Riesz bases of exponentials for multi-tiling measures

Let $G$ be a closed subgroup of ${\mathbb R}^d$ and let $\nu$ be a Borel probability measure admitting a Riesz basis of exponentials with frequency sets in the dual group $G^{\perp}$. We form a multi-tiling measure $\mu = \mu_1+...+\mu_N$ where $\mu_i$ is translationally equivalent to $\nu$ and different $\mu_i$ and $\mu_j$ have essentially disjoint support. We obtain some necessary and sufficient conditions for $\mu$ to admit a Riesz basis of exponentials . As an application, the square boundary, after a rotation, is a union of two fundamental domains of $G = {\mathbb Z}\times {\mathbb R}$ and can be regarded as a multi-tiling measure. We show that, unfortunately, the square boundary does not admit a Riesz basis of exponentials of the form as a union of translate of discrete subgroups ${\mathbb Z}\times \{0\}$. This rules out a natural candidate of potential Riesz basis for the square boundary.Comment: To appear in Sampling Theory, Signal Processing, and Data Analysi

Riesz basis of exponentials for multi-tiling measures

From literature in the study of Fourier series and spectrality in the weaker setting of Riesz bases and frames of exponentials, it is known that a multi-tile by a discrete lattice permits a Riesz basis of exponentials with a spectrum from a finite union of translates of the dual lattice. In this thesis, we generalize this result to a measure theoretic setting. Specifically, we suppose that a Borel measureνadmits an exponential Riesz basis in the lattice and we determine equivalent conditions such that an associated multi-tiling Borel measureμitself admits an exponential Riesz basis whose spectrum is a finite union of translates of the spectrum admitted byν. As an application, we show that the boundary of the square does not have such a basis but that a slight separation of the "top" and "bottom" triangular boundary halves does have such a basis. The result is also applied and works for singular multi-tiling measures such as fractal measures as well as for conventional multi-tiling constructions,e.g. a unit square covering R2.https://doi.org/10.46569/20.500.12680/qn59qb40

?2-Microglobulin Amyloid Fibril-Induced Membrane Disruption Is Enhanced by Endosomal Lipids and Acidic pH

Although the molecular mechanisms underlying the pathology of amyloidoses are not well understood, the interaction between amyloid proteins and cell membranes is thought to play a role in several amyloid diseases. Amyloid fibrils of ?2-microglobulin (?2m), associated with dialysis-related amyloidosis (DRA), have been shown to cause disruption of anionic lipid bilayers in vitro. However, the effect of lipid composition and the chemical environment in which ?2m-lipid interactions occur have not been investigated previously. Here we examine membrane damage resulting from the interaction of ?2m monomers and fibrils with lipid bilayers. Using dye release, tryptophan fluorescence quenching and fluorescence confocal microscopy assays we investigate the effect of anionic lipid composition and pH on the susceptibility of liposomes to fibril-induced membrane damage. We show that ?2m fibril-induced membrane disruption is modulated by anionic lipid composition and is enhanced by acidic pH. Most strikingly, the greatest degree of membrane disruption is observed for liposomes containing bis(monoacylglycero)phosphate (BMP) at acidic pH, conditions likely to reflect those encountered in the endocytic pathway. The results suggest that the interaction between ?2m fibrils and membranes of endosomal origin may play a role in the molecular mechanism of ?2m amyloid-associated osteoarticular tissue destruction in DRA

Beta2-Microglobulin Amyloid Fibrils Are Nanoparticles That Disrupt Lysosomal Membrane Protein Trafficking and Inhibit Protein Degradation by Lysosomes.

Fragmentation of amyloid fibrils produces fibrils that are reduced in length but have an otherwise unchanged molecular architecture. The resultant nanoscale fibril particles inhibit the cellular reduction of the tetrazolium dye 3-(4,5-dimethylthi-azol-2-yl)-2,5-diphenyltetrazolium bromide (MTT), a substrate commonly used to measure cell viability, to a greater extent than unfragmented fibrils. Here we show that the internalization of β2-microglobulin (β2m) amyloid fibrils is dependent on fibril length, with fragmented fibrils being more efficiently internalized by cells. Correspondingly, inhibiting the internalization of fragmented β2m fibrils rescued cellular MTT reduction. Incubation of cells with fragmented β2m fibrils did not, however, cause cell death. Instead, fragmented β2m fibrils accumulate in lysosomes, alter the trafficking of lysosomal membrane proteins, and inhibit the degradation of a model protein substrate by lysosomes. These findings suggest that nanoscale fibrils formed early during amyloid assembly reactions or by the fragmentation of longer fibrils could play a role in amyloid disease by disrupting protein degradation by lysosomes and trafficking in the endolysosomal pathway

Recent Developments in Fluorescence Correlation Spectroscopy for Diffusion Measurements in Planar Lipid Membranes

Fluorescence correlation spectroscopy (FCS) is a single molecule technique used mainly for determination of mobility and local concentration of molecules. This review describes the specific problems of FCS in planar systems and reviews the state of the art experimental approaches such as 2-focus, Z-scan or scanning FCS, which overcome most of the artefacts and limitations of standard FCS. We focus on diffusion measurements of lipids and proteins in planar lipid membranes and review the contributions of FCS to elucidating membrane dynamics and the factors influencing it, such as membrane composition, ionic strength, presence of membrane proteins or frictional coupling with solid support

The molecular lifecycle of amyloid – Mechanism of assembly, mesoscopic organisation, polymorphism, suprastructures, and biological consequences

The formation of a diverse range of amyloid structures from normally soluble proteins and peptides is a hallmark of devastating human disorders as well as biological functions. The current molecular understanding of the amyloid lifecycle reveals four processes central to their growth and propagation: primary nucleation, elongation, secondary nucleation and division. However, these processes result in a wide range of cross-β packing and filament arrangements, including diverse assemblies formed from identical monomeric precursors with the same amino acid sequences. Here, we review current structural and mechanistic understanding of amyloid self-assembly, and discuss how mesoscopic, i.e. micrometre to nanometre, organisation of amyloid give rise to suprastructural features that may be the key link between the polymorphic amyloid structures and the biological response they elicit. A greater understanding of the mechanisms governing suprastructure formation will guide future strategies to combat amyloid associated disorders and to use and control the amyloid quaternary structure in synthetic biology and materials applications