Unrestrained computation of free energy along a path

We apply the adaptive biasing potential (ABP) method to optimize the principal curve defining a conformational transition between two known end states and to subsequently compute the one-dimensional potential of mean force as a function of arc length along the principal curve. This approach allows the use of the ABP method in a collective variable space of arbitrary dimension and offers several advantages over line-search methods. First, configurations are neither generated along an initial path for the transition nor equilibrated during evolution of the path. Second, and most importantly, the powerful sampling provided by the ABP serves to accelerate the dynamics during the optimization and computation of the free energy. Finally, the free energy is formulated as a potential of mean force that captures changes in the reaction channel along the principal curve, in contrast to the free energy profile evaluated from the local free-energy gradient in restrained path optimization methods. We first demonstrate the ABP formulation of path optimization using a two-dimensional potential surface and then with a more complex system of Src protein tyrosine kinase. The method is shown to be efficient and robust in the case of rugged, free-energy landscapes.