CAREER: Identifiability and Inference for Phylogenetic Networks using Applied Algebraic Geometry

University Of Hawaii, Honolulu

Investigators

Abstract

Phylogenetic tree and network reconstructions lead not only to a better understanding of our natural world, but also have applications in other fields, such as conservation and epidemiology. For example, taking into account phylogenetic diversity in the restoration of natural vegetation can lead to restorations that establish quicker and are heartier, while understanding the phylogenetics of pathogens can aid in back-tracing the spread of a disease and guiding epidemiological interventions. While trees are a natural choice for representing evolution combinatorially, by restricting to the class of trees, it is possible to miss more complicated events such as hybridization and horizontal gene transfer. For more complete descriptions, phylogenetic networks, directed acyclic graphs, are increasingly becoming more common in evolutionary biology. This project focuses on phylogenetic networks and their parent models, phylogenetic mixture models, and will use applied algebraic geometry, in particular, algebraic statistics, to develop novel techniques for their inference. The project also has research training of graduate students and educational activities that introduce algebraic statistics and algebraic biology to undergraduate and K-12 students. In addition to hosting and running a series of one-week long graduate student research workshops in algebraic biology with follow-up small group collaborations, the PI will design a data-centered undergraduate mathematical biology course and organize annual collaborative and problem-based mathematical events for 6th graders with island-focused challenges. Since network-based Markov models and their mixtures are specified with a polynomial parameterization, they are amenable to analysis using algebraic and geometric methods. This project will utilize their algebraic properties to establish identifiability, a property necessary for meaningful statistical inference, and then utilize their geometric properties to develop a procedure for network inference. Since the corresponding model varieties are toric varieties, secant varieties, and determinental varieties, this project will not only develop new inference methods for biologists, but will also lay the foundation and motivation for a new class of problems for those working algebraic geometry and non-linear algebra. This award is jointly funded by the Mathematical Biology Program of DMS and the Cluster of Systematics and Biodiversity Sciences at the Division of Environmental Biology. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.