Sampling of rare events using hidden restraints

A method to enhance sampling of rare events is presented. It makes use of distance or dihedral-angle restraints to overcome an energy barrier separating two metastable states or to stabilize a transition state between the two metastable states. In order not to perturb these metastable end states themselves, a prefactor is introduced into the restraining energy function, which smoothly increases the weight of this function from zero to one at the transition state or on top of the separating energy barrier and then decreases the weight again to zero at the final state. The method is combined with multi-configurational thermodynamic integration and applied to two biomolecular systems, which were difficult to treat using standard thermodynamic integration. As first example the free energy difference of a cyclic alpha-aminoxy-hexapeptide-ion complex upon changing the ion from Cl- to Na+ was calculated. A large conformational rearrangement of the peptide was necessary to accommodate this change. Stabilizing the transition state by (hidden) restraints facilitates that. As a second example, the free energy difference between the C-4(1) and the C-1(4) conformation of beta-D-glucopyranoside was calculated. In unrestrained simulations the change from the C-4(1) into the C-1(4) conformation was never observed because of the high energy barrier separating the two states. Using (hidden) restraints, the transition from the C-4(1) into the C-1(4) state and back could be enforced without perturbing the end states. As comparison, for the same transitions the potential of mean force as obtained by using dihedral-angle constraints is provided.