Differential Geometry and Polymer Conformation. 2. Development of a Conformational Distance Function

In paper 1, we described a differential-geometric representation of protein backbone structure which reflects the influence of both short- and medium-range interactions. In this paper, we continue our development of this representation, particularly of its application to the comparison of protein structures. Initially, two mathematical features of the representation are discussed. They are (a) the dependence of the curvature (ki) and torsion (ti) on the sign of pp+i (where p is the scalar product of two nonperpendicular unit vectors at the ith and (i- l)th a-carbon atoms, respectively), and the conditions under which pp+1 changes sign, and (b) the existence (and size and shape) of a gap in the (k, t) plane, centered on the axis. Both of these features lead to discontinuities in and r, analogous to those which exist in systems with periodic boundary conditions. Application of these results to the distribution of residues in a large protein sample in (k, t) space reveals the existence of a continuum of bend structures, ranging from aR-like bends through flat structures to aL-like bends. A corrected treatment of chain handedness is then given, thereby greatly increasing the utility of the differential-geometric method for determining chain handedness. A discussion is given of the inversion of the (k, t) representation to recover C” coordinates, and results of such an inversion are presented for bovine pancreatic trypsin inhibitor. We then develop a function to represent the conformational distance between structures represented by any two points in the (k, t) plane. The utility of this distance function is demonstrated by comparison with a new superposition method for backbone segments composed of four C” atoms which requires no computational optimization of superposition. The differential geometric comparison method is applied to two specific cases: (a) the comparison of reduced and oxidized cytochrome c, treated preliminarily in paper 1, and (b) the comparison of oxidized cytochrome c and glyceraldehyde phosphate dehydrogenase. The results demonstrate the validity and utility of the differential-geometric comparison method. This method is then used to illustrate the detection of conformational similarity between two portions of a protein molecule, the test case being two domains of ferredoxin. Finally, a general discussion of the differential-geometric representation is given. It is pointed out that this representation is complementary to other representations of protein structure in current use, in that it operates on a length scale (that of four C” atoms) not conveniently treated by these others. It is therefore capable of revealing structural features that are not transparently evident in other representations. In addition, it is noted that comparison of the (k, t) and (ΦΨ) representations reveals a type of degeneracy in protein folding in which a given type of structure on one length scale can be attained in several ways through combinations of structures on smaller length scales.