Geometry, thermodynamics, and protein

We derive a new continuous free energy formula for protein folding. We obtain the formula first by adding hydrophobic effect to a classical free energy formula for cavities in water. We then obtain the same formula by geometrically pursuing the structure that fits best the well-known global geometric features of native structures of globular proteins: 1. high density; 2. small surface area; 3. hydrophobic core; 4. forming domains for long polypeptide chains. Conformations of a protein are presented as an all atom CPK model P = ∪_{i = 1}^N B (x_i, r_i) where each atom is a ball B (x_i, r_i). All conformations satisfy generally defined steric conditions. For each conformation P of a globular protein, there is a closed thermodynamic system Ω_P ⊃ P bounded by the molecular surface M_P. Both methods derive the same free energy aV (P) + bA (P) + cW (P), where a, b, c > 0, V (P), A (P), and W (P) are volume of Ω_P, area of M_P, and area of the hydrophobic surface W_P ⊂ M_P, which quantifies hydrophobic effect. Minimizing W (P) is sufficient to produce statistically significant native like secondary structures and hydrogen bonds in the proteins we simulated.