In several applications, the data consists of an mn matrixA and it is of interest to find an approximation D of a specified rank k to A where, k is much smaller than m and n.Traditional methods like the Singular Value Decomposition (SVD) help us find the "best" such approximation. However, these methods take time polynomial in m; n which is often too prohibitive.In this paper, we develop an algorithm which is qualitatively faster provided we may sample the entries of the matrix according to a natural probability distribution. Indeed, in the applications such sampling is possible.Our main result is that we can find the description of a matrix D of rank at most k so that jjA , D jj F min D;rankDk jjA , Djj F + "jjAjj F holds with probability at least 1 , . (For any matrix M, jjMjj 2 F denotes the sum of the squares of all the entries of M.) The algorithm takes time polynomial in k;1="; log1= only, independent of m; n.
We provide our detailed, standardized in vitro protocol for culture and differentiation of human retinal pigment epithelial (RPE) cells into a highly polarized, functional monolayer. Disruption of polarized RPE function plays an important role in the pathogenesis of common blinding disorders of the retina. The availability of this polarized RPE monolayer allows for reproducible evaluation of RPE function, modeling of RPE dysfunction in retinal disease, and in vitro evaluation of novel therapies. The protocol, which takes approximately 6 weeks, describes the culture of RPE from human fetal donor eyes, and the differentiation of these cells into a polarized monolayer with high transepithelial resistance, and morphologic characteristics that mimic the RPE monolayer in vivo. By modifying the procedure for initial isolation of pure RPE cells, and culture conditions used in existing protocols, we have established a standardized protocol that provides highly reproducible RPE monolayers from the same donor eye.
Abstract.We consider the problem of partitioning a set of m points in the n-dimensional Euclidean space into k clusters (usually m and n are variable, while k is fixed), so as to minimize the sum of squared distances between each point and its cluster center. This formulation is usually the objective of the k-means clustering algorithm (Kanungo et al. (2000)). We prove that this problem in NP-hard even for k = 2, and we consider a continuous relaxation of this discrete problem: find the k-dimensional subspace V that minimizes the sum of squared distances to V of the m points. This relaxation can be solved by computing the Singular Value Decomposition (SVD) of the m × n matrix A that represents the m points; this solution can be used to get a 2-approximation algorithm for the original problem. We then argue that in fact the relaxation provides a generalized clustering which is useful in its own right.Finally, we show that the SVD of a random submatrix-chosen according to a suitable probability distributionof a given matrix provides an approximation to the SVD of the whole matrix, thus yielding a very fast randomized algorithm. We expect this algorithm to be the main contribution of this paper, since it can be applied to problems of very large size which typically arise in modern applications.
αB Crystallin is a chaperone protein with anti-apoptotic and anti-inflammatory functions and has been identified as a biomarker in age-related macular degeneration. The purpose of this study was to determine whether αB crystallin is secreted from retinal pigment epithelial (RPE) cells, the mechanism of this secretory pathway and to determine whether extracellular αB crystallin can be taken up by adjacent retinal cells and provide protection from oxidant stress. We used human RPE cells to establish that αB crystallin is secreted by a non-classical pathway that involves exosomes. Evidence for the release of exosomes by RPE and localization of αB crystallin within the exosomes was achieved by immunoblot, immunofluorescence, and electron microscopic analyses. Inhibition of lipid rafts or exosomes significantly reduced αB crystallin secretion, while inhibitors of classic secretory pathways had no effect. In highly polarized RPE monolayers, αB crystallin was selectively secreted towards the apical, photoreceptor-facing side. In support, confocal microscopy established that αB crystallin was localized predominantly in the apical compartment of RPE monolayers, where it co-localized in part with exosomal marker CD63. Severe oxidative stress resulted in barrier breakdown and release of αB crystallin to the basolateral side. In normal mouse retinal sections, αB crystallin was identified in the interphotoreceptor matrix. An increased uptake of exogenous αB crystallin and protection from apoptosis by inhibition of caspase 3 and PARP activation were observed in stressed RPE cultures. αB Crystallin was taken up by photoreceptors in mouse retinal explants exposed to oxidative stress. These results demonstrate an important role for αB crystallin in maintaining and facilitating a neuroprotective outer retinal environment and may also explain the accumulation of αB crystallin in extracellular sub-RPE deposits in the stressed microenvironment in age-related macular degeneration. Thus evidence from our studies supports a neuroprotective role for αB crystallin in ocular diseases.
Age-related macular degeneration (AMD) is a multi-factorial disease that is the leading cause of irreversible and severe vision loss in the developed countries. It has been suggested that the pathogenesis of dry AMD involves impaired protein degradation in retinal pigment epithelial cells (RPE). RPE cells are constantly exposed to oxidative stress that may lead to the accumulation of damaged cellular proteins, DNA and lipids and evoke tissue deterioration during the aging process. The ubiquitin-proteasome pathway and the lysosomal/autophagosomal pathway are the two major proteolytic systems in eukaryotic cells. NRF-2 (nuclear factor-erythroid 2-related factor-2) and PGC-1α (peroxisome proliferator-activated receptor gamma coactivator-1 alpha) are master transcription factors in the regulation of cellular detoxification. We investigated the role of NRF-2 and PGC-1α in the regulation of RPE cell structure and function by using global double knockout (dKO) mice. The NRF-2/PGC-1α dKO mice exhibited significant age-dependent RPE degeneration, accumulation of the oxidative stress marker, 4-HNE (4-hydroxynonenal), the endoplasmic reticulum stress markers GRP78 (glucose-regulated protein 78) and ATF4 (activating transcription factor 4), and damaged mitochondria. Moreover, levels of protein ubiquitination and autophagy markers p62/SQSTM1 (sequestosome 1), Beclin-1 and LC3B (microtubule associated protein 1 light chain 3 beta) were significantly increased together with the Iba-1 (ionized calcium binding adaptor molecule 1) mononuclear phagocyte marker and an enlargement of RPE size. These histopathological changes of RPE were accompanied by photoreceptor dysmorphology and vision loss as revealed by electroretinography. Consequently, these novel findings suggest that the NRF-2/PGC-1α dKO mouse is a valuable model for investigating the role of proteasomal and autophagy clearance in the RPE and in the development of dry AMD.
Subretinal fibrosis is a result of a wound healing response that follows choroidal neovascularization in neovascular age-related macular degeneration (nAMD). Although anti-vascular endothelial growth factor therapy has become a standard treatment that improves visual acuity in many nAMD patients, unsuccessful treatment outcomes have often been attributed to the progression of subretinal fibrosis. In this review, we summarize the cellular and extracellular components of subretinal fibrous membranes and also discuss the possible molecular mechanisms including the functional involvement of growth factors and the inflammatory response in the process. Moreover, we present an murine animal model of subretinal fibrosis that might facilitate greater understanding of the pathophysiology and the development of novel therapeutic strategies for the inhibition of subretinal fibrosis in nAMD.
Oxidative stress-induced damage to the retinal pigment epithelium (RPE) is considered to be a key factor in age-related macular degeneration (AMD) pathology. RPE cells are constantly exposed to oxidative stress that may lead to the accumulation of damaged cellular proteins, lipids, nucleic acids, and cellular organelles, including mitochondria. The ubiquitin-proteasome and the lysosomal/autophagy pathways are the two major proteolytic systems to remove damaged proteins and organelles. There is increasing evidence that proteostasis is disturbed in RPE as evidenced by lysosomal lipofuscin and extracellular drusen accumulation in AMD. Nuclear factor-erythroid 2-related factor-2 (NFE2L2) and peroxisome proliferator-activated receptor gamma coactivator-1 alpha (PGC-1α) are master transcription factors in the regulation of antioxidant enzymes, clearance systems, and biogenesis of mitochondria. The precise cause of RPE degeneration and the onset and progression of AMD are not fully understood. However, mitochondria dysfunction, increased reactive oxygen species (ROS) production, and mitochondrial DNA (mtDNA) damage are observed together with increased protein aggregation and inflammation in AMD. In contrast, functional mitochondria prevent RPE cells damage and suppress inflammation. Here, we will discuss the role of mitochondria in RPE degeneration and AMD pathology focused on mtDNA damage and repair, autophagy/mitophagy signaling, and regulation of inflammation. Mitochondria are putative therapeutic targets to prevent or treat AMD.
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