On Dynamics Related to a Class of Numeration Systems

 In this paper, we investigate the class of numeration systems and we study the associated dynamical systems, called odometers. It is shown that these odometers are measure-theoretically isomorphic to rank one transformations on the unit interval, constructed by a cutting-stacking method. Furthermore, a symbolic coding leads to isomorphic shift systems arising from substitutions. Some skew products of the odometers by cocycles related to the sum of digits are shown to be ergodic.     

A procedure to calculate torsion¶of elliptic curves over ℚ

We present an algorithm which uses the analytic parameterization of elliptic curves to rapidly calculate torsion subgroups, and calculate its running time. This algorithm is much faster than the “traditional” Lutz–Nagell algorithm used by most computer algebra systems to calculate torsion subgroups. Received: 7 August 1997 / Revised version: 28 November 1997     

Single-molecule magnet behavior with a single metal center enhanced through peripheral ligand modifications

Bis(imino)pyridine pincer ligands in conjunction with two isothiocyanate ligands have been used to prepare two mononuclear Co(II) complexes. Both complexes have a distorted square-pyramidal geometry with the Co(II) centers lying above the basal plane. This leads to significant spin-orbit coupling for the d(7) Co(II) ions and consequently to slow relaxation of the magnetization that is characteristic of Single-Molecule Magnet (SMM) behavior.     

Dependence of Self-Assembled Peptide Hydrogel Network Structure on Local Fibril Nanostructure

Physically cross-linked, fibrillar hydrogel networks are formed by the self-assembly of β-hairpin peptide molecules with varying degrees of strand asymmetry. The peptide registry in the self-assembled state can be used as a design element to generate fibrils with twisting, nontwisting, or laminated morphology. The mass density of the networks varies significantly, and can be directly related to the local fibrillar morphology as evidenced by small angle neutron scattering (SANS) and in situ substantiation using cryogenic transmission electron microscopy (cryo-TEM) under identical concentrations and conditions. Similarly, the density of the network is dependent on changes in the peptide concentration. Bulk rheological properties of the hydrogels can be correlated to the fibrillar nanostructure, with the stiffer, laminated fibrils forming networks with a higher G' as compared to the flexible, singular fibrillar networks.     

CHARMM force field parameters for simulation of reactive intermediates in native and thio-substituted ribozymes

Force field parameters specifically optimized for residues important in the study of RNA catalysis are derived from density-functional calculations, in a fashion consistent with the CHARMM27 all-atom empirical force field. Parameters are presented for residues that model reactive RNA intermediates and transition state analogs, thio-substituted phosphates and phosphoranes, and bound Mg(2+) and di-metal bridge complexes. Target data was generated via density-functional calculations at the B3LYP/6-311++G(3df,2p)// B3LYP/6-31++G(d,p) level. Partial atomic charges were initially derived from CHelpG electrostatic potential fitting and subsequently adjusted to be consistent with the CHARMM27 charges. Lennard-Jones parameters were determined to reproduce interaction energies with water molecules. Bond, angle, and torsion parameters were derived from the density-functional calculations and renormalized to maintain compatibility with the existing CHARMM27 parameters for standard residues. The extension of the CHARMM27 force field parameters for the nonstandard biological residues presented here will have considerable use in simulations of ribozymes, including the study of freeze-trapped catalytic intermediates, metal ion binding and occupation, and thio effects.     

Morphologies of microphase-separated conformationally asymmetric diblock copolymers

The equilibrium morphological behavior of a series of conformationally asymmetric linear diblock copolymers is characterized via small-angle X-ray scattering (SAXS) and transmission electron microscopy (TEM). The linear diblock molecules of polyisoprene and poly(t-butylmethacrylate) (PtBMA) are prepared anionically over a range of PtBMA volume fractions 0.17 to 0.85. Solution light-scattering experiments are performed on PtBMA homopolymer at theta conditions, and the results were compared with PI data in the literature in order to characterize the degree of conformational asymmetry between the respective blocks. This conformational asymmetry is quantified by an ε of 0.75. The experimental results are compared with morphological behavior calculated utilizing self-consistent mean field theory for a diblock system with ε = 0.75. At middle to high PtBMA volume fractions, ϕ_PtBMA > 0.30, the experimental morphological behavior agrees well with the calculated behavior; the microphase boundaries are slightly shifted to higher volume fractions of the PtBMA block due to its larger Kuhn length. At ϕ_PtBMA < 0.30, however, discrepancies are found in the volume fraction dependence of experimentally determined morphological behavior and that calculated theoretically. Interestingly, extremely well-ordered cylindrical microstructures were observed for PI cylinder domains embedded in PtBMA matrices; these samples were prepared by solvent casting with no treatment, such as shearing, to enhance long-range order. These well-ordered cylinder structures contrast with PtBMA cylinders in a PI matrix on the opposite side of the phase diagram that have very poor long-range order. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 2629–2643, 1997     

Benchmark calculations of proton affinities and gas-phase basicities of molecules important in the study of biological phosphoryl transfer

Benchmark calculations of proton affinities and gas-phase basicities of molecules most relevant to biological phosphoryl transfer reactions are presented and compared with available experimental results. The accuracy of proton affinity and gas-phase basicity results obtained from several multi-level model chemistries (CBS-QB3, G3B3, and G3MP2B3) and density-functional quantum models (PBE0, B1B95, and B3LYP) are assessed and compared. From these data, a set of empirical bond enthalpy, entropy, and free energy corrections are introduced that considerably improve the accuracy and predictive capability of the methods. These corrections are applied to the prediction of proton affinity and gas-phase basicity values of important biological phosphates and phosphoranes for which experimental data does not currently exist. Comparison is made with results from semiempirical quantum models that are commonly employed in hybrid quantum mechanical/molecular mechanical simulations. Data suggest that the design of improved semiempirical quantum models with increased accuracy for relative proton affinity values is necessary to obtain quantitative accuracy for phosphoryl transfer reactions in solution, enzymes, and ribozymes.     

Unique toroidal morphology from composition and sequence control of triblock copolymers

The mechanism by which the unique toroidal supramolecular assemblies were formed for triblock copolymers of acrylic acid (AA), methyl acrylate (MA), and styrene (S), PAA99-b-PMA73-b-PS66, was probed in this study by investigating the influences of the block copolymer compositions and sequences. Two triblock copolymers, PAA99-b-PMA73-b-PS66 and PAA99-b-PS76-b-PMA62, and two diblock copolymers, PAA99-b-PMA155 and PAA99-b-PS133, were studied under experimental solution-state conditions that involved a range of solvent/nonsolvent (tetrahydrofuran/water) compositions, each in the presence of 2,2'-(ethylenedioxy)bis(ethylamine). The resulting morphologies were determined by transmission electron microscopy. The failures to afford toroidal supramolecular assemblies from both diblock copolymers having comparable lengths of the total hydrophobic chain segment, either entirely PMA or entirely PS, and from the triblock copolymer having a reversed connection sequence for the hydrophobic (PMA and PS) segments demonstrate the unique self-assembly behaviors of triblock copolymers and the importance of the block copolymer sequence.     

Pseudorotation barriers of biological oxyphosphoranes: a challenge for simulations of ribozyme catalysis

Pseudorotation reactions of biologically relevant oxyphosphoranes were studied by using density functional and continuum solvation methods. A series of 16 pseudorotation reactions involving acyclic and cyclic oxyphosphoranes in neutral and monoanionic (singly deprotonated) forms were studied, in addition to pseudorotation of PF5. The effect of solvent was treated by using three different solvation models for comparison. The barriers to pseudorotation ranged from 1.5 to 8.1 kcal mol(-1) and were influenced systematically by charge state, apicophilicity of ligands, intramolecular hydrogen bonding, cyclic structure and solvation. Barriers to pseudorotation for monoanionic phosphoranes occur with the anionic oxo ligand as the pivotal atom, and are generally lower than for neutral phosphoranes. The OCH3 groups were observed to be more apicophilic than OH groups, and hence pseudorotations that involve axial OCH3/equatorial OH exchange had higher reaction and activation free energy values. Solvent generally lowered barriers relative to the gas-phase reactions. These results, together with isotope 18O exchange experiments, support the assertion that dianionic phosphoranes are not sufficiently long-lived to undergo pseudorotation. Comparison of the density functional results with those from several semiempirical quantum models highlight a challenge for new-generation hybrid quantum mechanical/molecular mechanical potentials for non-enzymatic and enzymatic phosphoryl transfer reactions: the reliable modeling of pseudorotation processes.