Hummer and Szabo-like potential of mean force estimator for bidirectional nonequilibrium pulling experiments/simulations

In the framework of single-molecule pulling experiments, the system is typically driven out of equilibrium by a time-dependent external potential V(t) acting on a collective coordinate such that the total Hamiltonian is the sum of V(t) and the Hamiltonian in the absence of external perturbation. Nonequilibrium work theorems such as Jarzynski equality and Crooks fluctuation theorem have been devised to recover free energy differences between states of this extended system. However, one is often more interested in the potential of mean force of the unperturbed Hamiltonian, i.e., in the effective potential dictating the equilibrium distribution of the collective coordinate in the absence of the external potential. In this respect, Hummer and Szabo proposed an algorithm to estimate the desired free energy differences when pulling experiments are performed in only one direction of the process (Proc. Natl. Acad. Sci. USA 2001, 98, 3658). In this paper, we present a potential of mean force estimator of the unperturbed system that exploits the work measured in both forward and backward directions of the process. The method is based on the reweighting technique of Hummer and Szabo and on the Bennett acceptance ratio. Using Brownian-dynamics simulations on a double-well free energy surface, we show that the estimator works satisfactorily in any pulling situation, from nearly equilibrium to strongly dissipative regimes. The method is also applied to the unfolding/refolding process of decaalanine, a system vastly used to illustrate and to test nonequilibrium methodologies. A thorough comparative analysis with another bidirectional potential of mean force estimator (Minh, D. D. L.; Adib, A. B. Phys. Rev. Lett. 2008, 100, 180602) is also presented. The proposed approach is well-suited to recover free energy profiles from single-molecule bidirectional-pulling experiments such as those performed by optical tweezers or atomic force microscopes.