Section 17.2.3.1. Geometric representation

Geometric construction encompasses dots, lines and surfaces described by lists of three-dimensional coordinates and connectivity or by analytic or parametric expressions that can generate such information for rendering. Basically, geometric rendering involves a projection of the 3-D geometry onto a two-dimensional viewing plane using matrix transformations that account for the viewpoint, perspective and clipping within the viewing volume. For dots and lines, the computation may end there; the only depth information in the rendering might be geometric perspective. Additional depth information can be added by `atmospheric perspective' or depth cueing, where the brightness or colour is modulated by the depth values of the points (Fig. 17.2.3.3). Surface representations permit additional three-dimensional cues such as occlusion and shape-from-shading. Occlusion, or `hidden surface removal', and atmospheric perspective depend on maintaining depth information for all of the picture elements (pixels) in screen space. Such `depth-buffer' algorithms provide visibility information for a given viewpoint. Hardware z-buffers facilitate such calculations in the graphics pipeline. Lighting cues, such as shading, are attained by approximating the ambient, diffuse and specular reflectance of the geometry using Lambert's law. Because typical surfaces are composed of polyhedral facets, interpolation schemes are used to produce smooth shaded representations. The most common technique used for molecular graphics is known as Gouraud shading (Gouraud, 1971), which interpolates the shaded colour values assigned at the vertices across the polyhedral face (Fig. 17.2.3.4). Phong shading (Phong, 1975), a more accurate but costly technique, interpolates the values of the normals of the facets to produce a more realistic rendering. Shading templates for specific geometries, such as spheres, can give very smooth results without having to resort to large polyhedral descriptions for each sphere. In the past, this approach was implemented in the graphics hardware design, resulting in very fast sphere rendering for molecular applications. With the advent of consumer-level 3-D graphics these specialized features have become increasingly rare. Shadows may also provide useful three-dimensional cues in viewing molecular objects, but may also be confusing when they provide too much visual contrast or clutter. Ray tracing is a general technique for producing a complete reflectance and shadow rendering of a three-dimensional scene. It can, however, be very costly in computational time, since every light ray in the final image must be iteratively traced back to its source. Faster approximations for shadow rendering have been implemented that work well for molecular scenes (Gwilliam & Max 1989; Lauher, 1990).

Figure 17.2.3.3| top | pdf | Simple bonding diagram of a DNA structure (PDB code: 140D) (Mujeeb et al., 1993). On the left all lines are of equal intensity. On the right the lines are depth-cued to show which parts of the structure are closer to the viewer.

Figure 17.2.3.4| top | pdf | CPK representation of the same DNA structure as in Fig. 17.2.3.3. The model on the left uses highly tessellated spheres for the atoms, while the one on the right uses a coarser tessellation. The Gouraud shading model produces some lighting artifacts, such as the star-shaped highlights, which are most apparent on the right-hand figure. This is due to colour interpolation between the facet vertices.

A number of useful surface representations have been developed that describe the interaction of a molecule with the surrounding solvent. Perhaps the most widely used are the solvent-accessible surface (Lee & Richards, 1971) and the molecular surface (Richards, 1977; Connolly, 1983; Sanner et al., 1996), sometimes referred to as the Connolly or solvent-excluded surface (Fig. 17.2.3.5). For large molecules, such as proteins, which have many atoms buried from solvent, these surfaces have proven to be important in studying molecular interactions. They not only help to visualize the complementarity of interacting molecules, but they are also important in quantifying the entropic changes associated with solvent effects upon binding.

Figure 17.2.3.5| top | pdf | Solvent-excluded surface of the DNA structure using a water probe radius of 1.5 Å. The figure on the left shows a depth-cued dot surface, while the figure on the right shows a Gouraud-shaded triangulated surface.

Surface representations have opened up the possibilities of displaying a large variety of computed or experimental molecular properties by mappings onto the surface using colour coding. Electrostatic potential, hydrophobicity, sequence conservation, surface shape and any other characteristic of the molecule that can be projected onto the surface can be colour coded and displayed. Typically, this is accomplished by colouring the vertices of the surface mesh using a colour mapping or scale and interpolating the colour across the polygonal faces of the mesh. Since colour values are interpolated between vertices, this can produce unwanted colour artifacts if there are abrupt spatial changes in the properties displayed, or if the colour interpolation does not correspond to the property mapping (Fig. 17.2.3.6).

Figure 17.2.3.6| top | pdf | Hydrophobicity mapped onto a molecular surface. A spherical-harmonic approximation of the actin monomer solvent-excluded surface is shown. (a) Vertex colouring of a medium-mesh tessellated surface. The hydrophobicity colour scale is shown above. Notice that the colours blend between vertices, producing colour artifacts in relationship to the property scale. (b) The same medium-mesh representation as in (a) but using a property-based (one-dimensional) texture map, applying the same colour scale. Notice that the boundaries between the colours are distinct, even when they intersect vertices. Here the property value is interpolated. (c) A coarser mesh showing the same texture-mapping technique used in (b). Since the properties are only sampled at the vertices of the mesh, the finer details of the mapping are lost at this coarse triangulation. (d) A two-dimensional texture map created as a `Mercator-like' projection in spherical coordinates (θ, φ) from the same hydrophobicity scale used in (a)–(c). (e) The 2-D texture map shown in (d) mapped onto the medium-mesh actin surface. Notice that the linear nature of the interpolation seen in (b) using the same mesh is no longer present. (f) The same 2-D texture map applied to the coarse-mesh surface of actin. Notice that, unlike in (c), the detail of the texture map is preserved independent of the mesh.

Another method for projecting information on a surface is texture mapping, an approach that is analogous to applying an image `decal' onto the surface. In this approach, instead of assigning colours to the surface vertices, indices are assigned which serve as coordinates into the image to be mapped. Thus, a great amount of detail may be displayed on a surface mesh that has relatively few polygons describing the geometry. Texture mapping has been used extensively in highly interactive graphics, such as flight simulators and video games, since transformation of the geometry tends to be the computational bottleneck. Since texture mapping requires an indexing scheme that relates an image to a set of geometric vertices on the molecular surface, one needs a rational way of producing such a map. For one-dimensional texture maps, this is relatively easily accomplished by assigning the texture index of each vertex to an appropriate property scale (Teschner et al., 1994) (Fig. 17.2.3.6). This approach, however, is still tied to the level of triangulation. The more general two-dimensional or location-based surface texture mapping requires a global scheme for assigning texture indices. While the original molecular surface geometry does not lend itself directly to this type of texture mapping, recent analytical approximations to these surfaces, such as spherical-harmonics-based molecular surfaces (Duncan & Olson, 1993), provide simple hierarchical meshing schemes that can be easily texture mapped by using a `Mercator'-like projection between the image and the molecular surface (Duncan & Olson, 1995) (Fig. 17.2.3.6).