A feedforward digital logic circuit is first translated into an equivalent dual-rail logic circuit and then translated into an equivalent seesaw DNA circuit. Visual DSD code and Mathematica code are generated for analysing and simulating the seesaw DNA circuit, and DNA sequences are generated for constructing the circuit. Bottom right diagram introduces the notations of seesaw circuits: black numbers indicate identities of nodes. The locations and values of red numbers indicate the identities of distinct DNA species and their relative initial concentrations, respectively.

(a) Gate diagram and truth table of a digital logic circuit that computes the transition rules 110 and 124 of elementary cellular automata. (b) Seesaw gate diagram of the equivalent DNA strand displacement circuit. Each seesaw node connected to a dual-rail input implements input fan-out. Each pair of seesaw nodes labelled and implements a dual-rail AND and OR gate, respectively. Each pair of dual-rail AND and OR gates implements an AND, OR or NAND gate in the original logic circuit. Each dual-rail output is converted to a fluorescence signal through a reporter, indicated as a half node with a zigzag arrow. Each circle and dot inside a seesaw node indicates a double-stranded threshold and gate molecule, respectively. Each dot on a wire indicates a single-stranded fuel molecule. (c) Simulations of the DNA strand displacement circuit using the previously developed model for purified seesaw circuits. Trajectories and their corresponding outputs have matching colours. Overlapping trajectories were shifted to be visible. Dotted and solid lines indicate dual-rail outputs that represent logic OFF and ON, respectively. For example, when input LCR=001, meaning L⁰, C⁰ and R¹ were introduced at a high concentration and L¹, C¹ and R⁰ at a low concentration, two output trajectories R124⁰ and R110¹ reached an ON state and the other two output trajectories R124¹ and R110⁰ remained in an OFF state, indicating that the output was computed to be 0 and 1 for rule 124 and 110, respectively. Simulations were performed at 1 × =50 nM—the compiler recommended standard concentration for large-scale purified seesaw circuits.

(a) Simulations and (b) experimental data of digital signal restoration. (c) Estimating effective threshold concentration by fitting simulations to the data obtained. (d) OR and AND logic gates constructed using adjusted nominal threshold concentrations. (e) Estimating effective gate concentration. Data show steady-state fluorescence level. 1 × =100 nM. Here and in later figures, all output signals in the data were normalized using the minimum fluorescence signal (the first data point) of an OFF trajectory as 0 and the maximum fluorescence signal (the average of the last five data points) of an ON trajectory as 1.

(a) Logic circuit diagram, seesaw circuit diagram and experimental data of a two-layer logic circuit. (b) Measuring the effective concentrations of the gate species. Three independent circuits were used to measure the effective concentrations of two gates fully triggered by x₁ and x₂, respectively, comparing with the effective concentration of x₃ (using signal strand w_18,53). (c) Experimental data of the two-layer logic circuit using adjusted nominal gate concentration. 1 × =100 nM.

Logic circuit diagram, seesaw circuit diagram and experimental data of a two-layer logic circuit with (a) two upstream OR gates connected to a downstream AND gate and (b) two upstream AND gates connected to a downstream OR gate. Nominal concentrations shown in grey and black indicate adjustments made in a previous step and in this step, respectively. Small insets of experimental data show the circuit behaviours before adjustments. 1 × =100 nM.

Insets show how the flowchart was used to construct the rule 110 sub-circuit. Y (yes) and N (no) highlighted in orange in the flowchart indicate the situations encountered and decisions made while building the rule 110 sub-circuit. 1 × =100 nM.

(a) Fluorescence kinetics data of the two pairs of dual-rail outputs. 1 × =100 nM. All DNA sequences are listed in . (b) Comparing the ideal logic circuit behaviour (left) with the DNA circuit behaviour (right). Each of the circuit outputs is illustrated by an array of 7 × 8 cells, representative of eight cellular automata generations on a torus with starting configuration (0,0,0,1,0,0,0). The arrays for the DNA circuit were plotted using the output values at 24 h from the data. The ideal logic circuit behaviour corresponds to an image of a black dog with a white background for R124¹, an inverted image for R124⁰ and their mirror images for R110¹ and R110⁰, respectively.

(a) Populations of signal, gate and threshold molecules without and with synthesis errors in the marked locations. r=0.01. (b) Example reactions that involve DNA strands without and with synthesis errors. ∀i, j, k, x and y.

(a) Simulations of the rule 110–124 circuit using the previously developed model for purified seesaw circuits, predicting that the circuit should yield desired outputs in roughly 8 h (shown as dotted lines) and the undesired reactions will take over in 24 h. (b) Simulations using the new model for unpurified seesaw circuits, predicting that the circuit should yield desired outputs in roughly 24 h. k_f=2 × 10⁶ M⁻¹ s⁻¹, k_s=5 × 10⁴ M⁻¹ s⁻¹, k_l=10 M⁻¹ s⁻¹, k_rf=26 s⁻¹, k_rs=1.3 s⁻¹. 1 × =100 nM.