The Shapley Value of Phylogenetic Trees

Every weighted tree corresponds naturally to a cooperative game that we call a "tree game"; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley value on the edge weights of the tree, and we also compute a null space basis of M. Both depend on the "split counts" of the tree. Finally, we characterize the Shapley value on tree games by four axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games.Comment: References added, and a section (calculating the Shapley value of a tree game from its subtrees) was removed for length reasons (request of referee) and may appear in another paper. 16 pages; related work at http://www.math.hmc.edu/~su/papers.html. Journal of Mathematical Biology, to appear. The original article is available at http://www.springerlink.co

The Shapley value of phylogenetic trees

Every weighted tree corresponds naturally to a cooperative game that we call a tree game; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley value on the edge weights of the tree, and we also compute a null space basis of M. Finally, we characterize the Shapley value on tree games by five axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games. We also include a brief discussion of the core of tree games.

Constructing Phylogenetic Trees from Subsplits

Phylogenetic trees represent theoretical evolutionary relationships among various species. Mathematically they can be described as weighted binary trees and the leaves represent the taxa being compared. One major problem in mathematical biology is the reconstruction of these trees. We already know that trees on the leaf set X can be uniquely constructed from splits, which are bipartitions of X. The question I explore in this thesis is whether reconstruction of a tree is possible from subsplits, or partial split information. The major result of this work is a constructive algorithm which allows us to determine whether a given set of subsplits will realize a tree and, if so, what the tree looks like

Hedging our bets: the expected contribution of species to future phylogenetic diversity

If predictions for species extinctions hold, then the `tree of life' today may be quite different to that in (say) 100 years. We describe a technique to quantify how much each species is likely to contribute to future biodiversity, as measured by its expected contribution to phylogenetic diversity. Our approach considers all possible scenarios for the set of species that will be extant at some future time, and weights them according to their likelihood under an independent (but not identical) distribution on species extinctions. Although the number of extinction scenarios can typically be very large, we show that there is a simple algorithm that will quickly compute this index. The method is implemented and applied to the prosimian primates as a test case, and the associated species ranking is compared to a related measure (the `Shapley index'). We describe indices for rooted and unrooted trees, and a modification that also includes the focal taxon's probability of extinction, making it directly comparable to some new conservation metrics.Comment: 19 pages, 2 figure

Computing evolutionary distinctiveness indices in large scale analysis

We present optimal linear time algorithms for computing the Shapley values and 'heightened evolutionary distinctiveness' (HED) scores for the set of taxa in a phylogenetic tree. We demonstrate the efficiency of these new algorithms by applying them to a set of 10,000 reasonable 5139-species mammal trees. This is the first time these indices have been computed on such a large taxon and we contrast our finding with an ad-hoc index for mammals, fair proportion (FP), used by the Zoological Society of London's EDGE programme. Our empirical results follow expectations. In particular, the Shapley values are very strongly correlated with the FP scores, but provide a higher weight to the few monotremes that comprise the sister to all other mammals. We also find that the HED score, which measures a species' unique contribution to future subsets as function of the probability that close relatives will go extinct, is very sensitive to the estimated probabilities. When they are low, HED scores are less than FP scores, and approach the simple measure of a species' age. Deviations (like the Solendon genus of the West Indies) occur when sister species are both at high risk of extinction and their clade roots deep in the tree. Conversely, when endangered species have higher probabilities of being lost, HED scores can be greater than FP scores and species like the African elephant Loxondonta africana, the two solendons and the thumbless bat Furipterus horrens can move up the rankings. We suggest that conservation attention be applied to such species that carry genetic responsibility for imperiled close relatives. We also briefly discuss extensions of Shapley values and HED scores that are possible with the algorithms presented here

Global Priorities for Conserving the Evolutionary History of Sharks, Rays, and Chimaeras

In an era of accelerated biodiversity loss and limited conservation resources, systematic prioritization of species and places is essential. In terrestrial vertebrates, evolutionary distinctness has been used to identify species and locations that embody the greatest share of evolutionary history. We estimate evolutionary distinctness for a large marine vertebrate radiation on a dated taxon-complete tree for all 1,192 chondrichthyan fishes (sharks, rays and chimaeras) by augmenting a new 610-species molecular phylogeny using taxonomic constraints. Chondrichthyans are by far the most evolutionarily distinct of all major radiations of jawed vertebrates—the average species embodies 26 million years of unique evolutionary history. With this metric, we identify 21 countries with the highest richness, endemism and evolutionary distinctness of threatened species as targets for conservation prioritization. On average, threatened chondrichthyans are more evolutionarily distinct—further motivating improved conservation, fisheries management and trade regulation to avoid significant pruning of the chondrichthyan tree of life