UC Berkeley

We present two applications of non-linear algebra to biology. Our first application is to the analysis of gene expression data from Arabidoposis roots. In Chapter 2, we present a method forcomputing non-negative roots to certain systems of polynomials. This algorithm is based on a generalization of the Expectation-Maximization and Iterative Proportional Fitting from statistics. In Chapter 3, this method is applied to a model for gene expression coming from roots of the Arabidopsis plant. Variation in gene expression is one method in which different tissue types develop different functional characteristics. Our model for gene expression in these roots is non-linear and so we apply the method from Chapter 2.

Our second application is the use of secant varieties to study mixture models for the distribution of single-nucleotide polymorphisms in genes. In Chapter 4, we give generators for the defining ideal of the first 5 secant varieties of P^2 x P^{n-1} embedded by O(1,2). Our equations come from a generalization of the flattening of a tensor, which we call an exterior flattening. The study of secant varieties is a classical subject in algebraic geometry, which has recently been connected to applications in algebraic statistics. In Chapter 5, this connection is used in order to understand how single-nucleotide polymorphisms occur within human genes. Because genes perform important functions, there is selective pressure against mutations which affect the gene's behavior. As we show, these selective pressures are closely tied to the nature by which genetic sequence codes for the constituent amino acids of a protein.