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= union or logical sum(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 Omega(P) supersetP 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 Omega(P), area of M(P), and area of the hydrophobic surface W(P) subsetM(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.