The Exploration of Phylogenetic Tree Space through Combinatorics, Statistics and Geometry

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The research proposal I am developing aims to help biologists understand the properties of the space ofall phylogenetic trees. I propose to combine three approaches to tackle this problem,combinatorial, statistical and geometrical.I have already made some good progress through a combinatorial approach to the problem that I have started with Persi Diaconis. We have built a bijection between the space of all phylogenetic trees and the space of perfect matchings. This has provided new distances between trees, ways of doing random walks on tree space, and a new approach to the $NP$ complete optimization procedures involved in building phylogenetic trees.The embedding of tree estimation in a statistical framework allows one to see trees as a special type of parameter to be estimated. The next step needed is the ability to construct confidence regions. My first work on this subject was done with Brad Efron and Betz Halloran a few years ago and started me on this project.Currently biologists validate the tree estimated by perturbing the data and then summarizing this simplyby associating p-values to the branches of the tree. Communication between statisticians and biologistshas a difficult history in this field. One school of taxonomists has been encouraging the use of maximum likelihood for the estimation of the tree.Other taxonomists have been using parsimony as their criterion, and have been highly critical ofwhat they called the `statistical approach'. In fact their own method is statistical also, but the parsimony method is a nonparametric approach, whereas the Maximum Likelihood is a parametric one. The MLE model is finite dimensional, however, statisticians know that if one allows the number of parameters to increase as this number becomes larger than the amount of information contained in the data, the method becomes nonparametric. Making this interpretation known to biologists would be a worthwhile enterprise, the consequence would be that nonparametric-parametric discussion in statistics could enlighten the parsimony-MLE debate.Understanding the geometry of the tree-space would help understand how to combine trees obtained from perturbed data or different genes on the same set of species. I hope to continue a collaboration I have started with a combinatorialist, Louis Billera and a topologist, Karen Vogtmann on a more mathematical understanding of tree space that would allow for a more natural definition of distances between points in the `tree polytope'. This topological study of tree space is crucial in the definition of a notion of neighborhood that would then allow a definition of continuity in the estimation function.The notion of confidence region could then be defined, and it would also be possible to combine trees from built from different datasets or to combine trees with other data (biogeographic, for instance). Finally I will diffuse my matlab tree analysis programs and preliminary results through a web site `Geometry of Tree Space'. This POWRE project is supported by the MPS Office of Multidisciplinary Activities (OMA).