciation

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Phylogenetic trees induced by speciation-extinction models
  5. Different classes of speciation-extinction models
  6. Uncovering speciation and extinction dynamics
  7. Uncovering speciation and extinction dynamics based on (ranked) tree shapes
  8. Uncovering speciation and extinction dynamics based on reconstructed trees
  9. Future challenges
  10. References

Phylogenetic trees of only extant species contain information about the underlying speciation and extinction pattern. In this review, I provide an overview over the different methodologies that recover the speciation and extinction dynamics from phylogenetic trees. Broadly, the methods can be divided into two classes: (i) methods using the phylogenetic tree shapes (i.e. trees without branch length information) allowing us to test for speciation rate variation and (ii) methods using the phylogenetic trees with branch length information allowing us to quantify speciation and extinction rates. I end the article with an overview on limitations, open questions and challenges of the reviewed methodology.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Phylogenetic trees induced by speciation-extinction models
  5. Different classes of speciation-extinction models
  6. Uncovering speciation and extinction dynamics
  7. Uncovering speciation and extinction dynamics based on (ranked) tree shapes
  8. Uncovering speciation and extinction dynamics based on reconstructed trees
  9. Future challenges
  10. References

A phylogenetic tree represents the evolutionary relationship between species. Tips correspond to extant species and branching events correspond to speciation events (Fig. 1, middle). Classically such phylogenetic trees were reconstructed using morphological data from different species, based on the evolutionary concept that morphology of closely related species shares more characteristics than morphology of distantly related species. The increasing amount of sequence data allowed the reconstruction of many more species phylogenies, based on the evolutionary concept that genetic sequences of closely related species share more characteristics than genetic sequences of distantly related species.

Phylogenetic trees not only inform systematists about the evolutionary relationship of species, but also allow us to infer evolutionary dynamics of the considered species clade using comparative methods (Harvey & Pagel, 1991). In this review, I will focus on recent methodological developments towards understanding speciation and extinction dynamics based on phylogenetic trees. Despite rarely being termed this way, such methods are also comparative methods as the phylogenetic relationships between species are used to understand the speciation and extinction dynamics.

Inferring speciation and extinction dynamics allows us to aim at identifying the main factors that led to the biodiversity we observe today. In particular, recent methods allow us to determine the impact of the environment on speciation and extinction, as well as to identify species traits and characteristics that have a selective advantage on a macroevolutionary scale. Furthermore, the amount by which competition among species limits speciation and fosters extinction may be determined.

Speciation and extinction dynamics are inferred by fitting macroevolutionary models to phylogenetic trees, that is, by determining which model explains the data best, with sophisticated fitting methods providing a quantification for the parameters of the model. In this review article, I will first define a general model for speciation and extinction and explain how it gives rise to phylogenetic trees. Thereafter, the main body of the article will show advances in fitting such models to empirical phylogenetic trees. There exists a wide range of fitting methodology; however, all of the methods essentially aim at identifying the model which gives rise to phylogenetic trees most similar to the empirical phylogeny. I will discuss how such model fitting allows us to test macroevolutionary hypotheses as well as to quantify macroevolutionary parameters (the speciation and extinction rates). Technical challenges for the future are highlighted in little ‘Outlook’ paragraphs. The article ends with putting the methods into a broader macroevolutionary perspective, discussing current limitations and identifying main future directions necessary for improving our macroevolutionary understanding.

Phylogenetic trees induced by speciation-extinction models

  1. Top of page
  2. Abstract
  3. Introduction
  4. Phylogenetic trees induced by speciation-extinction models
  5. Different classes of speciation-extinction models
  6. Uncovering speciation and extinction dynamics
  7. Uncovering speciation and extinction dynamics based on (ranked) tree shapes
  8. Uncovering speciation and extinction dynamics based on reconstructed trees
  9. Future challenges
  10. References

The general speciation-extinction model

In general, a macroevolutionary model for speciation and extinction is formulated as follows. The stochastic process starts with a single species at some time inline image in the past. The species gives rise to new species with some rate of speciation and goes extinct with some rate of extinction. Speciation may be symmetric or asymmetric. Symmetric speciation means that following a speciation event, both daughter species are assigned age 0. Asymmetric speciation means that following a speciation event, one of the daughter species inherits the age of the mother species and the second daughter species is assigned age 0.

The process is stopped after time inline image, the present, resulting in a phylogenetic tree on extant and extinct species (Fig. 1, left). This tree is called the complete phylogenetic tree. Pruning all extinct and nonsampled extant species results in a phylogenetic tree on a sample of extant species (Fig. 1, middle). This tree is called the reconstructed phylogenetic tree (Nee et al., 1994b). The oldest speciation event in the reconstructed tree is called the root. Its age inline image is the crown age of the tree, whereas inline image is the stem age of the tree.

A speciation-extinction model typically gives rise to a range of different reconstructed trees, of which some trees may be much more likely than other trees. Thus, a speciation-extinction model induces a distribution on reconstructed trees. The reconstructed tree is a model for the empirical trees obtained based on data of a set of extant species. When fitting the model to an empirical tree, the model which gives rise to reconstructed trees ‘most similar’ to the empirical trees is chosen to be the best model fit. Of course, there are many possibilities for defining ‘most similar’, and different ways are discussed in the section ‘Uncovering speciation and extinction dynamics’.

Branch lengths in reconstructed trees are in units of calendar time. Thus, to compare reconstructed trees with empirical trees, the empirical trees require branch lengths in units of calendar time (i.e. a unit of, for example, millions of years). When inferring empirical phylogenies from molecular data, branch lengths are in evolutionary units, that is, correspond to the amount of genetic change along a branch. To translate the evolutionary unit into a calendar time unit, a molecular clock is required. Unless a strict molecular clock can be assumed, further fossil calibrations need to be applied.

Due to these additional efforts and difficulties when inferring calendar time phylogenies, often only phylogenies in evolutionary time are provided. To fit speciation-extinction models to such empirical phylogenies, we need to consider reconstructed trees induced by the model without branch lengths. A reconstructed tree without branch lengths is called a tree shape (also frequently called a tree topology). A tree shape only reflects relatedness between species without any time information.

A reconstructed tree without branch lengths but preserving the order of internal vertices is called a ranked tree shape (Fig. 1, right). The particular ordering of internal vertices in a ranked tree shape is called ranking. Empirical trees may be ranked tree shapes if information (e.g. based on fossils) is known about the relative order of speciation events, but potentially inferred times of speciation events are too noisy and thus neglected in any further analyses.

image

Figure 1. Left: complete phylogenetic tree with five sampled species (sp1–sp5). Middle: reconstructed phylogenetic tree obtained by pruning the nonsampled and extinct lineages from the complete phylogenetic tree. Right: ranked phylogenetic tree shape obtained from the reconstructed phylogenetic tree by suppressing branch lengths but keeping the order of internal vertices.

Download figure to PowerPoint

In the main part of this review article (Section ‘Uncovering speciation and extinction dynamics’), I will discuss methods for extracting speciation and extinction dynamics from tree shapes and ranked tree shapes, followed by first highlighting the limitations when not considering branch lengths and second presenting methods using the tree shapes together with branch length, that is, using all information of a reconstructed phylogeny. I will finish the article highlighting that including fossil data in addition to the reconstructed phylogeny is a promising future direction for obtaining an improved macroevolutionary understanding.

Different classes of speciation-extinction models

  1. Top of page
  2. Abstract
  3. Introduction
  4. Phylogenetic trees induced by speciation-extinction models
  5. Different classes of speciation-extinction models
  6. Uncovering speciation and extinction dynamics
  7. Uncovering speciation and extinction dynamics based on (ranked) tree shapes
  8. Uncovering speciation and extinction dynamics based on reconstructed trees
  9. Future challenges
  10. References

Very general speciation and extinction dynamics can be modelled with the process introduced in the previous section. Note, though, that I assume no recombination, that is, no horizontal gene transfer or hybridization, between species. Then, the speciation-extinction models as introduced above only differ in the way how speciation and extinction rates vary through time and across species; in the most simple models, the rates are constant through time and across species. This model is termed the constant rate birth–death model and has been used widely as a null model (Nee et al., 1994b; Pybus & Harvey, 2000; Paradis, 2003). The popular Yule (1924) model is a constant rate model without extinction.

In particular, when considering large empirical clades over long evolutionary timescales, the assumption of constant rates becomes limiting, and the constant rate null model is often rejected (Pybus & Harvey, 2000; Weir, 2006; McPeek, 2008; Phillimore & Price, 2008). The models with rate variation can be divided into three classes, species-exchangeable models, species-speciation-exchangeable models and species-non-exchangeable models, for an overview see Table 1. The distinction of speciation-extinction models into four classes (constant rate and three nonconstant rate models) becomes important when fitting models to empirical trees. As I will discuss below, the (ranked) tree shapes under a constant rate model are exactly the same as the (ranked) tree shapes under a species-exchangeable model and a species-speciation-exchangeable model (i.e. the three classes of models give rise to the same distribution of tree shapes and ranked tree shapes), meaning we cannot distinguish between the models based on tree shapes or ranked tree shapes. I will discuss that species-non-exchangeable models may give rise to different ranked tree shape distributions, and such models may be identified using statistical tests based on ranked tree shapes for empirical data. I will now define the three speciation-extinction model classes allowing for rate variation in detail. I start with the most restricted model (containing the constant rate birth–death process) and generalize this class of models ending with the most general class of speciation-extinction models.

Table 1. The four classes of speciation-extinction models. The model classes become more general from top to bottom, that is, a model class contains all models from the model classes listed above. The top two model classes assume that all coexisting species are equivalent, while the species-speciation-exchangeable model only assumes exchangeability w.r.t. speciation (in particular, coexisting species may have different survival probabilities). The bottom model class further relaxes equivalence w.r.t. speciation (in particular coexisting species may have different speciation potentials). Thus, the bottom two model classes allow for modelling selection on a macroevolutionary level. Popular models within the particular model classes are provided in the middle column. The right column provides the r packages within which the particular models are available for parameter inference, based on the reconstructed trees. I want to highlight that statistical tests based on the ranked tree shape cannot distinguish between models from the top three model classes, as all these models induce a uniform distribution on ranked tree shapes
Model classExamplesImplementations
Constant rate birth–deathIn all packages below
Species-exchangeable

Environmental-dep. spec.

Environmental-dep. spec. & ext.

Diversity-dep. spec.

Diversity-dep. spec. & ext.

Laser

TreePar

Laser

DDD

TreePar

H. Morlon: coal. & birth–death approx.

Species-speciation-exchangeable

Deterministic-trait dep. ext. with asym. constant rate speciation

Non-heritable-trait dep. ext. with asym. time/diversity-dep. speciation

Lambert (2010): to be implemented

Species-non-exchangeable

Stochastic-trait dep. spec. & ext.

Hidden-trait dep. spec. & ext.

Deterministic-trait dep. spec. & ext.

Diversitree & Geiger

First, a model is called a species-exchangeable model if species are exchangeable (Aldous, 2001). This means that given k lineages, the next event (speciation or extinction) may depend on k and the prior numbers of lineages through time as well as on time, but not on the particular prior complete phylogeny. Now, given the next event is a speciation event, each species is equally likely being the one speciating, and given the next event is an extinction event, each species is equally likely to go extinct. Thus, under species-exchangeable models, the rate of speciation and extinction is the same at any point in time for all species, but may change, for example, through time (environmental dependence), or, for example, as a function of the number of species (diversity dependence).

We note that this model has the following property. Label the k − 1 coexisting lineages prior to a speciation event with 1,…,k − 1 and attach the additional species with label k. Similarly, label the k coexisting lineages after an extinction event with 1,…,k. The discrete speciation-extinction pattern descending these k lineages is the complete phylogenetic tree descending the k lineages, ignoring the particular time interval lengths between successive events. Now, under the species-exchangeable model, the discrete speciation-extinction pattern descending a speciation event is independent of which particular coexisting species speciates (i.e. where the species k is attached), as future dynamics are only influenced by past overall number of lineages through time. Analog, the discrete speciation-extinction pattern descending an extinction event is independent of which particular coexisting species goes extinct at the particular extinction event.

Second, a model is called a species-speciation-exchangeable model if species are exchangeable w.r.t. speciation, where I define exchangeable w.r.t. speciation as (i) at any point in time given a speciation event, each species is equally likely being the one speciating, and (ii) the discrete speciation-extinction pattern descending a speciation event is independent of which species was the one speciating.

Note that species-exchangeable models are a subset of species-speciation-exchangeable models. Further note that under a species-speciation-exchangeable model, given an extinction event, some coexisting species may be more likely to go extinct than others. An example of such a model fulfilling (i) and (ii) but not belonging to the species-exchangeable models is the asymmetric constant rate speciation and age-dependent extinction models. Under such a model, at each speciation event, all coexisting species are equally likely to be the one speciating in an asymmetric way, and extinction may depend on the age of a species. Obviously, condition (i) is fulfilled. Furthermore, condition (ii) is fulfilled, as following a speciation event, speciation-extinction patterns only depend on the ages of the coexisting species, which are in turn independent of which species speciated. With the same argument, one can show that asymmetric time- or diversity-dependent speciation models, in which extinction depends on time, diversity and/or an arbitrary trait, and the new species of age 0 obtains a trait value independent of the mother trait (i.e. the trait is nonheritable), belong to the species-speciation-exchangeable class of models.

Third, a species-non-exchangeable model is a model under which some coexisting species may be more likely to speciate at a given point in time than others (i.e. condition (i) is violated), or models where all coexisting species are equally likely to speciate but the speciation-extinction pattern descending a speciation event depends on the particular species speciating (i.e. condition (ii) is violated). The symmetric constant rate speciation and age-dependent extinction models are an example for species-non-exchangeable models. Under such a model, at each speciation event all coexisting species are equally likely to be the one speciating in a symmetric way and extinction may depend on the age of a species. Obviously, such a symmetric probability speciation and age-dependent extinction model fulfils condition (i), but violates condition (ii) as species ages depend on the particular species speciating, and thus, future extinction patterns depend on the particular species speciating. In general, the species-non-exchangeable models can acknowledge that the speciation dynamics may depend on a trait of the species (trait-dependence). A trait may be, for example, species size (stochastic trait), the substitution rate in a species population (hidden trait) or the age of a species (deterministic trait). I emphasize that trait-dependent models can be used for dynamics of speciation and extinction depending on a classic trait of a species (such as size), but also depending on other characteristics of a species such as geographic location or particular ecology of a species.

The reader may wonder why I did not introduce a class of species-extinction-exchangeable models, where species are exchangeable w.r.t. extinction, but not w.r.t. speciation. I highlight below that no mathematical and methodological unique features are obvious in this model class compared with species-non-exchangeable models (which do contain the species-extinction-exchangeable models). Thus, we only discuss the species-extinction-exchangeable models indirectly when discussing species-non-exchangeable models.

Incomplete species sampling

In many empirical data sets, not all species of a clade are sampled. To account for this, speciation-extinction models often assume uniform sampling, meaning each extant species is sampled and included into the phylogeny with the same probability. Besides the uniform sampling scheme, diversified sampling has been suggested as a model for how biologists include species into their data sets (Höhna et al., 2011). Under diversified sampling, n of the m extant species are chosen such that diversity, being defined as the sum of branch lengths, is maximized. The rationale for this model is that when resources for data collection or sequencing are limited, species within a clade which are as diverse as possible, may be considered preferentially.

Clades may further be incomplete on the species level, because only one species per higher taxon (e.g. family or genus) is sampled. Such incomplete phylogenies are called higher-level phylogenies. These trees will not be focus of the present review, but see Purvis & Agapow (2002) for the shape of higher-level phylogenies, and Paradis (2003) and Alfaro et al. (2009) for methods estimating speciation and extinction dynamics based on such trees and Stadler & Bokma (2013) for a discussion on how higher taxon definitions may bias speciation and extinction rate estimates.

Uncovering speciation and extinction dynamics

  1. Top of page
  2. Abstract
  3. Introduction
  4. Phylogenetic trees induced by speciation-extinction models
  5. Different classes of speciation-extinction models
  6. Uncovering speciation and extinction dynamics
  7. Uncovering speciation and extinction dynamics based on (ranked) tree shapes
  8. Uncovering speciation and extinction dynamics based on reconstructed trees
  9. Future challenges
  10. References

In the following, I will first review the methodology available for fitting speciation-extinction models to ranked tree shapes and tree shapes, that is, review the methodology determining which models explain the data best. Second, I will discuss the additional benefit of incorporating branch length information: I will discuss the methodology available for fitting speciation-extinction models to reconstructed trees, that is, review the methodology determining which models explain the data best, and quantifying parameters of the model. Third, I will emphasize that we can increase accuracy of our estimates by not only using tree shapes and branch lengths, but also taking fossil data into account when fitting speciation-extinction models to data, meaning we should start fitting speciation-extinction models to complete phylogenetic trees (i.e. phylogenies including fossil tips) rather than reconstructed trees or tree shapes. Although the inclusion of fossil data is not a focus of this review, potential directions are pointed out in the discussion.

Uncovering speciation and extinction dynamics based on (ranked) tree shapes

  1. Top of page
  2. Abstract
  3. Introduction
  4. Phylogenetic trees induced by speciation-extinction models
  5. Different classes of speciation-extinction models
  6. Uncovering speciation and extinction dynamics
  7. Uncovering speciation and extinction dynamics based on (ranked) tree shapes
  8. Uncovering speciation and extinction dynamics based on reconstructed trees
  9. Future challenges
  10. References

Tree shapes and ranked tree shapes obtained from empirical data can be used to identify patterns of different speciation and extinction dynamics in different parts of the tree. For example, accelerated speciation in one of the two clades subtending the root yields a highly unbalanced tree, meaning that one clade is much larger than the other. By looking at the distribution of tree shapes induced by a particular model, we can test whether an empirical tree shape is very unlikely and thus rejected under the particular model.

It has been recognized for decades that the constant rate birth–death model, that is, the simplest speciation-extinction null model, induces a uniform distribution on ranked tree shapes with n tips, that is, each ranked tree shape with n tips is equally likely (Edwards, 1970).

In this section, I want to discuss that many more models, namely all species-speciation-exchangeable models, induce a uniform distribution on ranked tree shapes with n tips. Thus, if we reject a constant rate birth–death model based on tree shape alone, this implies that rates of speciation must have varied across contemporaneous lineages or that the discrete speciation-extinction pattern descending a speciation event depended on the particular choice of species speciating at the speciation event. Furthermore, I will discuss that common schemes of incomplete sampling do not alter the uniform distribution on ranked tree shapes. As a consequence, hypothesis testing methods based on (ranked) tree shapes can be directly applied to incompletely sampled (ranked) tree shapes.

In the following, I will first characterize the models giving rise to a uniform distribution of ranked tree shapes and second show how to test the appropriateness of these models for given empirical (ranked) tree shapes.

Models inducing a uniform distribution on ranked tree shapes

First we assume that all extant species are included into the phylogenetic tree. At the end of this section, incomplete species sampling is discussed. Aldous (2001) showed that under complete species sampling, all species-exchangeable models induce the same distribution on ranked tree shapes with n tips, namely a uniform distribution. This result implies that even if diversification rates decline strongly through time (as observed in many studies), or accelerate near the present, tree shape itself will be unaffected provided that species are exchangeable.

Species-exchangeable models are not the only models inducing a uniform distribution on ranked tree shapes. In Lambert (2010), it is shown that also a non-exchangeable model, namely a species-speciation-exchangeable model with asymmetric constant speciation rate but age-dependent extinction rate, induces a uniform distribution on ranked tree shapes. Furthermore, in a species-speciation-exchangeable model where asymmetric speciation rate depends on time and/or number of species, and extinction rate depends on time, number of species and/or a non-heritable trait, a uniform distribution on ranked tree shapes is induced (A. Lambert & T. Stadler 2013, pers. comm.)

It turns out that all models in which species are exchangeable w.r.t. speciation (species-speciation-exchangeable models) induce a uniform distribution on ranked tree shapes. For the mathematically interested reader, I will now present a proof showing that each ranked tree shape on n species is equally likely under a species-speciation-exchangeable model.

The idea of such a proof is based on reversing time and tracing the coalescences of the n species. The coalescent events correspond to speciation events yielding extant species. Going back in time, each pair of species out of the inline image species pairs is equally likely to coalesce first as all coexisting species are equally likely to speciate and subsequent speciation-extinction patterns are independent of the particular species speciating. Using this argument iteratively yields that a particular leaf-labelled nonoriented ranked tree shape has probability inline image and thus that each ranked tree shape (which by definition is oriented) is equally likely with probability inline image.

Furthermore, the proof not only reveals that each ranked tree shape on n tips is equally likely, but also that for any fixed n − 1 speciation times, each ranked tree shape is equally likely, meaning that the ranked tree shape and the vector of speciation times are statistically independent. This property is very important for hypothesis testing: it means that we can perform a test based on ranked tree shape (e.g. runs statistic, see below) and a second test based on speciation times (e.g. γ statistic, see below) and combine the p-values of the test via multiplication, as the two tests are independent from each other.

A uniform distribution on ranked tree shapes with n tips induces a unique distribution on nonranked tree shapes with n tips: the probability of a tree shape is obtained through multiplying the probability of a ranked tree shape by the number of rankings of a possible tree shape. A formula for the number of rankings is, for example, given in Semple & Steel (2003), Corollary 2.4.3. As a consequence, all species-speciation-exchangeable models induce the same distribution on tree shape, and thus, any method performing tests based on tree shapes or ranked tree shapes cannot distinguish between species-speciation-exchangeable models. As it is often very convenient to work with uniform distributions, major focus is here given to ranked tree shapes.

Models not belonging to the species-speciation-exchangeable class can induce nonuniform ranked tree shape distributions. Violating condition (i) of the species-speciation-exchangeable definition can obviously lead to a nonuniform ranked tree shape distribution, for example, by considering a pure-birth model under which always one of the two species subtending the most recent speciation event is the one speciation next. This model yields a caterpillar tree (i.e. the tree where at each branching event, one of the two daughter clades consists of only one species) with probability 1. As mentioned above, a symmetric constant rate speciation and age-dependent extinction model is an example fulfilling condition (i) but not condition (ii) of the species-speciation-exchangeable definition. Violating (ii) can lead to a nonuniform distribution on ranked tree shapes. Consider, for example, a model under which each species dies at age 1. Symmetric speciation happens at times 0, inline image, inline image, and each coexisting species is equally likely to be the one speciating (i.e. condition (i) is fulfilled). All species die at time inline image. Now, consider the probability distribution of four-species trees. By considering all possible scenarios, it turns out that the balanced (noncaterpillar) tree has probability 1, whereas the unbalanced (caterpillar) tree has probability 0. This distribution is clearly different from the uniform distribution on ranked tree shapes.

I further highlight that compared with species-speciation-exchangeable models, species-extinction-exchangeable models do not necessarily induce a uniform distribution on ranked tree shapes. Exchangeable w.r.t. extinction is defined as (i) at any point in time given an extinction event each species is equally likely being the one going extinct, and (ii) the discrete speciation-extinction pattern descending an extinction event is independent of which species was the one going extinct (i.e. for any species extinction leading to k lineages, we can label the k lineages after an extinction event with 1,…,k, such that their descendants follow a speciation-extinction pattern independent of which particular species went extinct). Consider for example a model under which always one of the two species subtending the most recent speciation event in the tree is the one speciating next, and all species have a constant rate of extinction. This model belongs to the species-extinction-exchangeable class, as each coexisting species is equally likely to go extinct, and future speciation-extinction pattern is independent of which species was chosen to go extinct. However, the caterpillar tree is obtained with probability 1, which is clearly different from a uniform distribution on ranked tree shapes. Thus, we do not consider species-extinction-exchangeable models as a separate class but discuss them within the species-non-exchangeable models below.

Incomplete species sampling

Up to here, tree shapes that are obtained by sampling all extant species were considered, and next tree shapes obtained by incomplete species sampling are considered. I will now explain that uniform and diversified sampling in species-speciation-exchangeable models yields again a uniform distribution on ranked tree shapes, that is, any hypothesis testing method based on ranked tree shapes can be applied to incomplete phylogenies (given sampling is uniform or diversified).

Consider any speciation-extinction model that induces a uniform distribution on ranked tree shapes when sampling is complete. For such a speciation-extinction model and uniform sampling, the resulting ranked tree shape distribution on n sampled species is again a uniform distribution and independent of the speciation times (Stadler, 2008b).

Also for such a speciation-extinction model and diversified sampling, the resulting ranked tree shape distribution given n sampled species is again a uniform distribution and independent of the speciation times. The reason is that under diversified sampling, we prune say m − n of a total of m species, such that the m − n most recent coalescent events are pruned (going backward in time). The probability of the order of the remaining coalescent events can be calculated again as in the case of complete sampling, meaning we obtain a uniform distribution on ranked tree shapes.

Outlook

It remains an open task to characterize all speciation-extinction models and all sampling schemes giving rise to a uniform distribution on ranked tree shapes. Only such a full characterization allows us to specify all speciation-extinction models being rejected when performing an hypothesis test based on ranked tree shapes. In particular, it is unclear whether there is a model not belonging to the species-speciation-exchangeable model class that gives rise to a uniform distribution on ranked tree shapes.

Testing evolutionary hypotheses based on (ranked) tree shapes

Tree shapes and ranked tree shapes can be used to test evolutionary hypotheses. The aim is to determine whether an empirical tree shape is very unlikely/likely under a given model, and thus the model is rejected/not rejected. Classically, a uniform distribution on ranked tree shapes with n tips is used as a null model, meaning all species-speciation-exchangeable models (and additional models giving rise to a uniform distribution on ranked tree shapes, if such exist) are considered as one class of models, and it is tested whether the empirical tree deviates from this class of models.

Unranked tree shapes

As mentioned above, phylogenetic reconstruction methods infer the tree shape, but not necessarily the ranking. Thus, the initial methods for testing evolutionary hypotheses considered the tree shape without a ranking. Guyer & Slowinski (1991) tabulated frequencies of a number of 5-taxon tree shapes. However, for larger trees, the number of tree shapes is too big to be tabulated, and so-called summary statistics are used to compare the tree shape distribution induced by the models to the empirical tree shape. A summary statistic summarizes the tree shape in one number (or a few numbers), widely used summary statistics as the balance of the tree (Colless index, Colless, 1982), the average path lengths from the root to a tip in the tree (Sackin index, Sackin, 1972) or the number of internal vertices having exactly two tip descendants (cherries or neighbouring pairs, Steel & Penny, 1993). The Colless and Sackin index are available within the r package apTreeshape (Bortolussi et al., 2009), and the cherry statistic is available within the r package ape (Paradis et al., 2004). An overview over tree shape statistics and their properties is, for example, given in Mooers & Heard (1997), Agapow & Purvis (2002) and Chan & Moore (2002).

In hypothesis testing, the distribution of the tree shape statistic under an hypothesized model (typically a uniform distribution on ranked tree shapes) is determined. The tree shape statistic distribution is determined analytically or via simulations, depending on the summary statistic and the hypothesized model (Blum et al., 2006; Rosenberg, 2006). Second, the tree shape statistic of the empirical tree is calculated. If the empirical tree shape statistic is an outlier in the model tree shape statistic distribution, the model is rejected for the considered data.

Tree shape statistics have been widely used, and a main result is that empirical trees on average tend to be less balanced than trees induced by a model with a uniform distribution on ranked tree shapes (Guyer & Slowinski, 1991; Heard, 1992; Mooers, 1995; Mooers & Heard, 1997; Blum & François, 2006).

Ranked tree shapes

Ranked tree shapes become available if we infer calendar time-calibrated phylogenies, but then only use the order of speciation events and ignore calendar time (e.g. due to very noisy branch length estimates). The runs statistic (Ford et al., 2009) can detect departure from the uniform distribution on ranked tree shapes. As the runs statistic uses ranking information in addition to tree shape information, it increases the power of evolutionary hypothesis testing compared with the classic tree shape statistics.

As the classic statistics on tree shapes, the runs statistic can be used to distinguish between models giving rise to uniform ranked tree shape distributions (i.e. all the species-speciation-exchangeable models) and models with a different ranked tree shape distribution (i.e. models belonging to the species-non-exchangeable class).

Furthermore, the runs statistic can also be employed to distinguish between models within the species-non-exchangeable model class not giving rise to a uniform distribution on ranked tree shapes. Within this class, there exists a set of models that does not induce a uniform distribution on ranked tree shapes, but given a tree shape, each ranking is equally likely. A complete characterization of such models is given in Stadler (2008a), although it remains to specify an explicit macroevolutionary interpretation. An example of a macroevolutionary model giving rise to a uniform distribution of rankings given the tree shape (but not inducing a uniform distribution on ranked tree shapes) is the constant relative probability pure-birth model (Ford et al., 2009), where each branching point x is assigned a probability inline image with which a new species is attached to its left daughter clade compared with the right daughter clade.

In contrast, a clade within which adaptive radiations happened does not produce a uniform distribution on rankings given a tree shape. The part of the tree with the adaptive radiation has an access of internal vertices appearing sequentially in the ranking, as speciation happened unusually fast during the adaptive radiation.

The runs statistic can be used for testing for a uniform distribution on rankings given the tree shape. Thus, the runs statistic may identify adaptive radiations in different parts of the tree (or other processes giving rise to a nonuniform distribution on rankings given the tree shape). The method has been used to show that the radiation of ants during the Late Cretaceous to Early Eocene (which corresponds to the rise of angiosperms) was caused by a burst of all lineages rather than only a few lineages extant at that time (Ford et al., 2009).

Uncovering speciation and extinction dynamics based on reconstructed trees

  1. Top of page
  2. Abstract
  3. Introduction
  4. Phylogenetic trees induced by speciation-extinction models
  5. Different classes of speciation-extinction models
  6. Uncovering speciation and extinction dynamics
  7. Uncovering speciation and extinction dynamics based on (ranked) tree shapes
  8. Uncovering speciation and extinction dynamics based on reconstructed trees
  9. Future challenges
  10. References

The more recent methods for uncovering speciation and extinction dynamics are based on analysing empirical phylogenetic trees where branch lengths are given in calendar time. These methods aim at quantifying the speciation and extinction rates given the empirical phylogenetic tree. In this section, I first discuss how the speciation and extinction rates can be inferred assuming that the rates remained constant at all times. Second I discuss how to test departure from that null model, that is, to test for nonconstant rates. Third I show how to quantify speciation and extinction rates which may change, and present advances towards determining the cause of the rate changes (e.g. abiotic or biotic cause).

Estimating speciation and extinction rates under a model of constant rates

Magallon & Sanderson (2001) used the clade age and number of species in a clade to estimate the expected diversification rate (i.e. speciation rate – extinction rate), assuming a constant rate birth–death model for speciation and extinction: at all times, each species speciates with a constant speciation rate λ and goes extinct with a constant extinction rate μ. The diversification rate estimator relies on the property that lineages accumulate through time with a rate λ − μ. Based on the estimator by Magallon & Sanderson (2001), Bokma (2003) proposed a statistical test for investigating the appropriateness of the constant rate assumption. These methods are very useful if only clade age and number of species are known. However, if we have the phylogenetic tree with branching structure available, we want to use this information such that we can not only infer the diversification rate, but infer speciation and extinction rates separately.

Nee et al. (1994b) showed that based on a reconstructed tree on extant species, it is possible to estimate both speciation and extinction rates, despite no extinct species being included in the tree, assuming a constant rate birth–death model. Nee et al. (1994b) derived the likelihood of the rates λ and μ for a reconstructed tree. The likelihood is the probability density of the model giving rise to the observed reconstructed tree, given particular parameters λ and μ. Maximizing the likelihood over λ and μ yields the maximum-likelihood parameter estimates for the speciation and extinction rate. We recall that for a reconstructed tree on n tips, the likelihood only depends on the time of the n − 1 speciation events but not on the particular shape of the tree: for given speciation times, each ranked tree shape is equally likely (see previous section).

I will not go into details here about how the likelihood for a reconstructed tree is calculated in Nee et al. (1994b). Instead, I will discuss what the likelihood method does intuitively. When plotting the accumulation of lineages through time in a lineages-through-time plot (LTT plot, Harvey et al., 1994), where the horizontal axis displays time and the vertical axis (on a log scale) displays the number of species through time, then in expectation, the LTT plot is first a straight line with slope λμ and towards the present the slope changes to λ, see Fig. 2. This change in slope is also called the pull-of-the-present effect (Harvey et al., 1994). A maximum-likelihood method for estimating λ and μ determines parameters such that the LTT curves of their induced reconstructed trees are as ‘similar’ as possible to the empirical LTT curve.

This parameter fitting may be performed intuitively by fitting two straight lines using, for example, least square fits to the speciation times displayed in the LTT plot, and equating the estimated slopes to λ − μ and λ. A maximum-likelihood method is superior to such a simple fitting method as in addition a likelihood method statistically weights the speciation times represented in the LTT plot: the more lineages present in the LTT plot, the less likely are deviations from the expected model pattern due to stochastic fluctuations.

It is important for the following sections to note that the LTT plot under the constant rate birth–death process is a straight line for the extinction rate being zero (Fig. 2, middle curve) and a convex curve for a nonzero extinction rate (due to the pull-of-the-present effect; Fig. 2, top curve). If applying the constant rate birth–death model to a data set with a concave LTT plot (which may be the result of decelerating diversification; Fig. 2, bottom curve), then zero extinction is estimated as the concave plot is ‘more similar’ to the straight line LTT plot than to the convex LTT plot.

Incomplete sampling

The method of Nee et al. (1994b) has been used to determine the speciation and extinction rates for a number of data sets; however, extinction rate was estimated to be zero frequently, despite the fossil record suggesting frequent extinction (Purvis, 2008). Underestimating extinction can result from analysing incomplete phylogenies. Removing species at present removes very recent speciation events in the reconstructed tree, which in turn removes the recent upswing in the LTT plot, that is, removes the pull-of-the-present effect, and leads instead to a straight line or even concave LTT plot (Nee et al., 1994a; Pybus & Harvey, 2000). No pull-of-the-present effect results in a zero extinction rate estimate when using Nee's method (Stadler, 2009; Cusimano & Renner, 2010; Höhna et al., 2011).

image

Figure 2. Expected lineages-through-time plot under three different speciation-extinction models. Top line is a constant rate birth–death model with speciation rate λ and extinction rate μ. The slope increase from λμ to λ is called pull-of-the-present effect. The middle line is a pure-birth model with speciation rate λμ. Due to no extinction, the slope remains constant at λμ. The bottom line is a diversity-dependent pure-birth model. The slope decreases due to speciation rate decreasing with the number of species.

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Likelihood methods estimating speciation and extinction rates accounting for uniform species sampling (Yang & Rannala, 1997; FitzJohn et al., 2009; Stadler, 2009) and accounting for diversified species sampling (Höhna et al., 2011) have been introduced; such methods provide unbiased parameter estimates given the model assumptions are satisfied.

Detecting departure from the null model using the γ statistic

Using the method by Nee et al. (1994b), or the extensions including incomplete sampling, we can infer speciation and extinction rates, assuming these rates remained constant. Testing the hypothesis whether the rates actually were constant can be performed, for example, using the summary statistics based on tree shapes or ranked tree shapes introduced in the previous section. However, the constant rate birth–death process induces the same distribution on ranked tree shapes (and thus the same distribution on tree shapes), namely the uniform distribution, as any other species-speciation-exchangeable model such as an environmental- or diversity-dependent speciation and extinction model, or an asymmetric constant rate speciation and age-dependent extinction model.

Thus, to test the validity of the constant rate birth–death model with increased power, branch length information needs to be included. The γ statistic (Pybus & Harvey, 2000), summarizing the times of branching events into one single value, has been suggested for testing the constant rate birth–death model assumption. The γ statistic is expected to be normally distributed with mean zero under the Yule model (with a straight line LTT plot) and has a positive mean under the constant rate birth–death model with a nonzero extinction rate (with a convex LTT plot due to the pull-of-the-present effect). A negative γ value corresponding to a concave LTT plot indicates departure from the constant rate birth–death model. In particular, processes with a slowdown in diversification induce reconstructed trees with negative γ values. As the branching times are independent information from the ranked tree shape under a constant rate birth–death process, a tree shape statistic as well as the γ statistic can be combined to increase power of a test addressing the validity of the constant rate birth–death model.

A number of studies revealed a negative γ statistic for mammals and birds (Pybus & Harvey, 2000; Weir, 2006; Phillimore & Price, 2008), which corresponds to a concave LTT plot, meaning that these data sets yield zero extinction estimates. Accounting for incomplete sampling is not enough to obtain nonzero extinction rate estimates in general [e.g. some of the phylogenies in Phillimore & Price (2008) are almost complete], demonstrating the need for more complex models than the constant rate birth–death model.

Estimating speciation and extinction rate variation

The γ statistic is designed for testing departure from the null model assuming constant speciation and extinction rates, but cannot be used to distinguish between alternative complex hypotheses, such as between diversity-dependent speciation and a scenario of declining diversity (Quental & Marshall, 2011). Thus, to potentially discriminate between such hypotheses, the whole reconstructed tree rather than a single summary statistic (here γ) has to be considered.

In the following, I present recent studies calculating the likelihood of a reconstructed tree under such complex models for speciation and extinction (i.e. under species-exchangeable, species-speciation-exchangeable and species-non-exchangeable models), extending the constant rate birth–death model methodology introduced in Nee et al. (1994b). After determining the maximum-likelihood value under the simple and complex model, where the simple model is nested, that is, a special case of the complex model, we can perform a likelihood ratio test to determine whether the simple model is rejected in favour of the more complex model. Such a likelihood ratio test uses information of the whole reconstructed tree, rather than only summary statistics. In fact, as the likelihood ratio test is the test with the highest power among all possible tests (Neyman–Pearson lemma, Neyman & Pearson, 1933), this test is the recommended test for nested models. I want to emphasize that rejecting the simple model with a high significance tells us nothing about the fit of the complex model, besides it fitting better than the simple model. It may still be a very poor fit, so to identify whether the fit of the complex model is really good, we need to explore whether alternative, potentially more complex, scenarios fit better. When comparing two non-nested models, we cannot use likelihood ratio tests. In this case, we can employ the Akaike information criterion. The maximum-likelihood parameter estimates of the most plausible model quantify the speciation and extinction parameters suggested for the given data set.

In this section, speciation-extinction models accounting for the suggested main factors determining speciation and extinction rates are discussed (Stadler, 2011a):

First, I discuss models accounting for extrinsic factors determining speciation and extinction rates. An extrinsic factor may be abiotic, yielding environmental-dependent speciation and extinction rates, or biotic, yielding diversity-dependent speciation and extinction rates. Such models belong to the species-exchangeable class of models.

Second, I discuss models accounting for intrinsic factors determining extinction rates. An intrinsic factor may be age, and I discuss models with asymmetric constant rate speciation and age-dependent extinction rates, that is, models under which old species may be more or less prone to extinction than young species. Such models belong to the species-speciation-exchangeable class of models. I end with considering general models accounting for intrinsic factors, that is, models with general trait-dependent speciation and extinction rates. Such models belong to the species-non-exchangeable class of models.

Species-exchangeable models

Species-exchangeable models induce a uniform distribution on ranked tree shapes. Furthermore, for any set of n − 1 speciation times, each ranked tree shape is equally likely, meaning that speciation time distribution and ranked tree shape distribution are independent. Thus, a tree shape statistic may be used in addition to a likelihood ratio test (which only uses the speciation time information) to check for the validity of any of the models below.

I will now discuss methods for estimating speciation and extinction rates under two important species-exchangeable models accounting for extrinsic factors influencing speciation and extinction dynamics: first a model with environmental-dependent rates and second a model with diversity-dependent rates.

Environmental-dependent speciation and extinction rates

Speciation and extinction rates changing as a function of time model the response of macroevolutionary dynamics to environmental changes such as climatic changes or tectonic activity. Already Nee et al. (1994b) provided general equations describing the likelihood for a reconstructed phylogenetic tree under a model with speciation and extinction rates being arbitrary functions of time; however, the equations require the numerical solution of integrals and were not widely used (but see Turgeon et al. (2005) for the analyses of small reconstructed trees).

For large reconstructed trees, Paradis (1997) developed an approach detecting varying speciation and extinction rates through time by identifying changes in the slope of the LTT plot through time. The drawback of such a nonlikelihood-based approach is that it may confound a pull-of-the-present effect with a rate change: suppose we had a speciation rate λ and a extinction rate μ until time T in the past. Between time T and the present, suppose the speciation rate was λ/2 and the extinction rate was 0. Then, the LTT plot shows an initial slope of λ − μ, followed by a slope of λ (this is caused by a pull-of-the-present effect at time T) and by a slope of λ/2, meaning the LTT plot shows two slope changes, whereas only one rate change occurred.

To apply the likelihood equations of Nee et al. (1994b) to large reconstructed trees, an analytic solution of the integrals would be advantageous. Rabosky (2006) derived closed-form equations for a model with two time intervals, where in each interval, the rates are constant but may change arbitrary from one interval to the second. However, it turns out that the method, assuming independence of the two time intervals, implicitly assumes zero extinction in the later interval, as otherwise extinction in the later interval affects the earlier interval and thus violates the independence assumption. This limitation as well as the limitation of only two time intervals was relaxed in Stadler (2011b); a birth–death model with piecewise constant speciation and extinction rates allows us to estimate the speciation and extinction rates through time as well as the times of rate changes, based on a closed-form solution of the likelihood function. The method has been used to reject the hypothesis of accelerated mammalian diversification following the K/T-boundary at 65 Ma (Meredith et al., 2011; Stadler, 2011b). The method is available within the r package TreePar (Stadler, 2011b) as well as within Beast2 (http://beast2.cs.auckland.ac.nz).

Diversity-dependent speciation and extinction rates

Speciation and extinction rates changing as a function of the number of species accounts for diversity-dependent dynamics, with speciation rates decreasing and extinction rates increasing as the number of species increases.

Model without extinction. Both a linear and an exponential change in rates as a function of species have been suggested in Rabosky & Lovette (2008a). In this initial study, no extinction was considered. The method revealed that Dendroica warbler's diversification is driven by diversity-dependent speciation. The method for fitting diversity-dependent speciation models without extinction to reconstructed phylogenies is available within the r package laser (Rabosky, 2009).

Coalescent approximation. Morlon et al. (2010) introduced a coalescent-based framework to estimate speciation and extinction rates, which may vary as a continuous function through time. This seems like a framework belonging to the class of environmental-dependent models. However, the framework intends to test the hypothesis of exponential decreasing speciation rates or exponential increasing extinction rates through time. Such continuous rate changes following an exponential function seem unlikely to be caused by the environment, but are approximately predicted by the diversity-dependent model, and thus, the coalescent framework is approximating the diversity-dependent dynamics. Note that the coalescent framework cannot account for the speciation and extinction dynamics separately, but parameterizes an exponentially changing diversification rate (= speciation rate − extinction rate).

This framework was applied to a wide range of phylogenies (almost 300, McPeek, 2008) identifying a decay in speciation rates through time with diversity still expanding today though, raising the question whether (i) biodiversity is not limited or (ii) whether the limit has simply not been reached yet (Morlon et al., 2010). A third possibility is that the exponentially changing rates through time do not model the speciation and extinction dynamics very well.

Outlook

To properly account for diversity-dependent speciation and extinction dynamics in a coalescent framework, the rates should be varied as a function of the expected number of species through time predicted by the diversity-dependent model, rather than varied following an exponential function (or linear/constant as alternatives in the original paper).

Time-dependent birth–death model approximation. Instead of making use of the coalescent approximation, the likelihood for a reconstructed tree can also be calculated exact under a birth–death model with exponentially changing diversification rates (Rabosky & Lovette, 2008b; Morlon et al., 2011), essentially putting the equations with integrals (Nee et al., 1994b) into a numerical framework. As Morlon et al.'s (2010) coalescent approximation, these methods again approximate the density-dependent dynamics. However, compared with the coalescent, which only accounts for an exponentially changing diversification rate, the birth–death model explicitly distinguishes between speciation and extinction rates. Rabosky & Lovette (2008b) showed that varying speciation rates are necessary to obtain phylogenies with an early rapid lineage accumulation. Morlon et al. (2011) reconciled the cetacean fossil record with the molecular phylogenies. Again, only constant/linear/exponential speciation/extinction rate changes were considered, instead of considering rate changes that are expected under diversity-dependent models.

Exact likelihood method. In Etienne et al. (2012), the exact likelihood for a reconstructed phylogenetic tree under a diversity-dependent speciation model was derived. The method requires us to solve a system of differential equations numerically, where the number of equations equals the carrying capacity of the considered clade (i.e. the maximal number of species possible to exist simultaneously). The reason for the complexity of this exact method is that the dynamics of the species at any point in time are not independent of each other any more [as it was the case for environmental-dependent models, and the approximations in Morlon et al. (2010, 2011)], but speciation and extinction rates depend on the number of coexisting species. This problem of dependence is dealt with by calculating the likelihood for the reconstructed phylogeny allowing for all possible numbers of species at all times and then summing over all possibilities, employing a dynamic programming approach for efficiency. The approach is implemented in the r packages DDD (Etienne & Haegeman, 2012) and TreePar (Stadler, 2011b).

The method reveals that the previously considered Dendroica warbler phylogeny is not only better explained by diversity-dependent speciation [as already suggested by Rabosky & Lovette (2008a)] but also clearly shows signs of nonzero extinction, in contrast to zero extinction obtained using the constant rate birth–death process or the pure-birth diversity-dependent speciation model. As the Dendroica warblers lack a fossil record, no comparison with palaeontological data could be made. Thus, the Cenozoic macroperforate planktonic foraminifera phylogeny, which has a very good fossil record, was analysed, revealing that the extinction rate estimates based on the phylogeny agree very well with estimates based on the fossil record of foraminifera. Another three phylogenies (Plethodon salamanders, Heliconius butterflies and Cetacea) were furthermore considered, supporting a diversity-dependent model, but without strong signal for nonzero extinction (Etienne et al., 2012).

Outlook

The above methodologies assume equal strength of diversity dependence, that is, competition, among all species in the clade. As suggested in Ricklefs & Renner (2012), competition between species may occur at different scales. Thus, models accounting for different levels of competition within different taxonomic hierarchies need to be developed to start understanding the ecological limits in a quantitative way as well as to start understanding which biological communities are at equilibrium.

Species-speciation-exchangeable models

We recall that a model is species-speciation-exchangeable if the following two conditions are fulfilled: (i) at any point in time given a speciation event each species is equally likely being the one speciating, and (ii) the discrete speciation-extinction pattern descending a speciation event is independent of which species was the one speciating. Asymmetric constant rate speciation and age-dependent extinction models belong to the species-speciation-exchangeable class of models, and I briefly discuss in the following how to calculate the likelihood of a reconstructed tree under such a model. For other species-speciation-exchangeable models such as asymmetric constant rate speciation models with nonheritable trait-dependent extinction, no methods for calculating the likelihood of a reconstructed tree are available.

Asymmetric constant rate speciation and age-dependent extinction models

All methods presented above assumed that a species' risk of speciation and extinction is independent of its age. Venditti et al. (2009) indeed suggested that speciation is not age dependent, as they found that rare individual events may cause speciation instead of many small accumulated mutational changes causing a speciation event. However, species age may affect extinction risk, as well-adapted older species may be less flexible to adapt to a changing environment. Lambert (2010) introduced a mathematical framework calculating the likelihood for a reconstructed phylogenetic tree, assuming any species lifetime distribution and assuming a constant rate of asymmetric speciation. It turns out that the reconstructed trees induced by this model can be interpreted as a so-called point process (Popovic, 2004; Aldous & Popovic, 2005; Gernhard, 2008), resulting in particular in the uniform distribution on ranked tree shapes with n tips.

Outlook

Evaluating the likelihood for a reconstructed tree with asymmetric constant rate speciation and assuming general lifetime distributions require the numerical evaluation of inverse Laplace transforms (Lambert, 2010). A robust implementation will allow us to investigate whether older species are indeed more prone to extinction than younger species.

Species-non-exchangeable models

When considering a large number of reconstructed phylogenies, we observe less balanced trees than suggested by the uniform distribution on ranked tree shapes (Blum & François, 2006). In part, this might be due to biased incomplete sampling, but an additional explanation is required for the unbalanced completely sampled reconstructed trees. Unbalanced reconstructed trees may only be produced if the underlying model is not contained in the species-speciation-exchangeable class. Under species-non-exchangeable models, speciation and extinction rates may vary in any arbitrary way, typically modelled as a function of a species trait, where a trait may be, for example, species size (stochastic trait), substitution rate in a species population (hidden trait) or the age of a species (deterministic trait). For these models, it becomes important to distinguish between heritable traits and nonheritable traits and between symmetric and asymmetric speciation.

Very different approaches have to be employed for the three different types of traits: Stochastic and hidden traits are assumed to change according to some stochastic process (typically Brownian motion). While stochastic traits are observed for each tip in the reconstructed tree, hidden traits are not observed directly (e.g. we do not know the substitution rate of an extant species; however, we know its genetic sequence). Deterministic traits such as species age vary deterministically rather than stochastically (age is determined by time), facing yet different methodological challenges.

I will here discuss methods for heritable and nonheritable stochastic traits under symmetric and asymmetric speciation. To my knowledge, there is no method accounting for hidden traits. In the previous section, we discussed a nonheritable deterministic trait (age) governing extinction under an asymmetric constant rate speciation model. As no method relaxing the nonheritability, the asymmetric speciation or the constant rate speciation assumption was developed for determinstic traits, deterministic traits will not be discussed here.

Stochastic trait-dependent speciation and extinction rates

Alfaro et al. (2009) introduced a method assuming constant rates through time, with allowing these rates to change at specified branches. These branches with rate changes are identified using the Akaike information criterion. The method called Medusa is available within the r package geiger. Using Medusa, nine major radiations were found during the evolution of jawed vertebrates (Alfaro et al., 2009). A limitation of the method is that no explicit model for the change of rates is assumed (but rate changes are added on the basis of the Akaike information criterion), meaning that the extinct parts of the tree are implicitly assumed to have the same rates as their most recent ancestor in the reconstructed phylogeny.

An explicit model for rate changes was introduced in Maddison et al. (2007): each species is characterized by a considered trait being in state A or state B, with the speciation and extinction rates depending on this binary trait. Upon a speciation event, both descending species inherit the trait state from the ancestral species (i.e. symmetric speciation with a heritable trait is assumed). A constant rate is assumed for a species to change from state A to state B, and another constant rate to change from state B to A, meaning evolution of the trait is modelled as a gradual process. FitzJohn et al. (2009) extended this framework to incompletely sampled phylogenies. Magnuson-Ford & Otto (2012) and Goldberg & Igic (2012) in addition accounted for trait changes at speciation events (punctuated equilibrium); such approaches can also be used to account for asymmetric speciation and nonheritability. All these methods are available within the r package diversitree (Fitzjohn, 2009).

The methods provide a macroevolutionary perspective on the evolution of recombination. The above methods, with trait state A being selfing-compatible (SC) and trait state B being selfing-incompatible (SI), reveal that predominant SI in plants is caused by a significantly lower extinction rate for SI plants compared with SC plants (Goldberg et al., 2010; Goldberg & Igic, 2012). It should be emphasized that also the speciation rate for SI plants is lower compared with SC plants; however, as diversification (= speciation − extinction) rate is estimated positive in SI plants compared with negative in SC plants, the results predict an macroevolutionary advantage for species that recombine at reproduction.

These models were extended such that speciation and extinction rates may depend on more than two states as well as on time (Rabosky & Glor, 2010; FitzJohn, 2012). Goldberg et al. (2011;) adopted the methodology such that traits can be geographic locations. The difference to the original method is that a species may be assigned more than one trait, namely if it occurs in several geographic locations. Furthermore, instead of discrete traits, FitzJohn (2010) models the trait as a continuous variable, where traits change according to a Brownian motion process through time. The speciation and extinction rates are assumed to depend on the trait state and thus may also take any continuous value.

Outlook

The stochastic trait-dependent speciation-extinction models require the knowledge of the trait state for the extant species. To link hidden traits to macroevolution, such as linking microevolutionary substitution rates to macroevolutionary diversification rates, the methods have to be extended to account for hidden traits by incorporating trait-relevant information (e.g. sequence information if investigation substitution rates). Extending the model to deterministic trait-dependent speciation rates remains a open challenge, which is likely not being solved using the framework in Lambert (2010), as the employed ‘contour process representation’ for species-speciation-exchangeable models does not work anymore if speciation rates change as a function of the trait.

Future challenges

  1. Top of page
  2. Abstract
  3. Introduction
  4. Phylogenetic trees induced by speciation-extinction models
  5. Different classes of speciation-extinction models
  6. Uncovering speciation and extinction dynamics
  7. Uncovering speciation and extinction dynamics based on (ranked) tree shapes
  8. Uncovering speciation and extinction dynamics based on reconstructed trees
  9. Future challenges
  10. References

As reviewed in the previous sections, a large number of methods have been developed over the past few years to account for complex speciation and extinction dynamics.

First, methods being based on ranked tree shapes have been proposed to test the plausibility of the constant rate birth–death model (often termed equal-rates Markov model) or the even simpler Yule model. As shown here, rejection of this null model implies that all models within the species-speciation-exchangeable class are rejected simultaneously, as all these models induce a uniform distribution on ranked tree shapes.

Second, methods being based on the reconstructed tree with branch lengths have been proposed to quantify speciation and extinction rates and to test between different hypothesized speciation-extinction models (beyond only testing between species-speciation-exchangeable models and species-non-exchangeable models inducing nonuniform ranked tree shape distributions). The new methods can account for periods of declining diversity through environmental-dependent speciation and extinction rates, for the observed recent flattening of LTT plots through incomplete sampling and/or diversity-dependent rates and for the imbalance in empirical phylogenies through trait-dependent speciation rates. Closed-form solutions for the likelihood of the speciation and extinction rates given a reconstructed phylogenetic tree are known for the environmental-dependent model, while all other likelihood methods rely on numerical tools solving differential equations or inverse Laplace transforms. For an overview of available methodology, see Table 1.

I already mentioned a few rather technical challenges for the future about the specific models in the ‘Outlook’ paragraphs. Looking at the bigger picture, it will become essential to combine the approaches mentioned earlier. No clade will only be influenced by, for example, diversity-dependent diversification or only by, for example, trait-dependent diversification, but most likely by a combination of the different factors. Currently, the different hypotheses can only be tested in an exclusive way, either one hypothesis is supported or the other, or none. Although such exclusive statements may hold for the data sets considered in the studies presenting the new frameworks, most data sets will likely support a mix of factors. Formalizing one general model and inferring its parameters will reveal the importance of the different factors influencing speciation and extinction dynamics.

A general model will have limitations in inference as different speciation and extinction processes may lead to the same distribution on reconstructed phylogenetic trees, meaning that based on the reconstructed phylogenetic trees we cannot distinguish between such processes (Quental & Marshall, 2010). For example, simulations revealed that past mass extinction events leave the same fingerprint in reconstructed phylogenies as periods of stasis in diversification or adaptive radiations followed by a recent increase in diversification (Crisp & Cook, 2009; Stadler, 2011c). In such cases, sampling parameter space using Markov chain Monte Carlo (MCMC) methods may be useful to investigate the space of supported parameter combinations. Alternatively, contour plots of the likelihood surface may be considered, although this will be cumbersome as the general model has a high-dimensional parameter space. Recently, the MCMC method BayesRates (implemented in Python) for estimating time-dependent speciation and extinction rates became available (Silvestro et al., 2011), although the method still considers different models in an exclusive manner and tests them against each other using Bayes' factors. Investigating the full parameter space using MCMC methods allows us to identify alternative explanations for the data. To distinguish between such hypothesized explanations (e.g. mass extinction/diversification stasis), fossil data need to be considered additionally, see below.

The above methods attempt to estimate speciation and extinction rates as well as their changes. It has been widely recognized that although diversification (= speciation − extinction) can be calculated with high confidence, the turnover (extinction/speciation) is very hard to estimate: most if not all of the studies above observed this phenomenon. The diversification rate is already well informed by the final number of species as well as clade age, whereas turnover estimates require detailed information about the relative timing of speciation events. If simulated under the appropriate model, the turnover estimates seem to be accurate for large reconstructed trees though (Stadler, 2011b).

However, if we mis-specify the speciation-extinction model, it may become impossible to estimate the turnover (Rabosky, 2010). Rabosky (2010) showed that when diversification rates for each clade were drawn from a distribution of rates for simulations, but a constant rate birth–death model was fitted to the simulated data, turnover estimates were very biased. Thus, we should put increased effort into developing methods for inferring speciation and extinction rates under more complex speciation-extinction models, such that we can test the appropriateness of our models (e.g. using likelihood ratio tests) and identify aspects lacking in our models.

Developing phylogenetic methods including information from the fossil record will be a major step towards distinguishing between hypothesized models and obtaining more confined estimates about the turnover. So far, only the estimates based on the palaeontological data were compared with the estimates based on the neontological data (Morlon et al., 2011; Simpson et al., 2011). However, as the neontological data evolved from the palaeontological data, the two data sets should be considered simultaneously when inferring speciation and extinction dynamics. This could be done, for example, by using fossil counts as prior information about past biodiversity.

With the reconstruction of phylogenies including fossil tips (Ezard et al., 2011), we should aim for an extension of the phylogenetic methodology accounting for fossils being sampled and included in phylogenetic trees. Methodology developed for measurably evolving populations such as RNA viruses, where sampling in the past is common, can be adopted for analysing species phylogenies with fossils. In fact, in the mid-1990s, macroevolutionary phylogenetics was employed and extended to improve the understanding of viral evolution (Holmes et al., 1995; Nee et al., 1995), whereas in the future, we might do the reverse by using ideas from viral phylogenetics (Stadler, 2010; Stadler et al., 2012) to improve the understanding of macroevolution.

In addition to quantifying the turnover more precisely, new methods allowing us to simultaneously investigate neontological and palaeontological data sets have the potential to reveal whether the lineages leading to surviving species are significantly different from the lineages without any descendants (meaning our phylogenetic trees on extant species are a very biased sample) or whether survival of a lineage was a purely stochastic process.

All models discussed above assume certainty in species delimitation, which may not be appropriate for very recent speciation events. If we do not recognize diverging populations as separate species yet, we under-sample recent speciation events, leading to a flattened recent LTT plot, which again underestimates extinction. Etienne & Rosindell (2012) proposed a mechanism of protracted speciation that causes a flattening of the recent LTT plot. However, no likelihood equations are available yet for this model; thus, I propose to consider the phylogeny up to the time point for which we can be sure about the species delimitation and ignore the most recent past. According to Etienne & Rosindell (2012), such a time point might vary between 1 and 5 Ma depending on the organisms. The phylogenetic tree can be cut off at that time point, and the methods introduced above can be used with only a minor modification: we need to acknowledge the number of extant species at the cut-off time that went extinct before today through uniform incomplete sampling.

Last I want to point out that the discussed speciation-extinction models consider the species as a unit, although in fact species consist of a large population of individuals. Classically, we built a species tree based on genes from individuals in a population and then assume that a gene tree equals a species tree; however, this assumption does not always hold (Degnan & Rosenberg, 2009; Burbrink & Pyron, 2011). Recent methodological advances such as *Beast (Heled & Drummond, 2010) allow us to directly infer species trees based on genes from individuals; however, most species trees in the literature will in fact be gene trees. Not only the phylogenetic trees may be biased when ignoring the individuals and considering the species as a unit, but also speciation and extinction dynamics may be affected. As a species' extinction is the result of the last individual of its population dying, extinction is in fact a result of population size changes [as, for example, acknowledged in the unified neutral theory by Hubbell (2001)]. As long as population size changes within a species are quick compared with macroevolutionary dynamics, the approximation of treating a species as a unit may be very good; however, once processes happen at similar timescales, consequences of individual-based models have to be investigated, as, for example, in Davies et al. (2011).

I want close emphasizing that all methods for estimating macroevolutionary dynamics heavily rely on accurate phylogenetic trees. Thus, when reconstructing phylogenies based on data from extant species, appropriate methods and models together with additional data such as fossil data should be chosen carefully, and robustness of phylogenetic tree inference towards particular assumptions should be investigated. With high-quality phylogenies becoming available, and the development of general phylogenetic tools unifying dynamics at the macroevolutionary, microevolutionary and ecological scale, we have the unique opportunity to obtain a detailed understanding of the past processes and dynamics, which gave rise to the current biodiversity.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Phylogenetic trees induced by speciation-extinction models
  5. Different classes of speciation-extinction models
  6. Uncovering speciation and extinction dynamics
  7. Uncovering speciation and extinction dynamics based on (ranked) tree shapes
  8. Uncovering speciation and extinction dynamics based on reconstructed trees
  9. Future challenges
  10. References