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.

Functional transitions in myosin: Formation of a critical salt-bridge and transmission of effect to the sensitive tryptophan

For analyzing the mechanism of energy transduction in the “motor” protein, myosin, it is opportune both to model the structural change in the hydrolytic transition, ATP (myosin-bound) + H2O → ADP⋅Pi (myosin-bound) and to check the plausibility of the model by appropriate site-directed mutations in the functional system. Here, we made a series of mutations to investigate the role of the salt-bridge between Glu-470 and Arg-247 (of chicken smooth muscle myosin) that has been inferred from crystallography to be a central feature of the transition [Fisher, A. J., Smith, C. A., Thoden, J. B., Smith, R., Sutoh, K., Holden, H. M., & Rayment, I. (1995) Biochemistry 34, 8960–8972]. Our results suggest that whether in the normal, or in the inverted, direction an intact salt-bridge is necessary for ATP hydrolysis, but when the salt-bridge is in the inverted direction it does not support actin activation. Normally, fluorescence changes result from adding nucleotides to myosin; these signals are reported by Trp-512 (of chicken smooth muscle myosin). Our results also suggest that structural impairments in the 470–247 region interfere with the transmission of these signals to the responsive Trp.

Physiological astrocytic calcium levels stimulate glutamate release to modulate adjacent neurons

Astrocytes can release glutamate in a calcium-dependent manner and consequently signal to adjacent neurons. Whether this glutamate release pathway is used during physiological signaling or is recruited only under pathophysiological conditions is not well defined. One reason for this lack of understanding is the limited knowledge about the levels of calcium necessary to stimulate glutamate release from astrocytes and about how they compare with the range of physiological calcium levels in these cells. We used flash photolysis to raise internal calcium in astrocytes, while monitoring astrocytic calcium levels and glutamate, which evoked slow inward currents that were recorded electrophysiologically from single neurons grown on microislands of astrocytes. With this approach, we demonstrate that modest changes of astrocytic calcium, from 84 to 140 nM, evoke substantial glutamatergic currents in neighboring neurons (−391 pA), with a Hill coefficient of 2.1 to 2.7. Because the agonists glutamate, norepinephrine, and dopamine all raise calcium in astrocytes to levels exceeding 1.8 μM, these quantitative studies demonstrate that the astrocytic glutamate release pathway is engaged at physiological levels of internal calcium. Consequently, the calcium-dependent release of glutamate from astrocytes functions within an appropriate range of astrocytic calcium levels to be used as a signaling pathway within the functional nervous system.

Arabidopsis thaliana Responses to Mechanical Stimulation Do Not Require ETR1 or EIN21

Plants exposed to repetitive touch or wind are generally shorter and stockier than sheltered plants. These mechanostimulus-induced developmental changes are termed thigmomorphogenesis and may confer resistance to subsequent stresses. An early response of Arabidopsis thaliana to touch or wind is the up-regulation of TCH (touch) gene expression. The signal transduction pathway that leads to mechanostimulus responses is not well defined. A role for ethylene has been proposed based on the observation that mechanostimulation of plants leads to ethylene evolution and exogenous ethylene leads to thigmomorphogenetic-like changes. To determine whether ethylene has a role in plant responses to mechanostimulation, we assessed the ability of two ethylene-insensitive mutants, etr1–3 and ein2–1, to undergo thigmomorphogenesis and TCH gene up-regulation of expression. The ethylene-insensitive mutants responded to wind similarly to the wild type, with a delay in flowering, decrease in inflorescence elongation rate, shorter mature primary inflorescences, more rosette paraclades, and appropriate TCH gene expression changes. Also, wild-type and mutant Arabidopsis responded to vibrational stimulation, with an increase in hypocotyl elongation and up-regulation of TCH gene expression. We conclude that the ETR1 and EIN2 protein functions are not required for the developmental and molecular responses to mechanical stimulation.