Powers and Behaviors of Directed Self-assembly

In nature there are a variety of self-assembling systems occurring at varying scales which give rise to incredibly complex behaviors. Theoretical models of self-assembly allow us to gain insight into the fundamental nature of self-assembly independent of the specific physical implementation. In Winfree’s abstract tile assembly model (aTAM), the atomic components are unit square “tiles” which have “glues” on their four sides. Beginning from a seed assembly, these tiles attach one at a time during the assembly process in an asynchronous and nondeterministic manner. We can gain valuable insights into the nature of self-assembly by comparing different models of self-assembly which use fundamentally different mechanisms for local interactions. A powerful notion which allows us to compare models of self-assembly is simulation. The first result of this thesis examines the role of non-determinism in simulation. It shows that the universal simulation of directed aTAM systems requires undirectedness. A tile assembly model is said to be directed if it always assembles the same final assembly. We distinguish between two types of aTAM systems: cooperative systems and noncooperative systems. In cooperative aTAM systems, we are able to enforce that in order for a tile to attach to an assembly, the glues of a tile must match two or more glues of neighboring tiles. On the other hand, in non-cooperative aTAM systems, tiles are able to attach to an assembly provided that one of the tile’s glues match an exposed glue on the assembly. It is well known that the cooperative aTAM is computationally universal, and it is conjectured that the non-cooperative aTAM is not computationally universal. For our second result, we show that if we allow tiles to be polygons with six or more sides, then the class of non-cooperative systems is capable of universal computation. On the other hand, we show that the class of systems consisting of polygons with six or less sides is not capable of computing using any of the currently known methods. Acknowledgements I would like to give a special thanks to my adviser, Matthew Patitz. Matt was a great mentor both in research and outside of research. He taught me how to work hard and ingrained a work ethic in me which will aid me throughout my career. Matt introduced me to a community of great people who I had the pleasure of collaborating with. His support allowed me to travel around the world to collaborate with a wonderful group of people. In addition, I would like to thank Jacob Hendricks whose presence greatly improved my experience in graduate school. Jacob’s one of the most intelligent people that I have ever met. He had a profound impact on the way I approach problems and evaluate proofs. His work ethic inspired me to work harder, and we spent many nights in the lab together during marathon working sessions. It was inspiring to see his combination of brilliance and work ethic pay off when he received an offer for a tenure track position. Above all, he was a great friend and really made graduate school a lot more enjoyable. Finally, thanks to the entire self-assembly community. Dave Doty’s encyclopedic knowledge and elegant rigor inspired me to expand my knowledge and improved my proof writing abilities. Shinosuke Seki was a great mentor and always a welcoming host during my visits to Tokyo. Damien Woods is one of the most kind, genuine people that I have met and is always willing to listen to my ideas on new research directions. Dedication I would like to thank my parents for their support throughout my educational career.