Protein Folding Thermodynamics

The Folding Problem

Proteins spontaneously collapse from disordered polypeptide chains into precise three-dimensional structures. This process is driven by thermodynamics: the folded state minimizes the Gibbs free energy.

Two-State Model

Most small globular proteins fold cooperatively through a two-state equilibrium:

$\text{Unfolded} \rightleftharpoons \text{Folded}$

The free energy of folding is:

$\Delta G = \Delta H - T \Delta S$

• ΔH < 0: folding is enthalpically favorable (hydrogen bonds, van der Waals)
• ΔS < 0: folding reduces conformational entropy (unfavorable)
• The balance determines whether the protein is folded at a given temperature

Equilibrium Constant & Fraction Folded

$K_{\text{fold}} = \exp\left(-\frac{\Delta G}{RT}\right)$

$f_{\text{folded}} = \frac{K}{1 + K}$

Melting Temperature

At T_m, ΔG = 0, so ΔH = T_m · ΔS:

$T_m = \frac{\Delta H}{\Delta S}$

Urea Denaturation

Chemical denaturants like urea unfold proteins by weakening hydrophobic interactions. The linear extrapolation model describes how stability depends on denaturant concentration c:

$\Delta G_{\text{unfold}}(c) = \Delta G_{\text{unfold}}^{\text{H}_2\text{O}} - m \cdot c$

• ΔG_unfold > 0 means the protein is folded (unfolding is unfavorable)
• m = m-value (kJ mol⁻¹ M⁻¹): sensitivity of stability to denaturant
• C_m = midpoint concentration where ΔG_unfold = 0: C_m = ΔG_H₂O / m

The fraction folded given ΔG_unfold is:

$f_{\text{folded}} = \frac{1}{1 + \exp(-\Delta G_{\text{unfold}} / RT)}$

Your Task

Implement three functions:

1. fraction_folded(delta_H_kJ_mol, delta_S_J_mol_K, T_K) — Fraction of folded protein (ΔH in kJ/mol, ΔS in J/mol/K, T in K)
2. melting_temperature_K(delta_H_kJ_mol, delta_S_J_mol_K) — Melting temperature in Kelvin
3. fraction_folded_urea(c_urea_M, delta_G_H2O_kJ_mol, m_kJ_mol_M, T_K=298) — Fraction folded at given urea concentration (delta_G_H2O is ΔG_unfolding in water, positive = stable protein)