We present a structural model for amyloid fibrils formed by the 40-residue beta-amyloid peptide associated with Alzheimer's disease (Abeta(1-40)), based on a set of experimental constraints from solid state NMR spectroscopy. The model additionally incorporates the cross-beta structural motif established by x-ray fiber diffraction and satisfies constraints on Abeta(1-40) fibril dimensions and mass-per-length determined from electron microscopy. Approximately the first 10 residues of Abeta(1-40) are structurally disordered in the fibrils. Residues 12-24 and 30-40 adopt beta-strand conformations and form parallel beta-sheets through intermolecular hydrogen bonding. Residues 25-29 contain a bend of the peptide backbone that brings the two beta-sheets in contact through sidechain-sidechain interactions. A single cross-beta unit is then a double-layered beta-sheet structure with a hydrophobic core and one hydrophobic face. The only charged sidechains in the core are those of D23 and K28, which form salt bridges. Fibrils with minimum mass-per-length and diameter consist of two cross-beta units with their hydrophobic faces juxtaposed.

https://www.pnas.org/content/pnas/99/26/16742.full.pdf

A structural model for Alzheimer's β-amyloid fibrils based on experimental constraints from solid state NMR

Publisher Summary This chapter focuses on finite volume methods. The finite volume method is a discretization method that is well suited for the numerical simulation of various types (for instance, elliptic, parabolic, or hyperbolic) of conservation laws; it has been extensively used in several engineering fields, such as fluid mechanics, heat and mass transfer, or petroleum engineering. Some of the important features of the finite volume method are similar to those of the finite element method: it may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to robust schemes. The finite volume method is locally conservative because it is based on a “balance" approach: a local balance is written on each discretization cell that is often called “control volume;” by the divergence formula, an integral formulation of the fluxes over the boundary of the control volume is then obtained. The fluxes on the boundary are discretized with respect to the discrete unknowns.

http://materias.fi.uba.ar/7538/material/Otros/EymardEtAl%20-%20Finite%20Volume%20Methods.pdf

Finite Volume Methods

dynamics. We have arrived at an interesting concept, seeing solutions as continuous curves moving around in an infinite-dimensional metric space X (here, the function space L1(Ω)). Viewing solutions as continuous curves in a general space is the starting point of the abstract theory of differential equations, a way that we will travel quite often. In the so-called Abstract Dynamics it is typical to forget the variable x in the notation and look at the map t 7→ u(t) ∈ X, where u(t) is the abbreviated form for u(·, t). Remarks. (1) Note that the theorem allows to define the value u(t) of a limit solution (in particular, of a weak solution) u at any time t > 0 as a well-defined element of L1(Ω). Actually, in many cases, as when Φ is superlinear and f is bounded, it is an element of L∞(Ω). (2) If u0 and f are bounded the initial regularity is better. In that case the initial data are taken in the Lp sense: ũ(t) → ũ(0) in Lp(Ω), for every p 0; if u0 is continuous, then the convergence takes place uniformly in x as t → 0, see Section 7.5.1. (3) Unfortunately, there are no equivalent L1 estimates for the Dirichlet Problem with nonhomogeneous data g 6= 0. We end this subsection with a simple but very useful consequence. Corollary 6.3 Let u be a limit solution with data u0 ∈ L1(Ω) and f ∈ L1(Q). If t1 > 0, then ũ(x, t) = u(x, t + t1) is the limit solution with data ũ0(x) = u(x, t1) and forcing term f(x, t) = f(x, t + t1). This important result is immediate for the approximations. We leave the details to the reader. Remark. Let us note that any concept of limit solution depends on the type of admissible approximations and on the functional setting in which limits are taken. The definition we propose applies in the L1 setting. If needed, these solutions will be called L1-limit solutions. For an extension see Section 6.6. 6.2 Theory of very weak solutions The continuous dependence with respect to the L1-norm is a powerful property. It has allowed us to extend the existence result for weak solutions of the preceding section and

The Porous Medium Equation

An overview is given of theory and experiments on liquid crystal phases which appear in solutions of elongated colloidal particles or stiff polymers. The Onsager (1949) virial thecry for the isotropionematic transition of thin rodlike particles is treated comprehensively along with extensions to polydisperse solutions and soft interactions. Computer simulations of liquid crystal phases in hard particle fluids are summarized and used to assess the quality of statistical mechanical thwries for stiff panicles at higher dume haion-like the inclusion of higher Virial coefficients, yexpansion, scaled particle theory and density functional theory. Both computer simulations and density functional theory indicate formation of more highly ordered smectic phases. The range of experimental applicability h strongly widened by the extension of the viriai theory to wormlike chains by Khokhlov and Semenov (1981, 1982). Fmally, experimental results for a number of carefully studied, charged and uncharged colloids and polymers are summarized and wmpared to theoretical results. IE many cases the agreement is semi-quantitative.

/pdf/phase-transitions-in-lyotropic-colloidal-and-polymer-liquid-23qsyytblc.pdf

Phase transitions in lyotropic colloidal and polymer liquid crystals

The seven-residue peptide N-acetyl-Lys-Leu-Val-Phe-Phe-Ala-Glu-NH(2), called A beta(16-22) and representing residues 16-22 of the full-length beta-amyloid peptide associated with Alzheimer's disease, is shown by electron microscopy to form highly ordered fibrils upon incubation of aqueous solutions. X-ray powder diffraction and optical birefringence measurements confirm that these are amyloid fibrils. The peptide conformation and supramolecular organization in A beta(16-22) fibrils are investigated by solid state (13)C NMR measurements. Two-dimensional magic-angle spinning (2D MAS) exchange and constant-time double-quantum-filtered dipolar recoupling (CTDQFD) measurements indicate a beta-strand conformation of the peptide backbone at the central phenylalanine. One-dimensional and two-dimensional spectra of selectively and uniformly labeled samples exhibit (13)C NMR line widths of <2 ppm, demonstrating that the peptide, including amino acid side chains, has a well-ordered conformation in the fibrils. Two-dimensional (13)C-(13)C chemical shift correlation spectroscopy permits a nearly complete assignment of backbone and side chain (13)C NMR signals and indicates that the beta-strand conformation extends across the entire hydrophobic segment from Leu17 through Ala21. (13)C multiple-quantum (MQ) NMR and (13)C/(15)N rotational echo double-resonance (REDOR) measurements indicate an antiparallel organization of beta-sheets in the A beta(16-22) fibrils. These results suggest that the degree of structural order at the molecular level in amyloid fibrils can approach that in peptide or protein crystals, suggest how the supramolecular organization of beta-sheets in amyloid fibrils can be dependent on the peptide sequence, and illustrate the utility of solid state NMR measurements as probes of the molecular structure of amyloid fibrils. A beta(16-22) is among the shortest fibril-forming fragments of full-length beta-amyloid reported to date, and hence serves as a useful model system for physical studies of amyloid fibril formation.

Amyloid Fibril Formation by Aβ16-22, a Seven-Residue Fragment of the Alzheimer's β-Amyloid Peptide, and Structural Characterization by Solid State NMR†

General integral and series expressions are derived for the intensities of sidebands observed in the magic angle spectra of inhomogeneously broadened I=1/2 systems The expressions are evaluated for a wide range of shift parameters and the results used to construct graphical and numerical methods for extracting the principal values of chemical shift tensors from the intensities of just a few sidebands The methods are illustrated by application to 31P spectra of barium diethyl phosphate The results agree well with previous single crystal measurements

Sideband intensities in NMR spectra of samples spinning at the magic angle

A salient difficulty in describing the phase behavior of rodlike particles with large axial ratios or high packing densities has been the solution of the nonlinear integral equation for the orientation distribution function. Approximations that do well for small axial ratios and low packing densities become increasingly unsatisfactory as the axial ratio or the packing density increases. In this report, we demonstrate that solutions of high accuracy may be obtained relatively efficiently by an iterative procedure that converges particularly rapidly for systems with high packing densities and large axial ratios. We show that, for polydisperse as well as monodisperse systems, each iteration moves in a direction of locally decreasing free energy. Although it does not necessarily follow that the net free energy change in a discrete step will be negative, all cases examined thus far generate successive distribution functions with monotonically decreasing free energy.

A highly convergent algorithm for computing the orientation distribution functions of rodlike particles

An approach to the investigation of molecular structures in disordered solids, using two‐dimensional (2D) nuclear magnetic resonance (NMR) exchange spectroscopy with magic angle spinning (MAS), is described. This approach permits the determination of the relative orientation of two isotopically labeled chemical groups within a molecule in an unoriented sample, thus placing strong constraints on the molecular conformation. Structural information is contained in the amplitudes of crosspeaks in rotor‐synchronized 2D MAS exchange spectra that connect spinning sideband lines of the two labeled sites. The theory for calculating the amplitudes of spinning sideband crosspeaks in 2D MAS exchange spectra, in the limit of complete magnetization exchange between the labeled sites, is presented in detail. A new technique that enhances the sensitivity of 2D MAS exchange spectra to molecular structure, called orientationally weighted 2D MAS exchange spectroscopy, is introduced. Symmetry principles that underlie the cons...

Investigation of molecular structure in solids by two‐dimensional NMR exchange spectroscopy with magic angle spinning

It will be proven that an exponential tridiagonal difference scheme, when applied with a uniform mesh of size h to: euXx + b(x)ux = f(x) for O O, b and f smooth, e in (0, 1], and u(O) and u(I) given, is uniformly second-order accurate (i.e., the maximum of the errors at the grid points is bounded by Ch2 with the constant C independent of h and e). This scheme was derived by El-Mistikawy and Werle by a C' patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point. The behavior of the approximate solution in between the grid points will be analyzed, and some numerical results will also be given.

/pdf/an-analysis-of-a-uniformly-accurate-difference-method-for-a-4g6wo1i86f.pdf

An analysis of a uniformly accurate difference method for a singular perturbation problem

Bounds are obtained for the derivatives of the solution of a turning point problem. These results suggest a modification of the El-Mistikawy Werle finite difference scheme at the turning point. A uniform error estimate is obtained for the resulting method, and illustrative numerical results are given. I. Introduction. We will examine the following two-point boundary value problem with Dirichlet data at the endpoints: (l.la) Ly= -eyx.(x) -p(x)yx(x) +q(x)y(x) =f(x) for -I <x< 1,

/pdf/a-priori-estimates-and-analysis-of-a-numerical-method-for-a-2cbvzoha8v.pdf

Alan E. Berger

Papers

Sideband intensities in NMR spectra of samples spinning at the magic angle

A highly convergent algorithm for computing the orientation distribution functions of rodlike particles

Investigation of molecular structure in solids by two‐dimensional NMR exchange spectroscopy with magic angle spinning

An analysis of a uniformly accurate difference method for a singular perturbation problem

A priori estimates and analysis of a numerical method for a turning point problem