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.

Whole-body and intravital optical imaging of angiogenesis in orthotopically implanted tumors

The development of drugs for the control of tumor angiogenesis requires a simple, accurate, and economical assay for tumor-induced vascularization. We have adapted the orthotopic implantation model to angiogenesis measurement by using human tumors labeled with Aequorea victoria green fluorescent protein for grafting into nude mice. The nonluminous induced capillaries are clearly visible against the very bright tumor fluorescence examined either intravitally or by whole-body luminance in real time. The orthotopic implantation model of human cancer has been well characterized, and fluorescence shadowing replaces the laborious histological techniques for determining blood vessel density. Intravital images of orthotopically implanted human pancreatic tumors clearly show angiogenic capillaries at both primary and metastatic sites. A quantitative time course of angiogenesis was determined for an orthotopically growing human prostate tumor periodically imaged intravitally in a single nude mouse over a 19-day period. Whole-body optical imaging of tumor angiogenesis was demonstrated by injecting fluorescent Lewis lung carcinoma cells into the s.c. site of the footpad of nude mice. The footpad is relatively transparent, with comparatively few resident blood vessels, allowing quantitative imaging of tumor angiogenesis in the intact animal. Capillary density increased linearly over a 10-day period as determined by whole-body imaging. Similarly, the green fluorescent protein-expressing human breast tumor MDA-MB-435 was orthotopically transplanted to the mouse fat pad, where whole-body optical imaging showed that blood vessel density increased linearly over a 20-week period. These powerful and clinically relevant angiogenesis mouse models can be used for real-time in vivo evaluation of agents inhibiting or promoting tumor angiogenesis in physiological microenvironments.

Quantitative measurement of intraorganelle pH in the endosomal-lysosomal pathway in neurons by using ratiometric imaging with pyranine.

Organelle acidification is an essential element of the endosomal-lysosomal pathway, but our understanding of the mechanisms underlying progression through this pathway has been hindered by the absence of adequate methods for quantifying intraorganelle pH. To address this problem in neurons, we developed a direct quantitative method for accurately determining the pH of endocytic organelles in live cells. In this report, we demonstrate that the ratiometric fluorescent pH indicator 8-hydroxypyrene-1,3,6-trisulfonic acid (HPTS) is the most advantageous available probe for such pH measurements. To measure intraorganelle pH, cells were labeled by endocytic uptake of HPTS, the ratio of fluorescence emission intensities at excitation wavelengths of 450 nm and 405 nm (F450/405) was calculated for each organelle, and ratios were converted to pH values by using standard curves for F450/405 vs. pH. Proper calibration is critical for accurate measurement of pH values: standard curves generated in vitro yielded artifactually low organelle pH values. Calibration was unaffected by the use of culture medium buffered with various buffers or different cell types. By using this technique, we show that both acidic and neutral endocytically derived organelles exist in the axons of sympathetic neurons in different steady-state proportions than in the cell body. Furthermore, we demonstrate that these axonal organelles have a bimodal pH distribution, indicating a rapid acidification step in their maturation that reduces the average pH of a fraction of the organelles by 2 pH units while leaving few organelles of intermediate pH at steady state. Finally, we demonstrate a spatial gradient or organelle pH along axons, with the relative frequency of acidic organelles increasing with proximity to the cell body.