Some uniform stability and convergence results for integral equations on the real line and projection methods for their solution

The paper considers second kind equations of the form (abbreviated x=y + K2x) in which and the factor z is bounded but otherwise arbitrary so that equations of Wiener-Hopf type are included as a special case. Conditions on a set are obtained such that a generalized Fredholm alternative is valid: if W satisfies these conditions and I − Kz, is injective for each z ε W then I − Kz is invertible for each z ε W and the operators (I − Kz)−1 are uniformly bounded. As a special case some classical results relating to Wiener-Hopf operators are reproduced. A finite section version of the above equation (with the range of integration reduced to [−a, a]) is considered, as are projection and iterated projection methods for its solution. The operators (where denotes the finite section version of Kz) are shown uniformly bounded (in z and a) for all a sufficiently large. Uniform stability and convergence results, for the projection and iterated projection methods, are obtained. The argument generalizes an idea in collectively compact operator theory. Some new results in this theory are obtained and applied to the analysis of projection methods for the above equation when z is compactly supported and k(s − t) replaced by the general kernel k(s,t). A boundary integral equation of the above type, which models outdoor sound propagation over inhomogeneous level terrain, illustrates the application of the theoretical results developed.

Carbon monoxide inhibition of Cav3.2 T-type Ca2+ channels reveals tonic modulation by thioredoxin

T-type Ca2+ channels play diverse roles in tissues such as sensory neurons, vascular smooth muscle, and cancers, where increased expression of the cytoprotective enzyme, heme oxygenase-1 (HO-1) is often found. Here, we report regulation of T-type Ca2+ channels by carbon monoxide (CO) a HO-1 by-product. CO (applied as CORM-2) caused a concentration-dependent, poorly reversible inhibition of all T-type channel isoforms (Cav3.1-3.3, IC50 ∼3 μM) expressed in HEK293 cells, and native T-type channels in NG108-15 cells and primary rat sensory neurons. No recognized CO-sensitive signaling pathway could account for the CO inhibition of Cav3.2. Instead, CO sensitivity was mediated by an extracellular redox-sensitive site, which was also highly sensitive to thioredoxin (Trx). Trx depletion (using auranofin, 2-5 μM) reduced Cav3.2 currents and their CO sensitivity by >50% but increased sensitivity to dithiothreitol ∼3-fold. By contrast, Cav3.1 and Cav3.3 channels, and their sensitivity to CO, were unaffected in identical experiments. Our data propose a novel signaling pathway in which Trx acts as a tonic, endogenous regulator of Cav3.2 channels, while HO-1-derived CO disrupts this regulation, causing channel inhibition. CO modulation of T-type channels has widespread implications for diverse physiological and pathophysiological mechanisms, such as excitability, contractility, and proliferation

Syk and Src family kinases regulate C-type lectin receptor 2 (CLEC-2)-mediated clustering of podoplanin and platelet adhesion to lymphatic endothelial cells

The interaction of C-type lectin receptor 2 (CLEC-2) on platelets with Podoplanin on lymphatic endothelial cells initiates platelet signaling events that are necessary for prevention of blood-lymph mixing during development. In the present study, we show that CLEC-2 signaling via Src family and Syk tyrosine kinases promotes platelet adhesion to primary mouse lymphatic endothelial cells at low shear. Using supported lipid bilayers containing mobile Podoplanin, we further show that activation of Src and Syk in platelets promotes clustering of CLEC-2 and Podoplanin. Clusters of CLEC-2-bound Podoplanin migrate rapidly to the center of the platelet to form a single structure. Fluorescence lifetime imaging demonstrates that molecules within these clusters are within 10 nm of one another and that the clusters are disrupted by inhibition of Src and Syk family kinases. CLEC-2 clusters are also seen in platelets adhered to immobilized Podoplanin using direct stochastic optical reconstruction microscopy. These findings provide mechanistic insight by which CLEC-2 signaling promotes adhesion to Podoplanin and regulation of Podoplanin signaling, thereby contributing to lymphatic vasculature development.