Polypeptides and polynucleotides are natural programmable biopolymers that can self-assemble into complex tertiary structures. We describe a system analogous to designed DNA nanostructures in which protein coiled-coil (CC) dimers serve as building blocks for modular de novo design of polyhedral protein cages that efficiently self-assemble in vitro and in vivo. We produced and characterized >20 single-chain protein cages in three shapes-tetrahedron, four-sided pyramid, and triangular prism-with the largest containing >700 amino-acid residues and measuring 11 nm in diameter. Their stability and folding kinetics were similar to those of natural proteins. Solution small-angle X-ray scattering (SAXS), electron microscopy (EM), and biophysical analysis confirmed agreement of the expressed structures with the designs. We also demonstrated self-assembly of a tetrahedral structure in bacteria, mammalian cells, and mice without evidence of inflammation. A semi-automated computational design platform and a toolbox of CC building modules are provided to enable the design of protein cages in any polyhedral shape.

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Abstract:We consider the class of I-graphs Iðn; j; kÞ, which is a generalization over the class of the generalized Petersen graphs. We study different properties of I-graphs, such as connectedness, girth, and whether they are bipartite or vertex-transitive. We give an efficient test for isomorphism of I-graphs and characterize the automorphism groups of I-graphs. Regular bipartite graphs with girth at least 6 can be considered as Levi graphs of some symmetric combinatorial configurations. We consider configurations that arise from bipartite I-graphs. Some of them can be realized in the plane as cyclic astral configurations, i.e., as geometric configurations with maximal isometric symmetry. #

The so-called leapfrog transformation that was first introduced for fullerenes (trivalent polyhedra with 12 pentagonal faces and all other faces hexagonal) is generalised to general polyhedra and maps on surfaces. All spherical polyhedra can be classified according to their leapfrog order. A polyhedron is said to be of Clar type if there exists a set of faces that cover each vertex exactly once. It is shown that a fullerene is of Clar type if and only if it is a leapfrog transform of another fullerene. Several basic transformations on maps are defined by means of which t h e leapfrog and other transformations can be accomplished.

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