Alfio Borzì

Single-molecule analysis of fluorescently labeled G-protein–coupled receptors reveals complexes with distinct dynamics and organization

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 β1-adrenergic receptor (β1AR), the β2-adrenergic receptor (β2AR), and the γ-aminobutyric acid (GABAB) receptor. These GPCRs showed very different degrees of di-/oligomerization, lowest for β1ARs (monomers/dimers) and highest for GABAB receptors (prevalently dimers/tetramers of heterodimers). The size of receptor complexes increased with receptor density as a result of transient receptor–receptor interactions. Whereas β1-/β2ARs were apparently freely diffusing on the cell surface, GABAB 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 GABAB receptor complexes. These data provide a spatiotemporal characterization of β1-/β2ARs and GABAB 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.

Computational Optimization of Systems Governed by Partial Differential Equations

This book fills a gap between theory-oriented investigations in PDE-constrained optimization and the practical demands made by numerical solutions of PDE optimization problems. The authors discuss computational techniques representing recent developments that result from a combination of modern techniques for the numerical solution of PDEs and for sophisticated optimization schemes. Computational Optimization of Systems Governed by Partial Differential Equations offers readers a combined treatment of PDE-constrained optimization and uncertainties and an extensive discussion of multigrid optimization. It provides a bridge between continuous optimization and PDE modeling and focuses on the numerical solution of the corresponding problems. Audience: This book is intended for graduate students working in PDE-constrained optimization and students taking a seminar on numerical PDE-constrained optimization. It is also suitable as an introduction for researchers in scientific computing with PDEs who want to work in the field of optimization and for those in optimization who want to consider methodologies from the field of numerical PDEs. It will help researchers in the natural sciences and engineering to formulate and solve optimization problems.

Multigrid Methods for PDE Optimization

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

Optimal Control Formulation for Determining Optical Flow

An optimal control formulation for determining optical flow is presented. The new framework differs from preceding approaches in that it does not require differentiation of the data and does combine optical flow with image reconstruction. It can be considered as a control-in-the-coefficients problem with a cost functional of tracking type. A numerical algorithm that solves the optimality system consisting of hyperbolic and elliptic partial differential equations is presented. Numerical experiments demonstrate the effectiveness of the optimal control approach.

Multigrid methods for parabolic distributed optimal control problems

Multigrid schemes that solve parabolic distributed optimality systems discretized by finite differences are investigated. Accuracy properties of finite difference approximation are discussed and validated. Two multigrid methods are considered which are based on a robust relaxation technique and use two different coarsening strategies: semicoarsening and standard coarsening. The resulting multigrid algorithms show robustness with respect to changes of the value of v, the weight of the cost of the control, is sufficiently small. Fourier mode analysis is used to investigate the dependence of the linear twogrid convergence factor on v and on the discretization parameters. Results of numerical experiments are reported that demonstrate sharpness of Fourier analysis estimates. A multigrid algorithm that solves optimal control problems with box constraints on the control is considered.

Accuracy and Convergence Properties of the Finite Difference Multigrid Solution of an Optimal Control Optimality System

The finite difference multigrid solution of an optimal control problem associated with an elliptic equation is considered. Stability of the finite difference optimality system and optimal-order error estimates in the discrete L2 norm and in the discrete H1 norm under minimum smoothness requirements on the exact solution are proved. Sharp convergence factor estimates of the two grid method for the optimality system are obtained by means of local Fourier analysis. A multigrid convergence theory is provided which guarantees convergence of the multigrid process towards weak solutions of the optimality system.

A Multigrid Scheme for Elliptic Constrained Optimal Control Problems

A multigrid scheme for the solution of constrained optimal control problems discretized by finite differences is presented. This scheme is based on a new relaxation procedure that satisfies the given constraints pointwise on the computational grid. In applications, the cases of distributed and boundary control problems with box constraints are considered. The efficient and robust computational performance of the present multigrid scheme allows to investigate bang-bang control problems.

A Fokker-Planck control framework for multidimensional stochastic processes

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.

Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems

The detailed implementation and analysis of a finite element multigrid scheme for the solution of elliptic optimal control problems is presented. A particular focus is in the definition of smoothing strategies for the case of constrained control problems. For this setting, convergence of the multigrid scheme is discussed based on the BPX framework. Results of numerical experiments are reported to illustrate and validate the optimal efficiency and robustness of the performance of the present multigrid strategy.

Optimal control of probability density functions of stochastic processes

Abstract A Fokker‐Planck framework for the formulation of an optimal control strategy of stochastic processes is presented. Within this strategy, the control objectives are defined based on the probability density functions of the stochastic processes. The optimal control is obtained as the minimizer of the objective under the constraint given by the Fokker‐Planck model. Representative stochastic processes are considered with different control laws and with the purpose of attaining a final target configuration or tracking a desired trajectory. In this latter case, a receding‐horizon algorithm over a sequence of time windows is implemented.