Geometry, thermodynamics, and protein

Abstract 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.