Genomics data analysis via spectral shape and topology.

Mapper, a topological algorithm, is frequently used as an exploratory tool to build a graphical representation of data. This representation can help to gain a better understanding of the intrinsic shape of high-dimensional genomic data and to retain information that may be lost using standard dimension-reduction algorithms. We propose a novel workflow to process and analyze RNA-seq data from tumor and healthy subjects integrating Mapper, differential gene expression, and spectral shape analysis. Precisely, we show that a Gaussian mixture approximation method can be used to produce graphical structures that successfully separate tumor and healthy subjects, and produce two subgroups of tumor subjects. A further analysis using DESeq2, a popular tool for the detection of differentially expressed genes, shows that these two subgroups of tumor cells bear two distinct gene regulations, suggesting two discrete paths for forming lung cancer, which could not be highlighted by other popular clustering methods, including t-distributed stochastic neighbor embedding (t-SNE). Although Mapper shows promise in analyzing high-dimensional data, tools to statistically analyze Mapper graphical structures are limited in the existing literature. In this paper, we develop a scoring method using heat kernel signatures that provides an empirical setting for statistical inferences such as hypothesis testing, sensitivity analysis, and correlation analysis.

Ciliated epithelia are key elements in the recruitment of bacterial partners in the squid-vibrio symbiosis.

The Hawaiian bobtail squid, Euprymna scolopes, harvests its luminous symbiont, Vibrio fischeri, from the surrounding seawater within hours of hatching. During embryogenesis, the host animal develops a nascent light organ with ciliated fields on each lateral surface. We hypothesized that these fields function to increase the efficiency of symbiont colonization of host tissues. Within minutes of hatching from the egg, the host's ciliated fields shed copious amounts of mucus in a non-specific response to bacterial surface molecules, specifically peptidoglycan (PGN), from the bacterioplankton in the surrounding seawater. Experimental manipulation of the system provided evidence that nitric oxide in the mucus drives an increase in ciliary beat frequency (CBF), and exposure to even small numbers of V. fischeri cells for short periods resulted in an additional increase in CBF. These results indicate that the light-organ ciliated fields respond specifically, sensitively, and rapidly, to the presence of nonspecific PGN as well as symbiont cells in the ambient seawater. Notably, the study provides the first evidence that this induction of an increase in CBF occurs as part of a thus far undiscovered initial phase in colonization of the squid host by its symbiont, i.e., host recognition of V. fischeri cues in the environment within minutes. Using a biophysics-based mathematical analysis, we showed that this rapid induction of increased CBF, while accelerating bacterial advection, is unlikely to be signaled by V. fischeri cells interacting directly with the organ surface. These overall changes in CBF were shown to significantly impact the efficiency of V. fischeri colonization of the host organ. Further, once V. fischeri has fully colonized the host tissues, i.e., about 12-24 h after initial host-symbiont interactions, the symbionts drove an attenuation of mucus shedding from the ciliated fields, concomitant with an attenuation of the CBF. Taken together, these findings offer a window into the very first interactions of ciliated surfaces with their coevolved microbial partners.

Simultaneous Estimation of Corneal Topography, Pachymetry, and Curvature.

Identification of objective criteria to correctly diagnose ectatic diseases of the cornea or to detect early stages of corneal ectasia is of great interest in ophthalmology and optometry. Metrics for diagnosis typically employed are curvature maps (axial/sagittal, tangential); elevation map of the anterior surface of the cornea with respect to a reference sphere; and pachymetry (thickness) map of the cornea. We present evidence that currently used curvature maps do not represent the actual curvatures (principal or mean) in a human cornea. A novel contribution of this paper is the computation of the true mean curvature over every point of a central region of the cornea. We show that the true mean curvature can accurately identify the location of the ectasia. We present a quartic smoothing spline algorithm for the simultaneous computation of elevation maps for anterior and posterior corneal surfaces, pachymetry, and true mean curvature. The input to the algorithm is data from a single measurement from imaging devices such as an anterior segment optical coherence tomographer or a Scheimpflug imager. We show that a different combination of metrics is useful for the diagnosis of existing ectasia (true mean curvature and anterior elevation map) as opposed to subclinical ectasia (pachymetry and posterior elevation map). We compare our results with existing algorithms, and present applications to a normal cornea, a forme fruste keratoconic cornea, and an advanced keratoconic cornea.