Dynamic maintenance and visualization of molecular surfaces
Discrete Applied Mathematics - Special issue: Computational molecular biology series issue IV
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Macromolecules such as proteins and DNA have complex spatial structures which are often important for their biological functions. The idea of molecular shape and shape complementarity play crucial roles in protein folding, conformational stability, molecular solubility, crystal packing, and docking. Various models based on a union of balls representation have been suggested for representing molecular shape, including the van der Waals model, the solvent accessible surface, and the molecular surface. In this thesis, we present several extensions to the theory of Alpha shapes applicable to the analysis of molecular shape. First, we define a ``pocket'''' of a molecule, which is intuitively a `depression'', `canyon'', or `cavity'' of a molecule. The definition is based on mathematical notions of relative distance. We give efficient algorithms for computing pockets and examples of their application. Secondly, we look at the issue of maintaining shape dynamically as a molecule changes over time. Topological analysis of the changing structure can yield information about the function of the molecule. We describe algorithms and their implementations for dynamically maintaining the Delaunay complex, the basis for shape analysis. These algorithms have been implemented, and experimental results are reported. Finally, we discuss techniques for modelling uniform growth of the atoms of a molecule. The solvent accessible and molecular surface models of a molecule are based on such growth, and the algorithms presented here efficiently compute these models for all probe sizes.