Symmetric wiener-hopf and Toeplitz operators

We explain the relationship between the principal function of A C^* algebra generated by a pair of unitary operators with commutator of one dimensional range and the deficiency spaces of symmetric Toeplitz operators defined by real unbounded symbols.     

Molecular scale imaging of F-actin assemblies immobilized on a photopolymer surface

A photo-immobilization based process is presented for direct imaging of hierarchical assemblies of biopolymers using atomic force microscopy (AFM). The technique was used to investigate the phase behavior of F-actin aggregates as a function of concentration of the divalent cation Mg2+. The data provided direct experimental evidence of a coil-on-coil (braided) structure of F-actin bundles formed at high Mg2+ concentrations. At intermediate Mg2+ concentrations, the data showed the first images of the two-dimensional nematic rafts discovered by recent x-ray studies and theoretical treatments.     

A united-residue force field for off-lattice protein-structure simulations. I. Functional forms and parameters of long-range side-chain interaction potentials from protein crystal data

A two-stage procedure for the determination of a united-residue potential designed for protein simulations is outlined. In the first stage, the long-range and local-interaction energy terms of the total energy of a polypeptide chain are determined by analyzing protein-crystal data and averaging the all-atom energy surfaces. In the second stage (described in the accompanying article), the relative weights of the energy terms are optimized so as to locate the native structures of selected test proteins as the lowest energy structures. The goal of the work in the present study is to parameterize physically reasonable functional forms of the potentials of mean force for side-chain interactions. The potentials are of both radial and anisotropic type. Radial potentials include the Lennard-Jones and the shifted Lennard-Jones potential (with the shift parameter independent of orientation). To treat the angular dependence of side-chain interactions, three functional forms of the potential that were designed previously to describe anisotropic systems are evaluated: Berne-Pechukas (dilated Lennard-Jones); Gay-Berne (shifted Lennard-Jones with orientation-dependent shift parameters); and Gay-Berne-Vorobjev (the same as the preceding one, but with one more set of variable parameters). These functional forms were used to parameterize, within a short-distance range, the potentials of mean force for side-chain pair interactions that are related by the Boltzmann principle to the pair correlation functions determined from protein-crystal data. Parameter determination was formulated as a generalized nonlinear least-squares problem with the target function being the weighted sum of squares of the differences between calculated and “experimental” (i.e., estimated from protein-crystal data) angular, radial-angular, and radial pair correlation functions, as well as contact free energies. A set of 195 high-resolution nonhomologous structures from the Protein Data Bank was used to calculate the “experimental” values. The contact free energies were scaled by the slope of the correlation line between side-chain hydrophobicities, calculated from the contact free energies, and those determined by Fauchere and Pliška from the partition coefficients of amino acids between water and n-octanol. The methylene group served to define the reference contact free energy corresponding to that between the glycine methylene groups of backbone residues. Statistical analysis of the goodness of fit revealed that the Gay-Berne-Vorobjev anisotropic potential fits best to the experimental radial and angular correlation functions and contact free energies and therefore represents the free-energy surface of side-chain-side-chain interactions most accurately. Thus, its choice for simulations of protein structure is probably the most appropriate. However, the use of simpler functional forms is recommended, if the speed of computations is an issue. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 849–873, 1997     

A remark on the spectral multiplicity of normal extensions of commuting subnormal operator tuples

The authors gratefully acknowledge the support of the National Science Foundation.     

Principal currents

An operator theoretic analog of a local Chern class is discussed for certain finitely generated operator algebras. This gives rise to an index theory for the maximal ideals of the algebra. The algebraic receiver for this index is the Grothendieck group of the algebra completed at a maximal ideal.This research was partially supported by grants from the National Science Foundation.     

Cardinal representatives

In the absence of the axiom of choice it is sometimes, but not always, possible to define the notion of cardinal number such that for anyx, x ≈ |x|.