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.

A molecular basis for nitric oxide sensing by soluble guanylate cyclase

Nitric oxide (NO) functions as a signaling agent by activation of the soluble isoform of guanylate cyclase (sGC), a heterodimeric hemoprotein. NO binds to the heme of sGC and triggers formation of cGMP from GTP. Here we report direct kinetic measurements of the multistep binding of NO to sGC and correlate these presteady state events with activation of enzyme catalysis. NO binds to sGC to form a six-coordinate, nonactivated, intermediate (kon > 1.4 × 108 M−1⋅s−1 at 4°C). Subsequent release of the axial histidine heme ligand is shown to be the molecular step responsible for activation of the enzyme. The rate at which this step proceeds also depends on NO concentration (k = 2.4 × 105 M−1⋅s−1 at 4°C), thus identifying a novel mode of regulation by NO. NO binding to the isolated heme domain of sGC was also rapid (k = 7.1 ± 2 × 108 M−1⋅s−1 at 4°C); however, no intermediate was observed. The data show that sGC acts as an extremely fast, specific, and highly efficient trap for NO and that cleavage of the iron-histidine bond provides the driving force for activation of sGC. In addition, the kinetic data indicate that transport or stabilization of NO is not necessary for effective signal transmission.

Assembly of a Novel Cartilage Matrix Protein Filamentous Network: Molecular Basis of Differential Requirement of von Willebrand Factor A Domains

Cartilage matrix protein (CMP) is the prototype of the newly discovered matrilin family, all of which contain von Willebrand factor A domains. Although the function of matrilins remain unclear, we have shown that, in primary chondrocyte cultures, CMP (matrilin-1) forms a filamentous network, which is made up of two types of filaments, a collagen-dependent one and a collagen-independent one. In this study, we demonstrate that the collagen-independent CMP filaments are enriched in pericellular compartments, extending directly from chondrocyte membranes. Their morphology can be distinguished from that of collagen filaments by immunogold electron microscopy, and mimicked by that of self-assembled purified CMP. The assembly of CMP filaments can occur from transfection of a wild-type CMP transgene alone in skin fibroblasts, which do not produce endogenous CMP. Conversely, assembly of endogenous CMP filaments by chondrocytes can be inhibited specifically by dominant negative CMP transgenes. The two A domains within CMP serve essential but different functions during network formation. Deletion of the A2 domain converts the trimeric CMP into a mixture of monomers, dimers, and trimers, whereas deletion of the A1 domain does not affect the trimeric configuration. This suggests that the A2 domain modulates multimerization of CMP. Absence of either A domain from CMP abolishes its ability to form collagen-independent filaments. In particular, Asp22 in A1 and Asp255 in A2 are essential; double point mutation of these residues disrupts CMP network formation. These residues are part of the metal ion–dependent adhesion sites, thus a metal ion–dependent adhesion site–mediated adhesion mechanism may be applicable to matrilin assembly. Taken together, our data suggest that CMP is a bridging molecule that connects matrix components in cartilage to form an integrated matrix network.