Tree-like structures are ubiquitous in nature. In particular, neuronal axons and dendrites have tree-like geometries that mediate electrical signaling within and between cells. Electrical activity in neuronal trees is typically modeled using coupled cable equations on multi-compartment representations, where each compartment represents a small segment of the neuronal membrane. The geometry of each compartment is usually defined as a cylinder or, at best, a surface of revolution based on a linear approximation of the radial change in the neurite. The resulting geometry of the model neuron is coarse, with non-smooth or even discontinuous jumps at the boundaries between compartments. We propose a hyperbolic approximation to model the geometry of neurite compartments, a branched, multi-compartment extension, and a simple graphical approach to calculate steady-state solutions of an associated system of coupled cable equations. A simple case of transient solutions is also briefly discussed.

Quantitative three-dimensional (3D) computed tomography (CT) imaging of living single cells enables orientation-independent morphometric analysis of the intricacies of cellular physiology. Since its invention, x-ray CT has become indispensable in the clinic for diagnostic and prognostic purposes due to its quantitative absorption-based imaging in true 3D that allows objects of interest to be viewed and measured from any orientation. However, x-ray CT has not been useful at the level of single cells because there is insufficient contrast to form an image. Recently, optical CT has been developed successfully for fixed cells, but this technology called Cell-CT is incompatible with live-cell imaging due to the use of stains, such as hematoxylin, that are not compatible with cell viability. We present a novel development of optical CT for quantitative, multispectral functional 4D (three spatial + one spectral dimension) imaging of living single cells. The method applied to immune system cells offers truly isotropic 3D spatial resolution and enables time-resolved imaging studies of cells suspended in aqueous medium. Using live-cell optical CT, we found a heterogeneous response to mitochondrial fission inhibition in mouse macrophages and differential basal remodeling of small (0.1 to 1 fl) and large (1 to 20 fl) nuclear and mitochondrial structures on a 20- to 30-s time scale in human myelogenous leukemia cells. Because of its robust 3D measurement capabilities, live-cell optical CT represents a powerful new tool in the biomedical research field.

Driven by an increasing number of studies demonstrating its relevance to a broad variety of disease states, the bioenergy production phenotype has been widely characterized at the bulk sample level. Its cell-to-cell variability, a key player associated with cancer cell survival and recurrence, however, remains poorly understood due to ensemble averaging of the current approaches. We present a technology platform for performing oxygen consumption and extracellular acidification measurements of several hundreds to 1,000 individual cells per assay, while offering simultaneous analysis of cellular communication effects on the energy production phenotype. The platform comprises two major components: a tandem optical sensor for combined oxygen and pH detection, and a microwell device for isolation and analysis of single and few cells in hermetically sealed sub-nanoliter chambers. Our approach revealed subpopulations of cells with aberrant energy production profiles and enables determination of cellular response variability to electron transfer chain inhibitors and ion uncouplers.

The histone deacetylase (HDAC) inhibitor vorinostat has received significant attention in recent years as an ‘epigenetic’ drug used to treat solid tumors. However, its mechanisms of action are not entirely understood, particularly with regard to its interaction with the aberrations in 3D nuclear structure that accompany neoplastic progression. We investigated the impact of vorinostat on human esophageal epithelial cell lines derived from normal, metaplastic (pre-cancerous), and malignant tissue. Using a combination of novel optical computed tomography (CT)-based quantitative 3D absorption microscopy and conventional confocal fluorescence microscopy, we show that subjecting malignant cells to vorinostat preferentially alters their 3D nuclear architecture relative to non-cancerous cells. Optical CT (cell CT) imaging of fixed single cells showed that drug-treated cancer cells exhibit significant alterations in nuclear morphometry. Confocal microscopy revealed that vorinostat caused changes in the distribution of H3K9ac-marked euchromatin and H3K9me3-marked constitutive heterochromatin. Additionally, 3D immuno-FISH showed that drug-induced expression of the DNA repair gene MGMT was accompanied by spatial relocation toward the center of the nucleus in the nuclei of metaplastic but not in non-neoplastic cells. Our data suggest that vorinostat’s differential modulation of 3D nuclear architecture in normal and abnormal cells could play a functional role in its anti-cancer action.

Cellular heterogeneity plays a pivotal role in a variety of functional processes in vivo including carcinogenesis. However, our knowledge about cell-to-cell diversity and how differences in individual cells manifest in alterations at the population level remains very limited mainly due to the lack of appropriate tools enabling studies at the single-cell level. We present a study on changes in cellular heterogeneity in the context of pre-malignant progression in response to hypoxic stress. Utilizing pre-malignant progression of Barrett’s esophagus (BE) as a disease model system we studied molecular mechanisms underlying the progression from metaplastic to dysplastic (pre-cancerous) stage. We used newly developed methods enabling measurements of cell-to-cell differences in copy numbers of mitochondrial DNA, expression levels of a set of mitochondrial and nuclear genes involved in hypoxia response pathways, and mitochondrial membrane potential. In contrast to bulk cell studies reported earlier, our study shows significant differences between metaplastic and dysplastic BE cells in both average values and single-cell parameter distributions of mtDNA copy numbers, mitochondrial function, and mRNA expression levels of studied genes. Based on single-cell data analysis, we propose that mitochondria may be one of the key factors in pre-malignant progression in BE.

Hydrophobic platinum(II)-5,10,15,20-tetrakis-(2,3,4,5,6-pentafluorophenyl)-porphyrin (PtTFPP) was physically incorporated into micelles formed from poly(ε-caprolactone)-block-poly(ethylene glycol) to enable the application of PtTFPP in aqueous solution. Micelles were characterized using dynamic light scattering (DLS) and atomic force microscopy (AFM) to show an average diameter of about 140 nm. PtTFPP showed higher quantum efficiency in micellar solution than in tetrahydrofuran (THF) and dichloromethane (CH₂Cl₂). PtTFPP in micelles also exhibited higher photostability than that of PtTFPP suspended in water. PtTFPP in micelles exhibited good oxygen sensitivity and response time. This study provided an efficient approach to enable the application of hydrophobic oxygen sensors in a biological environment.

Explicit solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation are applied to wave fields with invariant features, such as oscillating laser beams in a parabolic waveguide and spiral light beams in varying media. A similar effect of superfocusing of particle beams in a thin monocrystal film, harmonic oscillations of cold trapped atoms, and motion in magnetic field are also mentioned.

We discuss a method of constructing solutions of the initial value problem for diffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered. Examples include, but are not limited to the Fokker-Planck equation in physics, the Black-Scholes equation and the Hull-White model in finance.