Phylogenetic Biodiversity Metrics Should Account for Both Accumulation and Attrition of Evolutionary Heritage.

Phylogenetic metrics are essential tools used in the study of ecology, evolution and conservation. Phylogenetic diversity (PD) in particular is one of the most prominent measures of biodiversity and is based on the idea that biological features accumulate along the edges of phylogenetic trees that are summed. We argue that PD and many other phylogenetic biodiversity metrics fail to capture an essential process that we term attrition. Attrition is the gradual loss of features through causes other than extinction. Here we introduce "EvoHeritage", a generalization of PD that is founded on the joint processes of accumulation and attrition of features. We argue that while PD measures evolutionary history, EvoHeritage is required to capture a more pertinent subset of evolutionary history including only components that have survived attrition. We show that EvoHeritage is not the same as PD on a tree with scaled edges; instead, accumulation and attrition interact in a more complex non-monophyletic way that cannot be captured by edge lengths alone. This leads us to speculate that the one-dimensional edge lengths of classic trees may be insufficiently flexible to capture the nuances of evolutionary processes. We derive a measure of EvoHeritage and show that it elegantly reproduces species richness and PD at opposite ends of a continuum based on the intensity of attrition. We demonstrate the utility of EvoHeritage in ecology as a predictor of community productivity compared with species richness and PD. We also show how EvoHeritage can quantify living fossils and resolve their associated controversy. We suggest how the existing calculus of PD-based metrics and other phylogenetic biodiversity metrics can and should be recast in terms of EvoHeritage accumulation and attrition.

Phylogenetic Diversity Indices from an Affine and Projective Viewpoint.

Phylogenetic diversity indices are commonly used to rank the elements in a collection of species or populations for conservation purposes. The derivation of these indices is typically based on some quantitative description of the evolutionary history of the species in question, which is often given in terms of a phylogenetic tree. Both rooted and unrooted phylogenetic trees can be employed, and there are close connections between the indices that are derived in these two different ways. In this paper, we introduce more general phylogenetic diversity indices that can be derived from collections of subsets (clusters) and collections of bipartitions (splits) of the given set of species. Such indices could be useful, for example, in case there is some uncertainty in the topology of the tree being used to derive a phylogenetic diversity index. As well as characterizing some of the indices that we introduce in terms of their special properties, we provide a link between cluster-based and split-based phylogenetic diversity indices that uses a discrete analogue of the classical link between affine and projective geometry. This provides a unified framework for many of the various phylogenetic diversity indices used in the literature based on rooted and unrooted phylogenetic trees, generalizations and new proofs for previous results concerning tree-based indices, and a way to define some new phylogenetic diversity indices that naturally arise as affine or projective variants of each other or as generalizations of tree-based indices.

The robustness of phylogenetic diversity indices to extinctions.

Phylogenetic diversity indices provide a formal way to apportion evolutionary history amongst living species. Understanding the properties of these measures is key to determining their applicability in conservation biology settings. In this work, we investigate some questions posed in a recent paper by Fischer et al. (Syst Biol 72(3):606-615, 2023). In that paper, it is shown that under certain extinction scenarios, the ranking of the surviving species by their Fair Proportion index scores may be the complete reverse of their ranking beforehand. Our main results here show that this behaviour extends to a large class of phylogenetic diversity indices, including the Equal-Splits index. We also provide a necessary condition for reversals of Fair Proportion rankings to occur on phylogenetic trees whose edge lengths obey the ultrametric constraint. Specific examples of rooted phylogenetic trees displaying these behaviours are given and the impact of our results on the use of phylogenetic diversity indices more generally is discussed.

Quantifying the difference between phylogenetic diversity and diversity indices.

Phylogenetic diversity is a popular measure for quantifying the biodiversity of a collection Y of species, while phylogenetic diversity indices provide a way to apportion phylogenetic diversity to individual species. Typically, for some specific diversity index, the phylogenetic diversity of Y is not equal to the sum of the diversity indices of the species in Y. In this paper, we investigate the extent of this difference for two commonly-used indices: Fair Proportion and Equal Splits. In particular, we determine the maximum value of this difference under various instances including when the associated rooted phylogenetic tree is allowed to vary across all rooted phylogenetic trees with the same leaf set and whose edge lengths are constrained by either their total sum or their maximum value.

Spaces of Phylogenetic Diversity Indices: Combinatorial and Geometric Properties.

Biodiversity is a concept most naturally quantified and measured across sets of species. However, for some applications, such as prioritising species for conservation efforts, a species-by-species approach is desirable. Phylogenetic diversity indices are functions that apportion the total biodiversity value of a set of species across its constituent members. As such, they aim to measure each species' individual contribution to, and embodiment of, the diversity present in that set. However, no clear definition exists that encompasses the diversity indices in current use. This paper presents conditions that define diversity indices arising from the phylogenetic diversity measure on rooted phylogenetic trees. In this context, the diversity index 'score' given to a species represents a measure of its unique and shared evolutionary history as displayed in the underlying phylogenetic tree. Our definition generalises the diversity index notion beyond the popular Fair Proportion and Equal-Splits indices. These particular indices may now be seen as two points in a convex space of possible diversity indices, for which the boundary conditions are determined by the underlying shape of each phylogenetic tree. We calculated the dimension of the convex space associated with each tree shape and described the extremal points.

Effects of discordance between species and gene trees on phylogenetic diversity conservation.

Phylogenetic diversity indices such as the Fair Proportion (FP) index are frequently discussed as prioritization criteria in biodiversity conservation. They rank species according to their contribution to overall diversity by taking into account the unique and shared evolutionary history of each species as indicated by its placement in an underlying phylogenetic tree. Traditionally, phylogenetic trees were inferred from single genes and the resulting gene trees were assumed to be a valid estimate for the species tree, i.e., the "true" evolutionary history of the species under consideration. However, nowadays it is common to sequence whole genomes of hundreds or thousands of genes, and it is often the case that conflicting genealogical histories exist in different genes throughout the genome, resulting in discordance between individual gene trees and the species tree. Here, we analyze the effects of gene and species tree discordance on prioritization decisions based on the FP index. In particular, we consider the ranking order of taxa induced by (i) The FP index on a species tree, and (ii) The expected FP index across all gene tree histories associated with the species tree. On the one hand, we show that for particular tree shapes, the two rankings always coincide. On the other hand, we show that for all leaf numbers greater than or equal to five, there exist species trees for which the two rankings differ. Finally, we illustrate the variability in the rankings obtained from the FP index across different gene tree and species tree estimates for an empirical multilocus mammal data set.

Biodiversity, Shapley value and phylogenetic trees: some remarks.

This paper explores the main differences between the Shapley values of a set of taxa introduced by Haake et al. (J Math Biol 56:479-497, 2007. https://doi.org/10.1007/s00285-007-0126-2) and Fuchs and Jin (J Math Biol 71:1133-1147, 2015. https://doi.org/10.1007/s00285-014-0853-0), the latter having been found identical to the Fair Proportion index (Redding and Mooers in Conserv Biol 20:1670-1678, 2006. https://doi.org/10.1111/j.1523-1739.2006.00555.x). In line with Shapley (in: Kuhn, Tucker (eds) Contributions to to the theory of games, volume II, annals of mathematics studies 28, Princeton University Press, Princeton, 1953), we identify the cooperative game basis for each of these two classes of phylogenetic games and use them (i) to construct simple formulas for these two Shapley values and (ii) to compare these different approaches. Using the set of weights of a phylogenetic tree as a parameter space, we then discuss the conditions under which these two values coincide and, if they are not the same, revisit Hartmann's (J Math Biol 67:1163-1170, 2013. https://doi.org/10.1007/s00285-012-0585-y) convergence result. An example illustrates our main argument. Finally, we compare the species ranking induced by these two values. Considering the Kendall and the Spearman rank correlation coefficient, simulations show that these rankings are strongly correlated. These results are consistent with Wicke and Fischer (J Theor Biol 430:207-214, 2017. https://doi.org/10.1016/j.jtbi.2017.07.010), who reach similar conclusions with a different simulation method.

Correlation between Shapley values of rooted phylogenetic trees under the beta-splitting model.

In recent years, several different versions of the Shapley value have been introduced in phylogenetics for the purpose of ranking biodiversity data in order to decide whether to preserve the data or not. Two of these Shapley values are the rooted and unrooted Shapley value which have been compared with the fair proportion index since this index is easier to compute. In particular, it was proved for the former that it is identical with the fair proportion index and numerical data was presented by several authors that the latter is strongly correlated with the fair proportion index. In this paper, we will prove a theoretical result which supports this observation. More precisely, we will prove that in random phylogenetic trees under the [Formula: see text]-splitting model, the correlation coefficient between the unrooted Shapley value and the fair proportion index indeed tends to one for all [Formula: see text] with [Formula: see text]. We also present data which suggests that the convergence worsens as [Formula: see text] is approaching [Formula: see text].

Evolutionary isolation and phylogenetic diversity loss under random extinction events.

The extinction of species at the present leads to the loss of 'phylogenetic diversity' (PD) from the evolutionary tree in which these species lie. Prior to extinction, the total PD present can be divided up among the species in various ways using measures of evolutionary isolation (such as 'fair proportion' and 'equal splits'). However, the loss of PD when certain combinations of species become extinct can be either larger or smaller than the cumulative loss of the isolation values associated with the extinct species. In this paper, we show that for trees generated under neutral evolutionary models, the loss of PD under a null model of random extinction at the present can be predicted from the loss of the cumulative isolation values, by applying a non-linear transformation that is independent of the tree. Moreover, the error in the prediction provably converges to zero as the size of the tree grows, with simulations showing good agreement even for moderate sized trees (n=64).

Comparing the rankings obtained from two biodiversity indices: the Fair Proportion Index and the Shapley Value.

The Shapley Value and the Fair Proportion Index of phylogenetic trees have been frequently discussed as prioritization tools in conservation biology. Both indices rank species according to their contribution to total phylogenetic diversity, allowing for a simple conservation criterion. While both indices have their specific advantages and drawbacks, it has recently been shown that both values are closely related. However, as different authors use different definitions of the Shapley Value, the specific degree of relatedness depends on the specific version of the Shapley Value - it ranges from a high correlation index to equality of the indices. In this note, we first give an overview of the different indices. Then we turn our attention to the mere ranking order provided by either of the indices. We compare the rankings obtained from different versions of the Shapley Value for a phylogenetic tree of European amphibians and illustrate their differences. We then undertake further analyses on simulated data and show that even though the chance of two rankings being exactly identical (when obtained from different versions of the Shapley Value) decreases with an increasing number of taxa, the distance between the two rankings converges to zero, i.e., the rankings are becoming more and more alike. Moreover, we introduce our freely available software package FairShapley, which was implemented in Perl and with which all calculations have been performed.

Equality of Shapley value and fair proportion index in phylogenetic trees.

The Shapley value and the fair proportion index of phylogenetic trees have been introduced recently for the purpose of making conservation decisions in genetics. Moreover, also very recently, Hartmann (J Math Biol 67:1163-1170, 2013) has presented data which shows that there is a strong correlation between a slightly modified version of the Shapley value (which we call the modified Shapley value) and the fair proportion index. He gave an explanation of this correlation by showing that the contribution of both indices to an edge of the tree becomes identical as the number of taxa tends to infinity. In this note, we show that the Shapley value and the fair proportion index are in fact the same. Moreover, we also consider the modified Shapley value and show that its covariance with the fair proportion index in random phylogenetic trees under the Yule-Harding model and uniform model is indeed close to one.

Phylogenetic diversity and biodiversity indices on phylogenetic networks.

In biodiversity conservation it is often necessary to prioritize the species to conserve. Existing approaches to prioritization, e.g. the Fair Proportion Index and the Shapley Value, are based on phylogenetic trees and rank species according to their contribution to overall phylogenetic diversity. However, in many cases evolution is not treelike and thus, phylogenetic networks have been developed as a generalization of phylogenetic trees, allowing for the representation of non-treelike evolutionary events, such as hybridization. Here, we extend the concepts of phylogenetic diversity and phylogenetic diversity indices from phylogenetic trees to phylogenetic networks. On the one hand, we consider the treelike content of a phylogenetic network, e.g. the (multi)set of phylogenetic trees displayed by a network and the so-called lowest stable ancestor tree associated with it. On the other hand, we derive the phylogenetic diversity of subsets of taxa and biodiversity indices directly from the internal structure of the network. We consider both approaches that are independent of so-called inheritance probabilities as well as approaches that explicitly incorporate these probabilities. Furthermore, we introduce our software package NetDiversity, which is implemented in Perl and allows for the calculation of all generalized measures of phylogenetic diversity and generalized phylogenetic diversity indices established in this note that are independent of inheritance probabilities. We apply our methods to a phylogenetic network representing the evolutionary relationships among swordtails and platyfishes (Xiphophorus: Poeciliidae), a group of species characterized by widespread hybridization.