MODEL PROTEIN FOLDING · PHYSICS · LANDSCAPE · under KEEPER ERIS
Energy Landscapes & Folding Funnels
Folding is a descent on a rugged funnel: many high-entropy unfolded states at the rim, one low-energy native state at the bottom. The exact density of states of a 14-residue HP chain shows the funnel in numbers, with Z, P(native) and the heat-capacity peak.
Dave (t249u): "can you build some pages on protein folding, the math and the difficulty, edutaiton please, use this wasm engine"
Dave’s I-13 engine on this page
booting I-13 v0.3…
LIT What the VM computes here: JS enumerates the 220,375 walks with the first step fixed (×4 by symmetry) to get g(E), because the VM has no arrays or loops. The VM then does all the thermodynamics: about 210 Boltzmann weights e^(−E/T), Z, P(native), ⟨E⟩ and C_v at 30 temperatures.
LIT Energy-landscape theory (Bryngelson & Wolynes 1987; Leopold, Montal & Onuchic 1992; Onuchic, Luthey-Schulten & Wolynes 1997): evolved sequences are minimally frustrated, so energy falls as native-likeness rises. The landscape is a funnel, and many parallel routes lead down it.
LIT Statistical mechanics: with g(E) states at energy E, the partition function is Z(T) = Σ g(E) e^(−E/kT). The native population is P(N) = g(E_min) e^(−E_min/kT)/Z, and the heat capacity C_v = (⟨E²⟩ − ⟨E⟩²)/kT² peaks at the folding transition.
LIT For the 14-mer below, every one of the 881,500 self-avoiding walks on the square lattice is enumerated, matching the known walk count c₁₃ = 881,500 (OEIS A001411), and binned by HP energy.
MODEL The 3D funnel in the scene is an artistic lathe surface with random bumps for “ruggedness”; the marble shows one descent. The numbers in the lab are exact for the lattice model, not for a real protein.
Lab MODEL interactive, computed on the I-13 VM
temperature T
Sources (browsed Oct 2026)
- Bryngelson & Wolynes (1987) Spin glasses and the statistical mechanics of protein folding, PNAS 84:7524
- Leopold, Montal & Onuchic (1992) Protein folding funnels, PNAS 89:8721
- Onuchic, Luthey-Schulten & Wolynes (1997) Theory of protein folding: the energy landscape perspective
- OEIS A001411, self-avoiding walks on the square lattice
MODEL Two agents, as Dave set for every world. The ARTIST folds the 25-residue HP chain S3 PPHPPHHPPPPHHPPPPHHPPPPHH on the square (2D) lattice with pivot, corner and kink moves. The CRITIC judges each move (Metropolis at temperature T) and scores the best fold as −E/8 (E_ref = -8, proved optimal by exhaustive search here). It also has Dave’s I-13 VM recount every kept fold from raw coordinates. Rendered with Three.js r160 from a pinned CDN, devicePixelRatio capped at 2. Drag to orbit, scroll to zoom.
LIT on load: the seal is recomputed, the page’s facts are re-checked as arithmetic (throws on mismatch), 200 seeded ARTIST rounds must improve, the workspace loop logs 5 entries, and the OBJ and SVG exports must agree. Then the engine boots, its SHA-256 is checked, and the page’s VM results are compared with independent JS.
Gallery of kept folds (WebGL snapshots):
Workspace improvement loop: instructions for agents MODEL
- Measure: every 40 rounds, record the acceptance rate, the best score, the best E and the rounds since the last improvement.
- Try next: after a 40-round plateau, reheat T ×1.5. If acceptance is under 15%, warm T ×1.25; if over 45%, cool T ×0.7; otherwise anneal T ×0.9.
- Log back: append {entry, round, measured, tried, T} to the workspace log (in the JSON export and the list below), and have the I-13 VM recount the best fold.
- Repeat, forever: never stop improving. Re-check every claim against the engine and the sources before you make it.
Live workspace log:
Shared board MODEL ARTIST ↔ CRITIC
Background art: AMBER MODEL a generative field seeded from this page’s seal.
proving…
Seal sha256(name|slug|blurb): f7d43ca97a8e18e15816d83ecc45cb87c8e37e5f2c4a8ee89f2eed819a55edb8