A unified convention for biological assemblies with helical symmetry

Abstract

Assemblies with helical symmetry can be conveniently formulated in many distinct ways. Here, a new convention is presented which unifies the two most commonly used helical systems for generating helical assemblies from asymmetric units determined by X-ray fibre diffraction and EM imaging. A helical assembly is viewed as being composed of identical repetitive units in a one-or two-dimensional lattice, named 1-D and 2-D helical systems, respectively. The unification suggests that a new helical description with only four parameters [n 1 , n 2 , twist, rise], which is called the augmented 1-D helical system, can generate the complete set of helical arrangements, including coverage of helical discontinuities (seams). A unified four-parameter characterization implies similar parameters for similar assemblies, can eliminate errors in reproducing structures of helical assemblies and facilitates the generation of polymorphic ensembles from helical atomic models or EM density maps. Further, guidelines are provided for such a unique description that reflects the structural signature of an assembly, as well as rules for manipulating the helical symmetry presentation.

Structural resolution of inorganic nanotubes with complex stoichiometry

“…1 and to unit cell’s parameters (modulus of unit cell vectors and angle between them), allowing unit cell distortions from the perfect hexagonal cell. Indeed, helical symmetries 44 allow one to generate a whole nanotube structure with inner radius R i , outer radius R e , and period T from any planar unit cell. It should be noted here that since X-rays are rather insensitive to H atoms, we considered virtual atoms at the electronic center of charge of OH and CH 3 groups, with X-ray form factors equal to the weighted sum of those of their constituents.…”

“…It is also transferable to other large unit cell single-walled inorganic nanotubes currently synthesized and with potential applications in various fields. In the case of INTs, the unit cell is a slightly deformed hexagonal one but one should underline here that the use of helical symmetries imposes no constraint on the elementary cell 44 . Moreover, the role of the synthesis medium and the temperature, which is crucial in the synthesis of SWNTs in suspension 1 , 3 , is included in the WAXS fitting procedure we propose, through the choice of the internal and external radii as fitting parameters.…”

“…A different approach is proposed here. It is based on the reduction of the number of independent atomic positions to be fitted to WAXS diagrams, thanks to the use of helical symmetries 44 and to a semi-empirical energy minimization. The strategy enables us to determine the atomic structure of both (OH) 3 Al 2 O 3 SiCH 3 and (OH) 3 Al 2 O 3 GeCH 3 nanotubes.…”

In vitro assembly of the Rous Sarcoma Virus capsid protein into hexamer tubes at physiological temperature

“…The significance of these particular helical families arises from the close relationship between the CA tubes, and the planar sheets of CA hexamers characterized previously 20 . This relationship suggests an alternate description of the helical symmetry, involving the generating vectors of a planar lattice, and the circumferential vector, that describes the geometric relationship between a helical and planar lattice 82 , 85 , 86 .…”

The role of chirality and plastic crystallinity in the optical and mechanical properties of chlorosomes

“…The last two parameters-radius and chiral angle-define the chiral vector relating to the chiral indices (n 1 , n 2 ) as C = n 1 a + n 2 b. A key step in the building process is to convert the supramolecular chirality to a local twist angle a and rise distance h needed for the growth of unit cells in a chiral fashion (He and Scheres, 2017;Tsai and Nussinov, 2011).…”