Helices and helix packings derived from the {3, 3, 5} polytope

Abstract

The {3, 3, 5}-polytope is described and used as template for dense structures. Then larger structures are derived from this polytope, using disclinations. That needs a study of symmetries in this polytope. A discretised version of the Hopf fibration is presented and used in order to generate a family of new polytopes. It is possible to gathered vertices of these structures on several helices and then to consider geometrical relation between these helices. This study is govern by biological consideration of helix building molecules, but the final purpose is to have a geometrical tool to study geometrical relationship occurring between different helices or strands. This can occur for instance in protein folding studies.

An Amino Acid Packing Code for α-Helical Structure and Protein Design

“…The XY•H socket characterizes not only the packing innate to α-helical structure, but also the role that packing at the level of 2° structure has in establishing higher order 3° and 4° interactions. Although the XY•H socket motif is found in other models 19;24;29;32;33 as far back as Efimov 30;31 and notably Lim 27;28 , neither recognizes the socket as the primary motif to protein packing, but rather complicate the description of packing with more general combinations of other motifs. Because we had developed a precise vocabulary that exactly describes packing 1 , we could eliminate dependent packing groups that were redundant to the description of packing and derive that the single knob-socket motif describes α-helical packing.…”

“…18 More elaborate analyses of protein packing, including our own, consider multi-body arrangements of residues. 1;18–33 While these studies have generally found side-chain interactions to be broadly regular and tetrahedral, none so far has been able to develop a coherent description of protein packing. Another approach employs graph theory to organize protein interactions in hopes of identifying some common patterns across fold types.…”

Frustrated self-assembly of non-Euclidean crystals of nanoparticles

“…20 such (homochiral) terahelices organized in four nested toroidal shells form the 600-cell regular polytope (Fig. 2 f–i) which is a regular tiling of S 3 with tetrahedra 48 . The global structure of linked rings has the topology of the Hopf fibration 49 , 50 (Fig.…”

On the Structure of Icosahedral Keplerates and Their Derivatives

“…2a), leads to the formation of a helix consisting of interpenetrating icosahedra (Fig. 2b) [19]. The heavy line in Fig.…”