This is an independent 2D reformulation of energy-landscape-theory folding, built directly from the Bryngelson–Wolynes free energy G(Q) = enthalpy − T·entropy, not a flattened copy of a 3D scene. The reaction coordinate Q (fraction of native contacts, 0→1) drives an explicit free-energy functional that separates the funnel's enthalpic bias from the conformational entropy lost on folding:
G(Q) = −D·Q − T·S0·(1−Q) + A·(1−Q)·ξ(Q)
dQ = −M·∂G/∂Q·dt + √(2MT·dt)·η(t) (overdamped Langevin, reflecting at Q=0,1)
Tf = D / S0 (folding transition temperature)
The −D·Q term is the funnel's downhill bias toward the native state. The −T·S0·(1−Q) term is new here: it is the actual configurational entropy of the unfolded ensemble (many accessible conformations at low Q, one native structure at Q=1), and it stabilizes the unfolded state more strongly as T rises. Setting the two competing terms equal gives a real, derivable folding transition temperature Tf = D/S0 — below it the funnel wins and the ensemble folds; above it entropy wins and it stays unfolded. A·(1−Q)·ξ(Q) is ruggedness/frustration exactly as in energy-landscape theory, fading toward the native state.
- Temperature T — thermal noise; also sets how strongly entropy stabilizes the unfolded ensemble via the −T·S0·(1−Q) term.
- Ruggedness A — energetic frustration superimposed on the smooth funnel; higher values create kinetic traps and slower, more heterogeneous folding.
- Funnel steepness D — the enthalpic gap between unfolded and native states; raising it raises Tf and folds the ensemble faster and more reliably.
- Configurational entropy S0 — how much conformational freedom is lost on folding; raising it lowers Tf, since a bigger entropic penalty makes the unfolded ensemble harder to beat thermally.
- Funnel width in the diagram — drawn proportional to S0·(1−Q), the same entropy term that drives the dynamics, so the visual "narrowing funnel" is the actual entropy loss, not just decoration.
The small free-energy plot under the funnel shows G(Q) at the current T, A, D, S0 — watch it tilt from a monotonic downhill funnel (T ≪ Tf) toward a landscape that favors Q=0 (T ≫ Tf) as you raise temperature past the computed Tf. Time τ and energy are reduced Langevin units, not real seconds.