Search methods for tile sets in patterned DNA self-assembly

Peer reviewe

Binary pattern tile set synthesis is NP-hard

In the field of algorithmic self-assembly, a long-standing unproven conjecture has been that of the NP-hardness of binary pattern tile set synthesis (2-PATS). The $k$-PATS problem is that of designing a tile assembly system with the smallest number of tile types which will self-assemble an input pattern of $k$ colors. Of both theoretical and practical significance, $k$-PATS has been studied in a series of papers which have shown $k$-PATS to be NP-hard for $k = 60$, $k = 29$, and then $k = 11$. In this paper, we close the fundamental conjecture that 2-PATS is NP-hard, concluding this line of study. While most of our proof relies on standard mathematical proof techniques, one crucial lemma makes use of a computer-assisted proof, which is a relatively novel but increasingly utilized paradigm for deriving proofs for complex mathematical problems. This tool is especially powerful for attacking combinatorial problems, as exemplified by the proof of the four color theorem by Appel and Haken (simplified later by Robertson, Sanders, Seymour, and Thomas) or the recent important advance on the Erd\H{o}s discrepancy problem by Konev and Lisitsa using computer programs. We utilize a massively parallel algorithm and thus turn an otherwise intractable portion of our proof into a program which requires approximately a year of computation time, bringing the use of computer-assisted proofs to a new scale. We fully detail the algorithm employed by our code, and make the code freely available online

Small Tile Sets That Compute While Solving Mazes

We ask the question of how small a self-assembling set of tiles can be yet have interesting computational behaviour. We study this question in a model where supporting walls are provided as an input structure for tiles to grow along: we call it the Maze-Walking Tile Assembly Model. The model has a number of implementation prospects, one being DNA strands that attach to a DNA origami substrate. Intuitively, the model suggests a separation of signal routing and computation: The input structure (maze) supplies a routing diagram, and the programmer's tile set provides the computational ability. We ask how simple the computational part can be. We give two tiny tile sets that are computationally universal in the Maze-Walking Tile Assembly Model. The first has four tiles and simulates Boolean circuits by directly implementing NAND, NXOR and NOT gates. Our second tile set has 6 tiles and is called the Collatz tile set as it produces patterns found in binary/ternary representations of iterations of the Collatz function. Using computer search we find that the Collatz tile set is expressive enough to encode Boolean circuits using blocks of these patterns. These two tile sets give two different methods to find simple universal tile sets, and provide motivation for using pre-assembled maze structures as circuit wiring diagrams in molecular self-assembly based computing.ISSN:1868-896

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

Design Automation of Polyomino Set That Self-Assembles into a Desired Shape

The problem of finding the smallest DNA tile set that self-assembles into a desired pattern or shape is a research focus that has been investigated by many researchers. In this paper, we take a polyomino, which is a non-square element composed of several connected square units, as an element of assembly and consider the design problem of the minimal set of polyominoes that self-assembles into a desired shape. We developed a self-assembly simulator of polyominoes based on the agent-based Monte Carlo method, in which the potential energy among the polyominoes is evaluated and the simulation state is updated toward the direction to decrease the total potential. Aggregated polyominoes are represented as an agent, which can move, merge, and split during the simulation. In order to search the minimal set of polyominoes, two-step evaluation strategy is adopted, because of enormous search space including many parameters such as the shape, the size, and the glue types attached to the polyominoes. The feasibility of the proposed method is shown through three examples with different size and complexity

The PATS Problem : Search Methods and Reliability

This work studies an NP-hard combinatorial optimisation problem, the Pattern self-Assembly Tile set Synthesis (PATS) problem, which stems from the field of DNA self-assembly. In this problem, we are given a coloured rectangular pattern as input, and the task is to find a minimal set of unit square tiles that self-assemble that pattern in the abstract Tile Assembly Model (aTAM). We present two new search methods for the PATS problem: a heuristic algorithm that conducts a search in the lattice of partitions of the input grid, and a declarative approach that uses the Answer Set Programming (ASP) paradigm. The former is based on a previous algorithm by Göös and Orponen (DNA 2010), and performs better in finding relatively small solutions even for quite large input patterns. The latter proves to find the optimal solution quickly in cases where it is small. In addition to the search procedures, we develop a method for estimating the reliability of solutions to the PATS problem from a stochastic point of view. It turns out that tile sets found by our procedures, as well as small tile sets in general, have a higher probability of error-free assembly compared to those that can be found by previous methods

Algorithmic Assembly of Nanoscale Structures

The development of nanotechnology has become one of the most significant endeavors of our time. A natural objective of this field is discovering how to engineer nanoscale structures. Limitations of current top-down techniques inspire investigation into bottom-up approaches to reach this objective. A fundamental precondition for a bottom-up approach is the ability to control the behavior of nanoscale particles. Many abstract representations have been developed to model systems of particles and to research methods for controlling their behavior. This thesis develops theories on two such approaches for building complex structures: the self-assembly of simple particles, and the use of simple robot swarms. The concepts for these two approaches are straightforward. Self-assembly is the process by which simple particles, following the rules of some behavior-governing system, naturally coalesce into a more complex form. The other method of bottom-up assembly involves controlling nanoscale particles through explicit directions and assembling them into a desired form. Regarding the self-assembly of nanoscale structures, we present two construction methods in a variant of a popular theoretical model known as the 2-Handed Tile Self-Assembly Model. The first technique achieves shape construction at only a constant scale factor, while the second result uses only a constant number of unique particle types. Regarding the use of robot swarms for construction, we first develop a novel technique for reconfiguring a swarm of globally-controlled robots into a desired shape even when the robots can only move maximally in a commanded direction. We then expand on this work by formally defining an entire hierarchy of shapes which can be built in this manner and we provide a technique for doing so