Iterative solution of elliptic problems by approximate factorization

AbstractAn iterative method for the numerical solution of singularly perturbed second-order linear elliptic problems is presented. It is a defect correction iteration in which the approximate operator is the product of two first-order operators, which is readily inverted numerically. The approximate operator is generated by formal asymptotic factorization of the original operator. Hence this is a QUasi Analytic Defect correction iteration (QUAD). Both its continuous and discrete versions are analyzed in one dimension. The scheme is extended to a variety of two dimensional operators and it is analyzed for a model advection-diffusion equation. Numerical calculations show the effectiveness of the scheme over a wide range of values of the small parameter

Molecular definition of multiple sites of antibody inhibition of malaria transmission-blocking vaccine antigen Pfs25.

The Plasmodium falciparum Pfs25 protein (Pfs25) is a leading malaria transmission-blocking vaccine antigen. Pfs25 vaccination is intended to elicit antibodies that inhibit parasite development when ingested by Anopheles mosquitoes during blood meals. The Pfs25 three-dimensional structure has remained elusive, hampering a molecular understanding of its function and limiting immunogen design. We report six crystal structures of Pfs25 in complex with antibodies elicited by immunization via Pfs25 virus-like particles in human immunoglobulin loci transgenic mice. Our structural findings reveal the fine specificities associated with two distinct immunogenic sites on Pfs25. Importantly, one of these sites broadly overlaps with the epitope of the well-known 4B7 mouse antibody, which can be targeted simultaneously by antibodies that target a non-overlapping site to additively increase parasite inhibition. Our molecular characterization of inhibitory antibodies informs on the natural disposition of Pfs25 on the surface of ookinetes and provides the structural blueprints to design next-generation immunogens

Hypothetical biomolecular probe based on a genetic switch with tunable symmetry and stability

Background: Genetic switches are ubiquitous in nature, frequently associated with the control of cellular functions and developmental programs. In the realm of synthetic biology, it is of great interest to engineer genetic circuits that can change their mode of operation from monostable to bistable, or even to multistable, based on the experimental fine-tuning of readily accessible parameters. In order to successfully design robust, bistable synthetic circuits to be used as biomolecular probes, or understand modes of operation of such naturally occurring circuits, we must identify parameters that are key in determining their characteristics. Results: Here, we analyze the bistability properties of a general, asymmetric genetic toggle switch based on a chemical-reaction kinetic description. By making appropriate approximations, we are able to reduce the system to two coupled differential equations. Their deterministic stability analysis and stochastic numerical simulations are in excellent agreement. Drawing upon this general framework, we develop a model of an experimentally realized asymmetric bistable genetic switch based on the LacI and TetR repressors. By varying the concentrations of two synthetic inducers, doxycycline and isopropyl ??-D-1-thiogalactopyranoside, we predict that it will be possible to repeatedly fine-tune the mode of operation of this genetic switch from monostable to bistable, as well as the switching rates over many orders of magnitude, in an experimental setting. Furthermore, we find that the shape and size of the bistability region is closely connected with plasmid copy number. Conclusions: Based on our numerical calculations of the LacI-TetR asymmetric bistable switch phase diagram, we propose a generic work-flow for developing and applying biomolecular probes: Their initial state of operation should be specified by controlling inducer concentrations, and dilution due to cellular division would turn the probes into memory devices in which information could be preserved over multiple generations. Additionally, insights from our analysis of the LacI-TetR system suggest that this particular system is readily available to be employed in this kind of probe.clos

Space-time domain decomposition for parabolic problems

We analyze a space-time domain decomposition iteration, for a model advection diffusion equation in one and two dimensions. The discretization of this iteration is the block red-black variant of the waveform relaxation method, and our analysis provides new convergence results for this scheme. The asymptotic convergence rate is super-linear, and it is governed by the diffusion of the error across the overlap between subdomains. Hence, it depends on both the size of this overlap and the diffusion coefficient in the equation. However it is independent of the number of subdomains, provided the size of the overlap remains fixed. The convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases. The duration of the transient linear regime is proportional to the length of the time window. For advection dominated problems, the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases. Moreover, it is independent of the size of the time window and of the number of subdomains. Numerical calculations illustrate our analysis

Inner and outer iterations for the Chebyshev algorithm

We analyze the Chebyshev iteration in which the linear system involving the splitting matrix is solved inexactly by an inner iteration. We assume that the tolerance for the inner iteration may change from one outer iteration to the other. When the tolerance converges to zero, the asymptotic convergence rate is unaffected. Motivated by this result, we seek the sequence of tolerance values that yields the lowest cost. We find that among all sequences of slowly varying tolerances, a constant one is optimal. Numerical calculations that verify our results are shown. Our analysis is based on asymptotic methods, such as the W.K.B method, for linear recurrence equations and an estimate of the accuracy of the resulting asymptotic result. 1 Introduction The Chebyshev iterative algorithm [1] for solving linear systems of equations requires at each step the solution of a subproblem i.e. the solution of another linear system. We Phone (o): (818)-395-4550, FAX: (818)-578-0124, e-mail: giladi@ama..