On the design of nested iterations for elliptic difference equations

By Richardson extrapolation from 2h andh toh/2 the results of computations with coarse grids are used for the construction of an initial approximation on finer grid. After a rather small number of iterations high accuracy is obtained. Numerical results are given for the two-dimensional Laplace equation.     

A “multigrid” extension of theFFT for the numerical inversion of Fourier and Laplace Transforms

An algorithm for accurate numerical inversion of slowly convergent Fourier and Laplace Transforms is studied. It makes use of several equidistant grids with the same number of points, covering different symmetric intervals of the time and frequency axes. Typically, the number of operations per computed function value is about twice as large as for an ordinary FFT. The distribution of points is, however, for many applications much more adequate because, globally, the union of the grids is an approximately equidistant point set on a logarithmic scale.Dedicated to Gene H. Golub on the occasion of his 60'th birthday     

Slow internal dynamics in proteins: application of NMR relaxation dispersion spectroscopy to methyl groups in a cavity mutant of T4 lysozyme

Recently developed carbon transverse relaxation dispersion experiments (Skrynnikov, N. R.; et al. J. Am. Chem. Soc. 2001, 123, 4556-4566) were applied to the study of millisecond to microsecond time scale motions in a cavity mutant of T4 lysozyme (L99A) using methyl groups as probes of dynamics. Protein expressed in E. coli cells with (13)CH(3)-pyruvate as the sole carbon source contained high levels of (13)C enrichment at a total of 80 Val gamma, Leu delta, Ile gamma (2), Ala beta, and Met epsilon methyl positions with little extraneous incorporation. Data for 72 methyl groups were available for analysis. Dispersion profiles with large amplitudes were measured for many of these residues and were well fit to a two-state exchange model. The interconversion rates and populations of the states, obtained from fitting relaxation dispersion profiles of each individual probe, were remarkably homogeneous and data for nearly all methyl groups in the protein could be collectively fit to a single cooperative conformational transition. The present study demonstrates the general applicability of methyl relaxation dispersion measurements for the investigation of millisecond time scale protein motions at a large number of side-chain positions. Potential artifacts associated with the experiments are described and methods to minimize their effects presented. These experiments should be particularly well suited for probing dynamics in high molecular weight systems due to the favorable NMR spectroscopic properties of methyl groups.     

Some aspects of the model potential method

The theoretical justification of the model potential method is studied in some detail. The correct equations within the framework of Roothaan's open-shell scheme are derived and the approximations necessary to get a workable method are discussed. Analysis of the local part of the model potential suggests a new analytical form for it. The new expression is theoretically more consistent than the original one, and it can be determined in a more straightforward way. A basis set approximation, which is particularly suitable for approximate evaluation of two-electron integrals when only valence orbitals are involved, is discussed and tested with encouraging results. The ideas are tested on the Fe and I atoms.     

Probing slow time scale dynamics at methyl-containing side chains in proteins by relaxation dispersion NMR measurements: application to methionine residues in a cavity mutant of T4 lysozyme.

A relaxation dispersion-based NMR experiment is presented for the measurement and quantitation of micros-ms dynamic processes at methyl side-chain positions in proteins. The experiment measures the exchange contribution to the 13C line widths of methyl groups using a constant-time CPMG scheme. The effects of cross-correlated spin relaxation between dipole-dipole and dipole-CSA interactions as well as the effects of scalar coupling responsible for mixing of magnetization modes during the course of the experiment have been investigated in detail both theoretically and through simulations. It is shown that the complex relaxation properties of the methyl spin system do not complicate extraction of accurate exchange parameters as long as care is taken to ensure that appropriate magnetization modes are interchanged in the middle of the constant-time CPMG period. An application involving the measurement of relaxation dispersion profiles of methionine residues in a Leu99Ala substitution of T4 lysozyme is presented. All of the methionine residues are sensitive to an exchange event with a rate on the order of 1200 s(-1) at 20 degrees C that may be linked to a process in which hydrophobic ligands are able to rapidly bind to the cavity that is present in this mutant.     

A special stability problem for linear multistep methods

The trapezoidal formula has the smallest truncation error among all linear multistep methods with a certain stability property. For this method error bounds are derived which are valid under rather general conditions. In order to make sure that the error remains bounded ast , even though the product of the Lipschitz constant and the step-size is quite large, one needs not to assume much more than that the integral curve is uniformly asymptotically stable in the sense of Liapunov.The preparation of this paper was partly sponsored by the Office of Naval Research and the US Army Research Office (Durham). Reproduction in whole or in part is permitted for any purpose of the US Government.     

A randomly generated program for automatic identity checking

A method for making secret programs for automatic identity checking with very high demands for security, is described. The consistency of the sequence of digits, which identifies the person, is tested by a sequence of transformations, chosen by means of a stored secret random table. The method has been implemented for a system of unattended cash dispenser terminals, equipped with microprocessors.     

Two-parameter exponential-type basis functions for atomic calculations

It is found that ordinary STO s fall off too fast in the atomic region in many cases. A new type of basis set, which is more adaptable to the rather different requirements of the various atomic orbitals in an atom, is developed. The suggested functional form χ(r) = Nr^n-1 exp{?α[(βr + 1)^1/2 ? 1]} contains the STO s as a limiting case. Calculations on a series of atoms from H to Zn show that the new basis gives better results than STO s for equal basis set size. The necessary integrals do not present any problem to evaluate.     

Coordination behaviour of 2-guanidinobenzimidazole towards cobalt(II), nickel(II), copper(II) and zinc(II). An experimental and theoretical study

Summary The following coordination compounds derived from 2-guanidinobenzimidazole (2GB) (1); [Ni(2GB)₂]Cl₂· H₂O, (2); [Ni(2GB)₂]Br₂·3H₂O, (3); [Ni(2GB)₂-(NO₃)₂, (4); [Ni(2GB)₂](OAc)₂, (5); [Cu(2GB)Cl₂], (6); [Cu(2GB)Br₂], (7); [Cu(2GB)₂]Br₂·2H₂O, (8); [Cu(2GB)₂](NO₃)₂·H₂O, (9); [Cu(2GB)₂](OAc)₂· H₂O, (10); [Zn(2GB)Cl₂]·H₂O, (11); [Zn(2GB)Br₂]·H₂O, (12); [Co(2GB)Cl₂(H₂O)₂]·5H₂O, (13); [Co-(2GB)₂Cl₂]·3H₂O, (14); [Co(2GB)₂(H₂O)₂](NO₃)₂· 4H₂O, (15); and [Co(2GB)₂(H₂O)₂](OAc)₂, (16) have been synthesized and characterized by i.r. and electronic spectroscopy. In addition (6)–(10) were analysed by e.p.r. The X-ray diffraction structure of compound (4) was obtained. It crystallizes in the monoclinic system, C2/c (a = 22.511(7), b = 6.735(6) and c= 15.345(5)Å, =115.31(3)°, Z = 4, final R = 0.0360 and R _w = 0.0388 for 1167 observed independent reflections). The nickel(II) atom coordinates two ligands in a square-planar geometry through the imidazolic N(3) and the guanidino N(12).The probable ligand isomers involved in the coordination were determined by theoretical calculations, and the possible structures of the coordination compounds were investigated in order to verify that the experimentally proposed structures were stable. Two different types of coordination compounds were found. One, where the ligand is chelating through the imidazolic N(3) and the guanidino N(12), which is the case for most of the complexes [(2)–(13)]. With only one ligand in the coordination sphere, the structure was either tetrahedral (copper and zinc chloride and bromide complexes) or octahedral (cobalt). With two chelating 2GB units a square-planar geometry was stabilized [(2)–(5) and (8)–(10)]. The second type of coordination behaviour was observed in the cobalt compounds [(14)–(16)]. Here the ligand coordinates monodentate through the imidazolic N(3); the structure is tetrahedral.