Statistical Mechanics of Helix Bundles Using a Dynamic Programming Approach

Despite much study, biomolecule folding cooperativity is not well understood. There are quantitative models for helix-coil transitions and for coil-to-globule transitions, but no accurate models yet treat both chain collapse and secondary structure formation together. We develop here a dynamic programming approach to statistical mechanical partition functions of foldamer chain molecules. We call it the ascending levels model. We apply it to helix-coil and helix-bundle folding and cooperativity. For 14- to 50-mer Baldwin peptides, the model gives good predictions for the heat capacity and helicity versus temperature and urea. The model also gives good fits for the denaturation of Oas's three-helix bundle B domain of protein A (F13W*) and synthetic protein α<sub>3</sub><i>C</i> by temperature and guanidine. The model predicts the conformational distributions. It shows that these proteins fold with transitions that are two-state, although the transitions in the Baldwin helices are nearly higher order. The model shows that the recently developed three-helix bundle polypeptoids of Lee et al. fold <i>anti-cooperatively</i>, with a predicted value of Δ<i>H</i><sub>vH</sub>/Δ<i>H</i><sub>cal</sub> = 0.72. The model also predicts that two-helix bundles are unstable in proteins but stable in peptoids. Our dynamic programming approach provides a general way to explore cooperativity in complex foldable polymers.