Figure 3.
Relationship between cubic and dodecahedral particles constructed from similar trimeric building blocks and with quasi-equivalent contacts across the twofolds. Each sphere represents an E2p subunit. (A) A representation of the A. vinelandii cubic E2p core next to a violet cube C, with edge acube and vertex–center distance Rcube, such that the edge runs through the contacts between trimers. (B) Cube C inside a blue dodecahedron D with same length of edge, a, and same particle center. The threefolds of the cube coincide with a subset of the threefolds of the dodecahedron. The yellow cube E is expanded by a factor 1.62 compared with the purple cube C, such that 8 vertices of cube E coincide with 8 of the 20 vertices of the dodecahedron. (C) Cubes C and E, plus the dodecahedron D, viewed along a fourfold axis of the cubes coinciding with a twofold of the dodecahedron. (D) A schematic representation of the 24-meric A. vinelandii E2p cubic core is placed onto cube C. (E) The eight trimers of the original cube C moved outwards by a factor of 1.62 to reside on the vertices of the expanded yellow cube E as well as on the vertices of the dodecahedron D. (F) The eight trimers rotated by 37.7° about the threefold axes such that the quasi-equivalent contact surfaces face each other along a dodecahedral edge instead of a cubic edge (as was the case in Fig. 3E). (G) Twenty trimers generated by applying icosahedral symmetry operations onto one trimer of F. Note that the spheres do not quite touch each other after this operation (see also Fig. 4 B and C). (H) The dodecahedron D, with contacts quasi-equivalent to those in the cube C, shown in the same orientation as in B and decreased slightly in size compared with G such that the spheres touch each other (see also Fig. 4D).