A topological nomenclature for protein structure.

In recent years, technical advances in X-ray crystallography and multi-dimensional NMR, coupled with comparable advances in protein engineering and computing, have led to an unprecedented growth in the number of solved protein structures, their number at least doubling every few years. This embarrassment of riches has, in turn, led to an increasing interest in the molecular systematics of protein structures (Wodak, 1996). The remarkable commonalties of protein structure mean that the search for structural relationships is a rich and rewarding avenue of investigation. With only relatively few exceptions, the three-dimensional structure of proteins is characterized by patterns of repeating secondary structure: α-helices and β-sheets (Chothia and Finkelstein, 1990). Structural topology— the relationship between the sequential ordering of such secondary structure elements and their spatial organisation— is one of the principal means by which protein structures and their commonalities can be classified, categorized and compared. The topologies exhibited by β-sheets—the relationship between the sequential ordering of strands and their hydrogen-bonded connectedness in space—have been studied particularly well. Two systems of nomenclature have been proposed to describe β-sheet topology. The first and better known of these is that developed by Richardson (1977): it is based on following a path through the sequence order of strands and noting their separation within the β-sheet. Only the connections between strands which follow each other in the sequence are considered, each connection having three properties: the physical separation, within the sheet, of the two participating strands; whether the strands are parallel or antiparallel; and whether the connection involves going forward or backward in the topology of the sheet. The other, less widely known, approach is based on following a path through the connectedness of neighbouring strands and noting their sequence separation. Rather than following the sequence order of strands a depth-first path is traced through the sheet and the labelled connections express the sequence separation between physically adjacent strands, whether this goes backwards or forwards in the sequence and whether the strands are parallel or antiparallel. Several authors have applied graph theoretical methods to facilitate the study of β-sheet topologies. Koch et al. (1992) introduced the problem and formulated approaches to searching and comparing different topologies. They noted that protein structures are complex topological objects possessed of properties and characteristics not readily expressed by a consecutive