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

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

Search methods for tile sets in patterned DNA self-assembly

Peer reviewe

Self-Assembly of Tiles: Theoretical Models, the Power of Signals, and Local Computing

DNA-based self-assembly is an autonomous process whereby a disordered system of DNA sequences forms an organized structure or pattern as a consequence of Watson-Crick complementarity of DNA sequences, without external direction. Here, we propose self-assembly (SA) hypergraph automata as an automata-theoretic model for patterned self-assembly. We investigate the computational power of SA-hypergraph automata and show that for every recognizable picture language, there exists an SA-hypergraph automaton that accepts this language. Conversely, we prove that for any restricted SA-hypergraph automaton, there exists a Wang Tile System, a model for recognizable picture languages, that accepts the same language. Moreover, we investigate the computational power of some variants of the Signal-passing Tile Assembly Model (STAM), as well as propose the concept of {\it Smart Tiles}, i.e., tiles with glues that can be activated or deactivated by signals, and which possess a limited amount of local computing capability. We demonstrate the potential of smart tiles to perform some robotic tasks such as replicating complex shapes

Nascent nanocomputers: DNA self-assembly in O(1) stages

DNA self-assembly offers a potential for nanoscale microcircuits and computers. To make that potential possible requires the development of reliable and efficient tile assembly models. Efficiency is often achieved by minimizing tile complexity, as well as by evaluating the cost and reliability of the specific elements of each tile assembly model. We consider a 2D tile assembly model at temperature 1. The standard 2D tile assembly model at temperature 1 has a tile complexity of O(n) for the construction of exact, complete n x n squares. However, previous research found a staged tile assembly model achieved a tile complexity of O(1) to construct n x n squares, with O(logn) stages. Our staged tile assembly model achieves a tile complexity of O(logn) using only O(1) stages to construct n x n squares

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

Integrating DNA strand-displacement circuitry with DNA tile self-assembly

DNA nanotechnology has emerged as a reliable and programmable way of controlling matter at the nanoscale through the specificity of Watson–Crick base pairing, allowing both complex self-assembled structures with nanometer precision and complex reaction networks implementing digital and analog behaviors. Here we show how two well-developed frameworks, DNA tile self-assembly and DNA strand-displacement circuits, can be systematically integrated to provide programmable kinetic control of self-assembly. We demonstrate the triggered and catalytic isothermal self-assembly of DNA nanotubes over 10 µm long from precursor DNA double-crossover tiles activated by an upstream DNA catalyst network. Integrating more sophisticated control circuits and tile systems could enable precise spatial and temporal organization of dynamic molecular structures

Design, Synthesis and Analysis of Self-Assembling Triangulated Wireframe DNA Structures

The field of DNA nanotechnology offers a wide range of design strategies with which nanometer-sized structures with a desired shape, size and aspect ratio can be built. The most established techniques in the field rely on close-packed 'solid' DNA nanostructures produced with either the DNA origami or the single-stranded tile techniques. These structures depend on high-salt buffer solutions and require more material than comparable size hollow wireframe structures. This dissertation explores the construction of hollow wireframe DNA nanostructures composed of equilateral triangles. To achieve maximal material efficiency the design is restricted to use a single DNA double helix per triangle edge. As a proof of principle, the DNA origami technique is extended to produce a series of truss structures including the flat, tetrahedral, octahedral, or irregular dodecahedral truss designs. In contrast to close packed DNA origami designs these structures fold at low-salt buffer conditions. These structures have defined cavities that may in the future be used to precisely position functional elements such as metallic nanoparticles or enzymes. The design process of these structures is simplified by a custom design software. Next, the triangulated construction motif is extended to the single-stranded DNA tile technique. A collection of finite structures, as well as one-dimensional crystalline assemblies is explored. The ideal assembly conditions are determined experimentally and using molecular dynamics simulations. A custom design software is presented to simplify the design and handling of these structures. At last, the cost-effective prototyping of triangulated wireframe DNA origami structures is explored. This is achieved through the introduction of single-stranded “gap” regions along the triangle edges. These gap regions are then filled using a DNA polymerase rather than by synthetic oligonucleotides. This technique also allows the mechanical transformation of these structures, which is exemplified by the transition of a bent into a straight structure upon completion of the gap filling.:Abstract v Publications vii Acknowledgements ix Contents xi Chapter 1 A short introduction into DNA nanotechnology 1 1.1 Nanotechnology 1 1.1.1 Top down 1 1.1.2 Bottom up 3 1.2 Deoxyribonucleic acid (DNA) 4 1.3 DNA Nanotechnology 6 1.3.1 Tile based assembly 9 1.3.2 DNA origami and single-stranded tiles 10 1.3.3 Some applications of DNA nanotechnology 12 1.3.4 Wireframe structures 15 1.3.5 Computational tools and DNA nanotechnology. 17 Chapter 2 Motivation and objectives 19 Chapter 3 Design and Synthesis of Triangulated DNA Origami Trusses 20 3.1 Introduction 20 3.2 Results and Discussion 21 3.2.1 Design 21 3.2.2 Nomenclature and parameters of the tube structures 23 3.2.3 Gel electrophoreses analysis 25 3.2.4 Imaging of the purified structures 26 3.2.5 Optimizing the folding conditions 28 3.2.6 Comparison to vHelix 29 3.3 Conclusions 29 3.4 Methods 30 3.4.1 Standard DNA origami assembly reaction. 30 3.4.2 Gel purification. 30 3.4.3 AFM sample preparation. 31 3.4.4 TEM sample preparation. 31 3.4.5 Instructions for mixing the staple sets. 31 Chapter 4 Triangulated wireframe structures assembled using single-stranded DNA tiles 33 4.1 Introduction 33 4.2 Results and Discussion 35 4.2.1 Designing the structures 35 4.2.2 Synthesis of test structures 37 4.2.3 Molecular dynamics simulations of 6-arm junctions 38 4.2.4 Assembly of the finite structures 40 4.2.5 Influence of salt concentration and folding times 42 4.2.6 Molecular dynamics simulations of the rhombus structure 43 4.2.7 1D SST crystals 44 4.2.8 Controlling the crystal growth 46 4.3 Conclusions 48 4.4 Methods 49 4.4.1 SST Folding 49 4.4.2 Agarose Gel Electrophoresis 49 4.4.3 tSEM Characterization 49 4.4.4 AFM Imaging 49 4.4.5 AGE-Based Folding-Yield Estimation 49 4.4.6 Molecular Dynamics Simulations 50 Chapter 5 Structural transformation of wireframe DNA origami via DNA polymerase assisted gap-filling 52 5.1 Introduction 52 5.2 Results and Discussion 54 5.2.1 Design of the Structures 54 5.2.2 Folding of Gap-Structures 56 5.2.3 Inactivation of Polymerase. 57 5.2.4 Secondary Structures. 58 5.2.5 Folding Kinetics of Gap Origami. 60 5.3 Conclusions 61 5.4 Methods 62 5.4.1 DNA origami folding 62 5.4.2 Gap filling of the wireframe DNA origami structures 63 5.4.3 Agarose gel electrophoresis 63 5.4.4 PAGE gel analysis 63 5.4.5 tSEM characterization 64 5.4.6 AFM imaging 64 5.4.7 AGE based folding-yield estimation 64 5.4.8 Gibbs free energy simulation using mfold 65 5.4.9 List of sequence for folding the DNA origami triangulated structures 65 Chapter 6 Summary and outlook 67 Appendix 69 A.1 Additional figures from chapter 369 A.2 Additional figures from chapter 4 77 A.3 Additional figures from chapter 5 111 Bibliography 127 Erklärung 13

Biological Nanowires: Integration of the silver(I) base pair into DNA with nanotechnological and synthetic biological applications

Modern computing and mobile device technologies are now based on semiconductor technology with nanoscale components, i.e., nanoelectronics, and are used in an increasing variety of consumer, scientific, and space-based applications. This rise to global prevalence has been accompanied by a similarly precipitous rise in fabrication cost, toxicity, and technicality; and the vast majority of modern nanotechnology cannot be repaired in whole or in part. In combination with looming scaling limits, it is clear that there is a critical need for fabrication technologies that rely upon clean, inexpensive, and portable means; and the ideal nanoelectronics manufacturing facility would harness micro- and nanoscale fabrication and self-assembly techniques. The field of molecular electronics has promised for the past two decades to fill fundamental gaps in modern, silicon-based, micro- and nanoelectronics; yet molecular electronic devices, in turn, have suffered from problems of size, dispersion and reproducibility. In parallel, advances in DNA nanotechnology over the past several decades have allowed for the design and assembly of nanoscale architectures with single-molecule precision, and indeed have been used as a basis for heteromaterial scaffolds, mechanically-active delivery mechanisms, and network assembly. The field has, however, suffered for lack of meaningful modularity in function: few designs to date interact with their surroundings in more than a mechanical manner. As a material, DNA offers the promise of nanometer resolution, self-assembly, linear shape, and connectivity into branched architectures; while its biological origin offers information storage, enzyme-compatibility and the promise of biologically-inspired fabrication through synthetic biological means. Recent advances in DNA chemistry have isolated and characterized an orthogonal DNA base pair using standard nucleobases: by bridging the gap between mismatched cytosine nucleotides, silver(I) ions can be selectively incorporated into the DNA helix with atomic resolution. The goal of this thesis is to explore how this approach to “metallize” DNA can be combined with structural DNA nanotechnology as a step toward creating electronically-functional DNA networks. This work begins with a survey of applications for such a transformative technology, including nanoelectronic component fabrication for low-resource and space-based applications. We then investigate the assembly of linear Ag+-functionalized DNA species using biochemical and structural analyses to gain an understanding of the kinetics, yield, morphology, and behavior of this orthogonal DNA base pair. After establishing a protocol for high yield assembly in the presence of varying Ag+ functionalization, we investigate these linear DNA species using electrical means. First a method of coupling orthogonal DNA to single-walled carbon nanotubes (SWCNTs) is explored for self-assembly into nanopatterned transistor devices. Then we carry out scanning tunneling microscope (STM) break junction experiments on short polycytosine, polycationic DNA duplexes and find increased molecular conductance of at least an order of magnitude relative to the most conductive DNA analog. With an understanding of linear species from both a biochemical and nanoelectronic perspective, we investigate the assembly of nonlinear Ag+-functionalized DNA species. Using rational design principles gathered from the analysis of linear species, a de novo mathematical framework for understanding generalized DNA networks is developed. This provides the basis for a computational model built in Matlab that is able to design DNA networks and nanostructures using arbitrary base parity. In this way, DNA nanostructures are able to be designed using the dC:Ag+:dC base pair, as well as any similar nucleobase or DNA-inspired system (dT:Hg2+:dT, rA:rU, G4, XNA, LNA, PNA, etc.). With this foundation, three general classes of DNA tiles are designed with embedded nanowire elements: single crossover Holliday junction (HJ) tiles, T-junction (TJ) units, and double crossover (DX) tile pairs and structures. A library of orthogonal chemistry DNA nanotechnology is described, and future applications to nanomaterials and circuit architectures are discussed