Principles determining the structure of β-sheet barrels in proteins I. A theoretical analysis

The major feature of many proteins is a large β-sheet that twists and coils to form a closed structure in which the first strand is hydrogen bonded to the last: the β-sheet barrel. McLachlan classified barrels in terms of two integral parameters: the number of strands in the β-sheet, n, and the "shear number", S, a measure of the stagger of the strands in the β-sheet. He showed that the mean radius of a barrel and the extent to which strands are tilted relative to its axis are determined by the values of n and S. Here we show that the (n, S) values determine all the other general structural features of regular β-sheet barrels, in particular, optimal values of the twist and coiling angles that produce the closed β-sheet, the hyperboloidal shape and the arrangement of residues in the barrel interior. Consideration of the residue arrangements in the interiors of different potential barrel structures, and of side-chain volumes, suggest that barrels, in which the interiors are close packed by the residues in β-sheets with good geometries, have structures that correspond to one of only ten different combinations of n and S. In the accompanying paper, we demonstrate, by an analysis of all observed protein structures that contain β-sheet barrels and for which atomic co-ordinates are available, the validity of these theoretical results.