85,806 research outputs found

    Complexity of Self-assembled Shapes (Extended Abstract)

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    The connection between self-assembly and computation suggests that a shape can be considered the output of a self-assembly “program,” a set of tiles that fit together to create a shape. It seems plausible that the size of the smallest self-assembly program that builds a shape and the shape’s descriptional (Kolmogorov) complexity should be related. We show that under the notion of a shape that is independent of scale this is indeed so: in the Tile Assembly Model, the minimal number of distinct tile types necessary to self-assemble an arbitrarily scaled shape can be bounded both above and below in terms of the shape’s Kolmogorov complexity. As part of the proof of the main result, we sketch a general method for converting a program outputting a shape as a list of locations into a set of tile types that self-assembles into a scaled up version of that shape. Our result implies, somewhat counter-intuitively, that self-assembly of a scaled up version of a shape often requires fewer tile types, and suggests that the independence of scale in self-assembly theory plays the same crucial role as the independence of running time in the theory of computability

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

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    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

    Optimal Staged Self-Assembly of General Shapes

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    We analyze the number of tile types tt, bins bb, and stages necessary to assemble n×nn \times n squares and scaled shapes in the staged tile assembly model. For n×nn \times n squares, we prove O(logntbtlogtb2+loglogblogt)\mathcal{O}(\frac{\log{n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t}) stages suffice and Ω(logntbtlogtb2)\Omega(\frac{\log{n} - tb - t\log t}{b^2}) are necessary for almost all nn. For shapes SS with Kolmogorov complexity K(S)K(S), we prove O(K(S)tbtlogtb2+loglogblogt)\mathcal{O}(\frac{K(S) - tb - t\log t}{b^2} + \frac{\log \log b}{\log t}) stages suffice and Ω(K(S)tbtlogtb2)\Omega(\frac{K(S) - tb - t\log t}{b^2}) are necessary to assemble a scaled version of SS, for almost all SS. We obtain similarly tight bounds when the more powerful flexible glues are permitted.Comment: Abstract version appeared in ESA 201

    Self-Assembly of Arbitrary Shapes Using RNAse Enzymes: Meeting the Kolmogorov Bound with Small Scale Factor (extended abstract)

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    We consider a model of algorithmic self-assembly of geometric shapes out of square Wang tiles studied in SODA 2010, in which there are two types of tiles (e.g., constructed out of DNA and RNA material) and one operation that destroys all tiles of a particular type (e.g., an RNAse enzyme destroys all RNA tiles). We show that a single use of this destruction operation enables much more efficient construction of arbitrary shapes. In particular, an arbitrary shape can be constructed using an asymptotically optimal number of distinct tile types (related to the shape's Kolmogorov complexity), after scaling the shape by only a logarithmic factor. By contrast, without the destruction operation, the best such result has a scale factor at least linear in the size of the shape, and is connected only by a spanning tree of the scaled tiles. We also characterize a large collection of shapes that can be constructed efficiently without any scaling

    Controlling Defects in Nematic and Smectic Liquid Crystals Through Boundary Geometry

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    Liquid crystals (LCs), presently the basis of the dominant electronics display technology, also hold immense potential for the design of new self-assembling, self-healing, and smart responsive materials. Essential to many of these novel materials are liquid crystalline defects, places where the liquid crystalline order is forced to break down, replacing the LC locally with a higher-symmetry phase. Despite the energetic cost of this local melting, defects are often present at equilibrium when boundary conditions frustrate the material order. These defects provide micron-scale tools for organizing colloids, focusing light, and generating micropatterned materials. Manipulating the shapes of the boundaries thus offers a route to obtaining new and desirable self-assembly outcomes in LCs, but each added degree of complexity in the boundary geometry increases the complexity of the liquid crystal\u27s response. Therefore, conceptually minimal changes to boundary geometry are investigated for their effects on the self-assembled defect arrangements that result in nematic and smectic-A LCs in three dimensions as well as two-dimensional smectic LCs on curved substrates. In nematic LCs, disclination loops are studied in micropost confining environments and in the presence of sharp-edged colloidal inclusions, using both numerical modeling and topological reasoning. In both scenarios, sharp edges add new possibilities for the shape or placement of disclinations, permitting new types of colloidal self-assembly beyond simple chains and hexagonal lattices. Two-dimensional smectic LCs on curved substrates are examined in the special cases where the substrate curvature is confined to points or curves, providing an analytically tractable route to demonstrate how Gaussian curvature is associated with disclinations and grain boundaries, as well as these defects\u27 likely experimental manifestations. In three-dimensional smectic-A LCs, novel self-assembled arrangements of focal conic domains (FCDs) are shown to arise from geometric patterning or curvature in boundaries exhibiting so-called hybrid anchoring. These new arrangements allow control over both the packing of the FCDs and their eccentricities. In general, defect self-assembly behavior in LCs is shown to depend sensitively on the shapes of confining boundaries, colloidal inclusions, and substrates, and several broad, new geometrical principles for directing the assembly of nontrivial defect configurations are presented

    New Geometric Algorithms for Fully Connected Staged Self-Assembly

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    We consider staged self-assembly systems, in which square-shaped tiles can be added to bins in several stages. Within these bins, the tiles may connect to each other, depending on the glue types of their edges. Previous work by Demaine et al. showed that a relatively small number of tile types suffices to produce arbitrary shapes in this model. However, these constructions were only based on a spanning tree of the geometric shape, so they did not produce full connectivity of the underlying grid graph in the case of shapes with holes; designing fully connected assemblies with a polylogarithmic number of stages was left as a major open problem. We resolve this challenge by presenting new systems for staged assembly that produce fully connected polyominoes in O(log^2 n) stages, for various scale factors and temperature {\tau} = 2 as well as {\tau} = 1. Our constructions work even for shapes with holes and uses only a constant number of glues and tiles. Moreover, the underlying approach is more geometric in nature, implying that it promised to be more feasible for shapes with compact geometric description.Comment: 21 pages, 14 figures; full version of conference paper in DNA2
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