Algorithmic Self-Assembly of DNA Sierpinski Triangles

Abstract

Algorithms and information, fundamental to technological and biological organization, are also an essential aspect of many elementary physical phenomena, such as molecular self-assembly. Here we report the molecular realization, using two-dimensional self-assembly of DNA tiles, of a cellular automaton whose update rule computes the binary function XOR and thus fabricates a fractal pattern—a Sierpinski triangle—as it grows. To achieve this, abstract tiles were translated into DNA tiles based on double-crossover motifs. Serving as input for the computation, long single-stranded DNA molecules were used to nucleate growth of tiles into algorithmic crystals. For both of two independent molecular realizations, atomic force microscopy revealed recognizable Sierpinski triangles containing 100–200 correct tiles. Error rates during assembly appear to range from 1% to 10%. Although imperfect, the growth of Sierpinski triangles demonstrates all the necessary mechanisms for the molecular implementation of arbitrary cellular automata. This shows that engineered DNA self-assembly can be treated as a Turing-universal biomolecular system, capable of implementing any desired algorithm for computation or construction tasks.

Reducing Facet Nucleation during Algorithmic Self-Assembly

“…The strands for each lattice (50 nM per strand) were annealed from 90 to 10°C over 10 h, held at 10°C for 2 h, and then melted back to 90°C at the same rate. For both lattices, we observed ( Figure 12 in Supporting Information) a reversible transition between 45 and 70°C, where tile formation has been observed previously, 3,17,24 and a hysteretic transition at lower temperatures, where lattices formed. Formation of both lattices occurred around 16°C and melted between 25 and 37°C, indicating a significant kinetic barrier to homogeneous nucleation.…”

“…9 In principle, arbitrarily complex objects can be constructed using algorithmic self-assembly, 10 and objects including large finite-sized shapes and some circuit diagrams can be assembled from a small number of components. [11][12][13][14] Aperiodic one- 15,16 and two-dimensional 17,18 structures have been algorithmically self-assembled from programmable crystal monomers constructed from DNA tiles.…”

“…Unlike standard crystal growth, where nucleation on facets is part of the desired growth process, in a proper algorithmic self-assembly every tile attaches by two or more binding sites. Facet errors were identified as a major source of algorithmic self-assembly errors both in experimental 17 and in theoretical 20,21 studies. The overall rate of errors during assembly is the sum of the growth errors and the facet errors.…”

Two Computational Primitives for Algorithmic Self-Assembly: Copying and Counting

“…This is consistent with the 1-10% per-tile error rate observed in previous algorithmic self-assembly experiments with similar tiles. 17 It is also consistent with tube joining processes that have been observed in single-tile DAE-E tubes; 38 here, COPY tubes that nucleate with different bit patterns may also join end-to-end if sufficiently many sticky ends match, resulting in apparent copying errors.…”

“…This results in the set of four DAE-E molecules shown in Figure 1c. (Although this doublecrossover motif has been shown previously to have intrinsic curvature that encourages assemblies of tiles to roll up into tubes, 17,34 assemblies grown from long scaffolds in that work usually contained 5 to 15 layers of rule tiles, which we predicted would be sufficient for our investigations here.) To distinguish tiles representing 0's from tiles representing 1's, we decorated the latter with protruding hairpin motifs that provide topographic contrast when imaged by atomic force microscopy (AFM).…”