A discontinuous Galerkin finite-element method for a 1D prototype of the Boltzmann equation

To develop and analyze new computational techniques for the Boltzmann equation based on model or approximation adaptivity, it is imperative to have disposal of a compliant model problem that displays the essential characteristics of the Boltzmann equation and that admits the extraction of highly accurate reference solutions. For standard collision processes, the Boltzmann equation itself fails to meet the second requirement for d = 2, 3 spatial dimensions, on account of its setting in 2d, while for d = 1 the first requirement is violated because the Boltzmann equation then lacks the convergence-to-equilibrium property that forms the substructure of simplified models. In this article we present a numerical investigation of a new one-dimensional prototype of the Boltzmann equation. The underlying molecular model is endowed with random collisions, which have been fabricated such that the corresponding Boltzmann equation exhibits convergence to Maxwell–Boltzmann equilibrium solutions. The new Boltzmann model is approximated by means of a discontinuous Galerkin (DG) finite-element method. To validate the one-dimensional Boltzmann model, we conduct numerical experiments and compare the results with Monte-Carlo simulations of equivalent molecular-dynamics models.     

Direct Minimization of the least-squares spectral element functional – Part I: Direct solver

This paper describes an equivalent but improved least-squares formulation for the numerical approximation of the solution of partial differential equations. Instead of using variational analysis to impose the conditions for minimizing the residual, the residuals are minimized directly, thus leading to a method we will denote by Direct Minimization (DM). DM circumvents setting up the normal equations which consists of matrix–matrix multiplications. Matrix–matrix multiplications are expensive, may lead to loss of accuracy and destroy the sparsity pattern present in the original system. The condition number of the DM formulation is the square root of the condition number which would be obtained if variational analysis was employed. An element-by-element procedure will be presented which allows for parallelization of DM. A computational comparison between DM and the conventional least-squares formulation based on variational analysis will be presented.     

Crystal structure predictions for acetic acid

Possible crystal structures of acetic acid were generated, considering eight space groups and assuming one molecule in the asymmetric unit. Our grid-search method was compared with a Monte Carlo approach as implemented in the Biosym/MSI Polymorph Predictor. This revealed no sampling deficiencies. A large number of possible crystal structures were found (∼100 within only 5 kJ/mol), including the experimental structure. Energy minimizations were done with a united-atoms force field (GROMOS), an all-atoms force field (AMBER), and a potential that describes the electrostatic interactions with distributed multipoles (DMA). In all cases, the experimental structure had a low lattice energy. The number of hypothetical crystal structures was reduced considerably by removing space-group symmetry constraints, or by a primitive molecular dynamics shake-up. Nevertheless, sufficient structures of equal or lower energy compared with the experimental structure remained to suggest that other factors need to be considered for genuine structure prediction. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 459–474, 1998     

Virtual screening using protein-ligand docking: avoiding artificial enrichment

This study addresses a number of topical issues around the use of protein-ligand docking in virtual screening. We show that, for the validation of such methods, it is key to use focused libraries (containing compounds with one-dimensional properties, similar to the actives), rather than "random" or "drug-like" libraries to test the actives against. We also show that, to obtain good enrichments, the docking program needs to produce reliable binding modes. We demonstrate how pharmacophores can be used to guide the dockings and improve enrichments, and we compare the performance of three consensus-ranking protocols against ranking based on individual scoring functions. Finally, we show that protein-ligand docking can be an effective aid in the screening for weak, fragment-like binders, which has rapidly become a popular strategy for hit identification. All results presented are based on carefully constructed virtual screening experiments against four targets, using the protein-ligand docking program GOLD.     

Ab initio crystal structure predictions for flexible hydrogen‐bonded molecules. Part II. Accurate energy minimization

A method is described to perform ab initio energy minimization for crystals of flexible molecules. The intramolecular energies and forces are obtained directly from ab initio calculations, whereas the intermolecular contributions follow from a potential that had been parameterized earlier on highly accurate quantum‐chemical calculations. Glycol and glycerol were studied exhaustively as prototypes. Lists of hypothetical crystal structures were generated using an empirical force field, after which ab initio energy minimizations were performed for a few hundreds of these. The experimental crystal structures were found among the structures with lowest energy, provided that sufficiently large basis sets were used. Moreover, their crystal geometries were well reproduced. This approach enables a systematic comparison between the merits of force fields at various levels of sophistication. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 805–815, 2001     

Modular Medical Imaging Agents Based on Azide–Alkyne Huisgen Cycloadditions: Synthesis and Pre-Clinical Evaluation of 18F-Labeled PSMA-Tracers for Prostate Cancer Imaging

Since the seminal contribution of Rolf Huisgen to develop the [3+2] cycloaddition of 1,3-dipolar compounds, its azide–alkyne variant has established itself as the key step in numerous organic syntheses and bioorthogonal processes in materials science and chemical biology. In the present study, the copper(I)-catalyzed azide–alkyne cycloaddition was applied for the development of a modular molecular platform for medical imaging of the prostate-specific membrane antigen (PSMA), using positron emission tomography. This process is shown from molecular design, through synthesis automation and in vitro studies, all the way to pre-clinical in vivo evaluation of fluorine-18- labeled PSMA-targeting ‘F-PSMA-MIC’ radiotracers (t_1/2=109.7 min). Pre-clinical data indicate that the modular PSMA-scaffold has similar binding affinity and imaging properties to the clinically used [⁶⁸Ga]PSMA-11. Furthermore, we demonstrated that targeting the arene-binding in PSMA, facilitated through the [3+2]cycloaddition, can improve binding affinity, which was rationalized by molecular modeling. The here presented PSMA-binding scaffold potentially facilitates easy coupling to other medical imaging moieties, enabling future developments of new modular imaging agents.