Fast computation, rotation, and comparison of low resolution spherical harmonic molecular surfaces

Abstract

A procedure that rapidly generates an approximate parametric representation of macromolecular surface shapes is described. The parametrization is expressed as an expansion of real spherical harmonic basis functions. The advantage of using a parametric representation is that a pair of surfaces can be matched by using a quasi-Newton algorithm to minimize a suitably chosen objective function. Spherical harmonics are a natural and convenient choice of basis function when the task is one of search in a rotational search space. In particular, rotations of a molecular surface can be simulated by rotating only the harmonic expansion coefficients. This rotational property is applied for the first time to the 3-dimensional molecular similarity problem in which a pair of similar macromolecular surfaces are to be maximally superposed. The method is demonstrated with the superposition of antibody heavy chain variable domains. Special attention is given to computational efficiency. The spherical harmonic expansion coefficients are determined using fast surface sampling and integration schemes based on the tessellation of a regular icosahedron. Low resolution surfaces can be generated and displayed in under 10 s and a pair of surfaces can be maximally superposed in under 3 s on a contemporary workstation. ᮊ

Crystal Structure of 4-Chlorocatechol 1,2-Dioxygenase from the Chlorophenol-utilizing Gram-positive Rhodococcus opacus 1CP

“…The intermediate active site region is made up by residues Ala-53, Ile-74, by the backbones of Gly-76 and Pro-77, and particularly by Cys-224, which has been suggested previously to interact with the chlorine substituents of the aromatic ring of the substrate molecules (42). (62). Several amino acid substitutions are observed in the catalytic cavity, the most evident being Asp-52/Pro-76, Ala-53/Gly-77, Phe-78/Leu-109, and Cys-224/Ala-254 (Rho 1,2-CCD/Ac 1,2-CTD), whereas the other residues are basically conserved, although significant backbone and side chain shifts are detected.…”

“…Global structures superimpositions were carried out by utilizing the matching algorithm implemented into the HEX 4.1 program (62). Least squares fit of the active site regions was performed using the McLachlan algorithm as implemented in the program ProFit 2.2 (www.bioinf.org.uk/software/profit/) specifying as the fitting subset the four amino acid ligands to the catalytic iron ions (63).…”

Characterization of anisotropy in high angular resolution diffusion‐weighted MRI

“…Unfortunately, unlike the discrete Fourier transform (DFT), for which there exists a matrix decomposition that allows the fast Fourier transform (FFT), no such algorithm exists for the SHT. Although a variety of algorithms have been proposed for the computation of spherical harmonic coefficients (23,24), no clear algorithm has emerged as clearly superior, and the subject remains an area of active research. Therefore, for the present work the coefficients were determined by direct computation of Eq.…”

Complex Copulatory Behavior and the Proximate Effect of Genital and Body Size Differences on Mechanical Reproductive Isolation in the Millipede Genus Parafontaria

“…Spherical harmonics, an extension of the classic Fourier transform, represent a three-dimensional (3-D) shape in terms of a sum of 3-D sines and cosines on a sphere (Brechbühler et al 1995;Ritchie and Kemp 1999;Shen et al 2007). We calculated the spherical harmonic representation of each surface using the algorithms described by Shen and Makedon (2006).…”