Packing and covering immersion models of planar subcubic graphs

A graph $H$ is an immersion of a graph $G$ if $H$ can be obtained by some sugraph $G$ after lifting incident edges. We prove that there is a polynomial function $f:\Bbb{N}\times\Bbb{N}\rightarrow\Bbb{N}$, such that if $H$ is a connected planar subcubic graph on $h>0$ edges, $G$ is a graph, and $k$ is a non-negative integer, then either $G$ contains $k$ vertex/edge-disjoint subgraphs, each containing $H$ as an immersion, or $G$ contains a set $F$ of $f(k,h)$ vertices/edges such that $G\setminus F$ does not contain $H$ as an immersion

Heat Shock Proteins and Amateur Chaperones in Amyloid-Beta Accumulation and Clearance in Alzheimer’s Disease

The pathologic lesions of Alzheimer’s disease (AD) are characterized by accumulation of protein aggregates consisting of intracellular or extracellular misfolded proteins. The amyloid-β (Aβ) protein accumulates extracellularly in senile plaques and cerebral amyloid angiopathy, whereas the hyperphosphorylated tau protein accumulates intracellularly as neurofibrillary tangles. “Professional chaperones”, such as the heat shock protein family, have a function in the prevention of protein misfolding and subsequent aggregation. “Amateur” chaperones, such as apolipoproteins and heparan sulfate proteoglycans, bind amyloidogenic proteins and may affect their aggregation process. Professional and amateur chaperones not only colocalize with the pathological lesions of AD, but may also be involved in conformational changes of Aβ, and in the clearance of Aβ from the brain via phagocytosis or active transport across the blood–brain barrier. Thus, both professional and amateur chaperones may be involved in the aggregation, accumulation, persistence, and clearance of Aβ and tau and in other Aβ-associated reactions such as inflammation associated with AD lesions, and may, therefore, serve as potential targets for therapeutic intervention