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The Power Crust, Unions of Balls, and the Medial Axis Transform
 Computational Geometry: Theory and Applications
, 2000
"... The medial axis transform (or MAT) is a representation of an object as an infinite union of balls. We consider approximating the MAT of a threedimensional object, and its complement, with a finite union of balls. Using this approximate MAT we define a new piecewiselinear approximation to the objec ..."
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Cited by 197 (5 self)
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The medial axis transform (or MAT) is a representation of an object as an infinite union of balls. We consider approximating the MAT of a threedimensional object, and its complement, with a finite union of balls. Using this approximate MAT we define a new piecewiselinear approximation to the object surface, which we call the power crust. We assume that we are given as input a suficiently dense sample of points from the object surface. We select a subset of the Voronoi balls of the sample, the polar balls, as the union of balls representation. We bound the geometric error of the union, and of the corresponding power crust, and show that both representations are topologically correct as well. Thus, our results provide a new algorithm for surface reconstruction from sample points. By construction, the power crust is always the boundary of a solid, so we avoid the holefilling or manifold extraction steps used in previous algorithms. The union of balls representation and the power crust have corresponding piecewiselinear dual representations, which in some sense approximate the medial axis. We show a geometric relationship between these duals and the medial axis by proving that, as the sampling density goes to infinity, the set of poles, the centers of the polar balls, converge to the medial axis.
A new method to detect related function among proteins independent of sequence and fold
, 2002
"... A new method has been developed to detect functional relationships among proteins independent of a given sequence or fold homology. It is based on the idea that protein function is intimately related to the recognition and subsequent response to the binding of a substrate or an endogenous ligand in ..."
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Cited by 123 (6 self)
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A new method has been developed to detect functional relationships among proteins independent of a given sequence or fold homology. It is based on the idea that protein function is intimately related to the recognition and subsequent response to the binding of a substrate or an endogenous ligand in a wellcharacterized binding pocket. Thus, recognition of similar ligands, supposedly linked to similar function, requires conserved recognition features exposed in terms of common physicochemical interaction properties via the functional groups of the residues flanking a particular binding cavity. Following a technique commonly used in the comparison of small molecule ligands, generic pseudocenters coding for possible interaction properties were assigned for a large sample set of cavities extracted from the entire PDB and stored in the database Cavbase. Using a particular query cavity a series of related cavities of decreasing similarity is detected based on a clique detection algorithm. The detected similarity is ranked according to propertybased surface patches shared in common by the different clique solutions. The approach either retrieves protein cavities accommodating the same (e.g. cofactors) or closely related ligands or it extracts proteins exhibiting similar function in terms of a related catalytic mechanism. Finally the new method has strong potential to suggest alternative molecular skeletons in de novo design. The retrieval of molecular building blocks accommodated in a particular subpocket that shares similarity with the pocket in a protein studied by drug design can inspire the discovery of novel ligands.
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 89 (20 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Inferring Functional Relationships of Proteins from Local Sequence and Spatial Surface Patterns
 J. Mol. Biol
, 2003
"... es, and for further inquiries on evolutionary origins of structural elements important for protein function. q 2003 Elsevier Ltd. All rights reserved. Keywords: protein surface; surface pattern; protein function; pocket sequence; pocket shape *Corresponding author Introduction With rapid progres ..."
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Cited by 74 (15 self)
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es, and for further inquiries on evolutionary origins of structural elements important for protein function. q 2003 Elsevier Ltd. All rights reserved. Keywords: protein surface; surface pattern; protein function; pocket sequence; pocket shape *Corresponding author Introduction With rapid progress in the determination of protein structures, 1,2 protein structural analysis has become an important source of information for understanding functional roles of proteins. Conservation of protein structures often reveals very distant evolutionary relationships, which are otherwise difficult to detect by sequence analysis alone. Analysis of protein structure can provide insightful ideas about the biochemical functions and mechanisms of proteins (e.g. active sites, catalytic residues, and substrate interactions). 911 An important approach of studying protein structures is fold analysis. Identifying the correct tertiary fold of protein is often helpful for inferring protein funct
Deformable Smooth Surface Design
, 1999
"... A new paradigm for designing smooth surfaces is described. A finite set of points with weights specifies a closed surface in space referred to as skin. It consists of one or more components, each tangent continuous and free of selfintersections and intersections with other components. The skin var ..."
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Cited by 57 (11 self)
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A new paradigm for designing smooth surfaces is described. A finite set of points with weights specifies a closed surface in space referred to as skin. It consists of one or more components, each tangent continuous and free of selfintersections and intersections with other components. The skin varies continuously with the weights and locations of the points, and the variation includes the possibility of a topology change facilitated by the violation of tangent continuity at a single point in space and time. Applications of the skin to molecular modeling and to geometric deformation are discussed.
HelixHelix Packing and Interfacial Pairwise Interactions of Residues in Membrane Proteins
 J. Mol. Biol
, 2001
"... this paper. Here, we are interested in the packing of peptide chains, therefore all ligands and water molecules are removed before computation. Voids and pockets in the TM regions are identied and measured with a probe radius of 1.4 A using the CAST method ..."
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Cited by 48 (17 self)
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this paper. Here, we are interested in the packing of peptide chains, therefore all ligands and water molecules are removed before computation. Voids and pockets in the TM regions are identied and measured with a probe radius of 1.4 A using the CAST method
Dynamic Skin Triangulation
"... This paper describes an algorithm for maintaining an approximating triangulation of a deforming surface in R³. The surface is the envelope of an infinite family of spheres defined and controlled by a finite collection of weighted points. The triangulation adapts dynamically to changing shape, curvat ..."
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Cited by 48 (14 self)
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This paper describes an algorithm for maintaining an approximating triangulation of a deforming surface in R³. The surface is the envelope of an infinite family of spheres defined and controlled by a finite collection of weighted points. The triangulation adapts dynamically to changing shape, curvature, and topology of the surface.
Measuring proteins and voids in proteins
 In Proc. 28th Ann. Hawaii Int’l Conf. System Sciences
, 1995
"... Common geometric models for proteins and other molecules are the space filling diagram, the solvent accessible surface, and the molecular surface. We descnbe software that compules metric properties of these models, including volume and surface area. It also measures voids or empty space enclosed ..."
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Cited by 45 (12 self)
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Common geometric models for proteins and other molecules are the space filling diagram, the solvent accessible surface, and the molecular surface. We descnbe software that compules metric properties of these models, including volume and surface area. It also measures voids or empty space enclosed by the protein, and it keeps track of surface area contributions of individual atoms. The software is based on Jdimensional alpha complexes and on inclusionexclusion formulas with terms derived from the simplices in this complex. The so,ftware is available via anonymous ftp at ftp.ncsa.uiuc.edu. 1 Introduct ion The space filling dzagram, SF, introduced by Lee and Richards [ll], models a protein as the union of possibly overlapping spherical balls in R3, see figure 1. Each ball represents an atom and its size is determined by the van der Waals radius of the atom. A void is a piece *This work is supported by the National Science Foundation, under grant ASC9200301, the CISE postdoctoral fellow