Probabilistic Fréchet means for time varying persistence diagrams

© 2015, Institute of Mathematical Statistics. All rights reserved.In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In [23], Mileyko and his collaborators made the first study of the properties of the Fréchet mean in (Dp, Wp), the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fréchet mean of a finite set of diagrams always exists, but is not necessarily unique. The means of a continuously-varying set of diagrams do not themselves (necessarily) vary continuously, which presents obvious problems when trying to extend the Fréchet mean definition to the realm of time-varying persistence diagrams, better known as vineyards. We fix this problem by altering the original definition of Fréchet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself a persistence diagram determined using a perturbation of the input diagrams. This definition gives for each N a map (Dp)^N→ℙ(Dp). We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards.

Computational Methods for RNA Structure Validation and Improvement.

With increasing recognition of the roles RNA molecules and RNA/protein complexes play in an unexpected variety of biological processes, understanding of RNA structure-function relationships is of high current importance. To make clean biological interpretations from three-dimensional structures, it is imperative to have high-quality, accurate RNA crystal structures available, and the community has thoroughly embraced that goal. However, due to the many degrees of freedom inherent in RNA structure (especially for the backbone), it is a significant challenge to succeed in building accurate experimental models for RNA structures. This chapter describes the tools and techniques our research group and our collaborators have developed over the years to help RNA structural biologists both evaluate and achieve better accuracy. Expert analysis of large, high-resolution, quality-conscious RNA datasets provides the fundamental information that enables automated methods for robust and efficient error diagnosis in validating RNA structures at all resolutions. The even more crucial goal of correcting the diagnosed outliers has steadily developed toward highly effective, computationally based techniques. Automation enables solving complex issues in large RNA structures, but cannot circumvent the need for thoughtful examination of local details, and so we also provide some guidance for interpreting and acting on the results of current structure validation for RNA.

Cell Fate Specification and the Regulation of RNA-dependent DNA Methylation in the Arabidopsis Root Meristem

The Arabidopsis root apical meristem (RAM) is a complex tissue capable of generating all the cell types that ultimately make up the root. The work presented in this thesis takes advantage of the versatility of high-throughput sequencing to address two independent questions about the root meristem. Although a lot of information is known regarding the cell fate decisions that occur at the RAM, cortex specification and differentiation remain poorly understood. In the first part of this thesis, I used an ethylmethanesulfonate (EMS) mutagenized marker line to perform a forward genetics screen. The goal of this screen was to identify novel genes involved in the specification and differentiation of the cortex tissue. Mapping analysis from the results obtained in this screen revealed a new allele of BRASSINOSTEROID4 with abnormal marker expression in the cortex tissue. Although this allele proved to be non-cortex specific, this project highlights new technology that allows mapping of EMS-generated mutations without the need to map-cross or back-cross. In the second part of this thesis, using fluorescence activated cell sorting (FACS) coupled with high throughput sequencing, my collaborators and I generated single-base resolution whole genome DNA methylomes, mRNA transcriptomes, and smallRNA transcriptomes for six different populations of cell types in the Arabidopsis root meristem. We were able to discover that the columella is hypermethylated in the CHH context within transposable elements. This hypermethylation is accompanied by upregulation of the RNA-dependent DNA methylation pathway (RdDM), including higher levels of 24-nt silencing RNAs (siRNAs). In summary, our studies demonstrate the versatility of high-throughput sequencing as a method for identifying single mutations or to perform complex comparative genomic analyses.