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 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.