Path Finding on High-Dimensional Free Energy Landscapes

We present a method for determining the average transition path and the free energy along this path in the space of selected collective variables. The formalism is based upon a history-dependent bias along a flexible path variable within the metadynamics framework but with a trivial scaling of the cost with the number of collective variables. Controlling the sampling of the orthogonal modes recovers the average path and the minimum free energy path as the limiting cases. The method is applied to resolve the path and the free energy of a conformational transition in alanine dipeptide.

Figure 1

Dynamic trajectories that start in reactant state $A$ and end in product state $B$ tend to localize in a valley of the free energy landscape. While reconstructing the free energy along a guess path ${\mathbf{s}}_g(\sigma {)}_g$ to escape the stable states and sample barrier crossings, the guess path is evolved in the perpendicular (hyper) planes $S_{{\sigma }_g}$, to converge at the average transition path together with its free energy profile.

Figure 2

Left: time evolution of a transition path, ${\mathbf{s}}_g(\sigma {)}_g$, that started from a straight path between $C{7}_{\mathrm{eq}}$ and $C{7}_{\mathrm{ax}}$, shown on a background of the alanine dipeptide $(\varphi ,\psi )$ landscape {previously computed using (normal) metadynamics [21]}. Middle: evolution of the free energy profile along the path-collective variable ${\sigma }_g(\mathbf{z})$. Right: probability distribution along three hyperplanes, also indicated in the left panel, illustrating the poorly bound reaction valley at $B$.