This models a real class of DNA nanotechnology device: a hollow DNA-origami box with a hinged lid, held shut by a small number of duplex "lock" strands, that only opens once enough locks are removed by toehold-mediated strand displacement (Andersen et al., Nature 2009, DNA box with a controllable lid; Douglas et al., Science 2012, logic-gated nanorobot).
Each lock is a short DNA duplex with a single-stranded "toehold" overhang. A complementary trigger (key) strand binds the toehold, then branch-migrates through the duplex and displaces the original strand — releasing the lock. Held under trigger excess, each lock opens as a memoryless (pseudo-first-order) process:
P(lock still closed at t) = exp(−k_obs · t)
k_obs = k(b) · [T]
k(b) ≈ k_max · (1 − e^(−b/b0)) + k_leak
where b is toehold length (nt), [T] the trigger concentration, k_max ≈ 3×10⁶ M⁻¹s⁻¹ is the diffusion-limited rate measured for long toeholds (Zhang & Winfree, JACS 2009), and k_leak ≈ 1 M⁻¹s⁻¹ is the slow "leak" rate of toehold-free branch migration by spontaneous fraying. A ≈2× rate increase per +10 °C (branch migration is thermally activated) is applied to k_obs.
The lid only swings open once N locks required have all been displaced — an AND-gate logic exactly like the antigen-recognizing aptamer locks on real payload-carrying nanorobots. Once open, cargo particles escape the box and undergo ordinary Brownian diffusion, with diffusion coefficient set by the Stokes–Einstein relation:
D = k_B·T / (6π·η·r)
for a cargo particle of radius r ≈ 4 nm in water (η ≈ 0.89 mPa·s). Raising temperature speeds up both the unlocking kinetics and the post-release diffusion. The reaction clock runs at ×15 visualization speed so low-concentration/short-toehold runs (which are genuinely slow — minutes to hours at 1 nM with a 0 nt toehold) stay watchable.