Fig 1.
The binary counter tile set (Left) and a partial assembly (Right).
The assembler counts from 0, represented as n-bits in the seed, by adding 1 to the previous layer up to the full n-bit value 2n − 1.
Fig 2.
The three characteristic sets of assemblies in an assembly system.
The set of assembled patterns An contains some target patterns (An∩Pn) that are already assembled, the set of promising patterns that includes An∩Pn as well as nontarget patterns that will eventually become target patterns of larger size n, and the remaining error assemblies (
.)
Fig 3.
A tile set designed for a temperature 2 assembly system that uses cooperation to simultaneously detect nondeterministic input signals, red (R) and green (G), and from the designed point of cooperation output a deterministic signal to the blue (B) site.
For cooperative assembly, double edges indicate strength 2 glues, and single edges strength 1. When the system is run in noncooperative mode, all glues are strength 1. R and G propagate from the inputs r and g by a series of R and G tiles. The D and E tiles are placed nondeterministically at some point along the signal to enable attachment of the cooperation tile C. At the point of cooperation, the O tile binds cooperatively, and then, grows the deterministic blue signal by a series of B tiles (only the first one is shown here.) The number of tiles in the blue signal is constrained by the size of the assembly space L.
Fig 4.
The ideal RGB assembler is designed to grow two signals, one red from site R and one green from site G, toward a point of cooperation C.
The seed is an L × L square perimeter where inputs R and G and output B are to be placed. Cooperation is achieved at the point C, where the signals would “meet”to effect the AND and produce a deterministic signal that is recorded at output B.
Table 1.
Efficiency of a cooperative assembler at temperature τ = 2 and its noncooperative implementation at temperature τ = 1.
The substantial drop in the probability of assembly with respect to the target set and its efficiency show the lack of robustness of the original assembler. Nevertheless, the data shows that some cooperation is better than none for efficiency of the assembler.
Fig 5.
The RGB assembler running in (noncooperative) experimental conditions can produce a variety of error patterns, both before and after the designed point of cooperation.
Either the red, green, or blue signal may set the radius (the size of the pattern.) Here, both signals attempt to fake cooperation by nondeterministically attaching a cooperation tile and spawning erroneous color signals. Each possible red and green signal may do so, producing a large quantity of error patterns.
Fig 6.
Past the point of cooperation, these phantom signals may or may not block one another, compounding the rate of error patterns.
Table 2.
Strength and efficiency of DNA Origami assemblers.
Values are roughly estimated based on their Monte Carlo sampling of the assembled products, with p being estimated as the smallest fraction of well-formed subpatterns in the target pattern and q being estimated as the complement of his fidelity (fraction of correct origamis.) Due to incomplete information, these estimates disagree with the general perception that origami protocols are efficient assemblers.
Fig 7.
Seed tiles and transition probabilities of probabilistic cellular automaton DKCA assemblers (Left).
The tile set (Center) for ECA rule 122 uses tile concentrations and nondeterminism to weakly assemble a variety of patterns (without spontaneous generation) from random seeds, as illustrated on the Right.
Fig 8.
DKCA patterns generated from the same seed can be wildly different (Left/Center).
The DKCA phase diagram (Right) describes the observable behavior of the DKCA across a spectrum of probabilities p1 and p2, with typical non/percolating behavior of an ECA assembler (shown as filled/empty circles, respectively) consistent with the phase-transition DKCA behavior overall (the thresholding curve separates percolating from nonpercolating CA regions.)