Tile Self-Assembly

The goal of DNA computing is to perform computations using biological components rather than traditional electronic circuits. Molecules can combine to form larger structures if they contain complementary strands of DNA. These structures, called assemblies, can be interpreted as the output of a program. The inherently parallel nature of chemical reactions enables this model of computation to simultaneously explore all paths in a nondeterministic computation tree [Adl94]. In this project, we explore an abstraction of DNA computing called the Tile Assembly Model (TAM) [Win98], which is known to be Turing complete. The TAM models molecules as a set of square tiles that self-assemble to tile the plane in a square lattice. A program consists of a finite set of tile types (it is assumed that there is an unlimited number of tiles of each type available), which may be placed in any location on the lattice, but not rotated. Once a tile has been placed in the lattice, it cannot be moved or removed. The program begins with a single seed tile to which other tiles can bond. The program continues running until it is not possible to add any more tiles to the assembly. The resulting assembly is called a terminal assembly. Two tiles can bond depending on their types. A tile’s type is determined by the glue on each of its four sides. Each glue has a particular color and integral strength. A tile may bond with another tile if their adjacent sides have glues of the same color and strength at least 2. For example, in Figure 1a, tile b can attach to tile a because the glue that they share has strength 2. Tile c cannot attach because its bottom glue only has strength 1. In addition to two individual tiles bonding, a tile can bond to an existing assembly of multiple tiles if the sum of glue strengths of all matching glues is at least 2 (see Figure 1b). Mismatched glues do not prevent bonding; they simply do not affect the formation of new bonds. Glues may also be null; a null glue cannot bind two tiles regardless of the strength (a null glue is equivalent to a glue of strength 0). One way of creating a correct tile set is to simply create a specific set of glues for each position in the desired terminal assembly. However, this requires an extremely large number of different tile types and glues, which can be difficult to design in practice. Hence an important concern in designing tile sets is using a