Geometrical principles of homomeric β-barrels and β-helices: Application to modeling amyloid protofilaments.

Examples of homomeric β-helices and β-barrels have recently emerged. Here we generalize the theory for the shear number in β-barrels to encompass β-helices and homomeric structures. We introduce the concept of the "β-strip," the set of parallel or antiparallel neighboring strands, from which the whole helix can be generated giving it n-fold rotational symmetry. In this context, the shear number is interpreted as the sum around the helix of the fixed register shift between neighboring identical β-strips. Using this approach, we have derived relationships between helical width, pitch, angle between strand direction and helical axis, mass per length, register shift, and number of strands. The validity and unifying power of the method is demonstrated with known structures including α-hemolysin, T4 phage spike, cylindrin, and the HET-s(218-289) prion. From reported dimensions measured by X-ray fiber diffraction on amyloid fibrils, the relationships can be used to predict the register shift and the number of strands within amyloid protofilaments. This was used to construct models of transthyretin and Alzheimer β(40) amyloid protofilaments that comprise a single strip of in-register β-strands folded into a "β-strip helix." Results suggest both stabilization of an individual β-strip helix and growth by addition of further β-strip helices can involve the same pair of sequence segments associating with β-sheet hydrogen bonding at the same register shift. This process would be aided by a repeat sequence. Hence, understanding how the register shift (as the distance between repeat sequences) relates to helical dimensions will be useful for nanotube design.