Programming Self-Assembly of DNA Tiles

Abstract

Abstract. SKI# is a Turing complete, language for programming in the aTAM model. A compilation technique provides a mapping from SKI # into finite sets of DNA Tiles that self-assemble in the aTAM model. Though such sets are always finite, the number of Tiles may be relevant, the construction of the Tiles may be heavy and the self-assembly can produce wrong molecular growth. In this paper we discuss the construction of a DNA Universal Machine as an aTAM interpreter for the entire Combinatory Logic, comparing it with the compiler based approach. Finally, Consensus is considered as a case study in distributed programming in the aTAM model and a further step in the design of SKI # and in the expressivity of aTAM (Wang Tiling) compared to Combinatory Logic and π-calculus. Computing with DNA TilesThe abstract Tile Assembly Model (aTAM) [1] is (together with its variants) the basic model of DNA Tiles Self-Assembly which is that part of molecular computing where Self-Assembly applies to DNA Tiles. DNA Tiles are molecules of DNA that have a fixed 2D or 3D connection structure and according to this connection structure, these molecules Self-Assemble to form structures which may grow even, indefinitely. Moreover, aTAM is an extension of the Wang formalism of the grid tiling, hence the Tiles are 2D, 4-connection structures that can be (graphically) represented by unit squares, as in Figure 1. Also aTAM SelfAssembly matches the Wang tiling laws (Tiles cover a grid and adjacent Tiles have touching sides of the same color) provided that the bio-chemical stability is satisfied. The equivalence between the Wang formalism and the Turing formalism, as shown in Figure 4, leads to the notions of program and of computation in both aTAM and Wang formalism 1 . A program is a finite set of Tiles. A computation is a grid tiling, i.e. a cover, without holes of the (infinite) plan, using (infinitely) many copies of the Tiles of the program. Figure 3 shows (a part) of a computation where each rectangle represents one sub(program)-computation of the (main) program. Though TM is a milestone in computation, it is not so good for programming because it is lacking of expressivity (for instance, for the composition of functions and of independent computations). For this reason, other formalisms have been studied for providing languages for programming Self-Assembly of DNA Tiles. Given any computable problem, these languages must provide a development environment for formalizing a solution to the problem and obtaining an aTAM program that Self-Assembles the Tiles according to such a solution. In [2] we considered these languages and proposed the new language SKI # . 2 The calculus SKI # and its compilation into SKI-Tileis a Turing complete, language designed for programming in the aTAM model and its fundamental characteristic is that it is a proper subset of Combinatory Logic (CL), hence it does not require variable symbols for dealing with bound variables. As a consequence, it provides mechanisms for function definition, application, composition ...