Showing papers by "Herbert Edelsbrunner published in 2005"

Stability of persistence diagrams

Abstract: The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply only small changes in the diagram. We apply this result to estimating the homology of sets in a metric space and to comparing and classifying geometric shapes.

Experimental Validation of the Docking Orientation of Cdc25 with Its Cdk2−CycA Protein Substrate†

Abstract: Cdc25 phosphatases are key activators of the eukaryotic cell cycle and compelling anticancer targets because their overexpression has been associated with numerous cancers. However, drug discovery targeting these phosphatases has been hampered by the lack of structural information about how Cdc25s interact with their native protein substrates, the cyclin-dependent kinases. Herein, we predict a docked orientation for Cdc25B with its Cdk2-pTpY-CycA protein substrate by a rigid-body docking method and refine the docked models with full-scale molecular dynamics simulations and minimization. We validate the stable ensemble structure experimentally by a variety of in vitro and in vivo techniques. Specifically, we compare our model with a crystal structure of the substrate-trapping mutant of Cdc25B. We identify and validate in vivo a novel hot-spot residue on Cdc25B (Arg492) that plays a central role in protein substrate recognition. We identify a hot-spot residue on the substrate Cdk2 (Asp206) and confirm its interaction with hot-spot residues on Cdc25 using hot-spot swapping and double mutant cycles to derive interaction energies. Our experimentally validated model is consistent with previous studies of Cdk2 and its interaction partners and initiates the opportunity for drug discovery of inhibitors that target the remote binding sites of this protein-protein interaction.

Extraction and simplification of iso-surfaces in tandem

Abstract: The tandem algorithm combines the marching cube algorithm for surface extraction and the edge contraction algorithm for surface simplification in lock-step to avoid the costly intermediate step of storing the entire extracted surface triangulation. Beyond this basic strategy, we introduce refinements to prevent artifacts in the resulting triangulation, first, by carefully monitoring the amount of simplification during the process and, second, by driving the simplification toward a compromise between shape approximation and mesh quality. We have implemented the algorithm and used extensive computational experiments to document the effects of various design options and to further fine-tune the algorithm.

Inequalities for the curvature of curves and surfaces

Abstract: In this paper, we bound the difference between the total mean curvatures of two closed surfaces in R3 in terms of their total absolute curvatures and the Frechet distance between the volumes they enclose. The proof relies on a combination of methods from algebraic topology and integral geometry. We also bound the difference between the lengths of two curves using the same methods.

Surface tiling with differential topology

Abstract: A challenging problem in computer-aided geometric design is the decomposition of a surface into four-sided regions that are then represented by NURBS patches. There are various approaches published in the literature and implemented as commercially available software, but all fall short in either automation or quality of the result. At Raindrop Geomagic, we have recently taken a fresh approach based on concepts from Morse theory. This by itself is not a new idea, but we have some novel ingredients that make this work, one being a rational notion of hierarchy that guides the construction of a simplified decomposition sensitive to only the major critical points.

Inclusion-exclusion formulas from independent complexes

Abstract: Using inclusion-exclusion, we can write the indicator function of a union of finitely many balls as an alternating sum of indicator functions of common intersections of balls. We exhibit abstract simplicial complexes that correspond to minimal inclusion-exclusion formulas. They include the dual complex, as defined in [2], and are characterized by the independence of their simplices and by geometric realizations with the same underlying space as the dual complex.

Evaluating the quality of NMR structures by local density of protons

Abstract: Evaluating the quality of experimentally determined protein structural models is an essential step toward identifying potential errors and guiding further structural refinement. Herein, we report the use of proton local density as a sensitive measure to assess the quality of nuclear magnetic resonance (NMR) structures. Using 256 high-resolution crystal structures with protons added and optimized, we show that the local density of different proton types display distinct distributions. These distributions can be characterized by statistical moments and are used to establish local density Z-scores for evaluating both global and local packing for individual protons. Analysis of 546 crystal structures at various resolutions shows that the local density Z-scores increase as the structural resolution decreases and correlate well with the ClashScore (Word et al. J Mol Biol 1999;285(4):1711-1733) generated by all atom contact analysis. Local density Z-scores for NMR structures exhibit a significantly wider range of values than for X-ray structures and demonstrate a combination of potentially problematic inflation and compression. Water-refined NMR structures show improved packing quality. Our analysis of a high-quality structural ensemble of ubiquitin refined against order parameters shows proton density distributions that correlate nearly perfectly with our standards derived from crystal structures, further validating our approach. We present an automated analysis and visualization tool for proton packing to evaluate the quality of NMR structures.