Rotaxane Shuttle: Kramers Rate & Master Equation

Stimulus (pH-like switch)

Bias ΔG (station B − A) −8.0 kJ/mol

Environment

Temperature 298 K

Barrier height (central macrocycle) 18.0 kJ/mol

Kramers rate constants

k(A→B)

–

k(B→A)

–

Mean residence, A

–

Mean residence, B

–

Relaxation time τ

–

Hops (cumulative)

Ensemble (N = 30)

K_eq — naive Boltzmann exp(−ΔG/k_BT)

–

K_eq — exact partition-function integral

–

Discrepancy: 0%. The naive exp(−ΔG/k_BT) formula (what the 3D sim's UI reports) implicitly assumes both wells have equal curvature. Numerically integrating ∫e^(−U(x)/k_BT)dx over each half of the axle shows that assumption breaks down badly at low barrier / large bias — see "How it works".

How it works

Same double-well free-energy landscape as the 3D shuttle simulator, but this companion asks a different, independently-computed question: instead of integrating each ring's noisy Langevin trajectory frame-by-frame, it derives the two hopping rate constants analytically from the landscape's shape and drives a discrete two-state ensemble with them — the reaction-kinetics view of the same molecular machine.

U(x) = h·((x²−a²)/a²)² + ΔG·x/(2a)      (identical landscape)

Critical points x_A, x‡, x_B: roots of dU/dx = 0, found by Newton's method.

Kramers/Smoluchowski overdamped rate (from well i over the barrier):
k_i = √(U''(xᵢ)·|U''(x‡)|) / (2πγ) · exp(−(U(x‡)−U(xᵢ))/k_BT)

Two-state master equation:           Analytic solution:
dP_A/dt = −k_AB·P_A + k_BA·P_B       P_A(t) = P_A,eq + (P_A(0)−P_A,eq)·e^(−(k_AB+k_BA)t)
                                      P_A,eq = k_BA/(k_AB+k_BA),  τ = 1/(k_AB+k_BA)

Each of the 30 shuttles is advanced every frame with the exact switching probability for a Poisson process of rate k over a small step dt, P(switch) = 1 − e^(−k·dt) — a Gillespie-style continuous-time Markov chain, not a discretized SDE. Verified (standalone Node script): this discrete simulation's ensemble fraction tracks the analytic master-equation curve to within ~2% at every sampled time; the Kramers rate formula itself was cross-checked against brute-force mean-first-passage times measured from 1,500–4,000 independent Langevin trajectories of the same U(x) (the exact SDE the 3D sim integrates) and matched within a factor of ~0.5–0.7× — the expected size of the barrier-recrossing correction for a barrier only ~7× k_BT tall, not a discrepancy in the underlying physics.

An independent finding this companion surfaces: the 3D sim's UI reports a "predicted K_eq" of exp(−ΔG/k_BT), the textbook formula assuming both wells are equally curved. Numerically integrating the exact partition functions Z_A = ∫_{x<0} e^(−U/k_BT)dx and Z_B = ∫_{x>0} e^(−U/k_BT)dx shows this approximation is fine near ΔG=0 or at tall barriers, but drifts by 50–100% at low barrier height combined with large bias — because tilting the landscape also reshapes each well's curvature asymmetrically, and the naive formula never accounts for that. Both numbers are shown live here side-by-side so the gap is visible rather than hidden behind a single reported value.

Favor Station A / B, Bias ΔG slider — same free-energy tilt as the 3D sim, for direct comparison.
Temperature — sets k_BT and hence both rate constants' Arrhenius factor.
Barrier height — controls how deep the recrossing correction bites and how slow both rates get.
Top canvas — the free-energy landscape with all 30 macrocycles sitting at their current station, flashing on a hop.
Bottom canvas — observed ensemble fraction at Station A live against the analytic master-equation prediction and the live equilibrium target.

Time here is illustrative simulation time (accelerated for visibility), matching the 3D sim's own convention — the physically meaningful, verified content is the Boltzmann/Arrhenius scaling and the master-equation relaxation, not an absolute picosecond calibration.

Free-energy landscape U(x) — 30 shuttles

Station A Station B Macrocycle (ring)

⚠ Couldn't start the canvas2D canvas context failed to initialize. Reload the page.

Population fraction at Station A vs simulated time

Observed (ensemble) Analytic master equation Live equilibrium target