Minimal folding pathways for coarse-grained biopolymer fragments.

The minimal folding pathway or trajectory for a biopolymer can be defined as the transformation that minimizes the total distance traveled between a folded and an unfolded structure. This involves generalizing the usual Euclidean distance from points to one-dimensional objects such as a polymer. We apply this distance here to find minimal folding pathways for several candidate protein fragments, including the helix, the beta-hairpin, and a nonplanar structure where chain noncrossing is important. Comparing the distances traveled with root mean-squared distance and mean root-squared distance, we show that chain noncrossing can have large effects on the kinetic proximity of apparently similar conformations. Structures that are aligned to the beta-hairpin by minimizing mean root-squared distance, a quantity that closely approximates the true distance for long chains, show globally different orientation than structures aligned by minimizing root mean-squared distance.