Energy 0 εBest 0H–H contacts 0Rg 0kT/ε 0MC steps 0—FPS 0

Left: the chain on a cubic lattice (orange = hydrophobic H, blue = polar P, yellow sticks = H–H contacts). Right: the energy-landscape funnel; the glowing ball sits at the chain's current radius of gyration (schematic funnel, real chain). Drag to orbit, wheel or pinch to zoom.

Protein Folding Funnel

A hydrophobic-polar chain on a 3D lattice, sampled by Metropolis Monte Carlo. Educational toy model, not a real protein.

Sequence

Chain (27 residues)

Thermodynamics

Sampling

Live readouts

Energy against MC steps

Visited states: energy against Rg

Levinthal's paradox

Missions

Mission 5: estimate log10 of the number of self-avoiding conformations of the 27-bead chain (26 bonds)

Hint: the number grows like μ^n with μ ≈ 4.68 on the cubic lattice.

Experiments

Runs headless: 6 independent chains per temperature, up to 200,000 steps each, and reports the median steps to reach energy −16 or lower (chain A).

Independent numerical checks

Controls

Drag / one finger: orbit; wheel / pinch: zoom; three camera presets.
kT/ε is the temperature in units of the H–H contact energy. Low = near-greedy descent (can freeze in local minima on harder sequences such as B); high = random coil; in between lies the folding transition.
Annealing lowers kT/ε smoothly from 1.2 to 0.3 over about 1.2 million steps. Pivot, corner-flip and end moves are all proposed with symmetric probabilities.

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 used twice (self-avoiding walk). The energy is E = −ε · N_HH, where N_HH counts non-bonded H–H nearest-neighbour pairs. The hydrophobic effect (water prefers to hydrogen-bond with itself rather than organise around oily side chains) is the origin of this attraction; it is entropy-driven in the real solvent, here folded into a single energy ε. Real proteins have 20 amino-acid types, side-chain packing, hydrogen bonds, helices and sheets, none of which appear here (Lau and Dill 1989).

Metropolis Monte Carlo. A trial move to a new conformation is accepted with probability min(1, exp(−ΔE/kT)). The proposal (end move, corner flip, pivot of head or tail by a random cubic symmetry) is symmetric, so detailed balance holds and long-run visiting frequency is the Boltzmann distribution P ∝ exp(−E/kT). A Monte Carlo step is not a physical time; it only orders the search.

Levinthal's paradox. The number of self-avoiding walks grows as c_n ≈ A μⁿ n^γ−1 with μ ≈ 4.684 and γ ≈ 1.157 on the cubic lattice (A ≈ 1.2 here, approximate). Even at 10¹³ conformations per second (a generous Levinthal-style rate, my assumption), exhaustive search for 100 residues takes astronomically long, yet real proteins fold in milliseconds to seconds because the energy landscape is funnel-shaped: native-like contacts lower the energy, so the search is biased downhill.

The funnel. The funnel is schematic: its radius is the chain's radius of gyration R_g = √(⟨|r_i − r_cm|²⟩) normalised between the best-folded and fully extended chain, and its height follows h ∝ ρ^1.6. The scatter chart shows the real energy and R_g of the sampled states.

Simplifications and limits. The lowest energy per sequence ("best known") was found by repeated simulated annealing, not proven optimal, except the all-H 27-mer where −28 equals the maximal contact count of a 3×3×3 cube. Chain A reaches −18 in the runs used here. Conformations are on a lattice, so geometry is crude. Assets built with Blender; scene by Three.js.