Program Size and Temperature in Self-Assembly

Winfree’s abstract Tile Assembly Model is a model of molecular self-assembly of DNA complexes known as tiles, which float freely in solution and attach one at a time to a growing “seed” assembly based on specific binding sites on their four sides. We show that there is a polynomial-time algorithm that, given an n×n square, finds the minimal tile system (i.e., the system with the smallest number of distinct tile types) that uniquely self-assembles the square, answering an open question of Adleman et al. (Combinatorial optimization problems in self-assembly, STOC 2002). Our investigation leading to this algorithm reveals other positive and negative results about the relationship between the size of a tile system and its “temperature” (the binding strength threshold required for a tile to attach)

Optimal self-assembly of finite shapes at temperature 1 in 3D

Working in a three-dimensional variant of Winfree's abstract Tile Assembly Model, we show that, for an arbitrary finite, connected shape $X \subset \mathbb{Z}^2$, there is a tile set that uniquely self-assembles into a 3D representation of a scaled-up version of $X$ at temperature 1 in 3D with optimal program-size complexity (the "program-size complexity", also known as "tile complexity", of a shape is the minimum number of tile types required to uniquely self-assemble it). Moreover, our construction is "just barely" 3D in the sense that it only places tiles in the $z = 0$ and $z = 1$ planes. Our result is essentially a just-barely 3D temperature 1 simulation of a similar 2D temperature 2 result by Soloveichik and Winfree (SICOMP 2007)

Morphology and microstructure evolution of gold nanostructures in the limited volume porous matrices

The modern development of nanotechnology requires the discovery of simple approaches that ensure the controlled formation of functional nanostructures with a predetermined morphology. One of the simplest approaches is the self-assembly of nanostructures. The widespread implementation of self-assembly is limited by the complexity of controlled processes in a large volume where, due to the temperature, ion concentration, and other thermodynamics factors, local changes in diffusion-limited processes may occur, leading to unexpected nanostructure growth. The easiest ways to control the diffusion-limited processes are spatial limitation and localized growth of nanostructures in a porous matrix. In this paper, we propose to apply the method of controlled self-assembly of gold nanostructures in a limited pore volume of a silicon oxide matrix with submicron pore sizes. A detailed study of achieved gold nanostructures’ morphology, microstructure, and surface composition at different formation stages is carried out to understand the peculiarities of realized nanostructures. Based on the obtained results, a mechanism for the growth of gold nanostructures in a limited volume, which can be used for the controlled formation of nanostructures with a predetermined geometry and composition, has been proposed. The results observed in the present study can be useful for the design of plasmonic-active surfaces for surface-enhanced Raman spectroscopy-based detection of ultra-low concentration of different chemical or biological analytes, where the size of the localized gold nanostructures is comparable with the spot area of the focused laser beam. © 2020 by the authors. Licensee MDPI, Basel, Switzerland.3.1.5.1Ministry of Education and Science of the Russian Federation, Minobrnauka: К-2018-036, N 211Russian Foundation for Fundamental Investigations, RFFI: 19-32-50058European Commission, ECMinistry of Science and Technology, MOSTFunding: This research was funded by H2020-MSCA-RISE2017-778308-SPINMULTIFILM Project, the scientific– technical program, ‘Technology-SG’ [project number 3.1.5.1], Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST «MISiS» [№ К-2018-036], implemented by a governmental decree dated 16th of March 2013, N 211 and Russian Foundation for Fundamental Investigations [project number 19-32-50058].Acknowledgments: D.V.Y. greatly acknowledges the World Federation of Scientists National Scholarship Program. E.Yu.K., D.V.Y., V.D.B., and V.S. greatly acknowledge the European Union program Mobility Scheme for Targeted People-to-People-Contacts (MOST) for supporting research visits

Noncooperative algorithms in self-assembly

We show the first non-trivial positive algorithmic results (i.e. programs whose output is larger than their size), in a model of self-assembly that has so far resisted many attempts of formal analysis or programming: the planar non-cooperative variant of Winfree's abstract Tile Assembly Model. This model has been the center of several open problems and conjectures in the last fifteen years, and the first fully general results on its computational power were only proven recently (SODA 2014). These results, as well as ours, exemplify the intricate connections between computation and geometry that can occur in self-assembly. In this model, tiles can stick to an existing assembly as soon as one of their sides matches the existing assembly. This feature contrasts with the general cooperative model, where it can be required that tiles match on \emph{several} of their sides in order to bind. In order to describe our algorithms, we also introduce a generalization of regular expressions called Baggins expressions. Finally, we compare this model to other automata-theoretic models.Comment: A few bug fixes and typo correction

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

Intrinsic universality and the computational power of self-assembly

This short survey of recent work in tile self-assembly discusses the use of simulation to classify and separate the computational and expressive power of self-assembly models. The journey begins with the result that there is a single universal tile set that, with proper initialization and scaling, simulates any tile assembly system. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system. From there we find that there is no such tile set in the noncooperative, or temperature 1, model, proving it weaker than the full tile assembly model. In the two-handed or hierarchal model, where large assemblies can bind together on one step, we encounter an infinite set, of infinite hierarchies, each with strictly increasing simulation power. Towards the end of our trip, we find one tile to rule them all: a single rotatable flipable polygonal tile that can simulate any tile assembly system. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.Comment: In Proceedings MCU 2013, arXiv:1309.104