BackgroundOsteoclasts, cells responsible for bone resorption, contribute to the development of degenerative, metabolic and neoplastic bone diseases, which are often characterized by persistent changes in bone microenvironment. We aimed to investigate the dynamics of osteoclast formation and death in cultures that considerably exceeded the length of standard protocol and to design a mathematical model describing osteoclastogenesis.Methodology/Principal FindingsRAW 264.7 monocytic cells fuse to form multinucleated osteoclasts upon treatment with pro-resorptive cytokine RANKL. We have found that in long-term experiments (15–26 days), the dynamics of changes in osteoclast numbers was remarkably complex and qualitatively variable in different experiments. Whereas 19 of 46 experiments exhibited single peak of osteoclast formation, in 27 experiments we observed development of successive waves of osteoclast formation and death. Periodic changes in osteoclast numbers were confirmed in long-term cultures of mouse bone marrow cells treated with M-CSF and RANKL. Because the dynamics of changes in osteoclast numbers was found to be largely independent of monocytes, a two-species model of ordinary differential equations describing the changes in osteoclasts and monocytes was ineffective in recapitulating the oscillations in osteoclast numbers. Following experimental observation that medium collected from mature osteoclasts inhibited osteoclastogenesis in fresh cultures, we introduced a third variable, factor f, to describe osteoclast-derived inhibitor. This model allowed us to simulate the oscillatory changes in osteoclasts, which were coupled to oscillatory changes in the factor f, whereas monocytes changed exponentially. Importantly, to achieve the experimentally observed oscillations with increasing amplitude, we also had to assume that osteoclast presence stimulates osteoclast formation.Conclusions/SignificanceThis study identifies the critical role for osteoclast autocrine regulation in controlling long-term dynamic of osteoclast formation and death and describes the complementary roles for negative and positive feedback mediators in determining the sharp dynamics of activation and inactivation of osteoclasts.

Abstract. We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an n-dimensional flat torus T n , and the Fourier transform of squares of the eigenfunctions |ϕ λ | 2 of the Laplacian have uniform l n bounds that do not depend on the eigenvalue λ. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on T n+2 . We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an n-dimensional sphere S n (λ) of radius √ λ, and we use it in the proof.

Abstract. We establish uniform bounds for the solutions e it∆ u of the Schrödinger equation on arithmetic flat tori, generalising earlier results by J. Bourgain. We also study the regularity properties of weak- * limits of sequences of densities of the form |e it∆ un| 2 corresponding to highly oscillating sequences of initial data (un). We obtain improved regularity properties of those limits using previous results by N. Anantharaman and F. Macià on the structure of semiclassical measures for solutions to the Schrödinger equation on the torus.

Abstract. We derive comparison formulas relating the zeta-regularized determinant of an arbitrary self-adjoint extension of the Laplace operator with domain C ∞ c (X \ {P }) ⊂ L 2 (X) to the zeta-regularized determinant of the Laplace operator on X. Here, X is a compact Riemannian manifold of dimension 2 or 3; P ∈ X.

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