DNA Strand-Displacement Temporal Logic Circuits

Abstract

Molecular circuits capable of processing temporal information are essential for complex decision making in response to both the presence and history of a molecular environment. A particular type of temporal information that has been recognized to be important is the relative timing of signals. Here we demonstrate the strategy of temporal memory combined with logic computation in DNA strand-displacement circuits capable of making decisions based on specific combinations of inputs as well as their relative timing. The circuit encodes the timing information on inputs in a set of memory strands, which allows for the construction of logic gates that act on current and historical signals. We show that mismatches can be employed to reduce the complexity of circuit design and that shortening specific toeholds can be useful for improving the robustness of circuit behavior. We also show that a detailed model can provide critical insights for guiding certain aspects of experimental investigations that an abstract model cannot. We envision that the design principles explored in this study can be generalized to more complex temporal logic circuits and incorporated into other types of circuit architectures, including DNA-based neural networks, enabling the implementation of timing-dependent learning rules and opening up new opportunities for embedding intelligent behaviors into artificial molecular machines.

Figure 1

Concept and chemical reaction network implementation of temporal logic circuits. (a) Abstract circuit diagram, (b) truth table, (c) chemical reaction network implementation, and (d) simulations of a two-input temporal AND gate. c is the concentration of input signals A and B. Simulations of output signals Y and Z are shown as relative concentrations to c over time, where c = 100 nM, k_s = 0.002/s, and k_f = 2 × 10⁶ /M/s.

Figure 2

DNA strand-displacement implementation of a two-input temporal AND gate. (a) Reaction pathways. (b) Reporting mechanism. Zigzagged and straight lines indicate short toehold and long branch migration domains, respectively. Asterisks in domain names indicate sequence complementarity. Gray boxes highlight gate species composing the circuit. Colored boxes in signal strands highlight functional domains that participate in downstream reactions. (c) Simulations. c is the concentration of input signals A and B. Output signals Y and Z are shown as relative concentrations to c over time, where c = 100 nM, k_s = 10⁵/M/s (estimated strand displacement rate with a 5-nt toehold), and k_f = 2 × 10¹³/M²/s (estimated cooperative hybridization rate with a 7-nt toehold). Reporting reactions are not shown here but were included in simulations (Supplementary Note S2). Gates and reporters were in 20% and 50% excess compared to inputs, respectively.

Figure 3

Crosstalk between two reaction pathways. (a) Crosstalk reactions. Forward and backward reactions are indicated as filled and open arrows, respectively. (b) Simulations of desired reactions shown in Figure 2c together with crosstalk reactions shown here, where input concentration c = 100 nM, k_s2 = 10⁵/M/s, k_f2 = 2 × 10¹³/M²/s, k_s1 = k_s2/Δk, and k_f1 = k_f2/Δk. The darkest to lightest trajectories correspond to simulations with no difference and a 2-fold and a 5-fold difference between the two pairs of rate constants, respectively.

Figure 4

Characterization of circuit behavior. (a) Sequence design that reduces crosstalk by utilizing mismatches. A two-nucleotide clamp domain (colored in gray) is used for reducing undesired leak reactions between gates. Each toehold domain consists of a core sequence (colored in black) and a clamp. (b) Simulation and fluorescence kinetics data. All gates, reporters, and inputs were at 100, 150, and 90 nM, respectively. (c) Simulations with varying branch migration rates. A detailed model that includes binding (with rate k_f), toehold dissociation (with rate k_r), and branch migration (with rate k_i) steps. Two trimolecular reactions are shown here as examples, but all four trimolecular reactions in Figures 2c and 3b were converted to the detailed model in simulations (a full list of reactions and rate constants are shown in Supplementary Note S2).

Figure 5

Demonstration of the two-input temporal AND gate with varying toehold lengths. (a) Sequence design with varying lengths of the S* toehold on the two cooperative hybridization gates. (b) Simulation and fluorescence kinetics data of the circuit with a 5-nt S* toehold. All gates, reporters, and inputs were at 100, 150, and 90 nM, respectively. (c) Output concentrations in experiments with varying toehold lengths. Darker and lighter bars correspond to output concentrations immediately (within 5 min, when the first data point was collected) and 1 h after the second input has arrived, respectively. Dashed line marks the separation between ON and OFF states.

Figure 6

Varying time intervals between two inputs. Darker and lighter bars correspond to experiment and simulation of output concentration at 60 min after the second input was added, respectively. Δt = |t_A – t_B|. All gates, reporters, and inputs were at 100, 150, and 90 nM, respectively. Data were averaged over three independent experiments using varying qualities of the gate molecules (unpurified vs gel purified). Error bars indicate standard deviation of the mean. Dashed line marks the separation between ON and OFF states. An example set of kinetics data is shown in Figure S5.