Equalities for the Nonequilibrium Work Transferred from an External Potential to a Molecular System. Analysis of Single-Molecule Extension Experiments

Nonequilibrium work equalities are investigated for a system that experiences both an intrinsic potential energy function, V(x), and an externally controlled time-dependent potential energy function, u(x,t). The work, w T (t) = f dx t ' <(-)∂u(x t ' ,t')/∂x t ' , is transferred from the external potential to the bare system, defined by V(x), as the system moves along its trajectory, x t , in response to the forces generated by V(x) + u(x,t). As observed by Hummer and Szabo, this transferred work, w T (t), differs in several ways from the so-called accumulated work, w A (t) = f dt' ∂u(x t ' ,t')/∂t', considered by Jarzynski, which represents the work transferred from the external agent controlling u(x,t) into the composite system, defined by V(x) + u(x,t). The system is assumed to evolve according to an isothermal stochastic dynamics that maps the canonical distribution for the current potential function, V(x) + u(x,t), into itself in any arbitrarily short time interval. New nonequilibrium work equalities involving exp(-βw T (t)) are derived and compared with related equalities involving exp(-βw A (t)). Whenever the external potential initially vanishes, an average over all canonically weighted initial conditions and subsequent trajectories gives (exp(-βw T (t))) = 1.0, even though (w T (t)) is typically positive and time dependent. For a reversible process, (w r e v T) is the work required to achieve a particular fluctuated macrostate of the bare system and governs its relative probability. In the case of single-molecule extension experiments, the external potential function, u(z(x),t), typically depends on the system coordinates via an extensional parameter z(x). The relevant nonequilibrium work equality involving w T (t) = f z a z b dz t ' (-)∂u(z t ' ,t')/∂z t ' , for cases wherein both initial and final extensions are specified, is derived under conditions pertaining to the experiments of Liphardt et al. This new equality is shown to agree rather well with their experimental results. This new equality is also contrasted with the somewhat different relation conjectured by Liphardt et al., the conditions required for validity of the latter are discussed, and errors arising from its use are estimated to be small. A protocol is suggested to determine the free energy profile of the constrained bare system as a function of its extension, and a possible method to assess the adequacy of trajectory sampling simultaneously from the same nonequilibrium force vs extension data is also presented. Finally a connection between the nonequilibrium equality (involving exp(-βw T (t))) for extending the system from z a to z b and macroscopic thermodynamics is briefly discussed.