This is the same coarse-grained Zimm–Bragg helix-coil model as the 3D version, rendered two-dimensionally instead of as a folded backbone. Each residue is coil or helical; a helical run of 4+ consecutive residues closes one backbone i,i+4 hydrogen bond per extra residue, and the whole chain's free energy is:
ΔG_bond(T) = ΔH − TΔS (per i,i+4 H-bond, ΔH scaled by residue helix propensity)
ΔG_nuc(T) = −RT·ln(σ) (one-off cost to start a NEW helical run, σ ≈ 10⁻³)
E(state) = Σ over runs [ ΔG_nuc + bonds(run)·ΔG_bond ]
Metropolis Monte-Carlo proposes single-residue coil↔helix flips exactly as in the 3D model. Two panels replace the rotating 3D backbone:
- Space-time map (left) — every residue is one column; every completed Monte-Carlo sweep adds one new row at the top and pushes older rows down, so the strip is literally a history of the chain's own state, not a snapshot. Nucleation appears as an isolated helical cell that survives; cooperative growth appears as it widening sideways into a solid block far more often than a second, unrelated cell appears elsewhere — because ΔG_nuc makes starting fresh expensive.
- Melting curve (right) — a phase diagram, not a 3D-camera view: the gold line is the exact single-dominant-domain equilibrium ⟨Q⟩(T) obtained by summing the Boltzmann weight of every possible contiguous helical run over the current sequence at each temperature (multi-domain states are negligible because ΔG_nuc ≫ typical ΔG_bond, so this sum is effectively exact for σ ≈ 10⁻³). The teal dot is where the live Metropolis simulation actually sits, tracing recent temperature scans as it fades.
- Temperature — raises T in ΔG_bond, tipping the balance toward the entropically-favoured coil (thermal denaturation); this shifts both the live dot and the analytic curve together.
- Sequence — scales ΔH per residue by its helix propensity: alanine-like residues fold strongly, glycine-like residues barely fold, and a proline breaker can never enter the helical state (it never appears colored teal in the space-time map), splitting any run that reaches it into two independent domains.