The Rarefaction of Phylogenetic Diversity: Formulation, Extension and Application

David A. Nipperess

Abstract Like other measures of diversity, Phylogenetic Diversity (PD) increases monotonically and asymptotically with increasing sample size. This relationship can be described by a rarefaction curve tracing the expected PD for a given number of accumulation units. Accumulation units represent individual organisms, collections of organisms (e.g. sites), or even species (or equivalent), giving individualbased, sample-based and species-based curves respectively. The formulation for the exact analytical solution for the rarefaction of PD is given in an expanded form to demonstrate congruence with the classic formulation for the rarefaction of species richness. Rarefaction is commonly applied as a standardisation for diversity values derived from differing numbers of sampling units. However, the solution can be simply extended to create measures of phylogenetic evenness, phylogenetic betadiversity and phylogenetic dispersion, derived from individual-based, sample-based and species-based curves respectively. This extension, termed ∆PD, is simply the initial slope of the rarefaction curve and is related to entropy measures such as PIE (Probability of Interspecific Encounter) and Gini-Simpson entropy. The application of rarefaction of PD to sample standardisation and measurement of phylogenetic evenness, phylogenetic beta-diversity and phylogenetic dispersion is demonstrated. Future prospects for PD rarefaction include the recognition of evolutionary hotspots (independent of species richness), the basis for ecological theory such as phylogenyarea relationships, and the prediction of unseen biodiversity.

Phylogenetic Diversity (PD) is a simple, intuitive and effective measure of biodiversity. The PD of a set of taxa, represented as the tips of a phylogenetic tree, is the sum of the branch lengths connecting those taxa (Faith 1992). PD is a particularly flexible measure because it can be applied to any set of relationships among entities that can be reasonably portrayed as a tree. Thus, the tips do not, by necessity, need to represent species but could be higher taxa, Operational Taxonomic Units, Evolutionarily Significant Units, individual organisms or unique haplotypes. Further, the tree itself might not portray evolutionary relationships but instead be, for example, a cluster dendrogram portraying functional relationships among taxa (Petchey and Gaston 2002).

Since the original formulation by Faith (1992), PD has come to be not just a single measure equating to a phylogenetically weighted form of richness, but rather a general class of measures dealing with various aspects of alpha and beta-diversity (Faith 2013). The common feature of this class of measures is the summation of branch lengths rather than the counting of tips. By substituting branch segments (intervals between nodes on a phylogenetic tree) for species, and including a weighting for the length of that segment, it is possible to modify many of the classic measures of Species Diversity (SD) to a PD equivalent (Faith 2013). By this means, phylogenetically weighted measures of endemism (Faith et al. 2004; Rosauer et al. 2009), ecological resemblance (Ferrier et al. 2007; Nipperess et al. 2010), and entropy (Chao et al. 2010, and chapter “Phylogenetic Diversity Measures and Their Decomposition: A Framework Based on Hill Numbers”) have been developed, for example.

In its classic form, PD, like species richness, has the property of concavity (Lande 1996). That is, the addition of individuals or sets of individuals to a community can increase PD but never decrease it. Thus, just like species richness, PD increases monotonically with increasing sampling effort, creating a classic sampling curve that reaches an asymptote when all species (and branch segments) are represented (Fig. 1). Gotelli and Colwell (2001) recognise two general types of sampling curve, individuals-based and sample-based, that are distinguished by the units on the x-axis, representing either individual organisms or samples, respectively. Samples, in this context, are collections of individuals bounded in space and time, corresponding to the common ecological usage of the term. For PD, we can recognise a third type of sampling curve where the units on the x-axis are species or their equivalent (Fig. 1). Species, like samples, are also collections of individuals bounded, in this case, by some minimum degree of relatedness. Obviously, speciesbased sampling curves are meaningless when plotting species richness but have real value when plotting PD. For the purposes of generalisation, it is useful to be able to refer to these units (individuals, samples, species) with a single term. Chiarucci et al. (2008) used “accumulation units” to refer to individuals and samples. I extend this term to also include species as an additional unit of sampling effort in sampling curves. While these different units (individuals, samples, species) all measure

Fig. 1 Sampling curve showing the relationship between Phylogenetic Diversity (PD) and sampling depth. The level of sampling is measured in accumulation units of individuals, samples (collections of individuals) or species as required. PDN is the Phylogenetic Diversity of the full set of N accumulation units. Rarefaction is the process (indicated by unidirectional arrow) of randomly subsampling (rarefying) the pool of N accumulation units to a subset of size m and calculating the expected PD of that subset (PDm). ∆PD is the expected gain in PD between the first and second accumulation unit, and can be used as a measure of phylogenetic evenness, beta-diversity or dispersion, depending on the nature of the unit of accumulation

sampling effort in some sense, they are not equivalent and sampling curves derived from them must be interpreted differently in each case.

Beside the units by which sampling effort is measured, Gotelli and Colwell (2001) distinguished between “accumulation curves” and “rarefaction curves”, based on the process by which the sampling curve is calculated. An accumulation curve plots a single ordering of individuals or samples (or species) against a cumulatively calculated concave diversity measure. The jagged shape of the resulting curve is highly dependent on the, often arbitrary, order of the accumulation units. To resolve this problem, rarefaction curves instead plot the expected value of the diversity measure against the corresponding number of accumulation units. Rarefaction can be achieved using an algorithmic procedure of repeated random sub-sampling of the full set of accumulation units and calculating the mean diversity (Gotelli and Colwell 2001). However, Hurlbert (1971) and Simberloff (1972) showed that expected diversity can be calculated using an exact analytical solution, obviating the need for computer-intensive repeated sub-sampling. Initially, this solution was for individuals-based rarefaction curves, but it has since been shown that the same solution applies to sample-based rarefaction (Kobayashi 1974; Ugland et al. 2003; Mao et al. 2005; Chiarucci et al. 2008).

The original purpose of rarefaction was to allow the comparison of datasets with differing amounts of sampling effort (Sanders 1968). Assemblages can be compared “fairly” when rarefied to the same number of accumulation units (Gotelli and Colwell 2001). However, rarefaction has broader application than this single purpose. Depending of the unit of accumulation, the shape of the rarefaction curve provides information on ecological evenness (Olszewski 2004) and beta-diversity (Crist and Veech 2006). Rarefaction of species richness also forms the basis of estimators of species richness, including unseen species (Colwell and Coddington 1994). In the case of PD, species-based rarefaction curves also allow for a measure of phylogenetic dispersion (Webb et al. 2002), effectively the expected PD for some given number of species (Nipperess and Matsen 2013). A solution for the rarefaction of PD is therefore desirable as it will allow for these applications to be realised for phylogenetically explicit datasets.

Rarefaction of Phylogenetic Diversity, using an algorithmic solution of repeated sub-sampling, has now been done several times (see for example Lozupone and Knight 2008; Turnbaugh et al. 2009; Yu et al. 2012). However, an analytical solution for PD rarefaction, similar to that determined by Hurlbert (1971) for species richness, is preferable both because its results are exact (not dependent on the number of repeated subsamples) and substantially more computationally efficient. Nipperess and Matsen (2013) recently published just such a solution for both the mean and variance of PD under rarefaction. This solution is quite general, being applicable to rooted and unrooted trees, and even allowing partition of the tree into smaller components than the individual branch segments. As a result, the solution is given in a very generalised form and its relationship with classic rarefaction formula for species richness is not immediately clear.

In this chapter, I provide a detailed formulation for the exact analytical solution for expected (mean) Phylogenetic Diversity for a given amount of sampling effort. This formulation is for the specific but common case of a rooted phylogenetic tree where whole branch segments are selected under rarefaction. I use the same form of expression as used by Hurlbert (1971) to demonstrate the direct relationship between rarefaction of PD and rarefaction of species richness. I do not include a solution for variance of PD under rarefaction due to its complexity when given in this form and instead refer the reader to Nipperess and Matsen (2013). I extend this framework to show how the initial slope of the rarefaction curve (∆PD) can be used as a flexible measure of phylogenetic evenness, phylogenetic beta-diversity or phylogenetic dispersion, depending on the unit of accumulation. I apply PD rarefaction and the derived ∆PD measure to real ecological datasets to demonstrate its usefulness in addressing ecological questions. Finally, I discuss some future directions for the extension and application of PD rarefaction.

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