The adaptive immune system uses several strategies to generate a repertoire of T-and B-cell antigen receptors with sufficient diversity to recognize the universe of potential pathogens. In ␣ T cells, which primarily recognize peptide antigens presented by major histocompatibility complex molecules, most of this receptor diversity is contained within the third complementarity-determining region (CDR3) of the T-cell receptor (TCR) ␣ and  chains. Although it has been estimated that the adaptive immune system can generate up to 10 16 distinct ␣ pairs, direct assessment of TCR CDR3 diversity has not proved amenable to standard capillary electrophoresis-based DNA sequencing. We developed a novel experimental and computational approach to measure TCR CDR3 diversity based on single-molecule DNA sequencing, and used this approach to determine the CDR3 sequence in millions of rearranged TCR genes from T cells of 2 adults. We find that total TCR receptor diversity is at least 4-fold higher than previous estimates, and the diversity in the subset of CD45RO ؉ antigen-experienced ␣ T cells is at least 10-fold higher than previous estimates. These methods should prove valuable for assessment of ␣ T-cell repertoire diversity after hematopoietic cell transplantation, in states of congenital or acquired immunodeficiency, and during normal aging. (Blood. 2009;114:4099-4107)
The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract.We formulate the string gradient weighted moving finite element method (SGWMFE) for systems of PDEs in multiple dimensions. Then we illustrate implementation issues for the method using two dimensions. The method is applied successfully to solve the highly nonlinear unsteady porous medium equation in two dimensions and the results are compared to exact solutions. We proceed to present results for the method when applied to the Gray Scott chemical reaction diffusion model in two dimensions. Finally, we apply the method to the shallow water equations in two dimensions and compare the meshes produced with those produced using gradient weighted finite element methods (GWMFE). Our conclusion is that SGWMFE is an easily applied member of the group of moving finite element methods. Key words. moving finite elements, mesh adaptation AMS subject classifications. 74S05, 65M60, 65M50DOI. 10.1137/040619557 1. Introduction. This paper has two objectives. The first is to present the general formulation for a string gradient weighted moving finite element method (SGWMFE) in multiple space dimensions. The second is to demonstrate the implementation of SGWMFE in several nonlinear systems of PDEs, the porous medium equation, the Gray Scott equations, and the shallow water equations. Solutions of each problem can be found using other methods, and we are not suggesting or attempting to demonstrate that SGWMFE is better or worse than other moving (or nonmoving) finite element or discretization methods; rather, it is an implementation of gradient weighted moving finite elements (GWMFE) with an elegant general form that is straightforward to use.Moving finite element methods (MFE), originally introduced in [14], are well established to provide reliable solutions to unsteady systems of PDEs, often with less computational cost than fixed mesh methods. There is a wide literature dealing with moving mesh methods, both for finite elements and for finite differences. The text of [1] gives an extensive view of MFE and [5,11,15,16] describe recent applications of moving mesh methods. GWMFE were developed in detail in [8] and [9], and the string version, suggested in [12], was implemented in [17]. For other background on moving mesh methods, see [17], where there is an extended reference list on moving mesh methods including MFE, GWMFE, and moving finite difference type methods. Here our interest is however in developing details of SGWMFE for systems of PDEs in two or more dimensions. Using a projection matr...
When the distribution function of plasma particles stays close to some analytically known function, statistical noise inherent to Monte Carlo simulations can be greatly reduced by introducing this function as a control variate in the computation of the velocity moments. Such a method, even though it can be naturally applied to nonlinear simulations, has originally emerged from linearised simulations and is usually called the δf particle-in-cell (PIC) method. In the past, the method has been extended to also handle collisions. This resulted in a two weight scheme which is known to produce a pronounced weight growth problem which rapidly makes it inefficient as a control variate method for variance reduction. In this work we analyse the weight growth problem within a simple example, which allows us to overcome its pathological behaviour. We also introduce a new split algorithm based on switching the control variate for PIC simulations with collisions. A key element of our algorithm is a new weight smoothing operator which enables us to obtain a significant noise reduction both in the presence of collisions and in the deep nonlinear phase of PIC simulations.
Simulations are carried out for varying mesh sizes, and the numerical solutions are compared by computing errors in two ways. In the case of an analytic solution being available, the errors in the numerical solutions are computed directly from the analytic solution. In the case of no availability of an analytic solution, an approximation to the error is computed using a very fine mesh numerical solution as the reference solution. MSC:65M50, 35R37
Alzheimer's disease (AD) is a severe neurodegenerative disorder characterised by cognitive impairment and dementia. In the AD‐affected brain, microglia cells are up‐regulated and accumulate at senile plaques, the most prominent pathological feature of AD. In order to further study and predict the movement of activated microglia, we utilised their chemotactic properties. Specifically, we formulated the string gradient weighted moving finite element method for a system of partial differential equations in two dimensions, which includes nonlinear diffusion of a different variable found in chemotaxis models. The method was applied successfully to solve highly nonlinear chemorepulsion–chemorepellent models in two dimensions, and the results were compared with one‐dimensional results found previously in the literature. We conclude that the string gradient weighted moving finite element method is easily applied to chemotaxis models, in particular movement and aggregation of microglia, resulting in the ability to study the models extended in two dimensions efficiently. Our study highlights the feasibility and power of mathematical modelling to advance our understanding of pathophysiological processes in neurodegenerative diseases, including AD. Copyright © 2012 John Wiley & Sons, Ltd.
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