The inertia-equivalent ellipsoid: a link between atomic structure and low-resolution models of small globular proteins determined by small-angle X-ray scattering

Abstract

Low‐resolution three‐parameter models of the shape of a biopolymer in solution can be determined by a new indirect method from small‐angle X‐ray scattering without contrast‐variation experiments. The basic low‐resolution model employed is a triaxial ellipsoid – the inertia‐equivalent ellipsoid (IEE). The IEE is related to the tensor of inertia of a body and the eigenvalues and eigenvectors of this tensor can be calculated directly from the atomic coordinates and from the homogeneous solvent‐excluded body of a molecule. The IEE defines a mean molecular surface (like the sea level on earth) which models the molecular shape adequately if the IEE volume is not more than 30% larger than the dry volume of the molecule. Approximately 10 to 15% of the solvent‐excluded volume is outside the ellipsoid; the radii of gyration of the IEE and of the homogeneous molecular body are identical. The largest diameter of the IEE is about 5 to 15% (~0.2–0.8 nm) smaller than the maximum dimension of globular molecules with molecular masses smaller than 65000 daltons. From the scattering curve of a molecule in solution the IEE can be determined by a calibration procedure. 29 proteins of known crystal structure have been used as a random sample. Systematic differences between the axes of the IEE, calculated directly from the structure, and the axes of the scattering‐equivalent ellipsoids of revolution, estimated from the scattering curve of the molecule in solution, are used to derive correction factors for the axial dimensions. Distortions of model dimensions of 20 to 40% (up to 1 nm), caused by misinterpretation of scattering contributions from electron density fluctuations within the molecule, are reduced to a quarter by applying these correction factors to the axes of the scattering‐equivalent ellipsoids of revolution. In a computer experiment the axes of the inertia‐equivalent ellipsoids have been determined for a further nine proteins with the same accuracy. The automated estimation of the IEE from the scattering curve of a molecule in solution is realized by the Fortran77 program AUTOIEE.

Monte Carlo simulation studies of the size and shape of linear and star-branched polymers embedded in the tetrahedral lattice

“…Figures in squared lattice units for a variety of chain architectures in several spatial dimensions. Even, an experimental method based on smallangle X-ray scattering has been evaluated for the determination of equivalent ellipsoids of globular proteins 174) . In all cases, starting point of the description is a symmetric tensor of the second kind …”

A dynamic Monte Carlo algorithm for exploration of dense conformational spaces in heteropolymers

“…This could be very useful in alleviating attrition problems during the later stages of folding. Another utility of developing effective dynamic schemes for compact configurational spaces is that they could be used to determine the set of conformations consistent with experimental data about the shape and size of macromolecules obtained from small angle x-ray scattering 14 or hydrodynamic experiments. There has been very little work done in this area because of the difficulty with exploring dense configurations.…”

Frequency Dependent Non- Thermal Effects of Oscillating Electric Fields in the Microwave Region on the Properties of a Solvated Lysozyme System: A Molecular Dynamics Study

“…Another result related to the above picture concerns the impact of the field on the density of the first hydration layer. The volume of S1 water layer required for these calculations have been determined by approximating lysozyme as an ellipsoid [ 50 ]. A thickness of 3 Å has been assumed for S1, as determined by combined SAXS and SANS experiments [ 51 ].…”