Cross-section of the folding funnel, projections of the lattice chain and the energy–Rg phase plane. Same Metropolis HP model as the 3D lab. Educational toy model, not a real protein.
Samples a 7-bead chain (HPHHPHH) with this page's Monte Carlo at the chosen kT/ε and compares the measured contact-count distribution with an exact enumeration of all 16,926 conformations.
Runs headless: 6 independent chains per temperature, up to 200,000 steps each; median steps to reach energy −16 or lower (chain A).
The science
HP model. Each residue is hydrophobic (H) or polar (P) and sits on a point of the simple cubic lattice; consecutive residues are neighbours and no site is reused (self-avoiding walk). The energy is E = −ε · NHH, where NHH counts non-bonded H–H nearest-neighbour pairs. The hydrophobic effect is the origin of this attraction.
Metropolis Monte Carlo. A trial move is accepted with probability min(1, exp(−ΔE/kT)). The proposals (end move, corner flip, pivot of head or tail by a cubic symmetry) are symmetric, so detailed balance holds and long-run frequencies are Boltzmann: P(E) ∝ g(E) exp(−E/kT), where g(E) is the number of conformations with energy E. A Monte Carlo step is not physical time.
Levinthal's paradox. The number of self-avoiding walks grows as cn ≈ A μn nγ−1 with μ ≈ 4.684, γ ≈ 1.157 (A ≈ 1.2 here, approximate). At an assumed 1013 conformations per second, exhaustive search of 100 residues takes astronomically long, yet real proteins fold in milliseconds to seconds because the landscape is funnel-shaped.
The funnel and its slope. The cross-section is schematic: height h(ρ) = HF ρ1.6 with ρ = (Rg − Rg,min)/(Rg,max − Rg,min) and Rg = √⟨|ri − rcm|²⟩. The arrow shows −dh/dρ, the downhill slope. The phase plane shows the real energy and Rg of sampled states.
Simplifications and limits. Best-known energies come from repeated annealing, not proofs, except the all-H 27-mer where −28 equals the maximal contact count of a 3×3×3 cube. Lattice geometry is crude. The Levinthal rate 1013/s is an assumption.