Techniques for the Generation of Arbitrary Three-Dimensional Shapes in Tile-Based Self-Assembly Systems

A big challenge in nanorobotics is the construction of nanoscale objects. DNA is a bio-compatible tool to reliably and constructively create objects at the nanoscale. A possible tool to build nano-sized structures are tile-based self-assembly systems on the basis of DNA. It is challenging and time-consuming to efficiently design blueprints for the desired objects. This paper presents basic algorithms for the creation of tilesets for nxnxn-cubes in the aTAM model. Only few publications focus on three-dimensional DNA crystals. Three-dimensional shapes are likely to be of more use in nanorobotics. We present three variations: hollow cubes, cube-grids and filled cubes. The paper also presents a basic algorithm to create arbitrary, finite, connected, three-dimensional and predefined shapes at temperature 1, as well as ideas for more efficient algorithms. Among those are algorithms for spheres, ellipsoids, red blood cells and other promising designs. The algorithms and tilesets are tested/verified using a software that has been developed for the purpose of verifying three-dimensional sets of tiletypes and was influenced by the tool ISU TAS. Others can use the simulator and the algorithms to quickly create sets of tiletypes for their desired nanostructures. A long learning process may thus be omitted

Self-assembly: modelling, simulation, and planning

Samoskládání je proces, při kterém se kolekce neuspořádaných částic samovolně orientuje do uspořádaného vzoru nebo funkční struktury bez působení vnější síly, pouze za pomoci lokálních interakcí mezi samotnými částicemi. Tato teze se zaměřuje na teorii dlaždicových samoskládacích systémů a jejich syntézu. Nejdříve je představena oblast výzkumu věnující se dlaždičovým samoskládacím systémům, a poté jsou důkladně popsány základní typy dlaždicových skládacích systémů, kterými jsou abstract Tile Assembly Model (aTAM ), kinetic Tile Assembly Model (kTAM ), a 2-Handed Assembly Model (2HAM ). Poté jsou představeny novější modely a modely se specifickým použitím. Dále je zahrnut stručný popis původu teorie dlaždicového samoskládání společně s krátkým popisem nedávného výzkumu. Dále jsou představeny dva obecné otevřené problémy dlaždicového samoskládání s hlavním zaměřením na problém Pattern Self-Assembly Tile Set Synthesis (PATS), což je NP-těžká kombinatorická optimalizační úloha. Nakonec je ukázán algoritmus Partition Search with Heuristics (PS-H ), který se používá k řešení problému PATS. Následovně jsou demonstrovány dvě aplikace, které byly vyvinuty pro podporu výzkumu abstraktních dlaždicových skládacích modelů a syntézy množin dlaždic pro samoskládání zadaných vzorů. První aplikace je schopná simulovat aTAM a 2HAM systémy ve 2D prostoru. Druhá aplikace je řešič PATS problému, který využívá algoritmu PS-H. Pro obě aplikace jsou popsány hlavní vlastnosti a návrhová rozhodnutí, která řídila jejich vývoj. Nakonec jsou předloženy výsledky několika experimentů. Jedna skupina experimentů byla zaměřena na ověření výpočetní náročnosti vyvinutých algoritmů pro simulátor. Druhá sada experimentů zkoumala vliv jednotlivých vlastností vzorů na vlastnosti dlaždicových systémů, které byly získány syntézou ze vzorů pomocí vyvinutého řešiče PATS problému. Bylo prokázáno, že algoritmus simulující aTAM systém má lineární časovou výpočetní náročnost, zatímco algoritmus simulující 2HAM systém má exponenciální časovou výpočetní náročnost, která navíc silně závisí na simulovaném systému. Aplikace pro řešení syntézy množiny dlaždic ze vzorů je schopna najít relativně malé řešení i pro velké zadané vzory, a to v přiměřeném čase.Self-assembly is the process in which a collection of disordered units organise themselves into ordered patterns or functional structures without any external direction, solely using local interactions among the components. This thesis focuses on the theory of tile-based self-assembly systems and their synthesis. First, an introduction to the study field of tile-based self-assembly systems are given, followed by a thorough description of common types of tile assembly systems such as abstract Tile Assembly Model (aTAM ), kinetic Tile Assembly Model (kTAM ), and 2-Handed Assembly Model (2HAM ). After that, various recently developed models and models with specific applications are listed. A brief summary of the origins of the tile-based self-assembly is also included together with a short review of recent results. Two general open problems are presented with the main focus on the Pattern Self-Assembly Tile Set Synthesis (PATS) problem, which is NP-hard combinatorial optimisation problem. Partition Search with Heuristics (PS-H ) algorithm is presented as it is used for solving the PATS problem. Next, two applications which were developed to study the abstract tile assembly models and the synthesis of tile sets for pattern self-assembly are introduced. The first application is a simulator capable of simulating aTAM and 2HAM systems in 2D. The second application is a solver of the PATS problem based around the PS-H algorithm. Main features and design decisions are described for both applications. Finally, results from several experiments are presented. One set of experiments were focused on verification of computation complexity of algorithms developed for the simulator, and the other set of experiments studied the influences of the properties of the pattern on the tile assembly system synthesised by our implementation of PATS problem solver. It was shown that the algorithm for simulating aTAM systems have linear computation time complexity, whereas the algorithm simulating 2HAM systems have exponential computation time complexity, which strongly varies based on the simulated system. The synthesiser application is capable of finding a relatively small solution even for quite large input patterns in reasonable amounts of time

Approximate Self-Assembly of the Sierpinski Triangle

The Tile Assembly Model is a Turing universal model that Winfree introduced in order to study the nanoscale self-assembly of complex (typically aperiodic) DNA crystals. Winfree exhibited a self-assembly that tiles the first quadrant of the Cartesian plane with specially labeled tiles appearing at exactly the positions of points in the Sierpinski triangle. More recently, Lathrop, Lutz, and Summers proved that the Sierpinski triangle cannot self-assemble in the "strict" sense in which tiles are not allowed to appear at positions outside the target structure. Here we investigate the strict self-assembly of sets that approximate the Sierpinski triangle. We show that every set that does strictly self-assemble disagrees with the Sierpinski triangle on a set with fractal dimension at least that of the Sierpinski triangle (roughly 1.585), and that no subset of the Sierpinski triangle with fractal dimension greater than 1 strictly self-assembles. We show that our bounds are tight, even when restricted to supersets of the Sierpinski triangle, by presenting a strict self-assembly that adds communication fibers to the fractal structure without disturbing it. To verify this strict self-assembly we develop a generalization of the local determinism method of Soloveichik and Winfree

The 3D abstract Tile Assembly Model is Intrinsically Universal

In this paper, we prove that the three-dimensional abstract Tile Assembly Model (3DaTAM) is intrinsically universal. This means that there is a universal tile set in the 3DaTAM which can be used to simulate any 3DaTAM system. This result adds to a body of work on the intrinsic universality of models of self-assembly, and is specifically motivated by a result in FOCS 2016 showing that any intrinsically universal tile set for the 2DaTAM requires nondeterminism (i.e. undirectedness) even when simulating directed systems. To prove our result we have not only designed, but also fully implemented what we believe to be the first intrinsically universal tile set which has been implemented and simulated in any tile assembly model, and have made it and a simulator which can display it freely available

TAMScript - High Level Programming Interface for the abstract Tile Assembly Model

This paper describes a programming interface, TAMScript, for use with the PyTAS simulator. The interface allows for the dynamic generation of tile types as the simulation progresses, with the goal of reducing complexity for researchers. This paper begins with an introduction to the PyTAS software and a description of the 3D model which it simulates. Next, the changes made to support a dynamic generation scheme are detailed, and some of the potential benefits of this scheme are outlined. Then several of the example scripts which have been written using the TAMScript interface are reviewed. Finally, the potential for future research is discussed, laying out one intended approach for use with the interface

Simulation and Analysis of Self-Assembling Slat-Based DNA Ribbons

Though still in its infancy, the design of DNA crisscross slats presents great potential in the algorithmic self-assembly of DNA. The provision for higher levels of cooperativity allows for fewer errors through the natural proofreading of slat placement, leading to more robust assembly. Highly accurate simulations of self-assembling DNA squares have been achieved by following the kinetic Tile Assembly Model. Building on that foundation, this study seeks to calibrate the system parameters of a kinetic simulator for self-assembling DNA slats to match experimental results and to use those ranges of parameters to perform exploratory simulations of systems not yet tested in a lab setting. Novel systems include those with fewer unique slat types to analyze the trade-off between growth rate and accuracy of each assembly

Simulation and Analysis of Self-Assembling Slat-Based DNA Ribbons

Though still in its infancy, the design of DNA crisscross slats presents great potential in the algorithmic self-assembly of DNA. The provision for higher levels of cooperativity allows for fewer errors through the natural proofreading of slat placement, leading to more robust assembly. Highly accurate simulations of self-assembling DNA squares have been achieved by following the kinetic Tile Assembly Model. Building on that foundation, this study seeks to calibrate the system parameters of a kinetic simulator for self-assembling DNA slats to match experimental results and to use those ranges of parameters to perform exploratory simulations of systems not yet tested in a lab setting. Novel systems include those with fewer unique slat types to analyze the trade-off between growth rate and accuracy of each assembly

Algorithmic Temperature 1 Self-Assembly

We investigate the power of the Wang tile self-assembly model at temperature 1, a threshold value that permits attachment between any two tiles that share even a single bond. When restricted to deterministic assembly in the plane, no temperature 1 assembly system has been shown to build a shape with a tile complexity smaller than the diameter of the shape. Our work shows a sharp contrast in achievable tile complexity at temperature 1 if either growth into the third dimension or a small probability of error are permitted. Motivated by applications in nanotechnology and molecular computing, and the plausibility of implementing 3 dimensional self-assembly systems, our techniques may provide the needed power of temperature 2 systems, while at the same time avoiding the experimental challenges faced by those systems