Tile Assembly Model Research Articles

The Power of Nondeterminism in Self-Assembly

We investigate the role of nondeterminism in Winfree's abstract Tile Assembly Model (aTAM), which was conceived to model artificial self-assembling systems constructed from DNA. Of particular practical importance is to find tile systems that minimize resources such as the number of distinct tile types, each of which corresponds to a set of DNA strands that must be custom-synthesized in actual implementations of the aTAM. We seek to identify to what extent the use of nondeterminism in tile systems affects the resources required by such shape-building algorithms. We first show a molecular computability theoretic result: there is an infinite shape S that is uniquely assembled by a tile system but not by any deterministic tile system. We then show an analogous phenomenon in the finitary molecular complexity theoretic case: there is a finite shape S that is uniquely assembled by a tile system with c tile types, but every deterministic tile system that uniquely assembles S has more than c tile types. In fact we extend the technique to derive a stronger (classical complexity theoretic) result, showing that the problem of finding the minimum number of tile types that uniquely assemble a given finite shape is Sigma-P-2-complete. In contrast, the problem of finding the minimum number of deterministic tile types that uniquely assemble a shape was shown to be NP-complete by Adleman, Cheng, Goel, Huang, Kempe, Moisset de Espanes, and Rothemund (Combinatorial Optimization Problems in Self-Assembly, STOC 2002). The conclusion is that nondeterminism confers extra power to assemble a shape from a small tile system, but unless the polynomial hierarchy collapses, it is computationally more difficult to exploit this power by finding the size of the smallest tile system, compared to finding the size of the smallest deterministic tile system.

Solving Vertex Cover Problem Using DNA Tile Assembly Model

DNA tile assembly models are a class of mathematically distributed and parallel biocomputing models in DNA tiles. In previous works, tile assembly models have been proved be Turing-universal; that is, the system can do what Turing machine can do. In this paper, we use tile systems to solve computational hard problem. Mathematically, we construct three tile subsystems, which can be combined together to solve vertex cover problem. As a result, each of the proposed tile subsystems consists ofΘ(1) types of tiles, and the assembly process is executed in a parallel way (like DNA’s biological function in cells); thus the systems can generate the solution of the problem in linear time with respect to the size of the graph.

Direct Atomic Force Microscopy Observation of DNA Tile Crystal Growth at the Single-Molecule Level

While the theoretical implications of models of DNA tile self-assembly have been extensively researched and such models have been used to design DNA tile systems for use in experiments, there has been little research testing the fundamental assumptions of those models. In this paper, we use direct observation of individual tile attachments and detachments of two DNA tile systems on a mica surface imaged with an atomic force microscope (AFM) to compile statistics of tile attachments and detachments. We show that these statistics fit the widely used kinetic Tile Assembly Model and demonstrate AFM movies as a viable technique for directly investigating DNA tile systems during growth rather than after assembly.

Step-wise tile assembly with a constant number of tile types

In this paper, we study the step-wise tile assembly model introduced by Reif (in: Based computers III, vol 48 of DIMACS, 1999) in which shape is assembled in multiple steps. In each step the partially built structure is exposed to a new set of tiles. We show that an N × N square can be assembled in N steps using a constant number of tile types. In general, we show that a constant number of tile types (24) is sufficient to assemble a large class of shapes in n steps, where n is the number of tiles of the shape. This class includes all shapes obtained from any shape by scaling by a factor of 2, in which case only 14 tile types suffices. For general shapes, we show that the tile complexity of this model is related to the monotone connected node search number of a spanning tree of the shape assuming the number of steps is n.

Negative Interactions in Irreversible Self-assembly

This paper explores the use of negative (i.e., repulsive) interactions in the abstract Tile Assembly Model defined by Winfree. Winfree in his Ph.D. thesis postulated negative interactions to be physically plausible, and Reif, Sahu, and Yin studied them in the context of reversible attachment operations. We investigate the power of negative interactions with irreversible attachments, and we achieve two main results. Our first result is an impossibility theorem: after t steps of assembly, Ω(t) tiles will be forever bound to an assembly, unable to detach. Thus negative glue strengths do not afford unlimited power to reuse tiles. Our second result is a positive one: we construct a set of tiles that can simulate an s-space-bounded, t-time-bounded Turing machine, while ensuring that no intermediate assembly grows larger than O(s), rather than O(s⋅t) as required by the standard Turing machine simulation with tiles. In addition to the space-bounded Turing machine simulation, we show another example application of negative glues: reducing the number of tile types required to assemble “thin” (n×o(logn/loglogn)) rectangles.

Self assembly of rectangular shapes on concentration programming and probabilistic tile assembly models

Efficient tile sets for self assembling rectilinear shapes is of critical importance in algorithmic self assembly. A lower bound on the tile complexity of any deterministic self assembly system for an n × n square is [Formula: see text] (inferred from the Kolmogrov complexity). Deterministic self assembly systems with an optimal tile complexity have been designed for squares and related shapes in the past. However designing [Formula: see text] unique tiles specific to a shape is still an intensive task in the laboratory. On the other hand copies of a tile can be made rapidly using PCR (polymerase chain reaction) experiments. This led to the study of self assembly on tile concentration programming models. We present two major results in this paper on the concentration programming model. First we show how to self assemble rectangles with a fixed aspect ratio (α:β), with high probability, using Θ(α + β) tiles. This result is much stronger than the existing results by Kao et al. (Randomized self-assembly for approximate shapes, LNCS, vol 5125. Springer, Heidelberg, 2008) and Doty (Randomized self-assembly for exact shapes. In: proceedings of the 50th annual IEEE symposium on foundations of computer science (FOCS), IEEE, Atlanta. pp 85-94, 2009)-which can only self assembly squares and rely on tiles which perform binary arithmetic. On the other hand, our result is based on a technique called staircase sampling. This technique eliminates the need for sub-tiles which perform binary arithmetic, reduces the constant in the asymptotic bound, and eliminates the need for approximate frames (Kao et al. Randomized self-assembly for approximate shapes, LNCS, vol 5125. Springer, Heidelberg, 2008). Our second result applies staircase sampling on the equimolar concentration programming model (The tile complexity of linear assemblies. In: proceedings of the 36th international colloquium automata, languages and programming: Part I on ICALP '09, Springer-Verlag, pp 235-253, 2009), to self assemble rectangles (of fixed aspect ratio) with high probability. The tile complexity of our algorithm is Θ(log(n)) and is optimal on the probabilistic tile assembly model (PTAM)-n being an upper bound on the dimensions of a rectangle.

Tile Complexity of Approximate Squares

The standard Tile Assembly Model (TAM) of Winfree (Algorithmic self-assembly of DNA, Ph.D. thesis, 1998) is a mathematical theory of crystal aggregations via monomer additions with applications to the emerging science of DNA self-assembly. Self-assembly under the rules of this model is programmable and can perform Turing universal computation. Many variations of this model have been proposed and the canonical problem of assembling squares has been studied extensively.

Efficient 3-SAT algorithms in the tile assembly model

Self-assembly is a powerful process found in nature that guides simple objects assembling, on their own, into complex structures. Self-assembly is of interest to computer scientists because self-assembling systems can compute functions, assemble shapes, and guide distributed robotics systems. The tile assembly model is a formal mathematical model of self-assembly that allows the study of time and space complexities of self-assembling systems that lie at the heart of several molecular computer implementations and distributed computational software systems. These implementations and systems require efficient tile systems with small tilesets and fast execution times. The state of the art, however, requires vastly complex tile systems with large tilesets to implement fast algorithms. In this paper, I present \({\mathbb{S}}_{FS},\) a tile system that decides 3-SAT by creating \(O^{\star}(1.8393^n)\) nondeterministic assemblies in parallel, improving on the previous best known solution that requires \(\Uptheta(2^n)\) such assemblies. This solution directly improves the feasibility of building molecular 3-SAT solvers and efficiency of distributed software. I formally prove the correctness of the system, the number of required parallel assemblies, that the size of the system’s tileset is \(147 = \Uptheta(1),\) and that the assembly time is nondeterministic linear in the size of the input.

Tile Complexity of Linear Assemblies

Self-assembly is fundamental to both biological processes and nanoscience. Key features of self-assembly are its probabilistic nature and local programmability. These features can be leveraged to design better self-assembled systems. The conventional tile assembly model (TAM) developed by Winfree using Wang tiles is a powerful, Turing-universal theoretical framework which models varied self-assembly processes. A particular challenge in DNA nanoscience is to form linear assemblies or rulers of a specified length using the smallest possible tile set, where any tile type may appear more than once in the assembly. The tile complexity of a linear assembly is the cardinality of the tile set that produces it. These rulers can then be used as components for construction of other complex structures. While square assemblies have been extensively studied, many questions remain about fixed length linear assemblies, which are more basic constructs yet fundamental building blocks for molecular architectures. In this work, we extend TAM to take advantage of inherent probabilistic behavior in physically realized self-assembled systems by introducing randomization. We describe a natural extension to TAM called the probabilistic tile assembly model (PTAM). A restriction of the model, which we call the standard PTAM is considered in this report. Prior work in DNA self-assembly strongly suggests that standard PTAM can be realized in the laboratory. In TAM, a deterministic linear assembly of length $N$ requires a tile set of cardinality at least $N$. In contrast, we show various nontrivial probabilistic constructions for forming linear assemblies in PTAM with tile sets of sublinear cardinality, using techniques that differ considerably from existing assembly techniques. In particular, for any given $N$ we demonstrate linear assemblies of expected length $N$ with a tile set of cardinality $\Theta(\log N)$ using one pad per side of each tile. We prove a matching lower bound of $\Omega(\log N)$ on the tile complexity of linear assemblies of any given expected length $N$ in standard PTAM systems using one pad per side of each tile. We further demonstrate how linear assemblies can be modified to produce assemblies with sharp tail bounds on distribution of lengths by concatenating various assemblies together. In particular, we show that for infinitely many $N$ we can get linear assemblies with exponentially dropping tail distributions using $O(\log^3 N)$ tile types. We also propose a simple extension to PTAM called $\kappa$-pad systems in which we associate $\kappa$ pads with each side of a tile, allowing abutting tiles to bind when at least one pair of corresponding pads match. This gives linear assemblies of expected length $N$ with a $2$-pad (two pads per side of each tile) tile set of cardinality $\Theta(\frac{\log N}{\log \log N})$ for infinitely many $N$. We show that we cannot get smaller tile complexity by proving a lower bound of $\Omega(\frac{\log N}{\log \log N})$ for each $N$ on the cardinality of the $\kappa$-pad ($\kappa$-pads per side of each tile) tile set required to form linear assemblies of expected length $N$ in standard $\kappa$-pad PTAM systems for any positive integer $\kappa$. The techniques that we use for deriving these tile complexity lower bounds are notable as they differ from traditional Kolmogorov complexity based information theoretic methods used for lower bounds on tile complexity. Also, Kolmogorov complexity based lower bounds do not preclude the possibility of achieving assemblies of very small tile multiset cardinality for infinitely many $N$. In contrast, our lower bounds are stronger as they hold for every $N$, rather than for almost all $N$. All our probabilistic constructions are free from co-operative tile binding errors. Thus, for linear assembly systems, we have shown that randomization can be exploited to get large improvements in tile complexity at a small expense of precision in length.

Hierarchical self assembly of patterns from the Robinson tilings: DNA tile design in an enhanced Tile Assembly Model

We introduce a hierarchical self assembly algorithm that produces the quasiperiodic patterns found in the Robinson tilings and suggest a practical implementation of this algorithm using DNA origami tiles. We modify the abstract Tile Assembly Model, (aTAM), to include active signaling and glue activation in response to signals to coordinate the hierarchical assembly of Robinson patterns of arbitrary size from a small set of tiles according to the tile substitution algorithm that generates them. Enabling coordinated hierarchical assembly in the aTAM makes possible the efficient encoding of the recursive process of tile substitution.

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 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 (≈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.

Reducing Tile Complexity for the Self-assembly of Scaled Shapes Through Temperature Programming

This paper concerns the self-assembly of scaled-up versions of arbitrary finite shapes. We work in the multiple temperature model that was introduced by Aggarwal, Cheng, Goldwasser, Kao, and Schweller (Complexities for Generalized Models of Self-Assembly, SIAM J. Comput. 2005). The multiple temperature model is a natural generalization of Winfree’s abstract tile assembly model, where the temperature of a tile system is allowed to be shifted up and down as self-assembly proceeds. We first exhibit two constant-size tile sets in which scaled-up versions of arbitrary shapes self-assemble. Our first tile set has the property that each scaled shape self-assembles via an asymptotically “Kolmogorov-optimum” temperature sequence but the scaling factor grows with the size of the shape being assembled. In contrast, our second tile set assembles each scaled shape via a temperature sequence whose length is proportional to the number of points in the shape but the scaling factor is a constant independent of the shape being assembled. We then show that there is no constant-size tile set that can uniquely assemble an arbitrary (non-scaled, connected) shape in the multiple temperature model, i.e., the scaling is necessary for self-assembly. This answers an open question of Kao and Schweller (Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pp. 571–580, 2006), who asked whether such a tile set exists.

Integer Factorization Based on the Tile Assembly Model

Integer Factorization Based on the Tile Assembly Model

Breaking the EIGamal Algorithm in the Tile Assembly Model

This article has been removed: please see Elsevier Policy on Article Withdrawal (http://www.elsevier.com/locate/withdrawalpolicy).Further to an investigation the 2011 International Conference on Energy Systems and Electric Power (ESEP 2011), Energy Procedia, Volume 13 has been removed by the publisher due to insufficient assurances by the programme organisers that the professional ethical codes of publishing and standards were applied consistently. For author enquiries please contact: [email protected].

Distributed agreement in tile self-assembly

Laboratory investigations have shown that a formal theory of fault-tolerance will be essential to harness nanoscale self-assembly as a medium of computation. Several researchers have voiced an intuition that self-assembly phenomena are related to the field of distributed computing. This paper formalizes some of that intuition. We construct tile assembly systems that are able to simulate the solution of the wait-free consensus problem in some distributed systems. (For potential future work, this may allow binding errors in tile assembly to be analyzed, and managed, with positive results in distributed computing, as a blockage in our tile assembly model is analogous to a crash failure in a distributed computing model.) We also define a strengthening of the traditional consensus problem, to make explicit an expectation about consensus algorithms that is often implicit in distributed computing literature. We show that solution of this strengthened consensus problem can be simulated by a two-dimensional tile assembly model only for two processes, whereas a three-dimensional tile assembly model can simulate its solution in a distributed system with any number of processes.

The effect of malformed tiles on tile assemblies within the kinetic tile assembly model

Many different constructions of proofreading tile sets have been proposed in the literature to reduce the effect of deviations from ideal behaviour of the dynamics of the molecular tile self-assembly process. In this paper, we consider the effect on the tile assembly process of a different kind of non-ideality, namely, imperfections in the tiles themselves. We assume a scenario in which some small proportion of the tiles in a tile set are "malformed". We study, through simulations, the effect of such malformed tiles on the self-assembly process within the kinetic Tile Assembly Model (kTAM). Our simulation results show that some tile set constructions show greater error-resilience in the presence of malformed tiles than others. For example, the 2- and 3-way overlay compact proofreading tile sets of Reif et al. (DNA Computing 10, Lecture Notes in Computer Science, vol 3384. Springer, 2005) are able to handle malformed tiles quite well. On the other hand, the snaked proofreading tile set of Chen and Goel (DNA Computing 10, Lecture Notes in Computer Science, vol 3384. Springer, 2005) fails to form even moderately sized tile assemblies when malformed tiles are present. We show how the Chen---Goel construction may be modified to yield new snaked proofreading tile sets that are resilient not only to errors intrinsic to the assembly process, but also to errors caused by malformed tiles.

Factoring Integers in the Tile Assembly Model

Factoring Integers in the Tile Assembly Model

Self-assembly of decidable sets

The theme of this paper is computation in Winfree's Abstract Tile Assembly Model (TAM). We first review a simple, well-known tile assembly system (the construction) that is capable of universal computation. We then extend the wedge construction to prove the following result: if a set of natural numbers is decidable, then it and its complement's canonical two-dimensional representation self-assemble. This leads to a novel characterization of decidable sets of natural numbers in terms of self-assembly. Finally, we show that our characterization is robust with respect to various (restrictive) geometrical constraints.

Efficient algorithms for self assembling non-rectangular nano structures

Nano fabrication with biomolecular/DNA self assembly is a promising area of research. Building nano structures with self assembly is both efficient and inexpensive. Soloveichik and Winfree (SIAM J Comput 36(6):1544---1569, 2007) formalized a two dimensional (2D) tile assembly model based on Wang's tiling technique. Algorithms with an optimal tile complexity of $$\left(\Uptheta\left(\frac{\log(N)}{\log(\log(N))}\right)\right)$$ were proposed earlier to uniquely self assemble an N × N square (with a temperature of ? = 2) on this model. However efficient constructions to assemble arbitrary shapes are not known and have remained open. In this paper we present self assembling algorithms to assemble a triangle of base 2N ? 1 (units) and height N with a tile complexity of $$\Uptheta(\log(N)).$$ We also describe how this framework can be used to construct other shapes such as rhombus, hexagon etc.

Self-assembly of infinite structures: A survey

We survey some recent results related to the self-assembly of infinite structures in Winfree’s abstract Tile Assembly Model. These results include impossibility results, as well as the construction of novel tile assembly systems that produce computationally interesting shapes and patterns. Several open questions are also presented and motivated.