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.

Fluorescence in situ hybridization analysis of keratinocyte growth factor gene amplification and dispersion in evolution of great apes and humans

Keratinocyte growth factor (KGF) is a member of the fibroblast growth factor family. Portions of the gene encoding KGF were amplified during primate evolution and are present in multiple nonprocessed copies in the human genome. Nucleotide analysis of a representative sampling of these KGF-like sequences indicated that they were at least 95% identical to corresponding regions of the KGF gene. To localize these sequences to specific chromosomal sites in human and higher primates, we used fluorescence in situ hybridization. In human, using a cosmid probe encoding KGF exon 1, we assigned the location of the KGF gene to chromosome 15q15–21.1. In addition, copies of KGF-like sequences hybridizing only with a cosmid probe encoding exons 2 and 3 were localized to dispersed sites on chromosome 2q21, 9p11, 9q12–13, 18p11, 18q11, 21q11, and 21q21.1. The distribution of KGF-like sequences suggests a role for alphoid DNA in their amplification and dispersion. In chimpanzee, KGF-like sequences were observed at five chromosomal sites, which were each homologous to sites in human, while in gorilla, a subset of four of these homologous sites was identified; in orangutan two sites were identified, while gibbon exhibited only a single site. The chromosomal localization of KGF sequences in human and great ape genomes indicates that amplification and dispersion occurred in multiple discrete steps, with initial KGF gene duplication and dispersion taking place in gibbon and involving loci corresponding to human chromosomes 15 and 21. These findings support the concept of a closer evolutionary relationship of human and chimpanzee and a possible selective pressure for such dispersion during the evolution of higher primates.

The cluA− mutant of Dictyostelium identifies a novel class of proteins required for dispersion of mitochondria

The cluA gene of Dictyostelium discoideum encodes a novel 150-kDa protein. Disruption of cluA results in clustering of mitochondria near the cell center. This is a striking difference from normal cells, whose mitochondria are dispersed uniformly throughout the cytoplasm. The mutant cell populations also exhibit an increased frequency of multinucleated cells, suggesting an impairment in cytokinesis. Both phenotypes are reversed by transformation of cluA− cells with a plasmid carrying a constitutively expressed cluA gene. The predicted sequence of the cluA gene product is homologous to sequences encoded by open reading frames in the genomes of Saccharomyces cerevisiae and Caenorhabditis elegans, but not to any known protein. The only exception is a short region with some homology to the 42-residue imperfect repeats present in the kinesin light chain, which probably function in protein–protein interaction. These studies identify a new class of proteins that appear to be required for the proper distribution of mitochondria.

Adsorption from a one-dimensional lattice gas and the Brunauer–Emmett–Teller equation

An exact treatment of adsorption from a one-dimensional lattice gas is used to eliminate and correct a well-known inconsistency in the Brunauer–Emmett–Teller (B.E.T.) equation—namely, Gibbs excess adsorption is not taken into account and the Gibbs integral diverges at the transition point. However, neither model should be considered realistic for experimental adsorption systems.

Self-similar intermediate asymptotics for a degenerate parabolic filtration-absorption equation

The equation ∂tu = u∂xx2u − (c − 1)(∂xu)2 is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water-absorbing fissurized porous rock; therefore, we refer to this equation as a filtration-absorption equation. A family of self-similar solutions to this equation is constructed. Numerical investigation of the evolution of non-self-similar solutions to the Cauchy problems having compactly supported initial conditions is performed. Numerical experiments indicate that the self-similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state: identical zero. An open problem caused by the nonuniqueness of the solution of the Cauchy problem is discussed.

PMD fundamentals: Polarization mode dispersion in optical fibers

This paper reviews the fundamental concepts and basic theory of polarization mode dispersion (PMD) in optical fibers. It introduces a unified notation and methodology to link the various views and concepts in Jones space and Stokes space. The discussion includes the relation between Jones vectors and Stokes vectors, rotation matrices, the definition and representation of PMD vectors, the laws of infinitesimal rotation, and the rules for PMD vector concatenation.

Photosystem II single crystals studied by EPR spectroscopy at 94 GHz: The tyrosine radical Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{D}^{{\bullet}}}}\end{equation*}\end{document}

Electron paramagnetic resonance (EPR) spectroscopy at 94 GHz is used to study the dark-stable tyrosine radical Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{D}^{{\bullet}}}}\end{equation*}\end{document} in single crystals of photosystem II core complexes (cc) isolated from the thermophilic cyanobacterium Synechococcus elongatus. These complexes contain at least 17 subunits, including the water-oxidizing complex (WOC), and 32 chlorophyll a molecules/PS II; they are active in light-induced electron transfer and water oxidation. The crystals belong to the orthorhombic space group P212121, with four PS II dimers per unit cell. High-frequency EPR is used for enhancing the sensitivity of experiments performed on small single crystals as well as for increasing the spectral resolution of the g tensor components and of the different crystal sites. Magnitude and orientation of the g tensor of Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{D}^{{\bullet}}}}\end{equation*}\end{document} and related information on several proton hyperfine tensors are deduced from analysis of angular-dependent EPR spectra. The precise orientation of tyrosine Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{D}^{{\bullet}}}}\end{equation*}\end{document} in PS II is obtained as a first step in the EPR characterization of paramagnetic species in these single crystals.

BeF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{3}^{-}}}\end{equation*}\end{document} acts as a phosphate analog in proteins phosphorylated on aspartate: Structure of a BeF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{3}^{-}}}\end{equation*}\end{document} complex with phosphoserine phosphatase

Protein phosphoaspartate bonds play a variety of roles. In response regulator proteins of two-component signal transduction systems, phosphorylation of an aspartate residue is coupled to a change from an inactive to an active conformation. In phosphatases and mutases of the haloacid dehalogenase (HAD) superfamily, phosphoaspartate serves as an intermediate in phosphotransfer reactions, and in P-type ATPases, also members of the HAD family, it serves in the conversion of chemical energy to ion gradients. In each case, lability of the phosphoaspartate linkage has hampered a detailed study of the phosphorylated form. For response regulators, this difficulty was recently overcome with a phosphate analog, BeF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{3}^{-}}}\end{equation*}\end{document}, which yields persistent complexes with the active site aspartate of their receiver domains. We now extend the application of this analog to a HAD superfamily member by solving at 1.5-Å resolution the x-ray crystal structure of the complex of BeF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{3}^{-}}}\end{equation*}\end{document} with phosphoserine phosphatase (PSP) from Methanococcus jannaschii. The structure is comparable to that of a phosphoenzyme intermediate: BeF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{3}^{-}}}\end{equation*}\end{document} is bound to Asp-11 with the tetrahedral geometry of a phosphoryl group, is coordinated to Mg2+, and is bound to residues surrounding the active site that are conserved in the HAD superfamily. Comparison of the active sites of BeF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{3}^{-}}}\end{equation*}\end{document}⋅PSP and BeF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{3}^{-}}}\end{equation*}\end{document}⋅CeY, a receiver domain/response regulator, reveals striking similarities that provide insights into the function not only of PSP but also of P-type ATPases. Our results indicate that use of BeF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{3}^{-}}}\end{equation*}\end{document} for structural studies of proteins that form phosphoaspartate linkages will extend well beyond response regulators.

Radial and tangential dispersion patterns in the mouse retina are cell-class specific.

The retina is derived from a pseudostratified germinal zone in which the relative position of a progenitor cell is believed to determine the position of the progeny aligned in the radial axis. Such a developmental mechanism would ensure that radial arrays of cells which comprise functional units in the mature central nervous system are also clonally related. The present study has tested this hypothesis by using X chromosome-inactivation transgenic mosaic mice. We report that the retina shows a conspicuous distinction for clonally related neuroblasts of different laminar and functional fates: the rod photoreceptor, Müller, and bipolar cells are aligned in the radial axis, whereas the cone photoreceptor, horizontal, amacrine, and ganglion cells are tangentially displaced with respect to them. These results indicate that the dispersion of cell classes across the retinal surface is differentially constrained. Some classes of retinal neuroblast exhibit a significant tangential, as well as radial, component in their dispersion from the germinal zone, whereas others disperse only in the radial dimension. Consequently, the majority of radial columns within the mature retina must be derived from multiple progenitors. Because the cone photoreceptor, horizontal, amacrine, and ganglion cells establish nonrandom matrices in their cellular distributions within the respective retinal layers, tangential dispersion may be the means by which these matrices are constructed.