| The past few decades have witnessed the incredible advancements in modeling, digitizing and visualizing techniques for three-dimensional shapes. Those advancements led to an explosion in the number of three-dimensional models being created for design, manufacture, architecture, medical imaging, etc. At the same time, the structure, function, stability, and dynamics of proteins, subcellular structures, organelles, and multiprotein complexes have emerged as a leading interest in structural biology, another major source of large and complex geometric models. Geometric modeling not only provides visualizations of shapes for large biomolecular complexes but also fills the gap between structural information and theoretical modeling, and enables the understanding of function, stability, and dynamics.;We first propose, for tessellated volumes of arbitrary topology, a compact data structure that offers constant-time-complexity incidence queries among cells of any dimensions. Our data structure is simple to implement, easy to use, and allows for arbitrary, user-defined 3-cells such as prisms and hexahedra, while remaining highly efficient in memory usage compared to previous work. We also provide the analysis on its time complexity for commonly-used incidence and adjacency queries such as vertex and edge one-rings.;We then introduce a suite of computational tools for volumetric data processing, information extraction, surface mesh rendering, geometric measurement, and curvature estimation for biomolecular complexes. Particular emphasis is given to the modeling of Electron Microscopy Data Bank (EMDB) data and Protein Data Bank (PDB) data. Lagrangian and Cartesian representations are discussed for the surface presentation. Based on these representations, practical algorithms are developed for surface area and surface--enclosed volume calculation, and curvature estimation. Methods for volumetric meshing have also been presented. Because the technological development in computer science and mathematics has led to a variety of choices at each stage of the geometric modeling, we discuss the rationales in the design and selection of various algorithms. Analytical test models are designed to verify the computational accuracy and convergence of proposed algorithms. We selected six EMDB data and six PDB data to demonstrate the efficacy of the proposed algorithms in handling biomolecular surfaces and explore their capability of geometric characterization of binding targets. Thus, our toolkit offers a comprehensive protocol for the geometric modeling of proteins, subcellular structures, organelles, and multiprotein complexes.;Furthermore, we present a method for computing "choking" loops---a set of surface loops that describe the narrowing of the volumes inside/outside of the surface and extend the notion of surface homology and homotopy loops. The intuition behind their definition is that a choking loop represents the region where an offset of the original surface would get pinched. Our generalized loops naturally include the usual 2g handles/tunnels computed based on the topology of the genus-g surface, but also include loops that identify chokepoints or bottlenecks, i.e., boundaries of small membranes separating the inside or outside volume of the surface into disconnected regions. Our definition is based on persistent homology theory, which gives a measure to topological structures, thus providing resilience to noise and a well-defined way to determine topological feature size.;Finally, we explore the application of persistent homology theory in protein folding analysis. The extremely complex process of protein folding brings challenges for both experimental study and theoretical modeling. The persistent homology approach studies the Euler characteristics of the protein conformations during the folding process. More precisely, the persistence is measured by the variation of van der Waals radius, which leads to the change of protein 3D structures and uncovers the inter-connectivity. Our results on fullerenes demonstrate the potential of our geometric and topological approach to protein stability analysis. |
