Folding Funnel Energy Landscape

Landscape Parameters

Temperature T 0.60

Ruggedness (frustration) A 1.00

Funnel steepness D 6.0

Ensemble size N 150

Ensemble State

Folded (Q > 0.92)

Mean native contacts ⟨Q⟩

0.00

Elapsed time τ

0.0

Mean folding time τ_f

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How it works

Energy-landscape theory (Bryngelson & Wolynes; Onuchic & Wolynes) describes folding not as one fixed pathway but as diffusion of an ensemble of conformations across a free-energy surface, plotted against the reaction coordinate Q, the fraction of native contacts formed (Q=0 unfolded, Q=1 native).

H(Q,φ) = −D·Q + A·(1−Q)·n(Q,φ)
R(Q)   = R₀·(1−Q)^0.7        (conformational entropy ∝ shrinking radius)
dQ = −M·∂H/∂Q·dt + √(2MT·dt)·ξ(t)   (overdamped Langevin)

D is the energetic bias toward the native state — a "minimally frustrated" sequence has a smooth, funnel-shaped H(Q) that biases folding downhill, which is how real proteins fold in micro-to-milliseconds instead of needing to sample the astronomical number of conformations implied by a random search (Levinthal's paradox). A, the ruggedness amplitude, adds energetic frustration: bumps that can trap chains in local minima, exactly as competing native and non-native contacts do in a real polypeptide. n(Q,φ) is a bounded roughness field that fades toward Q=1, mirroring how conformational and energetic frustration both shrink near the native state.

Temperature T — thermal noise driving the random walk. Too low and chains freeze into local traps on the rugged surface (a folding "glass transition"); too high and thermal agitation overwhelms the funnel's downhill bias and folding stalls.
Ruggedness A — energetic frustration/roughness superimposed on the smooth funnel; higher values create more kinetic traps and slower, more heterogeneous folding times.
Funnel steepness D — the energy gap between unfolded and native states; a steeper funnel folds faster and more reliably at a given temperature.
Ensemble size N — number of independent conformations diffusing on the surface simultaneously, each an instanced sphere colour-coded by its own Q.

Time τ and energy here are in reduced, dimensionless Langevin units, not real seconds — the qualitative behaviour (funnel bias vs. thermal noise vs. ruggedness-induced trapping) is what energy-landscape theory actually predicts, not a claim about a specific protein's real folding rate.

Near-native (high Q) Unfolded (low Q)

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