Computing the Volume, Surface Area, Mean, and Gaussian Curvatures of Molecules and Their Derivatives

Abstract

Geometry is crucial in our efforts to comprehend the structures and dynamics of biomolecules. For example, volume, surface area, and integrated mean and Gaussian curvature of the union of balls representing a molecule are used to quantify its interactions with the water surrounding it in the morphometric implicit solvent models. The Alpha Shape theory provides an accurate and reliable method for computing these geometric measures. In this paper, we derive homogeneous formulas for the expressions of these measures and their derivatives with respect to the atomic coordinates, and we provide algorithms that implement them into a new software package, AlphaMol. The only variables in these formulas are the interatomic distances, making them insensitive to translations and rotations. AlphaMol includes a sequential algorithm and a parallel algorithm. In the parallel version, we partition the atoms of the molecule of interest into 3D rectangular blocks, using a kd-tree algorithm. We then apply the sequential algorithm of AlphaMol to each block, augmented by a buffer zone to account for atoms whose ball representations may partially cover the block. The current parallel version of AlphaMol leads to a 20-fold speed-up compared to an independent serial implementation when using 32 processors. For instance, it takes 31 s to compute the geometric measures and derivatives of each atom in a viral capsid with more than 26 million atoms on 32 Intel processors running at 2.7 GHz. The presence of the buffer zones, however, leads to redundant computations, which ultimately limit the impact of using multiple processors. AlphaMol is available as an OpenSource software.

Analyzing the Geometry and Dynamics of Viral Structures: A Review of Computational Approaches Based on Alpha Shape Theory, Normal Mode Analysis, and Poisson–Boltzmann Theories

“…The alpha shape theory is a comprehensive method for measuring unions of balls using the Voronoi decomposition of the union [ 89 , 98 , 99 , 100 , 101 , 102 , 103 , 104 , 105 , 106 ]. It is fast, robust, and amenable to the study of large biomolecular systems [ 103 , 106 , 107 ]. In the following, we briefly introduce the theory and show applications for measuring the envelope of the WNV.…”

“…The next two subsections provide solutions to these two issues. Note that more comprehensive presentations of those solutions can be found in References [ 100 , 103 , 106 ].…”

Exact Analytical Algorithm for the Solvent-Accessible Surface Area and Derivatives in Implicit Solvent Molecular Simulations on GPUs

“…One reliable and accurate method was developed based on the Alpha Complex construction by which the geometric descriptors of molecules can be computed through union of balls. The Alpha Complex is obtained with the atomic coordinates and radii information and can be applied to decompose the space into small segments through which the surface areas are computed by using the Gauss–Bonnet theorem or inclusion–exclusion formulas. − Recently, the extension to other geometric measures, such as volumes, mean, and Gaussian curvatures, was proposed in which the parallel implementation can achieve speed improvement over the serial version. However, the full implementation of such a method on GPUs is still unavailable because each step in the method requires substantial effort to utilize the parallel computing property to improve performance.…”