How round is a protein? Exploring protein structures for globularity using conformal mapping

Tools for Characterizing Proteins: Circular Variance, Mutual Proximity, Chameleon Sequences, and Subsequence Propensities

For the characterization of various aspects of protein structures, four useful concepts are discussed: chameleon sequences, circular variance, mutual proximity, and a subsequence-based foldability score. These concepts were used in estimating foldability of globular, intrinsically disordered and fold-switching proteins, properties of protein-protein interfaces, quantifying sphericity, helping to improve protein-protein docking scores, and estimating the effect of mutations on stability. A conjecture about the Achilles' heel of proteins is presented as well.

LMap: Shape-Preserving Local Mappings for Biomedical Visualization

Visualization of medical organs and biological structures is a challenging task because of their complex geometry and the resultant occlusions. Global spherical and planar mapping techniques simplify the complex geometry and resolve the occlusions to aid in visualization. However, while resolving the occlusions these techniques do not preserve the geometric context, making them less suitable for mission-critical biomedical visualization tasks. In this paper, we present a shape-preserving local mapping technique for resolving occlusions locally while preserving the overall geometric context. More specifically, we present a novel visualization algorithm, LMap, for conformally parameterizing and deforming a selected local region-of-interest (ROI) on an arbitrary surface. The resultant shape-preserving local mappings help to visualize complex surfaces while preserving the overall geometric context. The algorithm is based on the robust and efficient extrinsic Ricci flow technique, and uses the dynamic Ricci flow algorithm to guarantee the existence of a local map for a selected ROI on an arbitrary surface. We show the effectiveness and efficacy of our method in three challenging use cases: (1) multimodal brain visualization, (2) optimal coverage of virtual colonoscopy centerline flythrough, and (3) molecular surface visualization.

Minimum action principle and shape dynamics

In this paper, we propose a new method for computing a distance between two shapes embedded in three-dimensional space. Instead of comparing directly the geometric properties of the two shapes, we measure the cost of deforming one of the two shapes into the other. The deformation is computed as the geodesic between the two shapes in the space of shapes. The geodesic is found as a minimizer of the Onsager-Machlup action, based on an elastic energy for shapes that we define. Its length is set to be the integral of the action along that path; it defines an intrinsic quasi-metric on the space of shapes. We illustrate applications of our method to geometric morphometrics using three datasets representing bones and teeth of primates. Experiments on these datasets show that the variational quasi-metric we have introduced performs remarkably well both in shape recognition and in identifying evolutionary patterns, with success rates similar to, and in some cases better than, those obtained by expert observers.

Molecular Surface Maps

We present Molecular Surface Maps, a novel, view-independent, and concise representation for molecular surfaces. It transfers the well-known world map metaphor to molecular visualization. Our application maps the complex molecular surface to a simple 2D representation through a spherical intermediate, the Molecular Surface Globe. The Molecular Surface Map concisely shows arbitrary attributes of the original molecular surface, such as biochemical properties or geometrical features. This results in an intuitive overview, which allows researchers to assess all molecular surface attributes at a glance. Our representation can be used as a visual summarization of a molecule's interface with its environment. In particular, Molecular Surface Maps simplify the analysis and comparison of different data sets or points in time. Furthermore, the map representation can be used in a Space-time Cube to analyze time-dependent data from molecular simulations without the need for animation. We show the feasibility of Molecular Surface Maps for different typical analysis tasks of biomolecular data.

Landmark-free geometric methods in biological shape analysis

In this paper, we propose a new approach for computing a distance between two shapes embedded in three-dimensional space. We take as input a pair of triangulated genus zero surfaces that are topologically equivalent to spheres with no holes or handles, and construct a discrete conformal map f between the surfaces. The conformal map is chosen to minimize a symmetric deformation energy Esd(f) which we introduce. This measures the distance of f from an isometry, i.e. a non-distorting correspondence. We show that the energy of the minimizing map gives a well-behaved metric on the space of genus zero surfaces. In contrast to most methods in this field, our approach does not rely on any assignment of landmarks on the two surfaces. We illustrate applications of our approach to geometric morphometrics using three datasets representing the bones and teeth of primates. Experiments on these datasets show that our approach performs remarkably well both in shape recognition and in identifying evolutionary patterns, with success rates similar to, and in some cases better than, those obtained by expert observers.

Use of circular variance to quantify the deviation of a macromolecule from the spherical shape

It is shown that the extent of deviation of a molecular shape from spherical can be characterized by comparing the distribution of the circular variances, a measure originally proposed to quantify angular spread, of the vectors from each atom to the rest of the molecule to the circular variance of a collection of atoms filling the unit sphere. Different measures for quantifying the difference between distribution are proposed and compared.