Functions of lipid raft membrane microdomains at the blood-brain barrier

The blood-brain barrier (BBB) is a highly specialized structural and functional component of the central nervous system that separates the circulating blood from the brain and spinal cord parenchyma. Brain endothelial cells (BECs) that primarily constitute the BBB are tightly interconnected by multiprotein complexes, the adherens junctions and the tight junctions, thereby creating a highly restrictive cellular barrier. Lipid-enriched membrane microdomain compartmentalization is an inherent property of BECs and allows for the apicobasal polarity of brain endothelium, temporal and spatial coordination of cell signaling events, and actin remodeling. In this manuscript, we review the role of membrane microdomains, in particular lipid rafts, in the BBB under physiological conditions and during leukocyte transmigration/diapedesis. Furthermore, we propose a classification of endothelial membrane microdomains based on their function, or at least on the function ascribed to the molecules included in such heterogeneous rafts: (1) rafts associated with interendothelial junctions and adhesion of BECs to basal lamina (scaffolding rafts); (2) rafts involved in immune cell adhesion and migration across brain endothelium (adhesion rafts); (3) rafts associated with transendothelial transport of nutrients and ions (transporter rafts).

The block numerical range of analytic operator functions

We introduce the block numerical range Wn(L) of an operator function L with respect to a decomposition H = H1⊕. . .⊕Hn of the underlying Hilbert space. Our main results include the spectral inclusion property and estimates of the norm of the resolvent for analytic L . They generalise, and improve, the corresponding results for the numerical range (which is the case n = 1) since the block numerical range is contained in, and may be much smaller than, the usual numerical range. We show that refinements of the decomposition entail inclusions between the corresponding block numerical ranges and that the block numerical range of the operator matrix function L contains those of its principal subminors. For the special case of operator polynomials, we investigate the boundedness of Wn(L) and we prove a Perron-Frobenius type result for the block numerical radius of monic operator polynomials with coefficients that are positive in Hilbert lattice sense.