Differential operators on homogeneous spaces. I

Summary In this paper, we extend recent work of one of us [Br] to investigate an old problem of the other one [B2]. Given a connected semisimple complex Lie-groupG with Lie-algebrag, we study the representation of the enveloping algebra of by global differential operators on a complete homogeneous spaceX=G/P. It turns out that the kernelI _x of _X is the annihilator of a generalizedVerma-module. On the other hand, we study the associated graded ideal grI _x , and relate it to the geometry of a generalizedSpringer-resolution, that is a map of the cotangent-bundle ofX onto a nilpotent variety in , as studied e.g. in [BM1]. We prove, for instance, that grI _x is prime if and only if _X is birational with normal image. In general, we show that is prime. Equivalently, the associated variety ofI _x in is irreducible: In fact, it is the closure of theRichardson-orbit determined byP. For the homogeneous spaceY=G/(P, P), we prove that the analogous idealI _y has for associated variety the closure of theDixmier-sheet determined byP. From this main result, we derive as a corollary, that for any module induced from a finitedimensional LieP-module the associated variety of the annihilator is irreducible, proving an old conjecture [B2], 2.5. Finally, we give some applications to the study of associated varieties of primitive ideals.     

The geometry of two-dimensional symbols

We give a complex-analytic construction of the two-dimensional symbol of Parshin and Kato. Our approach leads to a new analytic formula for the symbol together with a direct proof of the reciprocity law it satisfies.     

Q-Dock: Low-resolution flexible ligand docking with pocket-specific threading restraints

The rapidly growing number of theoretically predicted protein structures requires robust methods that can utilize low-quality receptor structures as targets for ligand docking. Typically, docking accuracy falls off dramatically when apo or modeled receptors are used in docking experiments. Low-resolution ligand docking techniques have been developed to deal with structural inaccuracies in predicted receptor models. In this spirit, we describe the development and optimization of a knowledge-based potential implemented in Q-Dock, a low-resolution flexible ligand docking approach. Self-docking experiments using crystal structures reveals satisfactory accuracy, comparable with all-atom docking. All-atom models reconstructed from Q-Dock's low-resolution models can be further refined by even a simple all-atom energy minimization. In decoy-docking against distorted receptor models with a root-mean-square deviation, RMSD, from native of approximately 3 A, Q-Dock recovers on average 15-20% more specific contacts and 25-35% more binding residues than all-atom methods. To further improve docking accuracy against low-quality protein models, we propose a pocket-specific protein-ligand interaction potential derived from weakly homologous threading holo-templates. The success rate of Q-Dock employing a pocket-specific potential is 6.3 times higher than that previously reported for the Dolores method, another low-resolution docking approach.     

Čech cocycles for characteristic classes

We give general formulae for explicit ech cocycles representing characteristic classes of real and complex vector bundles, as well as for cocycles representing Chern-Simons classes of bundles with arbitrary connections. Our formulae involve integrating differential forms over moving simplices inside homogeneous spaces. An important feature of our cocycles is that they take integer values (as opposed to real or rational values). We find in particular a formula for the instanton number of a connection over a closed four-manifold with arbitrary structure group. For flat connections, our formulae recover and generalize those of Cheeger and Simons. The methods of this paper apply also to the purely geometric construction of the Quillen line bundle with its metric.The first author was supported in part by N.S.F. grant DMS-9203517.The second author was supported in part by N.S.F. grant DMS-9310433.     

Q‐DockLHM: Low‐resolution refinement for ligand comparative modeling

The success of ligand docking calculations typically depends on the quality of the receptor structure. Given improvements in protein structure prediction approaches, approximate protein models now can be routinely obtained for the majority of gene products in a given proteome. Structure‐based virtual screening of large combinatorial libraries of lead candidates against theoretically modeled receptor structures requires fast and reliable docking techniques capable of dealing with structural inaccuracies in protein models. Here, we present Q‐Dock^LHM, a method for low‐resolution refinement of binding poses provided by FINDSITE^LHM, a ligand homology modeling approach. We compare its performance to that of classical ligand docking approaches in ligand docking against a representative set of experimental (both holo and apo) as well as theoretically modeled receptor structures. Docking benchmarks reveal that unlike all‐atom docking, Q‐Dock^LHM exhibits the desired tolerance to the receptor's structure deformation. Our results suggest that the use of an evolution‐based approach to ligand homology modeling followed by fast low‐resolution refinement is capable of achieving satisfactory performance in ligand‐binding pose prediction with promising applicability to proteome‐scale applications. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Hydrophobic collapse in (in silico) protein folding

A model of hydrophobic collapse, which is treated as the driving force for protein folding, is presented. This model is the superposition of three models commonly used in protein structure prediction: (1) 'oil-drop' model introduced by Kauzmann, (2) a lattice model introduced to decrease the number of degrees of freedom for structural changes and (3) a model of the formation of hydrophobic core as a key feature in driving the folding of proteins. These three models together helped to develop the idea of a fuzzy-oil-drop as a model for an external force field of hydrophobic character mimicking the hydrophobicity-differentiated environment for hydrophobic collapse. All amino acids in the polypeptide interact pair-wise during the folding process (energy minimization procedure) and interact with the external hydrophobic force field defined by a three-dimensional Gaussian function. The value of the Gaussian function usually interpreted as a probability distribution is treated as a normalized hydrophobicity distribution, with its maximum in the center of the ellipsoid and decreasing proportionally with the distance versus the center. The fuzzy-oil-drop is elastic and changes its shape and size during the simulated folding procedure.     

Atomic scale oxidation of a complex system: O2/alpha-SiC(0001)-( 3 x 3)

The atomic scale oxidation of the alpha-SiC(0001)-(3 x 3) surface is investigated by atom-resolved scanning tunneling microscopy, core level synchrotron radiation based photoemission spectroscopy, and infrared absorption spectroscopy. The results reveal that the initial oxidation takes place through the relaxation of lower layers, away from the surface dangling bond, in sharp contrast to silicon oxidation.