2004 Mt. Baldy Conference on Geometry, Algebra, and Phylogenetic Trees: Abstracts
Bernd Sturmfels (UC-Berkeley): Phylogenetic Algebraic Geometry
Many widely used statistical models of evolution are algebraic varieties, that is, solutions sets of polynomial equations. We discuss this algebraic representation and its implications for the construction of maximum likelihood trees. The ensuing interaction between algebraic geometry and phylogenetics is a two-way street: computational biologists may be benefit from new algebraic tools, while algebraic geometers will find a rich source of open problems concerning objects reminiscent of classical projective varieties.
Michael Hendy (Massey University, New Zealand): Hadamard Conjugation and the Molecular Clock
Molecular phylogenetics is the art of inferring evolutionary trees (phylogenies) from comparative biological sequence data. As a mathematician, the challenge is to make a science of this art. Instances of that art include the reconstruction algorithms of Maximum Parsimony (MP) and Maximum Likelihood (ML).
A phylogeny has a time dimension, the molecular clock hypothesis asserts that there is a common rate of substitutional change along branches of the tree. It is common to ignore this hypothesis with MP and ML.
I will show how Hadamard conjugation was developed as a tool to analyse the accuracy of MP under a molecular clock, and how it has been useful for other applications, including the theory of invariants and in analysing ML.
David Bryant (McGill University, Canada): Probabilistic Models for Splits Graphs
Splits graphs are generalisations of trees that permit the representation of conflict and ambiguity. One of the biggest challenges in splits graphs is their interpretation. If we want to go beyond splits graphs as a simple means of representation (and I believe we should), we need some way of modelling evolution in terms of a splits graph. I will discuss recent progress in this area, and outline connections with graphical models and the Hadamard transform.
Susan Holmes (Stanford): Using Phylogenetic Trees to Analyse Microarrays
Microarrays provide data on what genes are transcribed at a given time and we follow cell development and differentiation by looking at the gene expression patterns. We will present cases when hierarchical clustering and phylogenetic trees can be used to anlalyse gene expression data and show how using the bootstrap and distances between trees we can capture information on which genes or groups of genes are responsible for parts of the tree.
This will be illustrated on data from a study on T-cells done in collaboration with Peter Lee from Stanford's School of Medicine.