Optimal Staged Self-Assembly of General Shapes

We analyze the number of tile types $t$, bins $b$, and stages necessary to assemble $n \times n$ squares and scaled shapes in the staged tile assembly model. For $n \times n$ squares, we prove $\mathcal{O}(\frac{\log{n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and $\Omega(\frac{\log{n} - tb - t\log t}{b^2})$ are necessary for almost all $n$. For shapes $S$ with Kolmogorov complexity $K(S)$, we prove $\mathcal{O}(\frac{K(S) - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and $\Omega(\frac{K(S) - tb - t\log t}{b^2})$ are necessary to assemble a scaled version of $S$, for almost all $S$. We obtain similarly tight bounds when the more powerful flexible glues are permitted.Comment: Abstract version appeared in ESA 201

Active Self-Assembly of Algorithmic Shapes and Patterns in Polylogarithmic Time

We describe a computational model for studying the complexity of self-assembled structures with active molecular components. Our model captures notions of growth and movement ubiquitous in biological systems. The model is inspired by biology's fantastic ability to assemble biomolecules that form systems with complicated structure and dynamics, from molecular motors that walk on rigid tracks and proteins that dynamically alter the structure of the cell during mitosis, to embryonic development where large-scale complicated organisms efficiently grow from a single cell. Using this active self-assembly model, we show how to efficiently self-assemble shapes and patterns from simple monomers. For example, we show how to grow a line of monomers in time and number of monomer states that is merely logarithmic in the length of the line. Our main results show how to grow arbitrary connected two-dimensional geometric shapes and patterns in expected time that is polylogarithmic in the size of the shape, plus roughly the time required to run a Turing machine deciding whether or not a given pixel is in the shape. We do this while keeping the number of monomer types logarithmic in shape size, plus those monomers required by the Kolmogorov complexity of the shape or pattern. This work thus highlights the efficiency advantages of active self-assembly over passive self-assembly and motivates experimental effort to construct general-purpose active molecular self-assembly systems

Optimization of supply diversity for the self-assembly of simple objects in two and three dimensions

The field of algorithmic self-assembly is concerned with the design and analysis of self-assembly systems from a computational perspective, that is, from the perspective of mathematical problems whose study may give insight into the natural processes through which elementary objects self-assemble into more complex ones. One of the main problems of algorithmic self-assembly is the minimum tile set problem (MTSP), which asks for a collection of types of elementary objects (called tiles) to be found for the self-assembly of an object having a pre-established shape. Such a collection is to be as concise as possible, thus minimizing supply diversity, while satisfying a set of stringent constraints having to do with the termination and other properties of the self-assembly process from its tile types. We present a study of what we think is the first practical approach to MTSP. Our study starts with the introduction of an evolutionary heuristic to tackle MTSP and includes results from extensive experimentation with the heuristic on the self-assembly of simple objects in two and three dimensions. The heuristic we introduce combines classic elements from the field of evolutionary computation with a problem-specific variant of Pareto dominance into a multi-objective approach to MTSP.Comment: Minor typos correcte

Limiting the valence: advancements and new perspectives on patchy colloids, soft functionalized nanoparticles and biomolecules

Limited bonding valence, usually accompanied by well-defined directional interactions and selective bonding mechanisms, is nowadays considered among the key ingredients to create complex structures with tailored properties: even though isotropically interacting units already guarantee access to a vast range of functional materials, anisotropic interactions can provide extra instructions to steer the assembly of specific architectures. The anisotropy of effective interactions gives rise to a wealth of self-assembled structures both in the realm of suitably synthesized nano- and micro-sized building blocks and in nature, where the isotropy of interactions is often a zero-th order description of the complicated reality. In this review, we span a vast range of systems characterized by limited bonding valence, from patchy colloids of new generation to polymer-based functionalized nanoparticles, DNA-based systems and proteins, and describe how the interaction patterns of the single building blocks can be designed to tailor the properties of the target final structures

Contrasting Geometric Variations of Mathematical Models of Self-assembling Systems

Self-assembly is the process by which complex systems are formed and behave due to the interactions of relatively simple units. In this thesis, we explore multiple augmentations of well known models of self-assembly to gain a better understanding of the roles that geometry and space play in their dynamics. We begin in the abstract Tile Assembly Model (aTAM) with some examples and a brief survey of previous results to provide a foundation. We then introduce the Geometric Thermodynamic Binding Network model, a model that focuses on the thermodynamic stability of its systems while utilizing geometrically rigid components (dissimilar to other thermodynamic models). We show that this model is computationally universal, an ability conjectured to be impossible in similar models with non-rigid components. We continue by introducing the Flexible Tile Assembly Model, a generalization of the 2D aTAM that allows bonds between tiles to flex and assemblies to therefore reconfigure. We show how systems in this model can deterministically assemble shapes that adhere to a number of certain restrictions. Finally, we introduce the Spatial abstract Tile Assembly Model, a variation of the 3D aTAM that restricts tiles from attaching without a diffusion path. We show that this model is intrinsically universal, a property of computational models to simulate themselves which has been shown for the 3D aTAM and other similar models. We conclude this thesis with a summary of the presented results, a brief impact analysis, and potential directions for future research within this area

Algorithmic Self-Assembly of DNA: Theoretical Motivations and 2D Assembly Experiments

Biology makes things far smaller and more complex than anything produced by human engineering. The biotechnology revolution has for the first time given us the tools necessary to consider engineering on the molecular level. Research in DNA computation, launched by Len Adleman, has opened the door for experimental study of programmable biochemical reactions. Here we focus on a single biochemical mechanism, the self-assembly of DNA structures, that is theoretically sufficient for Turing-universal computation. The theory combines Hao Wang?s purely mathematical Tiling Problem with the branched DNA constructions of Ned Seeman. In the context of mathematical logic, Wang showed how jigsaw-shaped tiles can be designed to simulate the operation of any Turing Machine. For a biochemical implementation, we will need molecular Wang tiles. DNA molecular structures and intermolecular interactions are particularly amenable to design and are sufficient for the creation of complex molecular objects. The structure of individual molecules can be designed by maximizing desired and minimizing undesired Watson-Crick complementarity. Intermolecular interactions are programmed by the design of sticky ends that determine which molecules associate, and how. The theory has been demonstrated experimentally using a system of synthetic DNA double-crossover molecules that self-assemble into two-dimensional crystals that have been visualized by atomic force microscopy. This experimental system provides an excellent platform for exploring the relationship between computation and molecular self-assembly, and thus represents a first step toward the ability to program molecular reactions and molecular structures

Tunable Digital Material Properties of 3D Voxel Printers

Digital materials are composed of many discrete voxels placed in a massively parallel layer deposition process, as opposed to continuous (analog) deposition techniques. We explore the material properties attainable using a voxel-based freeform fabrication process and simulate how the properties can be tuned for a wide range of applications. By varying the precision, geometry, and material of the individual voxels, we obtain continuous control over the density, elastic modulus, CTE, ductility, and failure mode of the material. Also, we demonstrate the effects of several hierarchical voxel “microstructures”, resulting in interesting properties such as negative poisson’s ratio. This implies that digital materials can exhibit widely varying properties in a single desktop fabrication process.Mechanical Engineerin