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For a set S of points in R d , an s-spanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε d ) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), well-separated pair decomposition, and approximate k-centers.

BackgroundThe primary cilium is an antenna-like, nonmotile structure that extends from the surface of most mammalian cell types and is critical for chemosensing and mechanosensing in a variety of tissues including cartilage, bone, and kidney. Flow-induced intracellular calcium ion (Ca2+) increases in kidney epithelia depend on primary cilia and primary cilium-localized Ca2+-permeable channels polycystin-2 (PC2) and transient receptor potential vanilloid 4 (TRPV4). While primary cilia have been implicated in osteocyte mechanotransduction, the molecular mechanism that mediates this process is not fully understood. We directed a fluorescence resonance energy transfer (FRET)-based Ca2+ biosensor to the cilium by fusing the biosensor sequence to the sequence of the primary cilium-specific protein Arl13b. Using this tool, we investigated the role of several Ca2+-permeable channels that may mediate flow-induced Ca2+ entry: PC2, TRPV4, and PIEZO1.ResultsHere, we report the first measurements of Ca2+ signaling within osteocyte primary cilia using a FRET-based biosensor fused to ARL13B. We show that fluid flow induces Ca2+ increases in osteocyte primary cilia which depend on both intracellular Ca2+ release and extracellular Ca2+ entry. Using siRNA-mediated knockdowns, we demonstrate that TRPV4, but not PC2 or PIEZO1, mediates flow-induced ciliary Ca2+ increases and loading-induced Cox-2 mRNA increases, an osteogenic response.ConclusionsIn this study, we show that the primary cilium forms a Ca2+ microdomain dependent on Ca2+ entry through TRPV4. These results demonstrate that the mechanism of mechanotransduction mediated by primary cilia varies in different tissue contexts. Additionally, we anticipate that this work is a starting point for more studies investigating the role of TRPV4 in mechanotransduction.Electronic supplementary materialThe online version of this article (doi:10.1186/s13630-015-0016-y) contains supplementary material, which is available to authorized users.

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The primary cilium is a solitary, immotile microtubule-based extension present on nearly every mammalian cell. This organelle has established mechanosensory roles in several contexts including kidney, liver, and the embryonic node. Mechanical load deflects the cilium, triggering biochemical responses. Defects in cilium function have been associated with numerous human diseases. Recent research has implicated the primary cilium as a mechanosensor in bone. In this review, we discuss the cilium, the growing evidence for its mechanosensory role in bone, and areas of future study. This article is part of a Special Issue entitled “Osteocytes”.

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Age-related changes in blood androgens and estrogens may contribute to the development or progression of frailty in men.