Characterization of near-miss connectivity-invariant homogeneous convex polyhedral cages

Near-Miss Symmetric Polyhedral Cages

Following the experimental discovery of several nearly symmetric protein cages, we define the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages made out of P-gons. We use group theory to parameterize the possible configurations and we minimize the irregularity of the P-gons numerically to construct all such polyhedral cages for P=6 to P=20 with deformation of up to 10%.

Shape-Morphing of an Artificial Protein Cage with Unusual Geometry Induced by a Single Amino Acid Change

Artificial protein cages are constructed from multiple protein subunits. The interaction between the subunits, notably the angle formed between them, controls the geometry of the resulting cage. Here, using the artificial protein cage, “TRAP-cage”, we show that a simple alteration in the position of a single amino acid responsible for Au(I)-mediated subunit–subunit interactions in the constituent ring-shaped building blocks results in a more acute dihedral angle between them. In turn, this causes a dramatic shift in the structure from a 24-ring cage with an octahedral symmetry to a 20-ring cage with a C2 symmetry. This symmetry change is accompanied by a decrease in the number of Au(I)-mediated bonds between cysteines and a concomitant change in biophysical properties of the cage.

Artificial Protein Cage with Unusual Geometry and Regularly Embedded Gold Nanoparticles

Artificial protein cages have great potential in a number of areas including cargo capture and delivery and as artificial vaccines. Here, we investigate an artificial protein cage whose assembly is triggered by gold nanoparticles. Using biochemical and biophysical methods we were able to determine both the mechanical properties and the gross compositional features of the cage which, combined with mathematical models and biophysical data, allowed the structure of the cage to be predicted. The accuracy of the overall geometrical prediction was confirmed by the cryo-EM structure determined to sub-5 Å resolution. This showed the cage to be nonregular but similar to a dodecahedron, being constructed from 12 11-membered rings. Surprisingly, the structure revealed that the cage also contained a single, small gold nanoparticle at each 3-fold axis meaning that each cage acts as a synthetic framework for regular arrangement of 20 gold nanoparticles in a three-dimensional lattice.

Characterization of near-miss connectivity-invariant homogeneous convex polyhedral cages

Following the discovery of a nearly symmetric protein cage, we introduce the new mathematical concept of a near-miss polyhedral cage (p-cage) as an assembly of nearly regular polygons with holes between them. We then introduce the concept of the connectivity-invariant p-cage and show that they are related to the symmetry of uniform polyhedra. We use this relation, combined with a numerical optimization method, to characterize some classes of near-miss connectivity-invariant p-cages with a deformation below 10% and faces with up to 17 edges.