Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

Differential fluorescence labeling of cysteinyl clusters uncovers high tissue levels of thionein

The isolation of thionein (T) from tissues has not been reported heretofore. T contains 20 cysteinyl residues that react with 7-fluorobenz-2-oxa-1,3-diazole-4-sulfonamide to form fluorescent adducts. In metallothionein (MT) the cysteinyl residues, which are bound to zinc, do not react. However, they do react in the presence of a chelating agent such as EDTA. The resultant difference in chemical reactivity provides a means to measure T in the absence of EDTA, (MT + T) in its presence, and, of course, MT by difference. The 7-fluorobenz-2-oxa-1,3-diazole-4-sulfonamide derivative of T can be isolated from tissue homogenates by HPLC and quantified fluorimetrically with a detection limit in the femtomolar range and a linear response over 3 orders of magnitude. Analysis of liver, kidney, and brain of rats reveals almost as much T as MT. Moreover, in contrast to earlier views, MT in tissue extracts appears to be less stable than T. The existence of T in tissues under normal physiological conditions has important implications for its function both in zinc metabolism and the redox balance of the cell.