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.

How the protease thrombin talks to cells

How does a protease act like a hormone to regulate cellular functions? The coagulation protease thrombin (EC 3.4.21.5) activates platelets and regulates the behavior of other cells by means of G protein-coupled protease-activated receptors (PARs). PAR1 is activated when thrombin binds to and cleaves its amino-terminal exodomain to unmask a new receptor amino terminus. This new amino terminus then serves as a tethered peptide ligand, binding intramolecularly to the body of the receptor to effect transmembrane signaling. The irreversibility of PAR1’s proteolytic activation mechanism stands in contrast to the reversible ligand binding that activates classical G protein-coupled receptors and compels special mechanisms for desensitization and resensitization. In endothelial cells and fibroblasts, activated PAR1 rapidly internalizes and then sorts to lysosomes rather than recycling to the plasma membrane as do classical G protein-coupled receptors. This trafficking behavior is critical for termination of thrombin signaling. An intracellular pool of thrombin receptors refreshes the cell surface with naïve receptors, thereby maintaining thrombin responsiveness. Thus cells have evolved a trafficking solution to the signaling problem presented by PARs. Four PARs have now been identified. PAR1, PAR3, and PAR4 can all be activated by thrombin. PAR2 is activated by trypsin and by trypsin-like proteases but not by thrombin. Recent studies with knockout mice, receptor-activating peptides, and blocking antibodies are beginning to define the role of these receptors in vivo.

Essential roles for four cytoplasmic intermediate filament proteins in Caenorhabditis elegans development

The structural proteins of the cytoplasmic intermediate filaments (IFs) arise in the nematode Caenorhabditis elegans from eight reported genes and an additional three genes now identified in the complete genome. With the use of double-stranded RNA interference (RNAi) for all 11 C. elegans genes encoding cytoplasmic IF proteins, we observe phenotypes for the five genes A1, A2, A3, B1, and C2. These range from embryonic lethality (B1) and embryonic/larval lethality (A3) to larval lethality (A1 and A2) and a mild dumpy phenotype of adults (C2). Phenotypes A2 and A3 involve displaced body muscles and paralysis. They probably arise by reduction of hypodermal IFs that participate in the transmission of force from the muscle cells to the cuticle. The B1 phenotype has multiple morphogenetic defects, and the A1 phenotype is arrested at the L1 stage. Thus, at least four IF genes are essential for C. elegans development. Their RNAi phenotypes are lethal defects due to silencing of single IF genes. In contrast to C. elegans, no IF genes have been identified in the complete Drosophila genome, posing the question of how Drosophila can compensate for the lack of these proteins, which are essential in mammals and C. elegans. We speculate that the lack of IF proteins in Drosophila can be viewed as cytoskeletal alteration in which, for instance, stable microtubules, often arranged as bundles, substitute for cytoplasmic IFs.

SPO21 Is Required for Meiosis-specific Modification of the Spindle Pole Body in Yeast

During meiosis II in the yeast Saccharomyces cerevisiae, the cytoplasmic face of the spindle pole body changes from a site of microtubule initiation to a site of de novo membrane formation. These membranes are required to package the haploid meiotic products into spores. This functional change in the spindle pole body involves the expansion and modification of its cytoplasmic face, termed the outer plaque. We report here that SPO21 is required for this modification. The Spo21 protein localizes to the spindle pole in meiotic cells. In the absence of SPO21 the structure of the outer plaque is abnormal, and prospore membranes do not form. Further, decreased dosage of SPO21 leaves only two of the four spindle pole bodies competent to generate membranes. Mutation of CNM67, encoding a known component of the mitotic outer plaque, also results in a meiotic outer plaque defect but does not block membrane formation, suggesting that Spo21p may play a direct role in initiating membrane formation.

Kinetics of self-assembling microtubules: an "inverse problem" in biochemistry.

Experimental time series for a nonequilibrium reaction may in some cases contain sufficient data to determine a unique kinetic model for the reaction by a systematic mathematical analysis. As an example, a kinetic model for the self-assembly of microtubules is derived here from turbidity time series for solutions in which microtubules assemble. The model may be seen as a generalization of Oosawa's classical nucleation-polymerization model. It reproduces the experimental data with a four-stage nucleation process and a critical nucleus of 15 monomers.

Kinetics of spindle pole body separation in budding yeast.

In the budding yeast Saccharomyces cerevisiae, the spindle pole body (SPB) serves as the microtubule-organizing center and is the functional analog of the centrosome of higher organisms. By expressing a fusion of a yeast SPB-associated protein to the Aequorea victoria green fluorescent protein, the movement of the SPBs in living yeast cells undergoing mitosis was observed by fluorescence microscopy. The ability to visualize SPBs in vivo has revealed previously unidentified mitotic events. During anaphase, the mitotic spindle has four sequential activities: alignment at the mother-daughter junction, fast elongation, translocation into the bud, and slow elongation. These results indicate that distinct forces act upon the spindle at different times during anaphase.