Numerical verification of the generalized Crooks nonequilibrium work theorem for non-Hamiltonian molecular dynamics simulations

The generalized Crooks theorem (GCT) for deterministic non-Hamiltonian molecular dynamics simulations [Phys. Rev. E 75, 050101 (2007)] connects the probabilities of nonequilibrium realizations switching the system between two thermodynamic states, to the partition functions of these states. In comparison to the "classical" Crooks nonequilibrium work theorem [J. Stat. Phys. 90, 1481 (1998)], which deals with realizations involving only mechanical work, the GCT also accounts for additional work resulting from changes of the intensive and extensive thermodynamic variables of the system. In this article we present a numerical verification of the GCT using a Lennard-Jones fluid model where two particles are subject to a time-dependent external potential. Moreover, in order to switch the system between different thermodynamic states, the temperature and the pressure (or volume), which are controlled through the Martyna-Tobias-Klein equations of motion [J. Chem. Phys. 101, 4177 (1994)], are also varied externally. The free energy difference between states characterized by different distances of the target particles is evaluated using both a standard methodology (pair radial distribution functions) and the GCT. In order to exploit the various options provided by the GCT approach, i.e., the possibility of temperature/pressure/volume changes during the realizations, the free energy difference is recovered via arbitrary thermodynamic cycles. In all tests, the GCT is quantitatively verified.     

Recovering the Crooks equation for dynamical systems in the isothermal-isobaric ensemble: a strategy based on the equations of motion

The Crooks equation [Eq. (10) in J. Stat. Phys. 90, 1481 (1998)] relates the work done on a system during a nonequilibrium transformation to the free energy difference between the final and the initial state of the transformation. Recently, the authors have derived the Crooks equation for systems in the canonical ensemble thermostatted by the Nose-Hoover or Nose-Hoover chain method [P. Procacci et al., J. Chem. Phys. 125, 164101 (2006)]. That proof is essentially based on the fluctuation theorem by Evans and Searles [Adv. Phys. 51, 1529 (2002)] and on the equations of motion. Following an analogous approach, the authors derive here the Crooks equation in the context of molecular dynamics simulations of systems in the isothermal-isobaric (NPT) ensemble, whose dynamics is regulated by the Martyna-Tobias-Klein algorithm [J. Chem. Phys. 101, 4177 (1994)]. Their present derivation of the Crooks equation correlates to the demonstration of the Jarzynski identity for NPT systems recently proposed by Cuendet [J. Chem. Phys. 125, 144109 (2006)].     

Escorted free energy simulations

We describe a strategy to improve the efficiency of free energy estimates by reducing dissipation in nonequilibrium Monte Carlo simulations. This strategy generalizes the targeted free energy perturbation approach [C. Jarzynski, Phys. Rev. E 65, 046122 (2002)] to nonequilibrium switching simulations, and involves generating artificial, "escorted" trajectories by coupling the evolution of the system to updates in external work parameter. Our central results are: (1) a generalized fluctuation theorem for the escorted trajectories, and (2) estimators for the free energy difference ΔF in terms of these trajectories. We illustrate the method and its effectiveness on model systems.     

A proof of Jarzynski's nonequilibrium work theorem for dynamical systems that conserve the canonical distribution

We present a derivation of the Jarzynski [Phys. Rev. Lett. 78, 2690 (1997)] identity and the Crooks [J. Stat. Phys. 90, 1481 (1998)] fluctuation theorem for systems governed by deterministic dynamics that conserves the canonical distribution such as Hamiltonian dynamics, Nose-Hoover dynamics, Nose-Hoover chains, and Gaussian isokinetic dynamics. The proof is based on a relation between the heat absorbed by the system during the nonequilibrium process and the Jacobian of the phase flow generated by the dynamics.     

Single-molecule dynamics of mechanical coiled-coil unzipping

We use atomic force microscopy (AFM) to mechanically unzip and rezip a double-stranded coiled-coil structure at varying pulling velocities. We find that force-extension traces exhibit hysteresis that grows with increasing pulling velocity. This shows that coiled-coil unzipping and rezipping do not occur in thermal equilibrium on our experimental time scale. We present a nonequilibrium simulation that fully reproduces the hysteresis effects, giving detailed insight into dynamics of coiled-coil folding. Using this model, we find that seed formation is responsible for the hysteresis. The seed consists of four alpha-helical turns on both strands of the coiled coil. To obtain equilibrium information from our nonequilibrium experiments, we used the Crooks fluctuation theorem (CFT) to calculate the equilibrium free energy of folding for all of the different pulling velocities. The paper presented here lays the groundwork for the study of self-assembly properties of many physiologically relevant coiled-coil structures at the single-molecule level.     

Accuracy and convergence of free energy differences calculated from nonequilibrium switching processes

The molecular-dynamics-based calculation of accurate free energy differences for biomolecular systems is a challenging task. Accordingly, convergence and accuracy of established equilibrium methods has been subject of many studies, often focusing at small test systems. In contrast, the potential of more recently proposed nonequilibrium methods, derived from the Jarzynski and Crooks equalities, has not yet fully been explored. Here, we compare the performance of these methods by calculating free energy differences for test systems at different levels of complexity and varying extent of the involved perturbations. We consider the interconversion of ethane into methanol, the switching of a tryptophane-sidechain in a tripeptide, and the binding of two different ligands to the globular protein snurportin 1. On the basis of our results, we suggest and assess a new nonequilibrium free energy method, Crooks Gaussian Intersection (CGI), which combines the advantages of existing methods. CGI is highly parallelizable and, for the test systems considered here, is shown to outperform the other studied equilibrium and nonequilibrium methods.     

Direct Observation of the Reversible Two‐State Unfolding and Refolding of an α/β Protein by Single‐Molecule Atomic Force Microscopy

Directly observing protein folding in real time using atomic force microscopy (AFM) is challenging. Here the use of AFM to directly monitor the folding of an α/β protein, NuG2, by using low‐drift AFM cantilevers is demonstrated. At slow pulling speeds (<50 nm s^?1), the refolding of NuG2 can be clearly observed. Lowering the pulling speed reduces the difference between the unfolding and refolding forces, bringing the non‐equilibrium unfolding–refolding reactions towards equilibrium. At very low pulling speeds (ca. 2 nm s^?1), unfolding and refolding were observed to occur in near equilibrium. Based on the Crooks fluctuation theorem, we then measured the equilibrium free energy change between folded and unfolded states of NuG2. The improved long‐term stability of AFM achieved using gold‐free cantilevers allows folding–unfolding reactions of α/β proteins to be directly monitored near equilibrium, opening the avenue towards probing the folding reactions of other mechanically important α/β and all‐β elastomeric proteins.     

Density-dependent analysis of nonequilibrium paths improves free energy estimates II. A Feynman-Kac formalism

The nonequilibrium fluctuation theorems have paved the way for estimating equilibrium thermodynamic properties, such as free energy differences, using trajectories from driven nonequilibrium processes. While many statistical estimators may be derived from these identities, some are more efficient than others. It has recently been suggested that trajectories sampled using a particular time-dependent protocol for perturbing the Hamiltonian may be analyzed with another one. Choosing an analysis protocol based on the nonequilibrium density was empirically demonstrated to reduce the variance and bias of free energy estimates. Here, we present an alternate mathematical formalism for protocol postprocessing based on the Feynmac-Kac theorem. The estimator that results from this formalism is demonstrated on a few low-dimensional model systems. It is found to have reduced bias compared to both the standard form of Jarzynski's equality and the previous protocol postprocessing formalism.     

Equilibrium free energy estimates based on nonequilibrium work relations and extended dynamics

Jarzynski's relation and the fluctuation theorem have established important connections between nonequilibrium statistical mechanics and equilibrium thermodynamics. In particular, an exact relationship between the equilibrium free energy and the nonequilibrium work is useful for computer simulations. In this paper, we exploit the fact that the free energy is a state function, independent of the pathway taken to change the equilibrium ensemble. We show that a generalized expression is advantageous for computer simulations of free energy differences. Several methods based on this idea are proposed. The accuracy and efficiency of the proposed methods are evaluated with a model problem.     

Bayesian estimates of free energies from nonequilibrium work data in the presence of instrument noise

The Jarzynski equality and the fluctuation theorem relate equilibrium free energy differences to nonequilibrium 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 been considered previously. Here, we present a Bayesian formalism for estimating free energy changes from nonequilibrium 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 atomic force microscopy, optical, and magnetic tweezers.     

Entropy-energy decomposition from nonequilibrium work trajectories

We derive expressions for the equilibrium entropy and energy changes in the context of the Jarzynski equality relating nonequilibrium work to equilibrium free energy. The derivation is based on a stochastic path integral technique that reweights paths at different temperatures. Stochastic dynamics generated by either a Langevin equation or a Metropolis Monte Carlo scheme are treated. The approach enables the entropy-energy decomposition from trajectories evolving at a single-temperature and does not require simulations or measurements at two or more temperatures. Both finite difference and analytical formulae are derived. Testing is performed on a prototypical model system and the method is compared with existing thermodynamic integration and thermodynamic perturbation approaches for entropy-energy decomposition. The new formulae are also put in the context of more general, dynamics-independent expressions that derive from either a fluctuation theorem or the Feynman-Kac theorem.     

Energy dissipation asymmetry in the non equilibrium folding/unfolding of the single molecule alanine decapeptide

We present non equilibrium molecular dynamics experiments of the unfolding and refolding of a single molecule alanine decapeptide in vacuo subject to a Nosé thermostat. Forward (unfolding) and reverse (refolding) work distribution are numerically calculated for various duration times of the non equilibrium experiments. Crooks theorem is accurately verified for all non equilibrium regimes and the time asymmetry of the process is measured using the recently proposed Jensen–Shannon divergence [E.H. Feng, G. Crooks, Phys. Rev. Lett. 101, 090602 (2008)]. Results on the alanine decapeptide are found similar to recent experimental data on m-RNA molecule in solution, thus evidencing the universal character of the Jensen–Shannon divergence. The patent non Markovianity of the process is rationalized by assuming that the observed forward and reverse distributions can be each described by a combination of two normal distributions satisfying the Crooks theorem, representative of two mutually exclusive linear events. Such bimodal approach reproduces with surprising accuracy the observed non Markovian work distributions.     

Free energy for protein folding from nonequilibrium simulations using the Jarzynski equality

The equilibrium free energy difference between two long-lived molecular species or "conformational states" of a protein (or any other molecule) can in principle be estimated by measuring the work needed to shuttle the system between them, independent of the irreversibility of the process. This is the meaning of the Jarzynski equality (JE), which we test in this paper by performing simulations that unfold a protein by pulling two atoms apart. Pulling is performed fast relative to the relaxation time of the molecule and is thus far from equilibrium. Choosing a simple protein model for which we can independently compute its equilibrium properties, we show that the free energy can be exactly and effectively estimated from nonequilibrium simulations. To do so, one must carefully and correctly determine the ensemble of states that are pulled, which is more important the farther from equilibrium one performs simulations; this highlights a potential problem in using the JE to extract the free energy from forced unfolding experiments. The results presented here also demonstrate that the free energy difference between the native and denatured states of a protein measured in solution is not always equal to the free energy profile that can be estimated from forced unfolding simulations (or experiments) using the JE.     

Crooks equation for steered molecular dynamics using a Nosé-Hoover thermostat

The Crooks equation [Eq. (10) in J. Stat. Phys. 90, 1481 (1998)], originally derived for microscopically reversible Markovian systems, relates the work done on a system during an irreversible transformation to the free energy difference between the final and the initial state of the transformation. In the present work we provide a theoretical proof of the Crooks equation in the context of constant volume, constant temperature steered molecular dynamics simulations of systems thermostated by means of the Nosé-Hoover method (and its variant using a chain of thermostats). As a numerical test we use the folding and unfolding processes of decaalanine in vacuo at finite temperature. We show that the distribution of the irreversible work for the folding process is markedly non-Gaussian thereby implying, according to Crooks equation, that also the work distribution of the unfolding process must be inherently non-Gaussian. The clearly asymmetric behavior of the forward and backward irreversible work distributions is a signature of a non-Markovian regime for the folding/unfolding of decaalanine.     

Free energy calculations,enhanced by a gaussian ansatz,for the “chemical work” distribution

The evaluation of the free energy is essential in molecular simulation because it is intimately related with the existence of multiphase equilibrium. Recently, it was demonstrated that it is possible to evaluate the Helmholtz free energy using a single statistical ensemble along an entire isotherm by accounting for the “chemical work” of transforming each molecule, from an interacting one, to an ideal gas. In this work, we show that it is possible to perform such a free energy perturbation over a liquid vapor phase transition. Furthermore, we investigate the link between a general free energy perturbation scheme and the novel nonequilibrium theories of Crook's and Jarzinsky. We find that for finite systems away from the thermodynamic limit the second law of thermodynamics will always be an inequality for isothermal free energy perturbations, resulting always to a dissipated work that may tend to zero only in the thermodynamic limit. The work, the heat, and the entropy produced during a thermodynamic free energy perturbation can be viewed in the context of the Crooks and Jarzinsky formalism, revealing that for a given value of the ensemble average of the “irreversible” work, the minimum entropy production corresponded to a Gaussian distribution for the histogram of the work. We propose the evaluation of the free energy difference in any free energy perturbation based scheme on the average irreversible “chemical work” minus the dissipated work that can be calculated from the variance of the distribution of the logarithm of the work histogram, within the Gaussian approximation. As a consequence, using the Gaussian ansatz for the distribution of the “chemical work,” accurate estimates for the chemical potential and the free energy of the system can be performed using much shorter simulations and avoiding the necessity of sampling the computational costly tails of the “chemical work.” For a more general free energy perturbation scheme that the Gaussian ansatz may not be valid, the free energy calculation can be expressed in terms of the moment generating function of the “chemical work” distribution. © 2014 Wiley Periodicals, Inc.     

The free energy of expansion and contraction: treatment of arbitrary systems using the Jarzynski equality

Thermodynamic integration, free energy perturbation, and slow change techniques have long been utilised in the calculation of free energy differences between two states of a system that has undergone some transformation. With the introduction of the Jarzynski equality and the Crooks relation, new approaches are possible. This paper investigates an important phenomenon - systems undergoing a change in volume/density - and derives both the Jarzynski equality and Crooks relation of such systems using a statistical mechanical approach. These results apply to systems with arbitrary particle interactions and densities. The application of this approach to the expansion/compression of particles confined within a vessel with a piston and within a periodic system is considered.     

Reciprocating motion on the nanoscale

This paper analyzes the confined motion of a Brownian particle fluctuating between two conformational states with different potential profiles and different position-dependent rate constants of the transitions, the fluctuations arising from both thermal (equilibrium) and external (nonequilibrium) noise. The model illustrates a mechanism to transduce, on the nanoscale, the energy of nonequilibrium fluctuations into mechanical energy of reciprocating motion. Expressions for the reciprocating velocity and the efficiency of energy conversion are derived. These expressions are treated in more detail in the slow-fluctuation (quasi-equilibrium) regime, by simple perturbation theory arguments, and in the fast fluctuation limit, in terms of the potential of mean force. A notable observation is that the generalized driving force of the reciprocating motion is caused by two sources: the energy contribution due to the difference between the potential profiles of the states and the entropic contribution due to the difference between the position-dependent rate constants. Two illustrative examples are presented, where one of the two sources can be ignored and an exact solution is allowed. Among other aspects, we also discuss the ways to construct a molecular motor based on the reciprocating engine.     

Fluctuation theorem and Onsager reciprocity relations

The Onsager and higher-order reciprocity relations are derived from a fluctuation theorem for nonequilibrium reactions ruled by the chemical master equation. The fluctuation theorem is obtained for the generating function of the macroscopic fluxes between chemiostats maintaining the system in a nonequilibrium steady state. The macroscopic affinities associated with the fluxes are identified by graph theory. The Yamamoto-Zwanzig formulas for the reaction constants are also derived from the fluctuation theorem.     

Equilibrium calculations of viscosity and thermal conductivity across a solid-liquid interface using boundary fluctuations

We calculate viscosity and thermal conductivity in systems of Lennard-Jones particles consisting of coexisting solid and liquid with different interface wetting properties using the recently developed equilibrium boundary fluctuation theory. We compare the slip length and equivalent liquid length obtained from these calculations with those obtained from nonequilibrium molecular dynamics. The equilibrium and nonequilibrium calculations of the slip length and the sum of the thermal equivalent lengths are in good agreement. We conclude that for both interfacial properties, the nonequilibrium simulations were probing the linear response. The significant dependence of the intrinsic equivalence length on the interfacial temperature difference used to generate the thermal gradient is explained as a consequence of the different thermodynamic states of the two interfaces.