A molecular beacon is a hairpin-shaped DNA probe with a fluorescent donor dye on one end and a dark quencher on the other. Closed and open are the two states of a real coupled thermodynamic equilibrium — stem melting combined with target hybridization — instead of the per-molecule stochastic model used in the 3D version, so what you see here is the exact steady-state solution of that equilibrium, recomputed live as you move the sliders:
Stem stability ΔG_close(T) = ΔH - T·ΔS (van't Hoff, T in kelvin)
Conformational Keq K_conf = exp(ΔG_close / RT)
Coupled binding F_open = K_conf·(1+K_hyb·[target]) / (1 + K_conf·(1+K_hyb·[target]))
FRET quenching E(r) = 1 / (1 + (r/R0)^6), F(r) = Fmax·(1-E(r))
This is a Monod–Wyman–Changeux-style two-state-with-ligand model: only the transiently open conformation can bind target, so binding pulls the whole equilibrium toward the open, fluorescent state. At zero target, F_open reduces to the pure stem-melting curve set by ΔH, ΔS and temperature; as target is added, the K_hyb·[target] term inflates the open population until it saturates near 1.
- Target concentration — the coupled-equilibrium term K_hyb·[target]; more target pulls more probes into the fluorescent open state, following a real Langmuir-type saturation.
- Temperature — sets ΔG_close(T) via the van't Hoff relation; below the melting point Tm (ΔH/ΔS ≈ 55 °C) the stem is favored, above it the open state dominates even without target.
- Ionic strength — real salt-correction (ΔTm ≈ 16.6·log₁₀[Na⁺]) that stabilizes the duplex at higher salt, shifting Tm and the effective hybridization constant upward, matching real PCR/hybridization buffer design.
The dose-response plot on the right traces fluorescence against log[target] at the current temperature and salt — a genuine sigmoid emerging directly from the free-energy calculation above, not a scripted curve. This is the same readout principle behind real-time PCR beacons, COVID-19 nucleic-acid tests and point-of-care DNA/RNA diagnostics.