DNA origami has shown that principles of molecular recognition can be used to reshape biomolecules into nonphysiological forms. The design and synthesis of a continuous, 12-helix polypeptide that spontaneously self-assembles into a defined tetrahedron now demonstrates that protein structures can be similarly manipulated. Protein structures evolved through a complex interplay of cooperative interactions, and it is still very challenging to design new protein folds de novo. Here we present a strategy to design self-assembling polypeptide nanostructured polyhedra based on modularization using orthogonal dimerizing segments. We designed and experimentally demonstrated the formation of the tetrahedron that self-assembles from a single polypeptide chain comprising 12 concatenated coiled coil–forming segments separated by flexible peptide hinges. The path of the polypeptide chain is guided by a defined order of segments that traverse each of the six edges of the tetrahedron exactly twice, forming coiled-coil dimers with their corresponding partners. The coincidence of the polypeptide termini in the same vertex is demonstrated by reconstituting a split fluorescent protein in the polypeptide with the correct tetrahedral topology. Polypeptides with a deleted or scrambled segment order fail to self-assemble correctly. This design platform provides a foundation for constructing new topological polypeptide folds based on the set of orthogonal interacting polypeptide segments.

The domination game played on a graph G consists of two players, Dominator and Staller, who alternate taking turns choosing a vertex from G such that whenever a vertex is chosen by either player, at least one additional vertex is dominated. Dominator wishes to dominate the graph in as few steps as possible, and Staller wishes to delay the process as much as possible. The game domination number $\gamma_g(G)$ (resp., $\gamma_g'(G)$) is the number of vertices chosen when Dominator (resp., Staller) starts the game. An imagination strategy is developed as a general tool for proving results on the domination game. We show that for any graph G, $\gamma(G)\leq\gamma_g(G)\leq2\gamma(G)-1$, and that all possible values can be realized. It is proved that for any graph G, $\gamma_g(G)-1\leq\gamma'_g(G)\leq\gamma_g(G)+2$, and that most of the possibilities for mutual values of $\gamma_g(G)$ and $\gamma_g'(G)$ can be realized. A connection with Vizing's conjecture is established, and a lower bound on the game domination number of an arbitrary Cartesian product is proved. Several problems and conjectures are also stated.

For a ny n ≥ 1 a nd a ny k ≥ 1, a graph S( n, k) is i ntroduced. Vertices of S( n, k) are n-tuples over {1, 2,. . . k} a nd two n-tuples are adjace nt if they are i n a certai n relatio n. These graph