January 1, 2003

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Publications in Computational Chemistry by Herbert Edelsbrunner et al. :

• On the definition and the construction of pockets in macromolecules, 1998.
• Analytic shape computation of macromolecules I: Molecular area and volume through alpha shape, 1998.
• Analytic shape computation of macromolecules II: Inaccessible cavities in proteins, 1998.

BibTeX references.

The shape of a protein is important for its functions. This includes the location and size of identifiable regions in its complement space. We formally define pockets as regions in the complement with limited accessibility from the outside. Pockets can be efficiently constructed by an algorithm based on alpha complexes. The algorithm is implemented and applied to proteins with known three-dimensional conformations.

Keywords. Combinatorial geometry and topology, algorithms, molecular biology; molecular modeling, docking, space filling and solvent accessible models, Voronoi cells, Delaunay simplices, alpha complexes.

1. Introduction

The motivation for the work reported in this paper is the apparent difficulty to talk in mathematically concrete terms about intuitive geometric concepts sometimes referred to as `depressions', `canyons', `cavities', and the like. In topology, the notions of homotopy and homology have long been used to define and study (perfect) holes of various types and dimensions. We are after a definition and study of imperfect holes, of regions people would instinctively refer to as holes although they are neither holes in the homotopical nor the homological sense.

The size and shape of macromolecules such as proteins and nucleic acids play an important role in their functions. Prior efforts to quantify these properties have been based on various discretization or tessellation procedures involving analytical or numerical computations. In this article, we present an analytically exact method for computing the metric properties of macromolecules based on the alpha shape theory. This method uses the duality between alpha complex and the weighted Voronoi decomposition of a molecule. We describe the intuitive ideas and concepts behind the alpha shape theory and the algorithm for computing areas and volumes of macromolecules. We apply our method to compute areas and volumes of a number of protein systems. We also discuss several difficulties commonly encountered in molecular shape computations and outline methods to overcome these problems.

The structures of proteins are well-packed, yet they contain numerous cavities which play key roles in accommodating small molecules, or enabling conformational changes. From high-resolution structures it is possible to identify these cavities. We have developed a precise algorithm based on alpha shapes for measuring space-filling-based molecular models (such as van der Waals, solvent accessible, and molecular surface descriptions). We applied this method for accurate computation of the surface area and volume of cavities in several proteins. In addition, all of the atoms/residues lining the cavities are identified. We use this method to study the structure and the stability of proteins, as well as to locate cavities that could contain structural water molecules in the proton transport pathway in the membrane protein bacteriorhodopsin.

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