Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle χn,k—the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to χn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n−k) × Rn−k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61–83] to map this tight frame isometrically to a tight frame for L2(Rn)—the k-plane ridgelets. This construction makes analysis of a function f ∈ L2(Rn) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for χn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.

Phototaxis away from blue light by an Escherichia coli mutant accumulating protoporphyrin IX.

The hemH gene of Escherichia coli encodes ferrochelatase (EC 4.99.1.1), the enzyme that catalyzes the last step in the production of heme, namely the synthesis of heme from protoporphyrin IX plus Fe2+. The behavioral responses to light were studied in E. coli carrying a hemH mutation. It was shown that the hemH mutant displayed a tumbling response upon illumination and a running response upon removal of the light. The most effect light to induce a tumbling response in the hemH mutant was blue light (396-450 nm). The chemotaxis machinery was needed for the light-induced tumbling response in the hemH mutant. The bacterial defect is an analog of the human inherited disease erythropoietic protoporphyria.

Microtubule release from the centrosome

Although microtubules (MTs) are generally thought to originate at the centrosome, a number of cell types have significant populations of MTs with no apparent centrosomal connection. The origin of these noncentrosomal MTs has been unclear. We applied kinetic analysis of MT formation in vivo to establish their mode of origin. Time-lapse fluorescence microscopy demonstrated that noncentrosomal MTs in cultured epithelial cells arise primarily by constitutive nucleation at, and release from, the centrosome. After release, MTs moved away from the centrosome and tended to depolymerize. Laser-marking experiments demonstrated that released MTs moved individually with their plus ends leading, suggesting that they were transported by minus end-directed motors. Released MTs were dynamic. The laser marking experiments demonstrated that plus ends of released MTs grew, paused, or shortened while the minus ends were stable or shortened. Microtubule release may serve two kinds of cellular function. Release and transport could generate the noncentrosomal MT arrays observed in epithelial cells, neurons, and other asymmetric, differentiated cells. Release would also contribute to polymer turnover by exposing MT minus ends, thereby providing additional sites for loss of subunits. The noncentrosomal population of MTs may reflect a steady-state of centrosomal nucleation, release, and dynamics.

Inactivating one hippocampus impairs avoidance of a stable room-defined place during dissociation of arena cues from room cues by rotation of the arena

Unilateral intrahippocampal injections of tetrodotoxin were used to temporarily inactivate one hippocampus during specific phases of training in an active allothetic place avoidance task. The rat was required to use landmarks in the room to avoid a room-defined sector of a slowly rotating circular arena. The continuous rotation dissociated room cues from arena cues and moved the arena surface through a part of the room in which foot-shock was delivered. The rat had to move away from the shock zone to prevent being transported there by the rotation. Unilateral hippocampal inactivations profoundly impaired acquisition and retrieval of the allothetic place avoidance. Posttraining unilateral hippocampal inactivation also impaired performance in subsequent sessions. This allothetic place avoidance task seems more sensitive to hippocampal disruption than the standard water maze task because the same unilateral hippocampal inactivation does not impair performance of the variable-start, fixed hidden goal task after procedural training. The results suggest that the hippocampus not only encodes allothetic relationships amongst landmarks, it also organizes perceived allothetic stimuli into systems of mutually stable coordinates. The latter function apparently requires greater hippocampal integrity.

Free energy reconstruction from nonequilibrium single-molecule pulling experiments

Laser tweezers and atomic force microscopes are increasingly used to probe the interactions and mechanical properties of individual molecules. Unfortunately, using such time-dependent perturbations to force rare molecular events also drives the system away from equilibrium. Nevertheless, we show how equilibrium free energy profiles can be extracted rigorously from repeated nonequilibrium force measurements on the basis of an extension of Jarzynski's remarkable identity between free energies and the irreversible work.

Ultrafast diffraction and structural dynamics: The nature of complex molecules far from equilibrium

Studies of molecular structures at or near their equilibrium configurations have long provided information on their geometry in terms of bond distances and angles. Far-from-equilibrium structures are relatively unknown—especially for complex systems—and generally, neither their dynamics nor their average geometries can be extrapolated from equilibrium values. For such nonequilibrium structures, vibrational amplitudes and bond distances play a central role in phenomena such as energy redistribution and chemical reactivity. Ultrafast electron diffraction, which was developed to study transient molecular structures, provides a direct method for probing the nature of complex molecules far from equilibrium. Here we present our ultrafast electron diffraction observations of transient structures for two cyclic hydrocarbons. At high internal energies of ≈4 eV, these molecules display markedly different behavior. For 1,3,5-cycloheptatriene, excitation results in the formation of hot ground-state structures with bond distances similar to those of the initial structure, but with nearly three times the average vibrational amplitude. Energy is redistributed within 5 ps, but with a negative temperature characterizing the nonequilibrium population. In contrast, the ring-opening reaction of 1,3-cyclohexadiene is shown to result in hot structures with a C—C bond distance of over 1.7 Å, which is 0.2 Å away from any expected equilibrium value. Even up to 400 ps, energy remains trapped in large-amplitude motions comprised of torsion and asymmetric stretching. These studies promise a new direction for studying structural dynamics in nonequilibrium complex systems.

Fluidity of the lipid domain of cell wall from Mycobacterium chelonae.

The mycobacterial cell wall contains large amounts of unusual lipids, including mycolic acids that are covalently linked to the underlying arabinogalactan-peptidoglycan complex. Hydrocarbon chains of much of these lipids have been shown to be packed in a direction perpendicular to the plane of the cell surface. In this study, we examined the dynamic properties of the organized lipid domains in the cell wall isolated from Mycobacterium chelonae grown at 30 degrees C. Differential scanning calorimetry showed that much of the lipids underwent major thermal transitions between 30 degree C and 65 degrees C, that is at temperatures above the growth temperature, a result suggesting that a significant portion of the lipids existed in a structure of extremely low fluidity in the growing cells. Spin-labeled fatty acid probes were successfully inserted into the more fluid part of the cell wall. Our model of the cell wall suggests that this domain corresponds to the outermost leaflet, a conclusion reinforced by the observation that labeling of intact cells produced electron spin resonance spectra similar to those of the isolated cell wall. Use of stearate labeled at different positions showed that the fluidity within the outer leaflet increased only slightly as the nitroxide group was placed farther away from the surface. These results are consistent with the model of mycobacterial cell wall containing an asymmetric lipid bilayer, with an internal, less fluid mycolic acid leaflet and an external, more fluid leaflet composed of lipids containing shorter chain fatty acids. The presence of the low-fluidity layer will lower the permeability of the cell wall to lipophilic antibiotics and chemotherapeutic agents and may contribute to the well-known intrinsic resistance of mycobacteria to such compounds.