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.
MethodsDesign, data-processing and modelling: DNA sequences were designed using our own algorithms based on sequence symmetry minimization implemented in Matlab and C (available at http://www.dna.caltech.edu/DNAdesign/). Curve fits for persistence length data and models for lattice strain energies were calculated in Matlab.Molecular models were constructed and visualized using a combination of NAMOT, RasMol, and PyMol (scripts and coordinates at http://www.dna.caltech.edu/SupplementaryMaterial/).The molecular models are for visualization only and have not been subjected to molecular dynamics calculations.DNA sample preparation: Lyophilized HPLC-or PAGEpurified DNA oligonucleotides were purchased from Integrated DNA Technologies (Coralville, IA), resuspended in water, quantitated by UV absorbance at 260 nm, and stored at -20• C. All samples were prepared in a 1X Tris-Acetate-EDTA (TAE) buffer with 12.5 mM magnesium acetate (pH=8.3). An equimolar mixture of strands (5 strands if one tile, 10 strands if two) was annealed from 95• C to 25• C (fluorescence microscopy) steps in a PCR machine (Eppendorf Mastercycler). For AFM, each strand was present at 200 nM, for fluorescence microscopy the total concentration of tiles was kept at 400 nM. For fluorescence microscopy, a single fluorescein-labeled strand was incorporated into each tile; the position of the dye was varied from the 5 end of the #3 strand to the 5 end of the #5 strand with no apparent effect. AFM of REp+SEp(3:FAM) was similar to that of REp+SEp.Preparation of PVP coated glass: Adapted from.1 Microscope slides and coverslips were washed in 1M NaOH for 1 hour, rinsed thoroughly with de-ionized (DI) water and immersed in 1% v/v acetic acid solution for 2 hours. Then, they were rinsed again with DI water and silanized in a 1% v/v 3-(trimethoxysilyl)propylmethacrylate (Aldrich) in 1% v/v acetic acid for 36 hours. For polymer coating, 500 mL of a 4% w/v Mw = 360, 000 poly(vinylpyrrolidone) (PVP, USB Corp.) solution with 2.5 mL of 10% w/w ammonium persulfate solution and 250 µl of N,N,N ,N -tetramethylethylenediamine (TEMED, Acros) was prepared. Slides and coverslips were incubated in the PVP solution at 80• C for 18 hours. They were then rinsed and stored in DI water. Coating was stable for at least 2 weeks.Preparation for fluorescence microscopy: Samples were left overnight at room temperature after annealing. Immediately prior to use, a PVP-coated microscope slide and coverslip were rinsed with ethanol and dried. Then, 2.6 µl of solution containing DNA tubes and oxygen scavenging system (0.035 mg/ml catalase, 0.2 mg/ml glucose oxidase, 4.5 mg/ml glucose, 5% β-mercaptoethanol) was deposited onto the slide, covered with the coverslip and sealed with epoxy or parafin. The distance between slide and coverslip was ≈5 µm and the thickness of sample solution was typically narrowed to ≈3 µm by the PVP coating.Fluorescence microscopy: Samples were imaged on an inverted microscope (IX 70, Olympus) with 100X/1.40 NA oil immersion and 40X/0.75 NA air objectives. Blue lig...
Abstract. In this paper we report the design and synthesis of DNA molecules (referred to as DNA tiles) with specific binding interactions that guide self-assembly to make one-dimensional assemblies shaped as lines, V's and X's. These DNA tile assemblies have been visualized by atomic force microscopy. The highly-variable distribution of shapes -e.g., the length of the arms of X-shaped assemblies -gives us insight into how the assembly process is occurring. Using stochastic models that simulate addition and dissociation of each type of DNA tile, as well as simplified models that more cleanly examine the generic phenomena, we dissect the contribution of accretion vs aggregation, reversible vs irreversible and seeded vs unseeded assumptions for describing the growth processes. The results suggest strategies for controlling self-assembly to make more uniformly-shaped assemblies.
Abstract. The ramification and qualification problems are two infamous, hard and ever present problems in databases and, more generally, in systems exhibiting a dynamic behavior. The ramification problem refers to determining the indirect effects of actions, whereas the qualification problem refers to determining the preconditions which must hold prior to the execution of an action. A solution to these problems in database systems permits reasoning about the dynamics of databases and allows proving consistency properties. These two problems become increasingly complex in temporal databases and no satisfactory solution has been proposed as of yet. In this paper, we describe these two problems in the context of temporal databases and we propose a solution of polynomial complexity based on the language of the Situation Calculus. This solution extends previous proposals for the solution of these problems in conventional (non-temporal) databases.
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