Section 17.2.3.2. Volumetric representation

Molecular properties are not confined to bonds and surfaces. These, in fact, are geometric constructs or abstractions of the time-dependent volumetric characteristics of molecules. In crystallography, electron density is the primary volumetric property to be visualized. Other derived or computed volumetric properties have become important to visualize as well, especially for macromolecules and their complexes. Electrostatic potential and field gradients help establish a molecule's effect at a distance, and a variety of volumetric atomic affinity potentials or grids (Goodford, 1985) can provide a picture of the types of molecular interactions that are energetically favoured.

Traditionally, electron density and other volumetric properties have been displayed as isocontour or isosurface representations, in which lines or surfaces of constant value are rendered in planes or in 3-D space to reveal characteristics of the volumetric property. Early computer-graphic pen plots of planar Fourier projections of electron density were usually sufficient to reveal atomic structure. As the molecules of study became larger and more complex, stacks of two-dimensional slices, creating three-dimensional isocontours, became necessary. The first computer representations of such 3-D isovalue surfaces were composed of three orthogonal 2-D plots – giving the impression of a `basket weave'. These plots depicted surface isocontours of the three-dimensional density, but had several problems from a computational and representational point of view. Since there were preferred directions of the contours (along the x, y and z axes), particular views were difficult to interpret. Additionally, the three orthogonal contours did not define a well formed triangulated geometric surface, so modern surface rendering techniques could not be applied directly. Moreover, the computation and recomputation of isosurfaces was relatively inefficient. An algorithm to compute directly the three-dimensional isosurface, called `marching cubes', was devised by Lorenson & Kline (1987) (Fig. 17.2.3.7). This algorithm speeded up the contouring process and enabled shaded surface representation of these surfaces. More recently, the re-computation of isosurfaces has been speeded up through the pre-computation of seed points that span all values of the volume. Using these seed points to flood-fill an isosurface of a given value reduces the contouring computation from a three-dimensional to a two-dimensional calculation. This enables the interactive modification of contour levels for even very large volumes (Bajaj et al., 1996)

Figure 17.2.3.7| top | pdf | Crystallographic electron-density isosurfaces, showing details of a protein iron–sulfur cluster. The surfaces are coloured by the gradient of the electron density, highlighting the iron and sulfur densities. Image by Michael Pique, The Scripps Research Institute.

While isocontours and isosurfaces have been the dominant modes of volumetric representation in molecular graphics, there has been a trend in scientific visualization to use alternative techniques, termed `direct volume rendering'. These methods bypass the construction of contours or surfaces to represent values within the volume, and instead use the scalar (or sometimes vector) values within the volume to produce an image directly. A general technique to accomplish this type of volumetric rendering is termed ray casting. If one considers a function that maps the scalar values of a volume into optical properties such as colour and opacity, one can simulate the passage of light rays through the volume, projecting the resulting rays onto the image plane. Given an appropriate transfer function or look-up table, the image represents the distribution of all of the values within the volume, circumventing the need to select only certain values as required for isocontouring. Such techniques have been used extensively in medical tomography (Höhne et al., 1989) and electron microscopy (Kremer et al., 1996; Hessler et al., 1996). Their use has also been explored in the rendering of volumetric properties of molecules (Goodsell et al., 1989). The images that are obtained by direct volume rendering tend to appear cloud-like, with soft edges. While this may be a `true' representation of the molecular characteristics, it is sometimes difficult to interpret visually. Techniques for imparting shading cues into these renderings by using gradient information in the volume has made this type of rendering more interpretable (Drebein et al., 1988). Another potential drawback to these methods is the cost of the computations. Since these methods require computing the effect of every element of the volume, the amount of computation scales as the cube of the linear dimension. There have been several clever software and hardware approaches to overcoming this problem. One novel hardware approach is to use three-dimensional texture mapping. By stacking texture-mapped planes to represent the colour and opacity of the volume, and using the hardware depth-buffer capabilities to compose the final image in the viewing plane, one can manipulate and render reasonable-size volumes (128³) at highly interactive rates. For molecular visualization, one would like to be able to represent both geometric and volumetric characteristics in the same rendering to visualize, for instance, model and data (Fig. 17.2.3.8). The three-dimensional texture-mapping approach enables this easily, since the planes upon which the volume data are mapped are in fact geometric. Other direct-volume rendering codes provide this capability as well.

Figure 17.2.3.8| top | pdf | A difference-electron-density map of a minor-groove drug binding in DNA. This image combines volumetric rendering of the electron density with a geometric model of the DNA molecule. Data courtesy of R. E. Dickerson, UCLA. Image by David Goodsell, The Scripps Research Institute.