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Register a new userBayesian estimates of free energies from nonequilibrium work data in the presence of instrument noise
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The Jarzynski equality and the fluctuation theorem relate equilibrium free energy differences to non-equilibrium measurements of the work. These relations extend to single-molecule experiments that have probed the finite-time thermodynamics of proteins and nucleic acids. The effects of experimental error and instrument noise have not previously been considered. Here, we present a Bayesian formalism for estimating free-energy changes from non-equilibrium work measurements that compensates for instrument noise and combines data from multiple driving protocols. We reanalyze a recent set of experiments in which a single RNA hairpin is unfolded and refolded using optical tweezers at three different rates. Interestingly, the fastest and farthest-from-equilibrium measurements contain the least instrumental noise, and therefore provide a more accurate estimate of the free energies than a few slow, more noisy, near-equilibrium measurements. The methods we propose here will extend the scope of single-molecule experiments; they can be used in the analysis of data from measurements with AFM, optical, and magnetic tweezers.
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We derive the optimal estimates of the free energies of an arbitrary number of thermodynamic states from nonequilibrium work measurements; the work data are collected from forward and reverse switching processes and obey a fluctuation theorem. The maximum likelihood formulation properly reweights all pathways contributing to a free energy difference, and is directly applicable to simulations and experiments. We demonstrate dramatic gains in efficiency by combining the analysis with parallel tempering simulations for alchemical mutations of model amino acids.
Extensions of statistical mechanics are routinely being used to infer free energies from the work performed over single-molecule nonequilibrium trajectories. A key element of this approach is the ubiquitous expression dW/dt=partial H(x,t)/ partial t which connects the microscopic work W performed by a time-dependent force on the coordinate x with the corresponding Hamiltonian H(x,t) at time t. Here we show that this connection, as pivotal as it is, cannot be used to estimate free energy changes. We discuss the implications of this result for single-molecule experiments and atomistic molecular simulations and point out possible avenues to overcome these limitations.
Jarzynskis nonequilibrium work relation can be understood as the realization of the (hidden) time-generator reciprocal symmetry satisfied for the conditional probability function. To show this, we introduce the reciprocal process where the classical probability theory is expressed with real wave functions, and derive a mathematical relation using the symmetry. We further discuss that the descriptions by the standard Markov process from an initial equilibrium state are indistinguishable from those by the reciprocal process. Then the Jarzynski relation is obtained from the mathematical relation for the Markov processes described by the Fokker-Planck, Kramers and relativistic Kramers equations.
Recent studies on the solvation of atomistic and nanoscale solutes indicate that a strong coupling exists between the hydrophobic, dispersion, and electrostatic contributions to the solvation free energy, a facet not considered in current implicit solvent models. We suggest a theoretical formalism which accounts for coupling by minimizing the Gibbs free energy of the solvent with respect to a solvent volume exclusion function. The resulting differential equation is similar to the Laplace-Young equation for the geometrical description of capillary interfaces, but is extended to microscopic scales by explicitly considering curvature corrections as well as dispersion and electrostatic contributions. Unlike existing implicit solvent approaches, the solvent accessible surface is an output of our model. The presented formalism is illustrated on spherically or cylindrically symmetrical systems of neutral or charged solutes on different length scales. The results are in agreement with computer simulations and, most importantly, demonstrate that our method captures the strong sensitivity of solvent expulsion and dewetting to the particular form of the solvent-solute interactions.
We show that the recently postulated non-standard definition of work proportional to force variation rather than to displacement [A. Imparato and L. Peliti, cond-mat arXiv:0706.1134v1] is thermodynamically inconsistent at both microscopic and macroscopic scales and leads to non-physical results, including free energy changes that depend on arbitrary parameters.
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