Truncation error estimation in the Discontinuous Galerkin Spectral Element Method

En esta tesis, el método de estimación de error de truncación conocido como restimation ha sido extendido de esquemas de bajo orden a esquemas de alto orden. La mayoría de los trabajos en la bibliografía utilizan soluciones convergidas en mallas de distinto refinamiento para realizar la estimación. En este trabajo se utiliza una solución en una única malla con distintos órdenes polinómicos. Además, no se requiere que esta solución esté completamente convergida, resultando en el método conocido como quasi-a priori T-estimation. La aproximación quasi-a priori estima el error mientras el residuo del método iterativo no es despreciable. En este trabajo se demuestra que algunas de las hipótesis fundamentales sobre el comportamiento del error, establecidas para métodos de bajo orden, dejan de ser válidas en esquemas de alto orden, haciendo necesaria una revisión completa del comportamiento del error antes de redefinir el algoritmo. Para facilitar esta tarea, en una primera etapa se considera el método conocido como Chebyshev Collocation, limitando la aplicación a geometrías simples. La extensión al método Discontinuouos Galerkin Spectral Element Method presenta dificultades adicionales para la definición precisa y la estimación del error, debidos a la formulación débil, la discretización multidominio y la formulación discontinua. En primer lugar, el análisis se enfoca en leyes de conservación escalares para examinar la precisión de la estimación del error de truncación. Después, la validez del análisis se demuestra para las ecuaciones incompresibles y compresibles de Euler y Navier Stokes. El método de aproximación quasi-a priori r-estimation permite desacoplar las contribuciones superficiales y volumétricas del error de truncación, proveyendo información sobre la anisotropía de las soluciones así como su ratio de convergencia con el orden polinómico. Se demuestra que esta aproximación quasi-a priori produce estimaciones del error de truncación con precisión espectral. ABSTRACT In this thesis, the τ-estimation method to estimate the truncation error is extended from low order to spectral methods. While most works in the literature rely on fully time-converged solutions on grids with different spacing to perform the estimation, only one grid with different polynomial orders is used in this work. Furthermore, a non timeconverged solution is used resulting in the quasi-a priori τ-estimation method. The quasi-a priori approach estimates the error when the residual of the time-iterative method is not negligible. It is shown in this work that some of the fundamental assumptions about error tendency, well established for low order methods, are no longer valid in high order schemes, making necessary a complete revision of the error behavior before redefining the algorithm. To facilitate this task, the Chebyshev Collocation Method is considered as a first step, limiting their application to simple geometries. The extension to the Discontinuous Galerkin Spectral Element Method introduces additional features to the accurate definition and estimation of the error due to the weak formulation, multidomain discretization and the discontinuous formulation. First, the analysis focuses on scalar conservation laws to examine the accuracy of the estimation of the truncation error. Then, the validity of the analysis is shown for the incompressible and compressible Euler and Navier Stokes equations. The developed quasi-a priori τ-estimation method permits one to decouple the interfacial and the interior contributions of the truncation error in the Discontinuous Galerkin Spectral Element Method, and provides information about the anisotropy of the solution, as well as its rate of convergence in polynomial order. It is demonstrated here that this quasi-a priori approach yields a spectrally accurate estimate of the truncation error.

Type-I intermittency with discontinuous reinjection probability density in a truncation model of the derivative nonlinear Schrödinger equation

In previous papers, the type-I intermittent phenomenon with continuous reinjection probability density (RPD) has been extensively studied. However, in this paper type-I intermittency considering discontinuous RPD function in one-dimensional maps is analyzed. To carry out the present study the analytic approximation presented by del Río and Elaskar (Int. J. Bifurc. Chaos 20:1185-1191, 2010) and Elaskar et al. (Physica A. 390:2759-2768, 2011) is extended to consider discontinuous RPD functions. The results of this analysis show that the characteristic relation only depends on the position of the lower bound of reinjection (LBR), therefore for the LBR below the tangent point the relation {Mathematical expression}, where {Mathematical expression} is the control parameter, remains robust regardless the form of the RPD, although the average of the laminar phases {Mathematical expression} can change. Finally, the study of discontinuous RPD for type-I intermittency which occurs in a three-wave truncation model for the derivative nonlinear Schrodinger equation is presented. In all tests the theoretical results properly verify the numerical data

Adaptation Strategies for Discontinuous Galerkin Spectral Element Methods by means of Truncation Error Estimations

In this work a p-adaptation (modification of the polynomial order) strategy based on the minimization of the truncation error is developed for high order discontinuous Galerkin methods. The truncation error is approximated by means of a truncation error estimation procedure and enables the identification of mesh regions that require adaptation. Three truncation error estimation approaches are developed and termed a posteriori, quasi-a priori and quasi-a priori corrected. Fine solutions, which are obtained by enriching the polynomial order, are required to solve the numerical problem with adequate accuracy. For the three truncation error estimation methods the former needs time converged solutions, while the last two rely on non-converged solutions, which lead to faster computations. Based on these truncation error estimation methods, algorithms for mesh adaptation were designed and tested. Firstly, an isotropic adaptation approach is presented, which leads to equally distributed polynomial orders in different coordinate directions. This first implementation is improved by incorporating a method to extrapolate the truncation error. This results in a significant reduction of computational cost. Secondly, the employed high order method permits the spatial decoupling of the estimated errors and enables anisotropic p-adaptation. The incorporation of anisotropic features leads to meshes with different polynomial orders in the different coordinate directions such that flow-features related to the geometry are resolved in a better manner. These adaptations result in a significant reduction of degrees of freedom and computational cost, while the amount of improvement depends on the test-case. Finally, this anisotropic approach is extended by using error extrapolation which leads to an even higher reduction in computational cost. These strategies are verified and compared in terms of accuracy and computational cost for the Euler and the compressible Navier-Stokes equations. The main result is that the two quasi-a priori methods achieve a significant reduction in computational cost when compared to a uniform polynomial enrichment. Namely, for a viscous boundary layer flow, we obtain a speedup of a factor of 6.6 and 7.6 for the quasi-a priori and quasi-a priori corrected approaches, respectively. RESUMEN En este trabajo se ha desarrollado una estrategia de adaptación-p (modificación del orden polinómico) para métodos Galerkin discontinuo de alto orden basada en la minimización del error de truncación. El error de truncación se estima utilizando el método tau-estimation. El estimador permite la identificación de zonas de la malla que requieren adaptación. Se distinguen tres técnicas de estimación: a posteriori, quasi a priori y quasi a priori con correción. Todas las estrategias requieren una solución obtenida en una malla fina, la cual es obtenida aumentando de manera uniforme el orden polinómico. Sin embargo, mientras que el primero requiere que esta solución esté convergida temporalmente, el resto utiliza soluciones no convergidas, lo que se traduce en un menor coste computacional. En este trabajo se han diseñado y probado algoritmos de adaptación de malla basados en métodos tau-estimation. En primer lugar, se presenta un algoritmo de adaptacin isótropo, que conduce a discretizaciones con el mismo orden polinómico en todas las direcciones espaciales. Esta primera implementación se mejora incluyendo un método para extrapolar el error de truncación. Esto resulta en una reducción significativa del coste computacional. En segundo lugar, el método de alto orden permite el desacoplamiento espacial de los errores estimados, permitiendo la adaptación anisotropica. Las mallas obtenidas mediante esta técnica tienen distintos órdenes polinómicos en cada una de las direcciones espaciales. La malla final tiene una distribución óptima de órdenes polinómicos, los cuales guardan relación con las características del flujo que, a su vez, depenen de la geometría. Estas técnicas de adaptación reducen de manera significativa los grados de libertad y el coste computacional. Por último, esta aproximación anisotropica se extiende usando extrapolación del error de truncación, lo que conlleva un coste computational aún menor. Las estrategias se verifican y se comparan en téminors de precisión y coste computacional utilizando las ecuaciones de Euler y Navier Stokes. Los dos métodos quasi a priori consiguen una reducción significativa del coste computacional en comparación con aumento uniforme del orden polinómico. En concreto, para una capa límite viscosa, obtenemos una mejora en tiempo de computación de 6.6 y 7.6 respectivamente, para las aproximaciones quasi-a priori y quasi-a priori con corrección.

Expression, stability, and membrane integration of truncation mutants of bovine rhodopsin

Premature termination of protein synthesis by nonsense mutations is at the molecular origin of a number of inherited disorders in the family of G protein-coupled seven-helix receptor proteins. To understand how such truncated polypeptides are processed by the cell, we have carried out COS-1 cell expression studies of mutants of bovine rhodopsin truncated at the first 1, 1.5, 2, 3, or 5 transmembrane segments (TMS) of the seven present in wild-type opsin. Our experiments show that successful completion of different stages in the cellular processing of the protein [membrane insertion, N-linked glycosylation, stability to proteolytic degradation, and transport from the endoplasmic reticulum (ER) membrane] requires progressively longer lengths of the polypeptide chain. Thus, none of the truncations affected the ability of the polypeptides to be integral membrane proteins. C-terminal truncations that generated polypeptides with fewer than two TMS resulted in misorientation and prevented glycosylation at the N terminus, whereas truncations that generated polypeptides with fewer than five TMS greatly destabilized the protein. However, all of the truncations prevented exit of the polypeptide from the ER. We conclude that during the biogenesis of rhodopsin, proper integration into the ER membrane occurs only after the synthesis of at least two TMS is completed. Synthesis of the next three TMS confers a gradual increase in stability, whereas the presence of more than five TMS is necessary for exit from the ER.

Functional separation of pre-rRNA processing steps revealed by truncation of the U3 small nucleolar ribonucleoprotein component, Mpp10

The U3 small nucleolar ribonucleoprotein (snoRNP) is required for three cleavage events that generate the mature 18S rRNA from the pre-rRNA. In Saccharomyces cerevisiae, depletion of Mpp10, a U3 snoRNP-specific protein, halts 18S rRNA production and impairs cleavage at the three U3 snoRNP-dependent sites: A0, A1, and A2. We have identified truncation mutations of Mpp10 that affect 18S rRNA synthesis and confer cold-sensitivity and slow growth. However, distinct from yeast cells depleted of Mpp10, the mutants carrying these truncated Mpp10 proteins accumulate a novel precursor, resulting from cleavage at only A0. The Mpp10 truncations do not alter association of Mpp10 with the U3 snoRNA, nor do they affect snoRNA or protein stability. Thus, the role in processing of the U3 snoRNP can be separated into cleavage at the A0 site, which occurs in the presence of truncated Mpp10, and cleavage at the A1/A2 sites, which occurs only with intact Mpp10. These results strongly argue for a role for Mpp10 in processing at the A1/A2 sites.

Rapid generation of incremental truncation libraries for protein engineering using α-phosphothioate nucleotides

Incremental truncation for the creation of hybrid enzymes (ITCHY) is a novel tool for the generation of combinatorial libraries of hybrid proteins independent of DNA sequence homology. We herein report a fundamentally different methodology for creating incremental truncation libraries using nucleotide triphosphate analogs. Central to the method is the polymerase catalyzed, low frequency, random incorporation of α-phosphothioate dNTPs into the region of DNA targeted for truncation. The resulting phosphothioate internucleotide linkages are resistant to 3′→5′ exonuclease hydrolysis, rendering the target DNA resistant to degradation in a subsequent exonuclease III treatment. From an experimental perspective the protocol reported here to create incremental truncation libraries is simpler and less time consuming than previous approaches by combining the two gene fragments in a single vector and eliminating additional purification steps. As proof of principle, an incremental truncation library of fusions between the N-terminal fragment of Escherichia coli glycinamide ribonucleotide formyltransferase (PurN) and the C-terminal fragment of human glycinamide ribonucleotide formyltransferase (hGART) was prepared and successfully tested for functional hybrids in an auxotrophic E.coli host strain. Multiple active hybrid enzymes were identified, including ones fused in regions of low sequence homology.

Construction of neocentromere-based human minichromosomes by telomere-associated chromosomal truncation

Neocentromeres (NCs) are fully functional centromeres that arise ectopically in noncentromeric regions lacking α-satellite DNA. Using telomere-associated chromosome truncation, we have produced a series of minichromosomes (MiCs) from a mardel(10) marker chromosome containing a previously characterized human NC. These MiCs range in size from ≈0.7 to 1.8 Mb and contain single-copy intact genomic DNA from the 10q25 region. Two of these NC-based Mi-Cs (NC-MiCs) appear circular whereas one is linear. All demonstrate stability in both structure and mitotic transmission in the absence of drug selection. Presence of a functional NC is shown by binding a host of key centromere-associated proteins. These NC-MiCs provide direct evidence for mitotic segregation function of the NC DNA and represent examples of stable mammalian MiCs lacking centromeric repeats.

Truncation of the class II beta-chain cytoplasmic domain influences the level of class II/invariant chain-derived peptide complexes.

Previous studies have established that antigen presenting cells (APC) expressing major histocompatibility complex class II beta chains with truncated cytoplasmic domains are impaired in their capacity to activate T cells. While it had been widely accepted that this impairment is due to a defect in class II cytoplasmic domain-dependent signal transduction, we recently generated transgenic mice expressing only truncated class II beta chains, and functional analyses of APC from these mice revealed signaling-independent defects in antigen presentation. Here, we demonstrate that T cells primed on such transgenic APC respond better to stimulation by APC expressing truncated beta chains than by wild-type APC. This finding suggests that APC expressing truncated class II beta chains are not inherently defective in their antigen presenting capacity but, rather, may differ from wild-type APC in the peptide antigens that they present. Indeed, analysis of the peptides bound to class II molecules isolated from normal and transgenic spleen cells revealed clear differences. Most notably, the level of class II-associated invariant chain-derived peptides (CLIP) is significantly reduced in cells expressing only truncated beta chains. Prior studies have established that CLIP and antigenic peptides compete for binding to class II molecules. Thus, our results suggest that the cytoplasmic domain of the class II beta chain affects antigen presentation by influencing the level of CLIP/class II complexes.

Motzkin decomposition of closed convex sets via truncation

A nonempty set F is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set C with a closed convex cone D. In that case, the sets C and D are called compact and conic components of F. This paper provides new characterizations of the Motzkin decomposable sets involving truncations of F (i.e., intersections of FF with closed halfspaces), when F contains no lines, and truncations of the intersection F̂ of F with the orthogonal complement of the lineality of F, otherwise. In particular, it is shown that a nonempty closed convex set F is Motzkin decomposable if and only if there exists a hyperplane H parallel to the lineality of F such that one of the truncations of F̂ induced by H is compact whereas the other one is a union of closed halflines emanating from H. Thus, any Motzkin decomposable set F can be expressed as F=C+D, where the compact component C is a truncation of F̂. These Motzkin decompositions are said to be of type T when F contains no lines, i.e., when C is a truncation of F. The minimality of this type of decompositions is also discussed.

Dynamics of motor nerve terminal remodeling unveiled using SNARE-cleaving botulinum toxins: the extent and duration are dictated by the sites of SNAP-25 truncation

Nerve sprouts emerge from motor nerve terminals following blockade of exo-endocytosis for more than 3 days by botulinum neurotoxin (BoNT), and form functional synapses, albeit temporary. Upon restoration of synaptic activity to the parent terminal 7 and 90 days after exposure to BoNT/F or A respectively, a concomitant retraction of the outgrowths was observed. BoNT/E caused short-term neuroparalysis, and dramatically accelerated the recovery of BoNT/A-paralyzed muscle by further truncation of SNAP-25 and its replenishment with functional full-length SNARE. The removal of 9 C-terminal residues from SNAP-25 by BoNT/A leads to persistence of the inhibitory product due to the formation of a nonproductive SNARE complex(es) at release sites, whereas deletion of a further 17 amino acids permits replenishment and a speedy recovery. (C) 2003 Elsevier Science (USA). All rights reserved.

RAD52 Y415X truncation polymorphism and epithelial ovarian cancer risk in Australian women

The RAD52 gene is involved in the homologous recombination repair pathway and is a plausible candidate ovarian cancer predisposition gene. We undertook a case-control comparison of 508 epithelial ovarian cancer cases (91 low malignant potential and 417 invasive) and 298 healthy controls to assess the RAD52 Y415X polymorphism as a risk factor for epithelial ovarian cancer in Australian women. Heterozygote frequencies of 2.6 and 4% were observed among cases and controls, respectively. The risk estimate was 0.55 (95%CI 0.24-1.24), suggesting that the RAD52 Y415X polymorphism is not associated with epithelial ovarian cancer in Australian women. (C) 2004 Elsevier Ireland Ltd. All rights reserved.

Truncation identities for the small polaron fusion hierarchy

We study a one-dimensional lattice model of interacting spinless fermions. This model is integrable for both periodic and open boundary conditions; the latter case includes the presence of Grassmann valued non-diagonal boundary fields breaking the bulk U(1) symmetry of the model. Starting from the embedding of this model into a graded Yang-Baxter algebra, an infinite hierarchy of commuting transfer matrices is constructed by means of a fusion procedure. For certain values of the coupling constant related to anisotropies of the underlying vertex model taken at roots of unity, this hierarchy is shown to truncate giving a finite set of functional equations for the spectrum of the transfer matrices. For generic coupling constants, the spectral problem is formulated in terms of a functional (or TQ-)equation which can be solved by Bethe ansatz methods for periodic and diagonal open boundary conditions. Possible approaches for the solution of the model with generic non-diagonal boundary fields are discussed.

A truncation in the Aryl Hydrocarbon Receptor of the CRL:WI(Han) rat does not affect the developmental toxicity of TCDD

The Aryl Hydrocarbon Receptor (AhR) is required for the toxicity of TCDD, and so the AhR of CRL:WI and CRL:WI(Han) rats was characterised. Western blot showed AhR proteins of ~110 and ~97 kDa in individual rats from both strains. The AhR cDNA from a CRL:WI(Han) rat with the ~110kDa protein revealed a sequence that was identical to that of the CRL:WI and SD rat. However, cloning of the AhR from a rat with the ~97kDa protein revealed a point mutation, and five variants encoding two C-terminally truncated variants of the AhR protein, arising from a point mutation in the intron/exon junction and consequent differential splicing. These C-terminally truncated variants were expressed and shown to give rise to a protein of ~97kDa; the recombinant AhR bound TCDD with an affinity that was not statistically different from the full-length protein. A single-nucleotide polymorphism (SNP) assay was developed, and showed that both alleles were represented in a Hardy-Weinberg equilibrium in samples of CRL:WI and CRL:WI(Han) populations; both alleles are abundant. Rats from two studies of TCDD developmental toxicity were genotyped, and the association with toxicity investigated using statistical analysis. There was no plausible evidence that the AhR allele had a significant effect on the toxic endpoints examined. These data show that the two AhR alleles are common in two strains of Wistar rat, and that the AhR alleles had no effect on TCDD-induced developmental toxicity in two independent studies.

Quasi-a priori truncation error estimation in the DGSEM

In this paper we show how to accurately perform a quasi-a priori estimation of the truncation error of steady-state solutions computed by a discontinuous Galerkin spectral element method. We estimate the spatial truncation error using the ?-estimation procedure. While most works in the literature rely on fully time-converged solutions on grids with different spacing to perform the estimation, we use non time-converged solutions on one grid with different polynomial orders. The quasi-a priori approach estimates the error while the residual of the time-iterative method is not negligible. Furthermore, the method permits one to decouple the surface and the volume contributions of the truncation error, and provides information about the anisotropy of the solution as well as its rate of convergence in polynomial order. First, we focus on the analysis of one dimensional scalar conservation laws to examine the accuracy of the estimate. Then, we extend the analysis to two dimensional problems. We demonstrate that this quasi-a priori approach yields a spectrally accurate estimate of the truncation error.

Morphologic and genomic characterisation of the koala Chlamydia pneumoniae strain

Chlamydia pneumoniae is a common human and animal pathogen associated with a wide range of upper and lower respiratory tract infections. In more recent years there has been increasing evidence to suggest a link between C. pneumoniae and chronic diseases in humans, including atherosclerosis, stroke and Alzheimer’s disease. C. pneumoniae human strains show little genetic variation, indicating that the human-derived strain originated from a common ancestor in the recent past. Despite extensive information on the genetics and morphology processes of the human strain, knowledge concerning many other hosts (including marsupials, amphibians, reptiles and equines) remains virtually unexplored. The koala (Phascolarctos cinereus) is a native Australian marsupial under threat due to habitat loss, predation and disease. Koalas are very susceptible to chlamydial infections, most commonly affecting the conjunctiva, urogenital tract and/or respiratory tract. To address this gap in the literature, the present study (i) provides a detailed description of the morphologic and genomic architecture of the C. pneumoniae koala (and human) strain, and shows that the koala strain is microscopically, developmentally and genetically distinct from the C. pneumoniae human strain, and (ii) examines the genetic relationship of geographically diverse C. pneumoniae isolates from human, marsupial, amphibian, reptilian and equine hosts, and identifies two distinct lineages that have arisen from animal-to-human cross species transmissions. Chapter One of this thesis explores the scientific problem and aims of this study, while Chapter Two provides a detailed literature review of the background in this field of work. Chapter Three, the first results chapter, describes the morphology and developmental stages of C. pneumoniae koala isolate LPCoLN, as revealed by fluorescence and transmission electron microscopy. The profile of this isolate, when cultured in HEp-2 human epithelial cells, was quite different to the human AR39 isolate. Koala LPCoLN inclusions were larger; the elementary bodies did not have the characteristic pear-shaped appearance, and the developmental cycle was completed within a shorter period of time (as confirmed by quantitative real-time PCR). These in vitro findings might reflect biological differences between koala LPCoLN and human AR39 in vivo. Chapter Four describes the complete genome sequence of the koala respiratory pathogen, C. pneumoniae LPCoLN. This is the first animal isolate of C. pneumoniae to be fully-sequenced. The genome sequence provides new insights into genomic ‘plasticity’ (organisation), evolution and biology of koala LPCoLN, relative to four complete C. pneumoniae human genomes (AR39, CWL029, J138 and TW183). Koala LPCoLN contains a plasmid that is not shared with any of the human isolates, there is evidence of gene loss in nucleotide salvage pathways, and there are 10 hot spot genomic regions of variation that were previously not identified in the C. pneumoniae human genomes. Sequence (partial-length) from a second, independent, wild koala isolate (EBB) at several gene loci confirmed that the koala LPCoLN isolate was representative of a koala C. pneumoniae strain. The combined sequence data provides evidence that the C. pneumoniae animal (koala LPCoLN) genome is ancestral to the C. pneumoniae human genomes and that human infections may have originated from zoonotic infections. Chapter Five examines key genome components of the five C. pneumoniae genomes in more detail. This analysis reveals genomic features that are shared by and/or contribute to the broad ecological adaptability and evolution of C. pneumoniae. This analysis resulted in the identification of 65 gene sequences for further analysis of intraspecific variation, and revealed some interesting differences, including fragmentation, truncation and gene decay (loss of redundant ancestral traits). This study provides valuable insights into metabolic diversity, adaptation and evolution of C. pneumoniae. Chapter Six utilises a subset of 23 target genes identified from the previous genomic comparisons and makes a significant contribution to our understanding of genetic variability among C. pneumoniae human (11) and animal (6 amphibian, 5 reptilian, 1 equine and 7 marsupial hosts) isolates. It has been shown that the animal isolates are genetically diverse, unlike the human isolates that are virtually clonal. More convincing evidence that C. pneumoniae originated in animals and recently (in the last few hundred thousand years) crossed host species to infect humans is provided in this study. It is proposed that two animal-to-human cross species events have occurred in the context of the results, one evident by the nearly clonal human genotype circulating in the world today, and the other by a more animal-like genotype apparent in Indigenous Australians. Taken together, these data indicate that the C. pneumoniae koala LPCoLN isolate has morphologic and genomic characteristics that are distinct from the human isolates. These differences may affect the survival and activity of the C. pneumoniae koala pathogen in its natural host, in vivo. This study, by utilising the genetic diversity of C. pneumoniae, identified new genetic markers for distinguishing human and animal isolates. However, not all C. pneumoniae isolates were genetically diverse; in fact, several isolates were highly conserved, if not identical in sequence (i.e. Australian marsupials) emphasising that at some stage in the evolution of this pathogen, there has been an adaptation/s to a particular host, providing some stability in the genome. The outcomes of this study by experimental and bioinformatic approaches have significantly enhanced our knowledge of the biology of this pathogen and will advance opportunities for the investigation of novel vaccine targets, antimicrobial therapy, or blocking of pathogenic pathways.