Rules for connectivity of secondary structure elements in protein: Two-layer αβ sandwiches.

In protein structures, the fold is described according to the spatial arrangement of secondary structure elements (SSEs: α-helices and β-strands) and their connectivity. The connectivity or the pattern of links among SSEs is one of the most important factors for understanding the variety of protein folds. In this study, we introduced the connectivity strings that encode the connectivities by using the types, positions, and connections of SSEs, and computationally enumerated all the connectivities of two-layer αβ sandwiches. The calculated connectivities were compared with those in natural proteins determined using MICAN, a nonsequential structure comparison method. For 2α-4β, among 23,000 of all connectivities, only 48 were free from irregular connectivities such as loop crossing. Of these, only 20 were found in natural proteins and the superfamilies were biased toward certain types of connectivities. A similar disproportional distribution was confirmed for most of other spatial arrangements of SSEs in the two-layer αβ sandwiches. We found two connectivity rules that explain the bias well: the abundances of interlayer connecting loops that bridge SSEs in the distinct layers; and nonlocal β-strand pairs, two spatially adjacent β-strands located at discontinuous positions in the amino acid sequence. A two-dimensional plot of these two properties indicated that the two connectivity rules are not independent, which may be interpreted as a rule for the cooperativity of proteins.