A β-amino acid modified heptapeptide containing a designed recognition element disrupts fibrillization of the amyloid β-peptide

We study the complex formation of a peptide betaAbetaAKLVFF, previously developed by our group, with Abeta(1–42) in aqueous solution. Circular dichroism spectroscopy is used to probe the interactions between betaAbetaAKLVFF and Abeta(1–42), and to study the secondary structure of the species in solution. Thioflavin T fluorescence spectroscopy shows that the population of fibers is higher in betaAbetaAKLVFF/Abeta(1–42) mixtures compared to pure Abeta(1–42) solutions. TEM and cryo-TEM demonstrate that co-incubation of betaAbetaAKLVFF with Abeta(1–42) causes the formation of extended dense networks of branched fibrils, very different from the straight fibrils observed for Abeta(1–42) alone. Neurotoxicity assays show that although betaAbetaAKLVFF alters the fibrillization of Abeta(1–42), it does not decrease the neurotoxicity, which suggests that toxic oligomeric Abeta(1–42) species are still present in the betaAbetaAKLVFF/Abeta(1–42) mixtures. Our results show that our designed peptide binds to Abeta(1–42) and changes the amyloid fibril morphology. This is shown to not necessarily translate into reduced toxicity.

An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included.

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the $L^2$ condition numbers for these formulations, and also on the norms of the classical acoustic single- and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number $k$, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like $k^{1/3}$ as $k\to\infty$, when the scatterer is a circle or sphere, it can grow as fast as $k^{7/5}$ for a class of `trapping' obstacles. In this paper we prove further bounds, sharpening and extending our previous results. In particular we show that there exist trapping obstacles for which the condition numbers grow as fast as $\exp(\gamma k)$, for some $\gamma>0$, as $k\to\infty$ through some sequence. This result depends on exponential localisation bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low $k$. In the second part of the paper we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper.

A high frequency boundary element method for scattering by convex polygons with impedance boundary conditions

We consider scattering of a time harmonic incident plane wave by a convex polygon with piecewise constant impedance boundary conditions. Standard finite or boundary element methods require the number of degrees of freedom to grow at least linearly with respect to the frequency of the incident wave in order to maintain accuracy. Extending earlier work by Chandler-Wilde and Langdon for the sound soft problem, we propose a novel Galerkin boundary element method, with the approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon. Theoretical analysis and numerical results suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency of the incident wave.

A moving-mesh finite element method and its application to the numerical solution of phase-change problems

A distributed Lagrangian moving-mesh finite element method is applied to problems involving changes of phase. The algorithm uses a distributed conservation principle to determine nodal mesh velocities, which are then used to move the nodes. The nodal values are obtained from an ALE (Arbitrary Lagrangian-Eulerian) equation, which represents a generalization of the original algorithm presented in Applied Numerical Mathematics, 54:450--469 (2005). Having described the details of the generalized algorithm it is validated on two test cases from the original paper and is then applied to one-phase and, for the first time, two-phase Stefan problems in one and two space dimensions, paying particular attention to the implementation of the interface boundary conditions. Results are presented to demonstrate the accuracy and the effectiveness of the method, including comparisons against analytical solutions where available.

Impact of earthworms on trace element solubility in contaminated mine soils amended with green waste compost

The common practice of remediating metal contaminated mine soils with compost can reduce metal mobility and promote revegetation, but the effect of introduced or colonising earthworms on metal solubility is largely unknown. We amended soils from an As/Cu (1150 mgAs kg−1 and 362 mgCu kg−1) and Pb/Zn mine (4550 mgPb kg−1 and 908 mgZn kg−1) with 0, 5, 10, 15 and 20% compost and then introduced Lumbricus terrestris. Porewater was sampled and soil extracted with water to determine trace element solubility, pH and soluble organic carbon. Compost reduced Cu, Pb and Zn, but increased As solubility. Earthworms decreased water soluble Cu and As but increased Pb and Zn in porewater. The effect of the earthworms decreased with increasing compost amendment. The impact of the compost and the earthworms on metal solubility is explained by their effect on pH and soluble organic carbon and the environmental chemistry of each element.

Changes in race-specific virulence in pseudomonas syringae pv. phaseolicola are Associated with a chimeric transposable element and rare deletion events in a plasmid-borne pathogenicity island

Virulence for bean and soybean is determined by effector genes in a plasmid-borne pathogenicity island (PAI) in race 7 strain 1449B of Pseudomonas syringae pv. phaseolicola. One of the effector genes, avrPphF, confers either pathogenicity, virulence, or avirulence depending on the plant host and is absent from races 2, 3, 4, 6, and 8 of this pathogen. Analysis of cosmid clones and comparison of DNA sequences showed that the absence of avrPphF from strain 1448A is due to deletion of a continuous 9.5-kb fragment. The remainder of the PAI is well conserved in strains 1448A and 1449B. The left junction of the deleted region consists of a chimeric transposable element generated from the fusion of homologs of IS1492 from Pseudomonas putida and IS1090 from Ralstonia eutropha. The borders of the deletion were conserved in 66 P. syringae pv. phaseolicola strains isolated in different countries and representing the five races lacking avrPphF. However, six strains isolated in Spain had a 10.5-kb deletion that extended 1 kb further from the right junction. The perfect conservation of the 28-nucleotide right repeat of the IS1090 homolog in the two deletion types and in the other 47 insertions of the IS1090 homolog in the 1448A genome strongly suggests that the avrPphF deletions were mediated by the activity of the chimeric mobile element. Our data strongly support a clonal origin for the races of P. syringae pv. phaseolicola lacking avrPphF.

Sequence analysis and distribution of an IS3-like insertion element isolated from Salmonella enteritidis

The nucleotide sequence of a 3 kb region immediately upstream of the sef operon operon of Salmonella enteritidis was determined. A 1230 base pair insertion sequence which shared sequence identity (> 75%) with members of the IS3 family was revealed. This element, designated IS1230, had almost identical (90% identity) terminal inverted repeats to Escherichia coli IS3 but unlike other IS3-like sequences lacked the two characteristic open reading frames which encode the putative transposase. S. enteritidis possessed only one copy of this insertion sequence although Southern hybridisation analysis of restriction digests of genomic DNA revealed another fragment located in a region different from the sef operon which hybridised weakly which suggested the presence of an IS1230 homologue. The distribution of IS1230 and IS1230-like elements was shown to be widespread amongst salmonellas and the patterns of restriction fragments which hybridised differed significantly between Salmonella serotypes and it is suggested that IS1230 has potential for development as a differential diagnostic tool.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

A high frequency $hp$ boundary element method for scattering by convex polygons

In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.

Gradient recovery in adaptive finite-element methods for parabolic problems

We derive energy-norm a posteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the �rst completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique.Our theoretical results are backed with extensive numerical experimentation aimed at (a) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (b) deriving an adaptive method based on our estimators. An extra novelty provided is an implementation of a coarsening error "preindicator", with a complete implementation guide in ALBERTA in the appendix.

A finite element method for nonlinear elliptic problems

We present a Galerkin method with piecewise polynomial continuous elements for fully nonlinear elliptic equations. A key tool is the discretization proposed in Lakkis and Pryer, 2011, allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretization method is that a recovered (finite element) Hessian is a byproduct of the solution process. We build on the linear method and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the Monge–Amp`ere equation and the Pucci equation.