Simulation of one dimensional staged DNA tile assembly by the signal-passing hierarchical TAM

The Tile Assembly Model, and its many variants, is one of the most fundamental algorithmic assembly formalism within DNA nanotechnology. Most of the research in this field is focused on the complexity of assembling different shapes and patterns. In many cases, the assembly process is intrinsically deterministic and the final product is unique, while the assembly process might evolve through several possible assembly strategies. In this study we consider the controlled assembly of one dimensional tile structures according to predefined assembly graphs. We provide algorithmic approaches for developing such controlled assembly protocols, using the signal-passing Tile Assembly Model, as well as probabilistic approaches for investigating the assembly of such tile-based one-dimensional structures. As a byproduct, we build a generalized TAS (tile assembly system) which generate specific non-local non-associative algebraic computations and we assamble n × n squares using only one tile, which is a better efficiency compared to the staged assembly model.