Types of inferences under DELINEATE

DELINEATE has three modes of inference, depending upon the goals of a study, the ancillary information a researcher has on the focal taxa, and/or the study design. For example, in most species delimitation applications, we very rarely have absolutely no knowledge of any species identities of the sampled individuals. Instead, we typically have some understanding of the species assignments of some of the individuals in a dataset, often by design, and it is the species identities of only a subset of the collected data that we are interested in actually inferring. In such cases, the “constrained” mode of inference would be applied in DELINEATE, rather than the “unconstrained” inference mode. Alternatively, the “tempo of speciation” mode of inference might be preferred if the focus of the study is on speciation dynamics, rather than delimitaion per se (see below).

Statistical model description and inference algorithm

We treat the data as a set of samples of sequences (D) for K loci from M populations (lineages), with N_m individuals sampled from population m. A population tree with branch lengths is inferred from these sequences using the “censored coalescent” or multispecies coalescent (MSC) model of [28]. In our current implementation or DELINEATE, the inference of the population tree under the MSC thus represents the first stage of analysis (Fig 2). Following the notation of [28], Λ represents the assignment of populations to species, and S is tree of population relationships, with branch lengths, estimated from multilocus data resulting from inference under the MSC model. The posterior probability of a particular partition given the data would require taking an integral over all possible trees of populations and all possible values of the other parameters of the model. While this could be approximated using Markov chain Monte Carlo methods, here we assume that species assignment is independent of the process of molecular evolution, once we condition on the lineage tree. Our approach focuses on calculating the likelihood of a species partition given a lineage tree and the speciation-completion rate, σ:

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Fig 2. Overview of speciation-based delimitation using DELINEATE.

Starting with genomic data (a), a lineage tree (b) is inferred under the multispecies coalescent (MSC) model using any of a number of programs, such as BP&P [28] or *BEAST [30]. The inferred lineages, which are consistent with a Wright-Fisher population under the MSC model (i.e., cannot be divided any further), are (c) organized into sets of one or more species in DELINEATE, with each possible organization (referred to as a “partition”) representing a different hypothesis about species boundaries. Partitions can range from a single species (i.e. all lineages assigned to the same species) to as many species as their are lineages (i.e., there are no population lineages, only different species). The probability of each of the different partitions is calculated and reported by DELINEATE, The partition with the highest probability is the maximum likelihood estimate, but investigators have at their disposable all the partitions in the 95% confidence interval as well if they wish to summarize support for particular results as well.

https://doi.org/10.1371/journal.pcbi.1008924.g002

To take uncertainty in the lineage tree into account, one could sample lineage trees from their posterior distribution (e.g., generated using software such as *BEAST), and average Pr(Λ∣σ, S) over this sample.

Information about the speciation-completion rate is inferrable from the lineage tree, , induced by the set of taxa the researcher has previously classified as the constrained leaf set (); we denote the species partition of this subset of leaves as . We assume that is a random sample of the full leafset to calculate as the likelihood for σ, otherwise, some sort of correction would be needed for an ascertainment bias.

We use a Poisson distribution to obtain the probability of 0 or ≥ 1 speciation completion events occurring along a branch given an instantaneous speciation-completion rate of σ. That is, given a branch with duration time of τ_i, then the probability of the speciation process not completing on this branch is given by: while the probability of the speciation process completing is given by:

Note, thus, that the conspecific/heterospecific status of an ancestor-descendant pair only depends on whether the number of speciation completion events is 0 or greater than 0. Despite only having to keep track of two states for each branch (0 vs > 0 speciation completion events), the number of possible configurations over the entire lineage tree () can be quite large: , where is the number of branches in a rooted tree. Fortunately, calculating the likelihood, only entails summing over those configurations of speciation completion events across branches that are compatible with all of the constraints implied by . For a fixed lineage tree and a moderate size of , this calculation is feasible via dynamic programming. Bookkeeping similar to Felsenstein’s pruning algorithm can be performed during a post-order traversal of the tree to determine the likelihood of the constrained leaf partition given a tree and σ (details of the dynamic programming algorithm used are provided in S1 Text.).

Note that if the researcher’s prior information on alpha taxonomy only consists of conspecific constraints, then because in that case the probability of all lineages belonging to the same species would have a probability of 1. Similarly, if the researcher’s constraints are all heterospecific constraints, then because this will assign a probability of 1 to a scenario in which every tip is its own species. Thus, one would expect a plausible estimate of σ only when the set of input constraints has at least one conspecific and at least one heterospecific constraint.

Performance assessments

The performance of each of the different modes of inference available within our full probabilistic model for species delimitation was evaluated using simulated data. These assessment include identifying the limitations of DELINEATE in addressing the various delimitation objectives discussed above. In all cases, post-inferential statistical analyses and visualization were done using the pandas [31], seaborn [32], and Matplotlib [33] Python libraries. The 95% confidence intervals were calculated by adding partitions in decreasing order of probability until 95% of the (constrained) probability is accounted for (i.e., the interval consists of the set of the highest-probability partitions, with the sum of probabilties equal to at least 0.95).

Software and data

Software for inference under the DELINEATE model is publically available at: https://github.com/jeetsukumaran/delineate This software is written in Python [35], and makes use of the NumPy [36], SciPy [37], and DendroPy [38] software libraries. The package is fully documentated, with documentation available online at https://jeetsukumaran.github.io/delineate/. The documentation includes a primer on background concepts as well as a fully worked empirical case example using a Lionepha dataset recently published by [39], illustrating analytical procedures and guides.

Scripts that were used generate data and analyze them for the performance tests are available S1 Data.

Accurate to inaccurate inference

The substantial accuracy of species assignments when the identities of a subset of lineages are provided contrasts strongly with the relatively poor performance of analyses using genetic data alone (i.e., without supplemental information), speaking to the limits of knowledge possible with genomic data used in isolation. However, even so, while actual species identity assignments may remain challenging without supplemental information, even in these cases inferences regarding the number of species are remarkably robust generally [Fig 3a]. In particular, these estimated species numbers are also markedly more reliable than those inferred under the MSC (see Fig 2a, [2], which shows the the MSC dramatically overestimates species numbers).

With respect to inferring the species status of unknown lineages, the delimitation model with the highest probability corresponded to the true partition under a broad parameter space, with two notable exceptions [Fig 3b]. First, accurately identifying the species status of all lineages is unlikely if half of those lineages have no information about them to constrain the inference procedure. Under such situations, the only property that can be reliably estimated is the number of species [Fig 3a]. Second, the probability of inferring the correct delimitation model depends upon the history of diversification itself. In particular, it is unlikely to correctly identify the species status of all lineages when the speciation-completion rate is very high. However, as we emphasized earlier, we simulated data over a parameter space designed to identify the theoretical limits, not just the promise, of DELINEATE. As such, the reported poor performance in specific areas of parameter space does not necessarily imply a limited utility of speciation-based delimitation in practice.

In practice, with more biologically realistic speciation-completion rates (e.g., half or tenth of the population isolation rates, as opposed to the relatively unrealistic cases where populations form at the same rate as species), and with even a little information in the form of known species assignments of some population lineages, the performance of DELINEATE increases dramatically. For example, accuracy approaches the 80%—90% range of simulated datasets in which the species status of all unknown lineages were correctly inferred. Note that in DELINEATE the user does not need to provide any information regarding the speciation process itself in the form of the speciation-completion rate parameter (σ): DELINEATE “learns” this from analysis of the data based on the known species assignments. Moreover, despite noise in the estimation of this parameter [Fig 4], estimation of the actual species delimitation model seems to still perform relatively well as long as the true rates themselves are not extreme (i.e., speciation-completion rates approaching parity with population isolation rates) [Fig 3].

Irrespective of whether the focus is on delimiting all, or just a subset, of lineages with unknown species status (or on speciation dynamics rather than delimitation per se; as discussed below), the study design under our new approach will differ from those in the past. In particular, investigators should adopt study designs that include lineages of known species identities, in addition to the lineages that they wish to assign to species, when they collect genetic data. That is, instead of restricting analysis to a set of genomic data collected in individuals in which we have no idea as to any of the species assignments, systematic studies should design analysis to span a broader context that includes at least some lineages of known species identities. Many species delimitation studies in fact do this routinely, as it is rather unusual for a system not to have any information about species identities for any of its lineages. This study design parallels those for analyses of divergence times in which the operational taxonomic units (i.e., the tips of the tree) are selected to include taxa for which calibration or fossil data is available. In the context of DELINEATE, the relationship of the number of species identities known a priori to accuracy is the simplest to understand, at least on a trivial level: the more information that we provide to the model, the better the model performs. In addition, it should be noted that the benefits of this information are not only in terms of informing the model, but also restricting parameter space in terms of the number of partitions to visit, thus speeding up computational times. However, computational time also becomes an important component the higher the total number of lineages. Although the amount of information about the speciation process that can be gleaned increases under such conditions, and allows for better inference about the delimitation model (see [Figs 3 and 4]), there is a computational trade-off. With larger datasets, the accuracy of inferences improves, but the number of partitions to be scored grows very quickly, making calculations infeasible when analyzing too many lineages.

Speciation dynamics

We note that the incorporation of an explicit speciation process opens new frontiers not only in species delimitation analysis, but also in macroevolutionary studies of diversity. Specifically, and in particular using the model applied in DELINEATE, the rate of development of species isolation mechanisms, as distinct from the rate of population isolation, can be directly estimated. This is a valuable evolutionary biology study objective in its own right [41]. But, in addition, this provides investigators with a framework for studying the linkages between population and species-level processes. For example, understanding why species diversity differs among geographic areas or among taxa requires an understanding of how diversity is generated and maintained. As such, speciation-based delimitation approaches like DELINEATE can be used to address such questions, including testing “museum” vs “cradle” models to explain the higher diversity in the tropics compared with temperate areas (i.e., the “museum” with lower extinction rates or the “cradle” with higher speciation rates in the tropics) [42–53]. Instead of just characterizing “cradle” areas as having higher speciation rates, with estimates of the speciation-completion rate, we can ask whether the higher diversity reflects higher rates of population isolation (perhaps due to complex or dynamic geographies) or higher rates of development of speciation isolation mechanisms (see [53]), because these two processes that affect the duration of speciation [29] are decoupled in our model. Just as importantly, evolutionary biologists [53–62] have long highlighted the need and importance for modeling speciation as an extended process as done by the PBD, and the modeling of lineage splitting (population isolation) and species development as two separate processes in the PBD has been shown to provide novel and important insight into understanding how diversity is generated and maintained [26, 29, 53]. The nuances and ramifications of these two different paths to higher speciation rates provide deeper insight into the evolutionary history of a system by building a better understanding of how patterns at evolutionary time-scales are shaped by mechanisms and processes at ecological time-scales [53, 61]. Distinguishing between high rates of population isolation versus development of speciation isolation mechanisms are also useful for analyzing some interesting modes of speciation, such as ephemeral or ecological speciation [59, 62]. Insights about the speciation-completion rate as estimated by DELINEATE may explain how macroevolutionary patterns are regulated by microevolutionary processes.

“Objective” species delimitation

DELINEATE allows for existing taxonomic knowledge, subjective or otherwise, to be incorporated into an objective species delimitation analysis. In contrast, when using the MSC alone for species delimitation, species boundaries are inferred algorithmically entirely from genomic data, without requiring any pre-existing taxonomic information. This might lead to the perception that the MSC is an entirely objective analysis in comparison with DELINEATE, as the MSC does not require or make use of any subjective information with regard to species status, concepts, or criteria. However, this characterization of the MSC is misleading.

The MSC adopts a single criteria for delimiting species boundaries: any and all detectable restrictions of gene flow. The criteria is subjective in the sense that it was not selected through an objective statistical optimization procedure, nor does it represent a scientific consensus regarding species boundaries that is universally accepted by all investigators for all systems. Furthermore, as it is a necessary assumption made when using the classical MSC alone for species delimitation, it remains an implict subjective choice even if it was not explicitly stated, understood, or put forward by the investigator. Thus, while the classical MSC model does indeed provide an objective approach to species delimitation, it does so under a specific subjective species criteria or concept, albeit perhaps one not always recognized by investigators. This subjective species boundary criteria might be valid for some systems. However, it is clearly invalid in many systems in nature—that is, in systems with multiple within-species population lineages (e.g., any species with population structure [10, 18, 20]), even if divergence occurs with gene flow (i.e., divergence with gene flow models based on the MSC, such as [63], are insufficient and will be misleading when applied to any system in nature in which there is detectable within-species population genetic structure).

Like the MSC, DELINEATE, too, provides for an objective species delimitation analysis under subjective criteria. However, unlike the MSC, this subjective criteria is not fixed and forced upon the study regardless of whether it is valid or not. Instead, DELINEATE allows for the criteria to vary based on the investigator’s particular understanding of what constitutes a species.

Distinguishing between species and population boundaries by modeling the speciation process

By conducting analyses that rely only on genetic data, with no other information to inform species delimitation (i.e., the unconstrained mode of inference in DELINEATE), our study speaks to the limits of knowledge and how much we can learn from genomic data alone. That is, with no information at all as to the species identities of any populations, while the true species partition is found within the 95% confidence interval, it is the best-preferred delimitation in only 8% of the cases. This is not a novel finding: such a non-integrated approach to species delimitation analysis—where a set of genomic data is used with no supporting or corroborating information and an algorithm is expected to “magically” [23] distinguish between populations and species boundaries to diagnose species—has always been problematical and unreliable [2–17, 20, 21].

Because we provide the true speciation-completion rate, estimation of the parameter itself is not compromising our model’s performance. As such, our work shows that the inherent limitation arises from distinguishing genetic structure associated with populations versus species. That is, the actual challenge for accurately delimiting species (as well as what makes the MSC an inadequate model for species delimitation) is the presence of restrictions in gene flow before speciation rather than gene flow after speciation. Yet, this issue has received very little attention (at least in theoretical treatments). Instead, a popular focus has been on gene flow after speciation (e.g., [64]), as if it is also the central problem with applications of the MSC for species delimitation. Hopefully this study will help dispel this misconception and future work can focus on how methods might provide robust inference by contending with genetic structure that arises before speciation. Genetic structure within populations before speciation is the fundamental impediment to more general genetic-based applications (e.g., [2–4, 9, 11, 20]), as well as for the new class of speciation-based delimitation models we introduce here.

We acknowledge that there are a number of limitations and simplifying assumptions with our approach as currently implemented in DELINEATE– e.g., we assume a constant fixed speciation-completion rate, and we face computational challenges with large numbers of lineages with unknown species identities. We found, for example, that analyses with more than 15 population lineages with unknown identities were the limit that could be executed without recourse to machines of 1TB or more of memory. Note that this number, 15, is specifically the number of population lineages with unknown species assignments; the entire analysis could easily consist of several hundred or more population lineages as long as most of these were of known species assignments. Given that the principal computational challenge in our current implementation is the requirement to enumerate all possible partitions, adopting any of the standard optimization heuristics such as hill-climbing for maximum likelihood estimation or various forms of MCMC for Bayesian estimation in future work should increase the efficiency of DELINEATE. With this increased efficiency, analysis of larger datasets are possible, and with the higher information content of these larger datasets, we are optimistic that the efficacy of DELINEATE will increase as well. This potentially provides an opening for more sophisticated modeling to capture the biological realities of diversification dynamics, such as differing speciation-completion rates across taxa.

Nevertheless, even with the current limitations, the big picture that emerges is this: the accuracy of species delimitation is improved with modeling of the speciation process. This modeling not only allows us to avoid conflating genetic structure within species with that between species [Fig 3a], it also allows us to ask and answer more sophisticated questions in macroevolutionary biology (see [Fig 4]). There are many different speciation processes that can be considered that will prove useful in this regard. For example, Morlon et al. [65] described 13 theoretical models and 8 empirical patterns of speciation. Our adoption of the protracted speciation model [26, 27, 29, 66] is, in fact, just one of this variety. We both hope and expect that other speciation models that better reflect either the realities of particular biological systems, or the perspectives of other investigators, will be incorporated into speciation-based delimitation approaches in the future.