DNA Computing by Self-Assembly

Information and algorithms appear to be central to biological organization and processes, from the storage and reproduction of genetic information to the control of developmental processes to the sophisticated computations performed by the nervous system. Much as human technology uses electronic microprocessors to control electromechanical devices, biological organisms use biochemical circuits to control molecular and chemical events. The engineering and programming of biochemical circuits, in vivo and in vitro, would transform industries that use chemical and nanostructured materials. Although the construction of biochemical circuits has been explored theoretically since the birth of molecular biology, our practical experience with the capabilities and possible programming of biochemical algorithms is still very young

Towards the Design of Heuristics by Means of Self-Assembly

The current investigations on hyper-heuristics design have sprung up in two different flavours: heuristics that choose heuristics and heuristics that generate heuristics. In the latter, the goal is to develop a problem-domain independent strategy to automatically generate a good performing heuristic for the problem at hand. This can be done, for example, by automatically selecting and combining different low-level heuristics into a problem specific and effective strategy. Hyper-heuristics raise the level of generality on automated problem solving by attempting to select and/or generate tailored heuristics for the problem at hand. Some approaches like genetic programming have been proposed for this. In this paper, we explore an elegant nature-inspired alternative based on self-assembly construction processes, in which structures emerge out of local interactions between autonomous components. This idea arises from previous works in which computational models of self-assembly were subject to evolutionary design in order to perform the automatic construction of user-defined structures. Then, the aim of this paper is to present a novel methodology for the automated design of heuristics by means of self-assembly

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

Designing the self-assembly of arbitrary shapes using minimal complexity building blocks

The design space for a self-assembled multicomponent objects ranges from a solution in which every building block is unique to one with the minimum number of distinct building blocks that unambiguously define the target structure. Using a novel pipeline, we explore the design spaces for a set of structures of various sizes and complexities. To understand the implications of the different solutions, we analyse their assembly dynamics using patchy particle simulations and study the influence of the number of distinct building blocks and the angular and spatial tolerances on their interactions on the kinetics and yield of the target assembly. We show that the resource-saving solution with minimum number of distinct blocks can often assemble just as well (or faster) than designs where each building block is unique. We further use our methods to design multifarious structures, where building blocks are shared between different target structures. Finally, we use coarse-grained DNA simulations to investigate the realisation of multicomponent shapes using DNA nanostructures as building blocks.Comment: 12 page

How crystals that sense and respond to their environments could evolve

An enduring mystery in biology is how a physical entity simple enough to have arisen spontaneously could have evolved into the complex life seen on Earth today. Cairns-Smith has proposed that life might have originated in clays which stored genomes consisting of an arrangement of crystal monomers that was replicated during growth. While a clay genome is simple enough to have conceivably arisen spontaneously, it is not obvious how it might have produced more complex forms as a result of evolution. Here, we examine this possibility in the tile assembly model, a generalized model of crystal growth that has been used to study the self-assembly of DNA tiles. We describe hypothetical crystals for which evolution of complex crystal sequences is driven by the scarceness of resources required for growth. We show how, under certain circumstances, crystal growth that performs computation can predict which resources are abundant. In such cases, crystals executing programs that make these predictions most accurately will grow fastest. Since crystals can perform universal computation, the complexity of computation that can be used to optimize growth is unbounded. To the extent that lessons derived from the tile assembly model might be applicable to mineral crystals, our results suggest that resource scarcity could conceivably have provided the evolutionary pressures necessary to produce complex clay genomes that sense and respond to changes in their environment

Verification in Staged Tile Self-Assembly

We prove the unique assembly and unique shape verification problems, benchmark measures of self-assembly model power, are $\mathrm{coNP}^{\mathrm{NP}}$-hard and contained in $\mathrm{PSPACE}$ (and in $\mathrm{\Pi}^\mathrm{P}_{2s}$ for staged systems with $s$ stages). En route, we prove that unique shape verification problem in the 2HAM is $\mathrm{coNP}^{\mathrm{NP}}$-complete.Comment: An abstract version will appear in the proceedings of UCNC 201

Combinatorial optimization problems in self-assembly

Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum running time for a "program" to assemble the shape.Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum running time for a "program" to assemble the shape. We prove that the first problem is NP-complete in general, and polynomial time solvable on trees and squares. In order to prove that the problem is in NP, we present a polynomial time algorithm to verify whether a given tile system uniquely produces a given shape. This algorithm is analogous to a program verifier for traditional computational systems, and may well be of independent interest. For the second problem, we present a polynomial time $O(\log n)$-approximation algorithm that works for a large class of tile systems that we call partial order systems