Circuit Simulation with Single-Stranded Tiles

The three focuses for when simulating a circuit using single-stranded tiles were managing the complexities of the design and experiments using an abstraction hierarchy, making the tile set reprogrammable, and designing effective DNA sequences.

Abstraction Hierarchy

At abstraction Level 1, an Iterated Boolean Circuit model was used as an abstract model of computation for the growth of SSTs into nanotubes. The IBC model has a repeatedly executed array of Boolean gates that are locally connected. An \(\mathit{n}\) -bit \(\mathit{l}\) -layer IBC has (\(\mathit{n}\) - 1)\(\mathit{l}\) gates that have two input and two output wires each and 2\(\mathit{l}\) boundary gates that have one input and one output each. The user specifies the logic function computed by each of the (\(\mathit{n}\) + 1)\(\mathit{l}\) gates. The values \(\mathit{n}\) = 6 and \(\mathit{l}\) = 1 were chosen by , which is capable of \(2^{44}\) unique circuits. To compute, the system iterates over the circuit layers repeatedly until reaching an chosen point or cycle. This model is computationally universal because it can simulate other models of computation including Turing machines.

Abstraction Level 2 used the abstract Tile Assembly Model by implementing each two input/two output gate using four abstract tiles, one for each pair of input bits. Kinetic barriers that prevent spontaneous nucleation were created by using glues that encode vertical gate position within the circuit along with bit values. The initial seed assembly that represents the input was cylindrical so the pattern that grows to represent the execution of the circuit was a tube.

Abstraction Level 3 was converting each abstract tile into four proofreading tiles. Using proofreading tiles substantially lowers error rates compared to directly implementing the abstract tiles. To make the tile set reprogrammable, this process was completed for all tiles needed to implement any 6-bit 1-layer IBC, totaling 355 tile types. To implement a chosen circuit, only the needed Boolean gate tiles are used.

At Abstraction Level 4, each proofreading tile was represented by one SST that had four domains to represent the four glues. This defined the SST binding-domain schematic and geometry. Additionally, input-adapter strands that were uniquely addressed to the correct seed position were used to implement the tube shape for the circuit's input bits.

Abstraction Level 5 was designing effective DNA sequences that were capable of implementing the circuit without being prone to undesired interactions that were accounted for previously, specifically spurious nucleation and tile-attachment errors. However, uniformity and specificity of glue-directed tile-attachment energies affect the effectiveness of the kinetic barrier and proofreading tiles.

Challenges

SSTs are floppy when added, so domains on the same strand could bind together and neighboring exposed domains on the nanotube could bind to each other. These bindings have to be undone during tile-attachment events. Furthermore, strands could bind together in solution, causing less and unequal concentrations of free monomers. Therefore ensuring uniform energies requires more than domain-binding energy uniformity and minimizing mismatching energies of tiles binding with one correct domain and one incorrect domain. All this was accounted for in the sequence design by creating a model based on NUPACK and ViennaRNA to predict nucleic secondary structures from thermodynamic measurements, and all other optimization problems were solved using a stochastic local search algorithm.

Unzipping strands that used strand displacement were used to remove the seam that opens the nanotubes so they can lie flat on the mica imaging surface with insides facing up. Guard strands that inactivated any remaining tiles were used to shut down the assembly and excess strands were washed away. For a clear AFM topographical marker, streptavidin was added to bind to the tiles that represent bit 1, which contained an upward-facing biotin molecule. The biotin modification weakened the binding energy of domains in which they appeared, so the ideal growth temperature depended on the particular bit sequence being copied. This was not ideal, so 1.1 kcal per mol was subtracted from the predicted binding strength of the biotin-modified domains.The biotin-modified sequences then bound stronger even without the biotin than the non-biotin-modified domains. Streptavidin labeling wasn't always complete either, but incomplete labels can be distinguished from incorrect ones based on the consistency of the next labels.

Systematic evaluation was needed to determine what tile concentration and temperature was best for seed growth without spontaneous nucleation, no matter the bit pattern. SST nanotubes with 16-helix circumferences were chosen by because it gave the most space for storing information. The ideal temperature range to avoid spontanteous nucleation was identified to be \(50.7^{o}\) C or \(50.8^{o}\) C. The nanotubes would be cooled from a starting temperature of \(90^{o}\) C at a rate of \(1^{o}\) C per minute until the ideal temperature was reached. Then the temperature would be held constant for more than 24 hours.

Results

All 355 tiles in the IBC tile set were tested for correctness with 21 circuits. No tiles had to be redesigned and the experimental conditions did not have to be adjusted. Tile-attachment error rates depended on the circuit but varied between 0.01\% and 0.07\%. The average number of seeds that grew nanotubes for all 21 circuits was 61.1\% and the overall tile-attachment error rate was 0.03\% \pm 0.0008.

The Sorting circuit layer (far left) on three different 6-bit inputs (h-j) is show as a Simulation (top) and an AFM image (bottom). Black (0) and white (1) dots represent the input bits. Two white triangles are used to highlight streptavidin labelling errors in h. Scale bars, 100nm.