Simulation of the abstract Tile Assembly Model Using Crisscross Slats

TL;DR

This work addresses the challenge of realizing efficient, error-robust self-assembly beyond tile-based models by introducing the abstract Slat Assembly Model (aSAM) and proving that it has the full theoretical power to simulate any abstract Tile Assembly Model (aTAM) system. The authors develop a formal intrinsic-simulation framework using macrotiles and a scale-factor map, and they construct explicit slat-based simulations for five progressively complex aTAM classes (zig-zag, standard, standard with across-the-gap, directed temperature-2, and the full aTAM). They quantify the simulation concision by giving concrete scale factors and maximum slat lengths, e.g., $2c \times 2c$ for zig-zag and $5c \times 5c$ for the full aTAM, demonstrating that slats can realize the computational power of tiles while offering potentially greater error resilience. The results pave the way for DNA-based slat designs that combine powerful algorithmic self-assembly with error tolerance, and the work includes software tools for design, simulation, and visualization. Overall, the paper establishes a rigorous theoretical bridge between tile-based and slat-based self-assembly with concrete, scalable constructions.

Abstract

Tile assembly systems in the abstract Tile Assembly Model (aTAM) are computationally universal and capable of building complex shapes, but DNA-based implementations encounter formidable error rates that stifle this theoretical potential. Slat-based self-assembly is a recent development wherein DNA forms long slats that combine together in 2 layers, rather than the aTAM's square tiles in a plane. While tiles tend to bind to 2 neighboring tiles at a time, slats may bind to dozens of other slats. Large slat-based DNA constructions have been implemented in the lab with incredible resilience to many of the errors that plague tile-based constructions, but these come at a cost as slat-based systems are often more difficult to design and simulate. Also, it has not been clear if slats, with their larger sizes and different geometries, have the same theoretical capabilities as tiles. Here we show that slats do, at least at scale. We give constructions showing that any aTAM system may be simulated by a system of slats and that these can be made more efficiently, using shorter slats and a smaller scale factor, when simulating simpler classes of systems. We consider 5 classes of aTAM systems with increasing complexity, from zig-zag systems to the full class of all aTAM systems, and show how they can be converted to equivalent slat systems. Zig-zag systems can be simulated by slats at only a $2c \times 2c$ scale (where $c$ is the freely chosen cooperativity of the slats), the full class of aTAM systems at only a $5c \times 5c$ scale, and intermediate classes using scales between these. Together, these results prove that slats have the full theoretical power of aTAM tiles while providing constructions compact enough to potentially provide designs for DNA-based implementations of slat systems that are both capable of powerful algorithmic self-assembly and possessing the strong error resilience of slats.

Figures (7)

• Figure 1: Examples of IO-marked tile type signatures: (Left) The light blue tile's signature is Input=(S,1),(E,1), Output=(N,1),(W,1). (Right) The yellow tile's signature is Input=(S,2), Output=(N,1),(E,1).
• Figure 2: An example zig-zag aTAM system that simulates a Turing machine. The seed tile is the rightmost of the bottom row. The first row (green) grows right to left. After growing upward by one tile, the second row grows left to right and extends one extra tile beyond the row below. Subsequent rows continue to alternate direction and extend in length by 1. Each row represents a configuration of the Turing machine with each tile representing a tape cell, the north glues representing the contents of each cell, and the red tiles showing the location of the simulated tape head and current state of the machine. If a row is growing in the direction in which the tape head needs to move after the last transition, that occurs. If it is growing in the opposite direction, the tape head and state remain the same for that row, and then the next row (which will be of alternating direction) simulates the head movement and state change.
• Figure 8: (a) Strength-2 macrotile template for a standard aTAM system. Cells are bounded by squares to show their functionality, and mark cell locations where output slat templates may be added to the macrotile. One of the marked cells may be designated as an input, and have its domains assigned such that they connect with those provided by the output slats of a neighboring macrotile whose output is of the same glue type. Cells are signified using the same color conventions as Figure \ref{['fig:cell-coloring']}. (b) Macrotile experiencing south strength-2 input. (b) Macrotile exhibiting south and west strength-1 inputs. Output slat templates are colored in accordance to Figure \ref{['fig:cell-coloring']}, and input domain locations are marked with a green box.
• Figure 9: (a)Left: Input slats for all directions. Center: Decision slats in the decision rows of a macrotile. Right: Output slats growing in all directions. (b) Left: a macrotile receiving inputs from all 4 of its neighbors. Red slats encode an incoming glue from the north, yellow from the east, green from the south, and blue from the west. Magenta slats attach non-deterministically to the glues presented by these slats and each encode a possible tile from $T$ to which this macrotile may resolve. Cyan slats decide a winner among the magenta slats. The remaining illustrations are example macrotiles which only receive input from 3 sides so the remaining side may act as an output.
• Figure 10: To make a set of IO-marked tile types whose collective behavior will be the same as an un-marked tile type, one IO-marked tile type is created for every minimal set of glues whose combined strength is $\ge \tau$ (minimal in that, if any individual glue was removed from the set, the combined strength would be $< \tau$), with those marked as input glues and the others as output glues. Two examples are shown. (Left) In the center, the un-marked tile type has a strength-1 glue on each side. Surrounding it are the six IO-marked tile types created from it, one for each possible pair of strength-1 glues marked as inputs. Note that this is the worst-case increase in tile complexity. (Right) In the center, the un-marked tile type has two strength-2 glues and two strength-1 glues. Surrounding it are the three IO-marked tile types created from it, one each with one of the strength-2 glues as the sole input, and the other with the pair of strength-1 glues as the inputs.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 6: IO TAS
• Definition 8: Zig-zag TAS
• Definition 9: Standard TAS
• Definition 10: Standard TAS with across-the-gap
• Definition 11: Equivalent Productions