One-​Dimensional Free-​Energy Profiles of Complex Systems: Progress Variables that Preserve the Barriers

We show that the balanced min.-​cut procedure introduced in PNAS 2004, 101, 14766 can be reinterpreted as a method for solving the constrained optimization problem of finding the min. cut among the cuts with a particular value of an additive function of the nodes on either side of the cut. Such an additive function (e.g., the partition function of the reactant region) can be used as a progress coordinate to det. a one-​dimensional profile (FEP) of the free-​energy surface of the protein-​folding reaction as well as other complex reactions. The algorithm is based on the network (obtained from an equil. mol. dynamics simulation) that represents the calcd. reaction behavior. The resulting FEP gives the exact values of the free energy as a function of the progress coordinate; i.e., at each value of the progress coordinate, the profile is obtained from the surface with the minimal partition function among the surfaces that divide the full free-​energy surface between two chosen end points. In many cases, the balanced min.-​cut procedure gives results for only a limited set of points. An approx. method based on pfold is shown to provide the profile for a more complete set of values of the progress coordinate. Applications of the approach to model problems and to realistic systems (β-​hairpin of protein G, LJ38 cluster) are presented.