DNA as a universal substrate for chemical kinetics

Abstract

Molecular programming aims to systematically engineer molecular and chemical systems of autonomous function and ever-increasing complexity. A key goal is to develop embedded control circuitry within a chemical system to direct molecular events. Here we show that systems of DNA molecules can be constructed that closely approximate the dynamic behavior of arbitrary systems of coupled chemical reactions. By using strand displacement reactions as a primitive, we construct reaction cascades with effectively unimolecular and bimolecular kinetics. Our construction allows individual reactions to be coupled in arbitrary ways such that reactants can participate in multiple reactions simultaneously, reproducing the desired dynamical properties. Thus arbitrary systems of chemical equations can be compiled into real chemical systems. We illustrate our method on the Lotka-Volterra oscillator, a limit-cycle oscillator, a chaotic system, and systems implementing feedback digital logic and algorithmic behavior.

Fig. 1.

Strand displacement molecular primitive. Domains are labeled by numbers, with * denoting Watson–Crick complementarity. Multiple elementary steps are indicated: (1) binding of toeholds 1 and 1^∗; (2) branch migration, a random walk process where domain 2 of strand 2–3 is partially displaced by domain 2 of strand 1–2; (3) the separation of toeholds 3 and 3^∗. (Inset) The single-reaction model of strand displacement.

Fig. 2.

Unimolecular module: DNA implementation of the formal unimolecular reaction X₁ → X₂ + X₃ with reaction index i. Orange boxes highlight the DNA species that correspond to the formal species X₁ (species identifier 1-2-3), X₂ (4-5-6), and X₃ (7-8-9). Domains identical or complementary to species identifiers for X₁, X₂, and X₃ are colored red, green, and blue, respectively. Black domains (10 and 11) are unique to this formal reaction. To reduce rate constant q_i, toehold domain may not be a full complement of domain 1. (A) Complex G_i undergoes a strand displacement reaction with strand X₁, with X₁ displacing strand O_i. (B) O_i displaces X₂ and X₃ from complex T_i. Without buffer cancellation, ; with buffer cancellation, . Reaction equations of type 2–3 are used in simulations (Figs. 5 and 6); simplified reaction equations 4–5 are useful for analysis.

Fig. 6.

Examples showing more complex behavior. In all the maximum strand displacement rate constant q_max = 10⁶ M^-1 s^-1 and the initial concentration of auxiliary species C_max = 10 μM. See Figs. S4 and S5 and SI Text for the rate constants used in A–D, as well as the details of C and D. Plots show the ideal CRN (Dashed lines) and the DNA reactions (Solid lines).

Fig. 5.

Lotka–Volterra chemical oscillator example. (A) The formal chemical reaction system to be implemented with original (unscaled) and scaled rate constants. Desired initial concentrations of X₁ and X₂ are 2 and 1 unscaled and 20 and 10 nM scaled. (B) Reactions modeling our DNA implementation. Each formal reaction corresponds to a set of DNA reactions as indicated. Species X₂ requires a buffering module because σ₂ < σ (σ = σ₁ = k₁ and σ₂ = 0). Maximum strand displacement rate constant q_max = 10⁶ M^-1 s^-1 and initial concentration of auxiliary species G_i, T_i, L_i, B_i, LS_j, and BS_j is C_max = 10 μM. Buffering-scaling factor γ^-1 = q_max(q_max - σ)^-1 = 2. The initial concentrations of strands X₁ and X₂ introduced into the system is γ^-120 nM = 40 nM and γ^-110 nM = 20 nM. (C) Plot of the concentrations of X₁ (Red curve) and X₂ (Green curve) for the ideal system (Dashed line) and the corresponding DNA species (Solid line).

Fig. 3.

Bimolecular module: DNA implementation of the formal bimolecular reaction X₁ + X₂ → X₃ with reaction index i. The black domain (12) is unique to this formal reaction. (A) X₁ reversibly displaces B_i from complex L_i producing complex H_i. (B) X₂ displaces O_i from complex H_i. Occurrence of reaction B precludes the backward reaction of A. (C) O_i displaces X₃ from complex T_i. Without buffer cancellation, q_i = k_i; with buffer cancellation, q_i = γ^-1k_i. Reaction equations of type 7–9 are used in simulations (Figs. 5 and 6); simplified reaction equations 10–11 are useful for analysis.

Fig. 4.

Buffering module: DNA implementation of the buffering module used to cancel out the buffering effect. A buffering module is needed for each formal species X_j for which σ_j < σ. The buffering module for species X₂ is shown. X₂ reversibly displaces BS₂ from complex LS₂ to produce complex HS₂ similarly to the first reaction of the bimolecular module. The black domain (13) is unique to this buffering module for species X₂. Set qs_j = γ^-1(σ - σ_j). Reaction equations 12 are used in simulations (Figs. 5 and 6); simplified reaction equations 13 are useful for analysis.