Principles of Protein Structural Organization - G. Schultz 1982

Models, Depiction, and Documentation of Protein Structures
Abstract Concepts of Chain Folding
Distance Maps Between Cα-atoms

Chain folding can be represented by C_a–C_a distance maps. The atomic coordinates of C_a that determine chain folding are completely defined by the distance map between C_a atoms, since all coordinates (except for three) can be easily derived trigonometrically from these distances*. On the mutual C_a distance map, points in certain regions lie extremely close to one another. Therefore, contour lines of equal distances can be used, similar to contours on an electron density map. In this way, chain folding can be represented using a two-dimensional graph.

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Fig. 7.11. Inter-residue contact map of Lactate dehydrogenase [407]. Distances between the Ca atoms of the i-th and j-th residues are represented as a triangular matrix. The results are presented in contour rather than numerical form; contours are drawn at 16, 12, 8, and 4 Å. Various structural elements are identified relative to the diagonal; the triangles correspond to four geometrically well-defined subdomain structures. Two triangles on the N-terminal side represent ßaßaß units.

Secondary structures are identified by characteristic regions on the distance map. a-Helices correspond to short distances (dense contour lines) along the diagonal, whereas parallel ß-pleated sheets correspond to short distances along lines parallel to the diagonal (Fig. 7.11). For antiparallel ß-pleated sheets, as well as reverse turns, regions of short distances running perpendicular to the diagonal are characteristic [406]. The arrangement of lines on the C_a distance map serves as a unique "fingerprint" of chain folding. Moreover, insertions and deletions only partially disrupt the overall pattern by adding or deleting a matrix row. Therefore, such maps can be used to compare chain folding [407].

* Given N∙(N + 1)/2 distances (i.e., equations) for 3(N — 1) unknown coordinates, the problem can be solved uniquely for three C_a atoms and becomes overdetermined for four or more C_a atoms.

Other dependencies. The plot of nearest-neighbor residue correlations in Fig. 5.14a is also nothing more than a very specific C_a distance plot. However, it cannot be used to retrieve the original coordinates.

Lewis et al. [202] plotted the distance between C_aⁱ and Cⁱ⁺³ as a function of the residue number i. Such a plot makes it possible to identify all polypeptide chain turns (Section 5.1); it also has the advantage of revealing turns that do not conform to the types listed by Venkatachalam [199]. However, in this case as well, it is impossible to reconstruct the original coordinates, since there are only N—3 distances for 3N—3 coordinates.

A plot of radial distances from a given point in the molecule, such as a Trp residue, may be of interest in relevant spectroscopic studies. In such a plot, the chain is unwrapped in a closed circle, and the distances from all C_aⁱ atoms to the selected point are plotted radially.

A schematic distance plot showing the list of contacts between atoms of different subunits in an oligomeric protein is presented in Fig. 5.18d.

All C_a distance plots discussed in this section can be easily programmed using C_a coordinates provided, for example, by the Protein Data Bank [390].