Section 22.1.1.1. Introduction

For geometric analysis, a protein consists of a set of points in three dimensions. This information corresponds to the actual data provided by the experiment, which are fundamentally of a geometric rather than chemical nature. That is, crystallography primarily tells one about the positions of atoms and perhaps an approximate atomic number, but not their charge or number of hydrogen bonds.

For the purposes of geometric calculation, each point has an assigned identification number and a position defined by three coordinates in a right-handed Cartesian system. (These coordinates will be based on the electron density for X-ray derived structures and on nuclear positions for those derived from neutron scattering. Each coordinate is usually assumed to have an accuracy between 0.5 and 1.0 Å.) Normally, only one additional characteristic is associated with each point: its size, usually measured by a van der Waals (VDW) radius. Furthermore, characteristics such as chemical nature and covalent connectivity, if needed, can be obtained from lookup tables keyed on the ID number.

Our model of a protein, thus, is the van der Waals envelope – the set of interlocking spheres drawn around each atomic centre. In brief, the geometric quantities of the model of particular concern in this section are its total surface area, total volume, the division of these totals among the amino-acid residues and individual atoms, and the description of the empty space (cavities) outside the van der Waals envelope. These values are then used in the analysis of protein structure and properties.

All the geometric properties of a protein (e.g. surfaces, volumes, distances etc.) are obviously interrelated. So the definition of one quantity, e.g. area, obviously impacts on how another, e.g. volume, can be consistently defined. Here, we will endeavour to present definitions for measuring protein volume, showing how they are related to various definitions of linear distance (VDW parameters) and surface. Further information related to macromolecular geometry, focusing on volumes, is available from http://www.molmovdb.org/geometry/ .