The weighted area derivative 
of a union of balls

 

Abstract

One of the driving forces of protein folding is hydrophobicity, which is measured as the weighted surface area exposed to the solvent. We develop geometric formulas for the weighted area and its derivative based on alpha complexes and the principle of inclusion-exclusion. Somewhat surprisingly, the formula for the derivative contains a component that favors rotational motion in some cases, which might be a rationalization for the experimental observation that the hydrophobic force seems to have a dominant influence on the motion during the initial phases of folding. We also report on a fast implementation of these formulas, which has been added in the inner loop of molecular dynamics computations. This is joint work with Robert Bryant (Mathematics at Duke) and Patrice Koehl and Michael Levitt (Structural Biology at Stanford).