Design formalism for DNA self-assembly of polyhedral skeletons using rigid tiles

We describe the half-lap model, a mathematical framework that captures the geometric constraints of rigid tiles that are branched junction molecules used as building blocks for tile-based DNA self-assembly. The model captures not only the combinatorial structures of the sets of cohesive ends on the tiles, but also the specific geometry of the inter-arm angles of the tiles and most critically the relative orientations of adhering tiles. We illustrate the functionality of the model by providing provably optimal DNA self-assembly strategies to construct Platonic and Archimedean 3-regular polyhedral skeletons and computing the minimum number of tile types and bond-edge types for each target structure. We further demonstrate the utility of the model by using it to analyze the benefits and limitations of palindromic rigid tiles. Moreover, we give explicit combinatorial and geometric descriptions of the tiles needed for each construction.