Anti-PTH monoclonal antibody, clone 5B3

We introduce the L-p spaces of measurable functions whose pth power is summable with respect to the uniform measure over the Levi-Civita field. These spaces are the counterparts of the real L-p spaces based upon the Lebesgue measure. Nevertheless, they lack some properties of the L-p spaces: for instance, the L-p spaces are not sequentially complete with respect to the p-norm. This motivates the study of the completions of the L-p spaces with respect to strong convergence, denoted by L-s(p). It turns out that the L-s(p) spaces are Banach spaces and that it is possible to define an inner product over L-s(2), thus making it a Hilbert space. Despite these positive results, these spaces are still not rich enough to represent every real continuous function. For this reason, we settle upon the representation of real measurable functions as sequences of measurable functions in R that weakly converge in measure. We also define a duality between measurable functions and representatives of continuous functions. This duality enables the study of some measurable functions that represent real distributions. We focus our discussion on the representatives of the Dirac distribution and on the well-known problem of the product between the Dirac and the Heaviside distribution, and we show that the solution obtained with measurable functions over R is consistent with the result obtained with other nonlinear generalized functions. (C) 2020 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

Parathyroid hormone (PTH) is crucial for bone remodeling. Intermittent PTH (1-34) administration stimulates osteogenesis and promotes bone formation; however, the possible targets and underlying mechanisms still remain unclear. In this study, functional links between PTH and Foxc1, a transcription factor reported to be predominant in skeletal development and formation, were indicated. We determined the impacts of Foxc1 on in vitro osteogenic differentiation and in vivo bone regeneration under intermittent PTH induction, and further explored its possible targets. We found that the expression level of Foxc1 was upregulated during osteogenic induction by intermittent PTH treatment, and the elevated expression of Foxc1 induced by PTH was inhibited by PTH1R silencing, while rescued by intermittent PTH supplement. By gain- and loss-of-function strategies targeting Foxc1 in MC3T3-E1 cells, we demonstrated that Foxc1 could promote in vitro osteogenic differentiation by intermittent PTH induction. Moreover, immunofluorescence analysis indicated the nuclear co-localization of Foxc1 with Runx2. Luciferase-reporter and chromatin immunoprecipitation analysis further confirmed that Foxc1 could bind to the P1 promoter region of Runx2 directly, which plays an indispensable part in osteogenic differentiation and bone mineralization. Meanwhile, we also revealed that Foxc1 could promote bone regeneration induced by intermittent PTH treatment in vivo. Taken together, this study revealed the role and mechanism of Foxc1 on in vitro osteogenic differentiation and in vivo bone regeneration in response of intermittent PTH treatment.