G-protein-coupled receptors (GPCRs) constitute the largest family of receptors and major pharmacological targets. Whereas many GPCRs have been shown to form di-/oligomers, the size and stability of such complexes under physiological conditions are largely unknown. Here, we used direct receptor labeling with SNAP-tags and total internal reflection fluorescence microscopy to dynamically monitor single receptors on intact cells and thus compare the spatial arrangement, mobility, and supramolecular organization of three prototypical GPCRs: the β 1 -adrenergic receptor (β 1 AR), the β 2 -adrenergic receptor (β 2 AR), and the γ-aminobutyric acid (GABA B ) receptor. These GPCRs showed very different degrees of di-/oligomerization, lowest for β 1 ARs (monomers/dimers) and highest for GABA B receptors (prevalently dimers/tetramers of heterodimers). The size of receptor complexes increased with receptor density as a result of transient receptor-receptor interactions. Whereas β 1 -/ β 2 ARs were apparently freely diffusing on the cell surface, GABA B receptors were prevalently organized into ordered arrays, via interaction with the actin cytoskeleton. Agonist stimulation did not alter receptor di-/oligomerization, but increased the mobility of GABA B receptor complexes. These data provide a spatiotemporal characterization of β 1 -/β 2 ARs and GABA B receptors at single-molecule resolution. The results suggest that GPCRs are present on the cell surface in a dynamic equilibrium, with constant formation and dissociation of new receptor complexes that can be targeted, in a ligand-regulated manner, to different cell-surface microdomains.live cell imaging | protein-protein interactions

This classic text focuses on elliptic boundary value problems in domains with nonsmooth boundaries and on problems with mixed boundary conditions. Its contents are essential for an understanding of the behavior of numerical methods for partial differential equations on two-dimensional domains with corners. It provides a careful and self-contained development of Sobolev spaces on nonsmooth domains, develops a comprehensive theory for second-order elliptic boundary value problems, and addresses fourth-order boundary value problems and numerical treatment of singularities. Readers need only a background in functional analysis to find the material accessible.

Transport of Bose-Einstein condensates in magnetic microtraps, controllable by external parameters such as wire currents or radio-frequency fields, is studied within the framework of optimal control theory (OCT). We derive from the Gross-Pitaevskii equation the optimality system for the OCT fields that allow to efficiently channel the condensate between given initial and desired states. For a variety of magnetic confinement potentials we study transport and wavefunction splitting of the condensate, and demonstrate that OCT allows to drastically outperfrom more simple schemes for the time variation of the microtrap control parameters.

Research on multigrid methods for optimization problems is reviewed. Optimization problems include shape design, parameter optimization, and optimal control problems governed by partial differential equations of elliptic, parabolic, and hyperbolic type.

An efficient framework for the optimal control of probability density functions (PDFs) of multidimensional stochastic processes is presented. This framework is based on the Fokker–Planck equation that governs the time evolution of the PDF of stochastic processes and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open-loop optimality systems in a receding-horizon control strategy. Many theoretical results concerning the forward and the optimal control problem are provided. In particular, it is shown that under appropriate assumptions the open-loop bilinear control function is unique. The resulting optimality system is discretized by the Chang–Cooper scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with a stochastic Lotka–Volterra model and a noised limit cycle model