Phosphate homeostasis is central to diverse physiologic processes including energy homeostasis, formation of lipid bilayers, and bone formation. Reduced phosphate levels due to excessive renal loss cause hypophosphatemic rickets, a disease characterized by prominent bone defects; conversely, hyperphosphatemia, a major complication of renal failure, is accompanied by parathyroid hyperplasia, hyperparathyroidism, and osteodystrophy. Here, we define a syndrome featuring both hypophosphatemic rickets and hyperparathyroidism due to parathyroid hyperplasia as well as other skeletal abnormalities. We show that this disease is due to a de novo translocation with a breakpoint adjacent to ␣-Klotho, which encodes a ␤-glucuronidase, and is implicated in aging and regulation of FGF signaling. Plasma ␣-Klotho levels and ␤-glucuronidase activity are markedly increased in the affected patient; unexpectedly, the circulating FGF23 level is also markedly elevated. These findings suggest that the elevated ␣-Klotho level mimics aspects of the normal response to hyperphosphatemia and implicate ␣-Klotho in the selective regulation of phosphate levels and in the regulation of parathyroid mass and function; they also have implications for the pathogenesis and treatment of renal osteodystrophy in patients with kidney failure.bone ͉ endocrinology ͉ genetics ͉ phosphorus T he maintenance of normal phosphate homeostasis is critical for diverse biochemical processes in vivo. Its importance is illustrated by disorders featuring hypophosphatemia due to excessive renal phosphate loss. Affected patients develop rickets, a disorder in which abnormal mineralization of bone and growth plate cartilage results in diminished bone strength, deformity, short stature, and bone pain. Rapidly growing bones of the lower extremities generally show the most striking abnormalities (1). Conversely, high phosphate levels may also have adverse physiologic effects. These are most pronounced in patients with chronic kidney disease (CKD) who develop hyperphosphatemia due to impaired renal clearance. These patients develop hyperparathyroidism and renal osteodystrophy.The kidney plays a predominant role in the regulation of serum phosphorus levels. The vast majority of phosphate reabsorption occurs in the proximal tubule and is mediated by the sodium-phosphate cotransporter type IIa (NaPi-IIa), with additional contributions from the related protein NaPi-IIc (2, 3). Parathyroid hormone (PTH) has been considered the major regulator of NaPi-IIa and NaPi-IIc density in the proximal tubular cell apical membrane (3, 4), promoting the rapid removal of NaPi-IIa from the membrane and its subsequent degradation (5). In addition, high phosphate levels are known to promote proliferation of parathyroid cells and enhance PTH secretion and mRNA stability; these effects have been speculated to be mediated by means of reduced serum Ca 2ϩ levels, but the mechanism is uncertain (6-9).More recently, FGF23 and its receptor have been recognized to play fundamental roles in phosphate homeos...

By group theoretical arguments, it can be shown that a wave function, if;"L, for a system of N particles corresponding to a total angular momentum quantum number, L, and a quantum number, j.I, referring to the z component of angular momentum may be written as a sum of terms:if;"L=~. DL(R)".*x. L .The DL(R)". are the representation coefficients for the Lth irreducible representation of the three-dimensional rotation group, and are functions of the three coordinates specifying the orientation of the system of particles in space. The X,.L are functions of the 3N-6 coordinates specifying the relative configuration of the N-particle system. The set of coupled differential equations for the functions, X"L, is obtained explicitly. The special case of the three-particle system is discussed in detail. The present treatment is more directly usable than the previous discussions since the basic equations do not involve implicit relationships between the variables.

The quantum mechanics of the collision of an atom with a diatomic molecule is discussed. This is an example of the scattering of particles by a nonspherical potential field. In the region in which the atom and molecule are not interacting, the wave function for the system can be written as the product of a wave function for the free atom and that of the free molecule. By an analysis of this function one can write the asymptotic form of the true wave function in the terms of the scattering coefficients, representing the angular dependence of the transition probabilities. In a previous paper, group theoretical arguments were used to separate the rotational coordinates from the n-particle Schroedinger equation. A set of coupled differential equations for the internal wave functions was derived. In the present paper, the scattering coefficients are obtained by an analysis of the solutions of the internal wave equations involving the three coordinates specifying the internal configuration. This set of equations is transformed into a set of coupled ordinary differential equations in which the coupling is the result of the transitions in the state of the molecule. The equations are solved by a perturbation method which is valid in the case of low energy collisions in which transitions occur relatively seldom. The angular transition probabilities are obtained explicitly in terms of integrals of the interaction potential and a set of phase shifts.

In calculating elastic constants of viscoelastic media from transverse wave propagation, it has been customary heretofore to assume a plane disturbance in a medium of infinite extent. Equations have now been derived for conditions which approach much more closely the experimental arrangement, namely, propagation from a finite source plate in a rectangular cell, and representative numerical calculations have been made. The new theory shows that the finite dimensions of cell and source should have very little influence on the wave-length as determined from observations of strain double refraction, but the attenuation should be perceptibly different from that of the plane wave case. Experimental data confirm these conclusions. Nevertheless, the simple plane theory is adequate within present experimental error for the majority of cases. Only in certain specified ranges of mechanical consistency will it be necessary to apply corrections taking into account the sizes of the cell and source.

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