The accessible surface of a macromolecule is a key determinant of its action. The definition(12) and computer implementation(13) of such surfaces have had a profound effect on subsequent visualizations of molecular form and function. Usually such surfaces are displayed on interactive graphics devices as dots(10) or triangles(14) and may be colour-coded to highlight chemical properties.
Splines have also been used to model molecular surfaces. Triangular surface patches were used to accurately model small molecule shapes(15). Spline ``contour lines'' have been constructed from dot surfaces to represent the shapes of proteins(16). A ``sphere'' of 64 patches collapsed onto the dot surface of a protein gave a smooth but ``blobby'' approximation of the shape (see Methods). The images are somewhat similar to those produced by modeling protein shape with spherical harmonics(17).
This method seeks not so much an exact surface construction, but an attractive approximate surface representation described with a minimal number of parameters. As mentioned in a previous paper(7), nucleic acid ribbons ``could be sectioned to show such information as base type, base tilt-angle or sugar conformation.'' This method further allows the display of key DNA surface features --- those involved in the ``reading'' of the DNA by enzymes and regulatory proteins.
B-DNA is about 20 A in diameter, with very close to 10 base pairs per helical repeat. The base pair planes are tilted about 6 degrees with respect to the helix axis and the spacing between adjacent pairs is about 3.4 A. The regularity of the structure allows one to infer the atomic positions from just the sequence, or alternatively from a reduced atom representation used in simulations where only three ``atoms'' are used per base(2). The latter case is fit by approximating the local helix axis.
The method is developed from ideal B-DNA. Although tRNA doesn't quite adopt this ideal configuration, the DNurbs based on the DNA fit rather well (Figure 7). Close inspection of the figure reveals a small section in the acceptor stem where the ribbon is not enclosed by the DNurbs.
Both the Bezier and B-spline surface loop approximations are specified with N u-values; however, Bezier curves require three times the number of control points. Also, the Bezier curve segment connections are not as smooth (C-1 vs C-2 continuity.) The Bezier form can be used if greater fidelity to the actual surface tangent is desired. For a more faithful representation of the surface, the non-planarity of the sugar rings in the minor groove should be taken into account. There are doubtless many variations on the theme of this work.
The data structures employed here could be used to create polygonal approximations such as triangular or quadrilateral meshes, but NURBS offer several advantages: 1) a surface patch is specified with a small number of parameters, 2) the NURBS are automatically decomposed into triangles which are a staple of rendering systems, 3) NURBS provide a particularly easy texture mapping. Such texturing represents the trend in evolving graphics standards.
The DNurbs offer a simplified, informative and aesthetically pleasing representation of complementary DNA. They may also offer a performance advantage on available hardware.
Steve Harvey had suggested the idea of a ``Barber's Pole'' representation of DNA several years ago. Charlie Bugg and NASA grant NAGW-813 and NCI grant CA-13148 have provided continuing encouragement and support.