Computation of protein geometry and its applications: Packing and function prediction

This chapter discusses geometric models of biomolecules and geometric constructs, including the union of ball model, the weigthed Voronoi diagram, the weighted Delaunay triangulation, and the alpha shapes. These geometric constructs enable fast and analytical computaton of shapes of biomoleculres (including features such as voids and pockets) and metric properties (such as area and volume). The algorithms of Delaunay triangulation, computation of voids and pockets, as well volume/area computation are also described. In addition, applications in packing analysis of protein structures and protein function prediction are also discussed.Comment: 32 pages, 9 figure

Alpha, Betti and the Megaparsec Universe: on the Topology of the Cosmic Web

We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of Betti numbers goes along the availability of fast algorithms to compute them. For continuous density fields, we determine the scale-dependence of Betti numbers by invoking the cosmologically familiar filtration of sublevel or superlevel sets defined by density thresholds. For the discrete galaxy distribution, however, the analysis is based on the alpha shapes of the particles. These simplicial complexes constitute an ordered sequence of nested subsets of the Delaunay tessellation, a filtration defined by the scale parameter, $\alpha$. As they are homotopy equivalent to the sublevel sets of the distance field, they are an excellent tool for assessing the topological structure of a discrete point distribution. In order to develop an intuitive understanding for the behavior of Betti numbers as a function of $\alpha$, and their relation to the morphological patterns in the Cosmic Web, we first study them within the context of simple heuristic Voronoi clustering models. Subsequently, we address the topology of structures emerging in the standard LCDM scenario and in cosmological scenarios with alternative dark energy content. The evolution and scale-dependence of the Betti numbers is shown to reflect the hierarchical evolution of the Cosmic Web and yields a promising measure of cosmological parameters. We also discuss the expected Betti numbers as a function of the density threshold for superlevel sets of a Gaussian random field.Comment: 42 pages, 14 figure

Shortest Geometric Paths Analysis in Structural Biology

The surface of a macromolecule, such as a protein, represents the contact point of any interaction that molecule has with solvent, ions, small molecules or other macromolecules. Analyzing the surface of macromolecules has a rich history but analyzing the distances from this surface to other surfaces or volumes has not been extensively explored. Many important questions can be answered quantitatively through these analyses. These include: what is the depth of a pocket or groove on the surface? what is the overall depth of the protein? how deeply are atoms buried from the surface? where are the tunnels in a protein? where are the pockets and what are their shapes? A single algorithm to solve one graph problem, namely Dijkstra’s shortest paths algorithm, forms the basis for algorithms to answer these many questions. Many distances can be measured, for instance the distance from the convex hull to the molecular surface while avoiding the interior of the surface is defined as Travel Depth. Alternatively, the distance from the surface to every atom can be measured, giving a measure of the Burial Depth of given residues. Measuring the minimum distance to the protein surface for all points in solvent, combined with topological guidance, allows tunnels to be located. Analyzing the surface from the deepest Travel Depth upwards allows pockets to be catalogued over the entire protein surface for additional shape analysis. Ligand binding sites in proteins are significantly deep, though this does not affect the binding affinity. Hyperthermostable proteins have a less deep surface but bury atoms more deeply, forming more spherical shapes than their mesophilic counterparts. Tunnels through proteins can be identified, for the first time tunnels that are winding or bifurcated can be analyzed. Pockets can be found all over the protein surface and these pockets can be tracked through time series, mutational series, or over protein families. All of these results are new and for the first time provide quantitative and statistical verification of some previous hypotheses about protein shape

Origin of Scaling Behavior of Protein Packing Density: A Sequential Monte Carlo Study of Compact Long Chain Polymers

Single domain proteins are thought to be tightly packed. The introduction of voids by mutations is often regarded as destabilizing. In this study we show that packing density for single domain proteins decreases with chain length. We find that the radius of gyration provides poor description of protein packing but the alpha contact number we introduce here characterize proteins well. We further demonstrate that protein-like scaling relationship between packing density and chain length is observed in off-lattice self-avoiding walks. A key problem in studying compact chain polymer is the attrition problem: It is difficult to generate independent samples of compact long self-avoiding walks. We develop an algorithm based on the framework of sequential Monte Carlo and succeed in generating populations of compact long chain off-lattice polymers up to length $N=2,000$. Results based on analysis of these chain polymers suggest that maintaining high packing density is only characteristic of short chain proteins. We found that the scaling behavior of packing density with chain length of proteins is a generic feature of random polymers satisfying loose constraint in compactness. We conclude that proteins are not optimized by evolution to eliminate packing voids.Comment: 9 pages, 10 figures. Accepted by J. Chem. Phy