An optical lens for focusing two pairs of points

We construct an optical lens in the (x, y)-plane which focuses two pairs of points, i.e., all the rays from a given point X _i are focused by the lens at a given point y _i , for i = 1, 2. The points X ₁, X ₂, Y ₁, Y ₂ lie on the x-axis and the lens has the form $$\left\{ {\gamma _{\text{1}} {\text{ }} + {\text{ }}f_{\text{1}} {\text{(}}y{\text{) }}\underline \leqslant {\text{ }}x{\text{ }}\underline \leqslant {\text{ }}\gamma _{\text{2}} {\text{ }} + {\text{ }}f_{\text{2}} {\text{(}}y{\text{)}},{\text{ }}\left| y \right|{\text{ }}\underline \leqslant {\text{ }}y_{\text{0}} } \right\}$$ where γ ₁, γ ₂ are given, and f _i (0) = 0, f _i (?y) = f _i (y). We then let X ₂ → X ₁, Y ₂ → Y ₁ and investigate the limiting lens. We show that this limit is generally not a symmetric lens, i.e., f ₁ + f ₂ ? 0.     

Calculation of Feynman diagrams with zero mass threshold from their small momentum expansion

A method of calculating Feynman diagrams from their small momentum expansion [1] is extended to diagrams with zero mass thresholds. We start from the asymptotic expansion in large masses [2] (applied to the case when all $M_i^2$ are large compared to all momenta squared). Using dimensional regularization, a finite result is obtained in terms of powers of logarithms (describing the zero-threshold singularity) times power series in the momentum squared. Surprisingly, these latter ones represent functions, which not only have the expected physical “second threshold” but have a branchcut singularity as well below threshold at a mirror position. These can be understood as pseudothresholds corresponding to solutions of the Landau equations. In the spacelike region the imaginary parts from the various contributions cancel. For the two-loop examples with one mass M, in the timelike region for q² ≈ M² we obtain approximations of high precision. This will be of relevance in particular for the calculation of the decay Z → bb?in the m _b = 0 approximation.     

Identifying the binding mode of a molecular scaffold

We describe a method for docking of a scaffold-based series and present its advantages over docking of individual ligands, for determining the binding mode of a molecular scaffold in a binding site. The method has been applied to eight different scaffolds of protein kinase inhibitors (PKI). A single analog of each of these eight scaffolds was previously crystallized with different protein kinases. We have used FlexX to dock a set of molecules that share the same scaffold, rather than docking a single molecule. The main mode of binding is determined by the mode of binding of the largest cluster among the docked molecules that share a scaffold. Clustering is based on our 'nearest single neighbor' method [J. Chem. Inf. Comput. Sci., 43 (2003) 208-217]. Additional criteria are applied in those cases in which more than one significant binding mode is found. Using the proposed method, most of the crystallographic binding modes of these scaffolds were reconstructed. Alternative modes, that have not been detected yet by experiments, could also be identified. The method was applied to predict the binding mode of an additional molecular scaffold that was not yet reported and the predicted binding mode has been found to be very similar to experimental results for a closely related scaffold. We suggest that this approach be used as a virtual screening tool for scaffold-based design processes.     

Comparison of azacyclic urea A-98881 as HIV-1 protease inhibitor with cage dimeric N-benzyl 4-(4-methoxyphenyl)-1,4- dihydropyridine as representative of a novel class of HIV-1 protease inhibitors: A molecular modeling study

The functional groups of cage dimeric N-alkyl substituted 3,5-bis(hydroxymethyl)-4-(4-methoxyphenyl)-1,4-dihydropyridines are similar to those of cyclic and azacyclic ureas that are potent inhibitors of HIV-1 protease of the dihydroxyethylene- and hydroxyethylene type, respectively. In the following study the conformity of common functional groups is investigated concerning their orientation in space as well as in the enzyme HIV-1 protease. Starting from X-ray crystal data of the centrosymmetric cage dimeric N-benzyl derivative with ester groups, the derivative with hydroxymethylene groups was built and a systematic conformational search was performed for the conformationally important torsion angles considering electrostatic and van der Waals interactions. From the huge number of conformations those comprising centrosymmetrical and C2-symmetrical energy minima were selected and minimized. The three remaining conformers were fitted to the azacyclic urea A-98881 selected from the HIV-1 protease enzyme- inhibitor complex using the centroids of the corresponding aromatic residues and additionally by the field fit option of the Advanced CoMFA module of SYBYL. Interestingly, the energetically most favourable one, which, additionally, possesses C2-symmetry like the active site cavity of HIV-1 protease, showed the best fit. Comparing the electrostatic potential (EP) of the latter with the EP of A-98881 the aromatic residues show excellent accordance. Slight differences in the extent of the EP were found in the areas of the hydroxymethylene groups of the cage dimer and the single hydroxy group as well as the urea carbonyl group of A- 98881, respectively. In order to compare the binding possibilities to the enzyme HIV-1 protease for the cage dimer and A-98881, their interaction fields with certain probes (CH3 for alkyl, NHamide, and carbonyl, O– of COO–), representing the decisive functional groups of the active site, have been calculated using GRID and projected into the enzyme placing the structures according to the position of A-98881 in the enzyme- inhibitor complex. The strongest calculated fields of the O– probe were found near Asp 25 for both structures. Another respective conformity consists in the overlap of the fields for the NHamide probe near Ile 50 and 50 for the investigated cage dimer and A-98881.     

Reinforcement problems for elliptic equations and variational inequalities

Summary Consider the Dirichlet problem for an elliptic equation in a domain , with coefficients having discontinuity on a surface . Suppose divides into _{1 2}(₂ the inner core), the thickness of ₁ is of order of magnitude , and the modulus of ellipticity in ₁ is of order magnitude ₁. The asymptotic behavior of the solution is studied as 0, ₁ 0, provided lim (/₁) exists. Other questions of this type are studied both for elliptic equations and for elliptic variational inequalities.The second author is partially supported by National Science Foundation Grant 7406375 A01. The third author is partially supported by National Science Foundation Grant MC575-21416 A01.     

Self-diffusion of polystyrene in solution 2. Discussion of experimental results on the basis of the reptation mechanism and entanglements

The expressions for polymer self-diffusion in semidilute solutions, theoretically derived from the reptation mechanism, the blob concept and scaling considerations, are discussed and compared against experimental data from the authors' investigations and the literature. In the nonentangled (from viscoelastic data) semidilute solution, the experimentally observed concentration and molar mass exponents are in fair agreement with those derived theoretically. However, a quantitative estimation shows that the experiments cannot be explained by reptation. Experiments with polymer mixtures also give strong evidence against reptation. It is concluded, that in the nonentangled semidilute solution, the polymer self-diffusion is more complicated than simple reptation. This is also supported by recently observed long-range density fluctuations or cluster formation in this concentration region detected by scattering techniques and NMR-PFGT. In the entangled semidilute solution, the self-diffusion data are in accordance with the reptation mechanism; reptation being within a tube having approximately 20 blobs between entanglements.