Genomic data analysis in tree spaces

Recently, an elegant approach in phylogenetics was introduced by Billera-Holmes-Vogtmann that allows a systematic comparison of different evolutionary histories using the metric geometry of tree spaces. In many problem settings one encounters heavily populated phylogenetic trees, where the large number of leaves encumbers visualization and analysis in the relevant evolutionary moduli spaces. To address this issue, we introduce tree dimensionality reduction, a structured approach to reducing large phylogenetic trees to a distribution of smaller trees. We prove a stability theorem ensuring that small perturbations of the large trees are taken to small perturbations of the resulting distributions. We then present a series of four biologically motivated applications to the analysis of genomic data, spanning cancer and infectious disease. The first quantifies how chemotherapy can disrupt the evolution of common leukemias. The second examines a link between geometric information and the histologic grade in relapsed gliomas, where longer relapse branches were specific to high grade glioma. The third concerns genetic stability of xenograft models of cancer, where heterogeneity at the single cell level increased with later mouse passages. The last studies genetic diversity in seasonal influenza A virus. We apply tree dimensionality reduction to 24 years of longitudinally collected H3N2 hemagglutinin sequences, generating distributions of smaller trees spanning between three and five seasons. A negative correlation is observed between the influenza vaccine effectiveness during a season and the variance of the distributions produced using preceding seasons' sequence data. We also show how tree distributions relate to antigenic clusters and choice of influenza vaccine. Our formalism exposes links between viral genomic data and clinical observables such as vaccine selection and efficacy.