A geometric construction determines all permissible strand arrangements of sandwich proteins

For a large class of proteins called sandwich-like proteins (SPs), the secondary structures consist of two beta-sheets packed face-to-face, with each beta-sheet consisting typically of three to five beta-strands. An important step in the prediction of the three-dimensional structure of a SP is the prediction of its supersecondary structure, namely the prediction of the arrangement of the beta-strands in the two beta-sheets. Recently, significant progress in this direction was made, where it was shown that 91% of observed SPs form what we here call "canonical motifs." Here, we show that all canonical motifs can be constructed in a simple manner that is based on thermodynamic considerations and uses certain geometric structures. The number of these structures is much smaller than the number of possible strand arrangements. For instance, whereas for SPs consisting of six strands there exist a priori 900 possible strand arrangements, there exist only five geometric structures. Furthermore, the few motifs that are noncanonial can be constructed from canonical motifs by a simple procedure.