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.

Geometry of circular vectors and pattern recognition of shape of a boundary

This paper deals with pattern recognition of the shape of the boundary of closed figures on the basis of a circular sequence of measurements taken on the boundary at equal intervals of a suitably chosen argument with an arbitrary starting point. A distance measure between two boundaries is defined in such a way that it has zero value when the associated sequences of measurements coincide by shifting the starting point of one of the sequences. Such a distance measure, which is invariant to the starting point of the sequence of measurements, is used in identification or discrimination by the shape of the boundary of a closed figure. The mean shape of a given set of closed figures is defined, and tests of significance of differences in mean shape between populations are proposed.

On the geometry of solutions of the quasi-geostrophic and Euler equations

We study solutions of the two-dimensional quasi-geostrophic thermal active scalar equation involving simple hyperbolic saddles. There is a naturally associated notion of simple hyperbolic saddle breakdown. It is proved that such breakdown cannot occur in finite time. At large time, these solutions may grow at most at a quadruple-exponential rate. Analogous results hold for the incompressible three-dimensional Euler equation.

A single point mutation in the pore region of the epithelial Na+ channel changes ion selectivity by modifying molecular sieving

The epithelial Na+ channel (ENaC) belongs to a new class of channel proteins called the ENaC/DEG superfamily involved in epithelial Na+ transport, mechanotransduction, and neurotransmission. The role of ENaC in Na+ homeostasis and in the control of blood pressure has been demonstrated recently by the identification of mutations in ENaC β and γ subunits causing hypertension. The function of ENaC in Na+ reabsorption depends critically on its ability to discriminate between Na+ and other ions like K+ or Ca2+. ENaC is virtually impermeant to K+ ions, and the molecular basis for its high ionic selectivity is largely unknown. We have identified a conserved Ser residue in the second transmembrane domain of the ENaC α subunit (αS589), which when mutated allows larger ions such as K+, Rb+, Cs+, and divalent cations to pass through the channel. The relative ion permeability of each of the αS589 mutants is related inversely to the ionic radius of the permeant ion, indicating that αS589 mutations increase the molecular cutoff of the channel by modifying the pore geometry at the selectivity filter. Proper geometry of the pore is required to tightly accommodate Na+ and Li+ ions and to exclude larger cations. We provide evidence that ENaC discriminates between cations mainly on the basis of their size and the energy of dehydration.

Conversion of agonist site to metal-ion chelator site in the β2-adrenergic receptor

Previously metal-ion sites have been used as structural and functional probes in seven transmembrane receptors (7TM), but as yet all the engineered sites have been inactivating. Based on presumed agonist interaction points in transmembrane III (TM-III) and -VII of the β2-adrenergic receptor, in this paper we construct an activating metal-ion site between the amine-binding Asp-113 in TM-III—or a His residue introduced at this position—and a Cys residue substituted for Asn-312 in TM-VII. No increase in constitutive activity was observed in the mutant receptors. Signal transduction was activated in the mutant receptors not by normal catecholamine ligands but instead either by free zinc ions or by zinc or copper ions in complex with small hydrophobic metal-ion chelators. Chelation of the metal ions by small hydrophobic chelators such as phenanthroline or bipyridine protected the cells from the toxic effect of, for example Cu2+, and in several cases increased the affinity of the ions for the agonistic site. Wash-out experiments and structure–activity analysis indicated, that the high-affinity chelators and the metal ions bind and activate the mutant receptor as metal ion guided ligand complexes. Because of the well-understood binding geometry of the small metal ions, an important distance constraint has here been imposed between TM-III and -VII in the active, signaling conformation of 7TM receptors. It is suggested that atoxic metal-ion chelator complexes could possibly in the future be used as generic, pharmacologic tools to switch 7TM receptors with engineered metal-ion sites on or off at will.

Localization of the C terminus of the assembly domain of hepatitis B virus capsid protein: Implications for morphogenesis and organization of encapsidated RNA

The capsid protein of hepatitis B virus, consisting of an “assembly” domain (residues 1–149) and an RNA-binding “protamine” domain (residues 150–183), assembles from dimers into icosahedral capsids of two different sizes. The C terminus of the assembly domain (residues 140–149) functions as a morphogenetic switch, longer C termini favoring a higher proportion of the larger capsids, it also connects the protamine domain to the capsid shell. We now have defined the location of this peptide in capsids assembled in vitro by engineering a mutant assembly domain with a single cysteine at its C terminus (residue 150), labeling it with a gold cluster and visualizing the cluster by cryo-electron microscopy. The labeled protein is unimpaired in its ability to form capsids. Our density map reveals a single undecagold cluster under each fivefold and quasi-sixfold vertex, connected to sites at either end of the undersides of the dimers. Considering the geometry of the vertices, the C termini must be more crowded at the fivefolds. Thus, a bulky C terminus would be expected to favor formation of the larger (T = 4) capsids, which have a greater proportion of quasi-sixfolds. Capsids assembled by expressing the full-length protein in Escherichia coli package bacterial RNAs in amounts equivalent to the viral pregenome. Our density map of these capsids reveals a distinct inner shell of density—the RNA. The RNA is connected to the protein shell via the C-terminal linkers and also makes contact around the dimer axes.

The importance of reactant positioning in enzyme catalysis: A hybrid quantum mechanics/molecular mechanics study of a haloalkane dehalogenase

Hybrid quantum mechanics/molecular mechanics calculations using Austin Model 1 system-specific parameters were performed to study the SN2 displacement reaction of chloride from 1,2-dichloroethane (DCE) by nucleophilic attack of the carboxylate of acetate in the gas phase and by Asp-124 in the active site of haloalkane dehalogenase from Xanthobacter autotrophicus GJ10. The activation barrier for nucleophilic attack of acetate on DCE depends greatly on the reactants having a geometry resembling that in the enzyme or an optimized gas-phase structure. It was found in the gas-phase calculations that the activation barrier is 9 kcal/mol lower when dihedral constraints are used to restrict the carboxylate nucleophile geometry to that in the enzyme relative to the geometries for the reactants without dihedral constraints. The calculated quantum mechanics/molecular mechanics activation barriers for the enzymatic reaction are 16.2 and 19.4 kcal/mol when the geometry of the reactants is in a near attack conformer from molecular dynamics and in a conformer similar to the crystal structure (DCE is gauche), respectively. This haloalkane dehalogenase lowers the activation barrier for dehalogenation of DCE by 2–4 kcal/mol relative to the single point energies of the enzyme's quantum mechanics atoms in the gas phase. SN2 displacements of this sort in water are infinitely slower than in the gas phase. The modest lowering of the activation barrier by the enzyme relative to the reaction in the gas phase is consistent with mutation experiments.

Crystal structures of the vitamin D receptor complexed to superagonist 20-epi ligands

The crystal structures of the ligand-binding domain (LBD) of the vitamin D receptor complexed to 1α,25(OH)2D3 and the 20-epi analogs, MC1288 and KH1060, show that the protein conformation is identical, conferring a general character to the observation first made for retinoic acid receptor (RAR) that, for a given LBD, the agonist conformation is unique, the ligands adapting to the binding pocket. In all complexes, the A- to D-ring moieties of the ligands adopt the same conformation and form identical contacts with the protein. Differences are observed only for the 17β-aliphatic chains that adapt their conformation to anchor the 25-hydroxyl group to His-305 and His-397. The inverted geometry of the C20 methyl group induces different paths of the aliphatic chains. The ligands exhibit a low-energy conformation for MC1288 and a more strained conformation for the two others. KH1060 compensates this energy cost by additional contacts. Based on the present data, the explanation of the superagonist effect is to be found in higher stability and longer half-life of the active complex, thereby excluding different conformations of the ligand binding domain.

Measuring the topology of the universe

Observations of microwave background fluctuations can yield information not only about the geometry of the universe but potentially about the topology of the universe. If the universe is negatively curved, then the characteristic scale for the topology of the universe is the curvature radius. Thus, if we are seeing the effects of the geometry of the universe, we can hope to soon see signatures of the topology of the universe. The cleanest signature of the topology of the universe is written on the microwave sky: There should be thousands of pairs of matched circles. These circles can be used to determine the precise topology and volume of the universe. Because we see hundreds of slices through the fundamental domain of the universe, we can use the microwave observations to reconstruct the initial conditions of the entire universe on the scale of a few megaparsecs.

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.

Faithful representation of similarities among three-dimensional shapes in human vision.

Efficient and reliable classification of visual stimuli requires that their representations reside a low-dimensional and, therefore, computationally manageable feature space. We investigated the ability of the human visual system to derive such representations from the sensory input-a highly nontrivial task, given the million or so dimensions of the visual signal at its entry point to the cortex. In a series of experiments, subjects were presented with sets of parametrically defined shapes; the points in the common high-dimensional parameter space corresponding to the individual shapes formed regular planar (two-dimensional) patterns such as a triangle, a square, etc. We then used multidimensional scaling to arrange the shapes in planar configurations, dictated by their experimentally determined perceived similarities. The resulting configurations closely resembled the original arrangements of the stimuli in the parameter space. This achievement of the human visual system was replicated by a computational model derived from a theory of object representation in the brain, according to which similarities between objects, and not the geometry of each object, need to be faithfully represented.

Self-assembly of a heteroduplex helicate from two different ligand strands and Cu(II) cations.

Cu(II) ions have been reacted with a 1/1 mixture of two linear ligands, one containing three 2,2'- bipyridine groups and the other three 2,2':6',2"-terpyridine groups. Absorption spectroscopy and fast atom bombardment mass spectrometry indicate the formation of a trinuclear complex containing one ligand of each kind. Determination of the crystal structure of this compound has confirmed that it is indeed a linear trinuclear complex in which two different ligands are wrapped in a helical fashion around the pentacoordinated metal ions. The central coordination geometry is trigonal bipyramidal; the two lateral Cu(II) ions are in a square pyramidal environment. Thus, a heteroduplex helicate is formed by the self-assembly of two different ligand strands and three specific metal ions induced by the coordination number and geometry of the latter. The self-assembly process may be considered to result from the reading of the steric and binding information present in the two ligands by Cu(II) ions through a pentacoordination algorithm. The same ligands have been shown earlier to yield homoduplex helicates from ions of tetrahedral and octahedral coordination geometry and strands of bidentate bipyridines and tridentate terpyridines, respectively. These two types of artificial double helical species may be related on one hand to the natural homoduplex nucleic acids and on the other hand to the DNA:RNA heteroduplex.

X-ray structure of a vanadium-containing enzyme: chloroperoxidase from the fungus Curvularia inaequalis.

The chloroperoxidase (EC 1.11.1.-) from the fungus Curvularia inaequalis belongs to a class of vanadium enzymes that oxidize halides in the presence of hydrogen peroxide to the corresponding hypohalous acids. The 2.1 A crystal structure (R = 20%) of an azide chloroperoxidase complex reveals the geometry of the catalytic vanadium center. Azide coordinates directly to the metal center, resulting in a structure with azide, three nonprotein oxygens, and a histidine as ligands. In the native state vanadium will be bound as hydrogen vanadate(V) in a trigonal bipyramidal coordination with the metal coordinated to three oxygens in the equatorial plane, to the OH group at one apical position, and to the epsilon 2 nitrogen of a histidine at the other apical position. The protein fold is mainly alpha-helical with two four-helix bundles as main structural motifs and an overall structure different from other structures. The helices pack together to a compact molecule, which explains the high stability of the protein. An amino acid sequence comparison with vanadium-containing bromoperoxidase from the seaweed Ascophyllum nodosum shows high similarities in the regions of the metal binding site, with all hydrogen vanadate(V) interacting residues conserved except for lysine-353, which is an asparagine.

Three-dimensional scroll waves of cAMP could direct cell movement and gene expression in Dictyostelium slugs.

Complex three-dimensional waves of excitation can explain the observed cell movement pattern in Dictyostelium slugs. Here we show that these three-dimensional waves can be produced by a realistic model for the cAMP relay system [Martiel, J. L. & Goldbeter, A. (1987) Biophys J. 52, 807-828]. The conversion of scroll waves in the prestalk zone of the slug into planar wave fronts in the prespore zone can result from a smaller fraction of relaying cells in the prespore zone. Further, we show that the cAMP concentrations to which cells in a slug are exposed over time display a simple pattern, despite the complex spatial geometry of the waves. This cAMP distribution agrees well with observed patterns of cAMP-regulated cell type-specific gene expression. The core of the spiral, which is a region of low cAMP concentration, might direct expression of stalk-specific genes during culmination.