Exact, E=0, Solutions for General Power-Law Potentials. I. Classical Orbits

For zero energy, $E=0$, we derive exact, classical solutions for {\em all} power-law potentials, $V(r)=-\gamma/r^\nu$, with $\gamma>0$ and $-\infty <\nu<\infty$. When the angular momentum is non-zero, these solutions lead to the orbits $\r(t)= [\cos \mu (\th(t)-\th_0(t))]^{1/\mu}$, for all $\mu \equiv \nu/2-1 \ne 0$. When $\nu>2$, the orbits are bound and go through the origin. This leads to discrete discontinuities in the functional dependence of $\th(t)$ and $\th_0(t)$, as functions of $t$, as the orbits pass through the origin. We describe a procedure to connect different analytic solutions for successive orbits at the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. Also, we explain why they all must violate the virial theorem. The unbound orbits are also discussed in detail. This includes the unusual orbits which have finite travel times to infinity and also the special $\nu = 2$ case.Comment: LaTeX, 27 pages with 12 figures available from the authors or can be generated from Mathematica instructions at end of the fil

Effect of age and cytoskeletal elements on the indentation-dependent mechanical properties of chondrocytes.

Articular cartilage chondrocytes are responsible for the synthesis, maintenance, and turnover of the extracellular matrix, metabolic processes that contribute to the mechanical properties of these cells. Here, we systematically evaluated the effect of age and cytoskeletal disruptors on the mechanical properties of chondrocytes as a function of deformation. We quantified the indentation-dependent mechanical properties of chondrocytes isolated from neonatal (1-day), adult (5-year) and geriatric (12-year) bovine knees using atomic force microscopy (AFM). We also measured the contribution of the actin and intermediate filaments to the indentation-dependent mechanical properties of chondrocytes. By integrating AFM with confocal fluorescent microscopy, we monitored cytoskeletal and biomechanical deformation in transgenic cells (GFP-vimentin and mCherry-actin) under compression. We found that the elastic modulus of chondrocytes in all age groups decreased with increased indentation (15-2000 nm). The elastic modulus of adult chondrocytes was significantly greater than neonatal cells at indentations greater than 500 nm. Viscoelastic moduli (instantaneous and equilibrium) were comparable in all age groups examined; however, the intrinsic viscosity was lower in geriatric chondrocytes than neonatal. Disrupting the actin or the intermediate filament structures altered the mechanical properties of chondrocytes by decreasing the elastic modulus and viscoelastic properties, resulting in a dramatic loss of indentation-dependent response with treatment. Actin and vimentin cytoskeletal structures were monitored using confocal fluorescent microscopy in transgenic cells treated with disruptors, and both treatments had a profound disruptive effect on the actin filaments. Here we show that disrupting the structure of intermediate filaments indirectly altered the configuration of the actin cytoskeleton. These findings underscore the importance of the cytoskeletal elements in the overall mechanical response of chondrocytes, indicating that intermediate filament integrity is key to the non-linear elastic properties of chondrocytes. This study improves our understanding of the mechanical properties of articular cartilage at the single cell level

c-Maf Transcription Factor Regulates ADAMTS-12 Expression in Human Chondrogenic Cells.

ObjectiveADAMTS (a disintegrin and metalloproteinase with thrombospondin type-1 motif) zinc metalloproteinases are important during the synthesis and breakdown of cartilage extracellular matrix. ADAMTS-12 is up-regulated during in vitro chondrogenesis and embryonic limb development; however, the regulation of ADAMTS-12 expression in cartilage remains unknown. The transcription factor c-Maf is a member of Maf family of basic ZIP (bZIP) transcription factors. Expression of c-Maf is highest in hypertrophic chondrocytes during embryonic development and postnatal growth. We hypothesize that c-Maf and ADAMTS-12 are co-expressed during chondrocyte differentiation and that c-Maf regulates ADAMTS-12 expression during chondrogenesis.DesignPromoter analysis and species alignments identified potential c-Maf binding sites in the ADAMTS-12 promoter. c-Maf and ADAMTS-12 co-expression was monitored during chondrogenesis of stem cell pellet cultures. Luciferase expression driven by ADAMTS-12 promoter segments was measured in the presence and absence of c-Maf, and synthetic oligonucleotides were used to confirm specific binding of c-Maf to ADAMTS-12 promoter sequences.ResultsIn vitro chondrogenesis from human mesenchymal stem cells revealed co-expression of ADAMTS-12 and c-Maf during differentiation. Truncation and point mutations of the ADAMTS-12 promoter evaluated in reporter assays localized the response to the proximal 315 bp of the ADAMTS-12 promoter, which contained a predicted c-Maf recognition element (MARE) at position -61. Electorphoretic mobility shift assay confirmed that c-Maf directly interacted with the MARE at position -61.ConclusionsThese data suggest that c-Maf is involved in chondrocyte differentiation and hypertrophy, at least in part, through the regulation of ADAMTS-12 expression at a newly identified MARE in its proximal promoter

Trafic ferroviaire au Simplon

En 1878, la ligne du Simplon est exploitée de Vallorbe à Brigue, via Lausanne et Saint-Maurice. La construction d’un tunnel au Simplon permettrait de relier la France à l’Italie, mais la réalisation d’un tel ouvrage implique l’accord de l’Italie. Une ligne, qui relierait la péninsule à la France et traverserait la Suisse, est tout à fait dans l’intérêt de l’Italie. En 1896, son gouvernement confie à la société Jura-Simplon-Bahn l’exploitation du futur tronçon sur sol italien. De son côté, la France cherche à empêcher la réalisation du tunnel du Simplon, car elle ne veut pas « offrir aux troupes italiennes, soutien potentiel de l’Allemagne, une voie d’accès commode vers la France ». Comme on se défie de l’armée suisse, dont on pense qu’elle ne résisterait pas véritablement à une invasion allemande, « l’intérêt français exigerait qu’il ne fût ouvert au travers des Alpes aucun passage, sinon ceux dont la France garderait au moins une porte ». Un des deux tubes du tunnel est mis en service le 1er juin 1906. Au début du XXIe siècle, la ligne du Simplon conserve son importance européenne. En 2007, la mise en service du tunnel de base du Lötschberg représentera une amélioration considérable.The Military Railway Traffic of Simplon. In 1878 the Simplon railway line was pushed through from Vallorbe to Brigue, via Lausanne and Saint-Maurice. The construction of a tunnel at Simplon allowed France to be connected to Italy, but carrying out such an enterprise implied that Italy agreed to it. A line that linked the Italian peninsula to France, and which traversed Switzerland, was entirely in Italy’s interests. In 1896 the Italian government entrusted the development of the impending section of the line on Italian territory to the Jura-Simplon-Bahn Railway Company. For its part, France strove to prevent the completion of the Simplon tunnel, for she ‘did not want to offer Italian troops, a potential reinforcement for Germany, an easy access route towards France’. As it was thought unlikely that the Swiss army would offer genuine resistance to a German invasion, ‘French interests required that there be no open passageway through the Alps, unless it’s be those where France was able to guard at least one of the gates’. One of the rail tunnel’s two tubes entered service on 1 June 1906. At the beginning of the 21st Century, the Simplon line retains its importance for Europe. In 2007 a considerable improvement will come about when the tunnel that begins at Lötschberg is opened for service.Nel 1878, la linea del Sempione è sfruttata da Vallorbe a Briga, via Losanna e Saint-Maurice. La costruzione del tunnel al Sempione permetterebbe di collegare la Francia all’Italia, ma la realizzazione di una tale opera implica l’accordo da parte dell’Italia. Una linea, che collegherebbe la penisola alla Francia e traverserebbe la Svizzera, è in tutto e per tutto nell’interesse dell’Italia. Nel 1896, il suo Governo affida alla Società Ferrovia del Giura-Sempione la gestione del futuro tratto sul suolo italiano. Dal canto suo, la Francia cerca di impedire la realizzazione del tunnel del Sempione, poiché non vuole « offrir aux troupes italiennes, soutien potentiel de l’Allemagne, une voie d’accès commode vers la France. » Non fidandosi dell’esercito svizzero, di cui si pensa in verità che non resisterebbe a un’invasione tedesca, « l’intérêt français exigerait qu’il ne fut ouvert au travers des Alpes aucun passage, sinon ceux dont la France garderait au moins une porte. » Uno dei due tubi della galleria è messo in servizio il 1° giugno 1906… All’inizio del XXI secolo, la linea del Sempione conserva la sua importanza europea. Nel 2007, la messa in servizio del tunnel di base del Lötschberg rappresenterà un miglioramento considerevole.1878 wir die Simplonlinie von Vallorbe nach Brig betrieben, via Lausanne und St. Maurice. Der Bau eines Tunnels am Simplon würde es erlauben, Frankreich mit Italien zu verbinden, aber die Verwirklichung eines solchen Bauwerkes erfordert das Einverständnis von Italien. Eine Linie welche die Halbinsel mit Frankreich verbinden und durch die Schweiz führen würde, ist ganz im Interesse von Italien. 1896 vertraut seine Regierung der Jura-Simplon-Bahn-Gesellschaft den Betrieb des zukünftigen Teilstückes auf italienischem Boden an. Seinerseits versucht Frankreich die Verwirklichung des Siplontunnels zu verhindern, da es den “italienischen Truppen, die Deutschland potenziell unterstützen, keinen bequemen Zugangsweg zu Frankreich anbieten will.” Da man der Schweizer Armee misstrauisch gegenübersteht und man glaubt, dass sie einer deutschen Invasion nicht wirklich Widerstand leisten werde, “würde das Interesse Frankreichs erfordern, dass durch die Alpen überhaupt kein Durchgang zu öffnen wäre, es sei denn solche, bei denen Frankreich wenigstens eine Türe beaufsichtigen könnte.” Eine der beiden Röhren des Tunnels wird am 1. Juni 1906 in Betrieb genommen... Zu Beginn des 21. Jahrhunderts behält die Simplonlinie ihre europäische Bedeutung. 2007 wird die Inbetriebsetzung des Lötschberg-Basistunnels eine beträchtliche Verbesserung brin

Trafic ferroviaire au Simplon

En 1878, la ligne du Simplon est exploitée de Vallorbe à Brigue, via Lausanne et Saint-Maurice. La construction d’un tunnel au Simplon permettrait de relier la France à l’Italie, mais la réalisation d’un tel ouvrage implique l’accord de l’Italie. Une ligne, qui relierait la péninsule à la France et traverserait la Suisse, est tout à fait dans l’intérêt de l’Italie. En 1896, son gouvernement confie à la société Jura-Simplon-Bahn l’exploitation du futur tronçon sur sol italien. De son côté, la France cherche à empêcher la réalisation du tunnel du Simplon, car elle ne veut pas « offrir aux troupes italiennes, soutien potentiel de l’Allemagne, une voie d’accès commode vers la France ». Comme on se défie de l’armée suisse, dont on pense qu’elle ne résisterait pas véritablement à une invasion allemande, « l’intérêt français exigerait qu’il ne fût ouvert au travers des Alpes aucun passage, sinon ceux dont la France garderait au moins une porte ». Un des deux tubes du tunnel est mis en service le 1er juin 1906. Au début du XXIe siècle, la ligne du Simplon conserve son importance européenne. En 2007, la mise en service du tunnel de base du Lötschberg représentera une amélioration considérable.The Military Railway Traffic of Simplon. In 1878 the Simplon railway line was pushed through from Vallorbe to Brigue, via Lausanne and Saint-Maurice. The construction of a tunnel at Simplon allowed France to be connected to Italy, but carrying out such an enterprise implied that Italy agreed to it. A line that linked the Italian peninsula to France, and which traversed Switzerland, was entirely in Italy’s interests. In 1896 the Italian government entrusted the development of the impending section of the line on Italian territory to the Jura-Simplon-Bahn Railway Company. For its part, France strove to prevent the completion of the Simplon tunnel, for she ‘did not want to offer Italian troops, a potential reinforcement for Germany, an easy access route towards France’. As it was thought unlikely that the Swiss army would offer genuine resistance to a German invasion, ‘French interests required that there be no open passageway through the Alps, unless it’s be those where France was able to guard at least one of the gates’. One of the rail tunnel’s two tubes entered service on 1 June 1906. At the beginning of the 21st Century, the Simplon line retains its importance for Europe. In 2007 a considerable improvement will come about when the tunnel that begins at Lötschberg is opened for service.Nel 1878, la linea del Sempione è sfruttata da Vallorbe a Briga, via Losanna e Saint-Maurice. La costruzione del tunnel al Sempione permetterebbe di collegare la Francia all’Italia, ma la realizzazione di una tale opera implica l’accordo da parte dell’Italia. Una linea, che collegherebbe la penisola alla Francia e traverserebbe la Svizzera, è in tutto e per tutto nell’interesse dell’Italia. Nel 1896, il suo Governo affida alla Società Ferrovia del Giura-Sempione la gestione del futuro tratto sul suolo italiano. Dal canto suo, la Francia cerca di impedire la realizzazione del tunnel del Sempione, poiché non vuole « offrir aux troupes italiennes, soutien potentiel de l’Allemagne, une voie d’accès commode vers la France. » Non fidandosi dell’esercito svizzero, di cui si pensa in verità che non resisterebbe a un’invasione tedesca, « l’intérêt français exigerait qu’il ne fut ouvert au travers des Alpes aucun passage, sinon ceux dont la France garderait au moins une porte. » Uno dei due tubi della galleria è messo in servizio il 1° giugno 1906… All’inizio del XXI secolo, la linea del Sempione conserva la sua importanza europea. Nel 2007, la messa in servizio del tunnel di base del Lötschberg rappresenterà un miglioramento considerevole.1878 wir die Simplonlinie von Vallorbe nach Brig betrieben, via Lausanne und St. Maurice. Der Bau eines Tunnels am Simplon würde es erlauben, Frankreich mit Italien zu verbinden, aber die Verwirklichung eines solchen Bauwerkes erfordert das Einverständnis von Italien. Eine Linie welche die Halbinsel mit Frankreich verbinden und durch die Schweiz führen würde, ist ganz im Interesse von Italien. 1896 vertraut seine Regierung der Jura-Simplon-Bahn-Gesellschaft den Betrieb des zukünftigen Teilstückes auf italienischem Boden an. Seinerseits versucht Frankreich die Verwirklichung des Siplontunnels zu verhindern, da es den “italienischen Truppen, die Deutschland potenziell unterstützen, keinen bequemen Zugangsweg zu Frankreich anbieten will.” Da man der Schweizer Armee misstrauisch gegenübersteht und man glaubt, dass sie einer deutschen Invasion nicht wirklich Widerstand leisten werde, “würde das Interesse Frankreichs erfordern, dass durch die Alpen überhaupt kein Durchgang zu öffnen wäre, es sei denn solche, bei denen Frankreich wenigstens eine Türe beaufsichtigen könnte.” Eine der beiden Röhren des Tunnels wird am 1. Juni 1906 in Betrieb genommen... Zu Beginn des 21. Jahrhunderts behält die Simplonlinie ihre europäische Bedeutung. 2007 wird die Inbetriebsetzung des Lötschberg-Basistunnels eine beträchtliche Verbesserung brin

Effect of alendronate on post-traumatic osteoarthritis induced by anterior cruciate ligament rupture in mice.

IntroductionPrevious studies in animal models of osteoarthritis suggest that alendronate (ALN) has antiresorptive and chondroprotective effects, and can reduce osteophyte formation. However, these studies used non-physiologic injury methods, and did not investigate early time points during which bone is rapidly remodeled prior to cartilage degeneration. The current study utilized a non-invasive model of knee injury in mice to investigate the effect of ALN treatment on subchondral bone changes, articular cartilage degeneration, and osteophyte formation following injury.MethodsNon-invasive knee injury via tibial compression overload or sham injury was performed on a total of 90 mice. Mice were treated with twice weekly subcutaneous injections of low-dose ALN (40&nbsp;μg/kg/dose), high-dose ALN (1,000&nbsp;μg/kg/dose), or vehicle, starting immediately after injury until sacrifice at 7, 14 or 56&nbsp;days. Trabecular bone of the femoral epiphysis, subchondral cortical bone, and osteophyte volume were quantified using micro-computed tomography (μCT). Whole-joint histology was performed at all time points to analyze articular cartilage and joint degeneration. Blood was collected at sacrifice, and serum was analyzed for biomarkers of bone formation and resorption.ResultsμCT analysis revealed significant loss of trabecular bone from the femoral epiphysis 7 and 14&nbsp;days post-injury, which was effectively prevented by high-dose ALN treatment. High-dose ALN treatment was also able to reduce subchondral bone thickening 56&nbsp;days post-injury, and was able to partially preserve articular cartilage 14&nbsp;days post-injury. However, ALN treatment was not able to reduce osteophyte formation at 56&nbsp;days post-injury, nor was it able to prevent articular cartilage and joint degeneration at this time point. Analysis of serum biomarkers revealed an increase in bone resorption at 7 and 14&nbsp;days post-injury, with no change in bone formation at any time points.ConclusionsHigh-dose ALN treatment was able to prevent early trabecular bone loss and cartilage degeneration following non-invasive knee injury, but was not able to mitigate long-term joint degeneration. These data contribute to understanding the effect of bisphosphonates on the development of osteoarthritis, and may support the use of anti-resorptive drugs to prevent joint degeneration following injury, although further investigation is warranted