Active Self-Assembly of Simple Units Using an Insertion Primitive

While computer science has given us a framework for determining the complexity and difficulty of solving computational problems, we do not yet have a theoretical framework for knowing what actions, behaviors, and life-like qualities can emerge from a given set of simple modular units. There has been much interest in developing models for programming active self-assembly processes in both the reconfigurable robotics community and the nanotechnology community. With respect to materials science and nanotechnology, the models proposed to date are either not yet implementable with our current understanding of synthetic chemistry or those that are implementable are limited to a set of features that do not capture the power of active components. Prior implementable models of molecular assembly only considered the passive behaviors of attaching and detaching from a complex. Inspired by the algorithmic tile assembly model [Winfree, 1996] and the graph grammar assembly model [Klavins et al., 2004], we describe a formal model for studying the complexity of self-assembled structures with active molecular components. In particular, we add an insertion primitive and we show a direct mapping of our model to a molecular implementation using DNA. We show that the expressive power of this language is stronger than regular languages, but at most as strong as context free grammars. Here, we explore the trade-off between the complexity of the system (in terms of the number of unit types), and the behavior of the system and speed of its assembly. We find that we can grow a line of any given length n in expected time O(log^3 n) using O(log^2 n) monomers. If we grow a line with k insertion rules, either the expected final length is infinite or the expected length at time t is at most (2t+2)^k^2, which is polynomial in t

Computing multi-scale organizations built through assembly

The ability to generate and control assembling structures built over many orders of magnitude is an unsolved challenge of engineering and science. Many of the presumed transformational benefits of nanotechnology and robotics are based directly on this capability. There are still significant theoretical difficulties associated with building such systems, though technology is rapidly ensuring that the tools needed are becoming available in chemical, electronic, and robotic domains. In this thesis a simulated, general-purpose computational prototype is developed which is capable of unlimited assembly and controlled by external input, as well as an additional prototype which, in structures, can emulate any other computing device. These devices are entirely finite-state and distributed in operation. Because of these properties and the unique ability to form unlimited size structures of unlimited computational power, the prototypes represent a novel and useful blueprint on which to base scalable assembly in other domains. A new assembling model of Computational Organization and Regulation over Assembly Levels (CORAL) is also introduced, providing the necessary framework for this investigation. The strict constraints of the CORAL model allow only an assembling unit of a single type, distributed control, and ensure that units cannot be reprogrammed - all reprogramming is done via assembly. Multiple units are instead structured into aggregate computational devices using a procedural or developmental approach. Well-defined comparison of computational power between levels of organization is ensured by the structure of the model. By eliminating ambiguity, the CORAL model provides a pragmatic answer to open questions regarding a framework for hierarchical organization. Finally, a comparison between the designed prototypes and units evolved using evolutionary algorithms is presented as a platform for further research into novel scalable assembly. Evolved units are capable of recursive pairing ability under the control of a signal, a primitive form of unlimited assembly, and do so via symmetry-breaking operations at each step. Heuristic evidence for a required minimal threshold of complexity is provided by the results, and challenges and limitations of the approach are identified for future evolutionary studies

Graph Transformations and Game Theory: A Generative Mechanism for Network Formation

Many systems can be described in terms of networks with characteristic structural properties. To better understand the formation and the dynamics of complex networks one can develop generative models. We propose here a generative model (named dynamic spatial game) that combines graph transformations and game theory. The idea is that a complex network is obtained by a sequence of node-based transformations determined by the interactions of nodes present in the network. We model the node-based transformations by using graph grammars and the interactions between the nodes by using game theory. We illustrate dynamic spatial games on a couple of examples: the role of cooperation in tissue formation and tumor development and the emergence of patterns during the formation of ecological networks