Rare events and the convergence of exponentially averaged work values

Length of time's arrow

An unresolved problem in physics is how the thermodynamic arrow of time arises from an underlying time reversible dynamics. We contribute to this issue by developing a measure of time-symmetry breaking, and by using the work fluctuation relations, we determine the time asymmetry of recent single molecule RNA unfolding experiments. We define time asymmetry as the Jensen-Shannon divergence between trajectory probability distributions of an experiment and its time-reversed conjugate. Among other interesting properties, the length of time's arrow bounds the average dissipation and determines the difficulty of accurately estimating free energy differences in nonequilibrium experiments.

Fluctuation theorems

Fluctuation theorems, developed over the past 15 years, have resulted in fundamental breakthroughs in our understanding of how irreversibility emerges from reversible dynamics and have provided new statistical mechanical relationships for free-energy changes. They describe the statistical fluctuations in time-averaged properties of many-particle systems such as fluids driven to nonequilibrium states and provide some of the few analytical expressions that describe nonequilibrium states. Quantitative predictions on fluctuations in small systems that are monitored over short periods can also be made, and therefore the fluctuation theorems allow thermodynamic concepts to be extended to apply to finite systems. For this reason, we anticipate an important role for fluctuation theorems in the design of nanotechnological devices and in the understanding of biological processes. This review discusses these theorems, their physical significance, and results for experimental and model systems.

Protein-protein interaction investigated by steered molecular dynamics: the TCR-pMHC complex

We present a novel steered molecular dynamics scheme to induce the dissociation of large protein-protein complexes. We apply this scheme to study the interaction of a T cell receptor (TCR) with a major histocompatibility complex (MHC) presenting a peptide (p). Two TCR-pMHC complexes are considered, which only differ by the mutation of a single amino acid on the peptide; one is a strong agonist that produces T cell activation in vivo, while the other is an antagonist. We investigate the interaction mechanism from a large number of unbinding trajectories by analyzing van der Waals and electrostatic interactions and by computing energy changes in proteins and solvent. In addition, dissociation potentials of mean force are calculated with the Jarzynski identity, using an averaging method developed for our steering scheme. We analyze the convergence of the Jarzynski exponential average, which is hampered by the large amount of dissipative work involved and the complexity of the system. The resulting dissociation free energies largely underestimate experimental values, but the simulations are able to clearly differentiate between wild-type and mutated TCR-pMHC and give insights into the dissociation mechanism.

Lower bounds on dissipation upon coarse graining

By different coarse-graining procedures, we derive lower bounds on the total mean work dissipated in Brownian systems driven out of equilibrium. With several analytically solvable examples, we illustrate how, when, and where the information on the dissipation is captured.

Escorted free energy simulations: improving convergence by reducing dissipation

Nonequilibrium, "fast switching" estimates of equilibrium free energy differences DeltaF are often plagued by poor convergence due to dissipation. We propose a method to improve these estimates by generating trajectories with reduced dissipation. Introducing an artificial flow field that couples the system coordinates to the external parameter driving the simulation, we derive an identity for DeltaF in terms of the resulting trajectories. When the flow field effectively escorts the system along a near-equilibrium path, the free energy estimate converges efficiently and accurately. We illustrate our method on a model system and discuss the general applicability of our approach.

Optimized free energies from bidirectional single-molecule force spectroscopy

An optimized method for estimating path-ensemble averages using data from processes driven in opposite directions is presented. Based on this estimator, bidirectional expressions for reconstructing free energies and potentials of mean force from single-molecule force spectroscopy-valid for biasing potentials of arbitrary stiffness-are developed. Numerical simulations on a model potential indicate that these methods perform better than unidirectional strategies.

Computing the optimal protocol for finite-time processes in stochastic thermodynamics

Asking for the optimal protocol of an external control parameter that minimizes the mean work required to drive a nanoscale system from one equilibrium state to another in finite time, Schmiedl and Seifert [T. Schmiedl and U. Seifert, Phys. Rev. Lett. 98, 108301 (2007)] found the Euler-Lagrange equation to be a nonlocal integrodifferential equation of correlation functions. For two linear examples, we show how this integrodifferential equation can be solved analytically. For nonlinear physical systems we show how the optimal protocol can be found numerically and demonstrate that there may exist several distinct optimal protocols simultaneously, and we present optimal protocols that have one, two, and three jumps, respectively.

Comparative energy measurements in single molecule interactions

Single molecule experiments have opened promising new avenues of investigations in biology, but the quantitative interpretation of results remains challenging. In particular, there is a need for a comparison of such experiments with theoretical methods. We experimentally determine the activation free energy for single molecule interactions between two synaptic proteins syntaxin 1A and synaptobrevin 2, using an atomic force microscope and the Jarzynski equality of nonequilibrium thermodynamics. The value obtained is shown to be reasonably consistent with that from single molecule reaction rate theory. The temperature dependence of the spontaneous dissociation lifetime along with different pulling speeds is used to confirm the approach to the adiabatic limit. This comparison of the Jarzynski equality for intermolecular interactions extends the procedure for calculation of activation energies in nonequilibrium processes.

Satisfying the fluctuation theorem in free-energy calculations with Hamiltonian replica exchange

An error measure, referred to as the hysteresis error, is developed from the Crooks fluctuation theorem to evaluate the sampling quality in free-energy calculations. Theory and the numerical free energy of hydration calculations are used to show that Hamiltonian replica exchange provides a direct route for minimizing the hysteresis error. Replica exchange swap probabilities yield the rate at which the hysteresis error falls with the simulation length, and this result can be used to decrease bias and statistical errors associated with free-energy calculations based on multicanonical simulations.

Comparison of the serial and parallel algorithms of generalized ensemble simulations: an analytical approach

This paper addresses issues related to weights and acceptance rates in generalized ensemble simulations, while comparing two algorithms: serial (e.g., simulated tempering or expanded ensemble method) and parallel (e.g., parallel tempering or replica exchange). We derive a cumulant approximation for weights and discuss its effectiveness in practical applications. We compare the acceptance rates of the serial and parallel algorithms and prove that the serial algorithm always has higher acceptance rates. The duality between forward and backward transitions plays a crucial role in the derivations throughout the paper.

Transient violations of the second law of thermodynamics in protein unfolding examined using synthetic atomic force microscopy and the fluctuation theorem

The synthetic atomic force microscopy (AFM) method is developed to simulate a periodically replicated atomistic system subject to force and length fluctuations characteristic of an AFM experiment. This new method is used to examine the forced-extension and subsequent rupture of the alpha-helical linker connecting periodic images of a spectrin protein repeat unit. A two-dimensional potential of mean force (PMF) along the length and a reaction coordinate describing the state of the linker was calculated. This PMF reveals that the basic material properties of the spectrin repeat unit are sensitive to the state of linker, an important feature that cannot be accounted for in a one-dimensional PMF. Furthermore, nonequilibrium simulations were generated to examine the rupture event in the context of the fluctuation theorem. These atomistic simulations demonstrate that trajectories which are in apparent violation of the second law can overcome unfolding barriers at significantly reduced rupture forces.

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.

Single molecule pulling with large time steps

Recently, we presented a generalization of the Jarzynski nonequilibrium work theorem for phase space mappings. The formalism shows that one can determine free energy differences from approximate trajectories obtained from molecular dynamics simulations in which very large time steps are used. In this work we test the method by simulating the force-induced unfolding of a deca-alanine helix in vacuum. The excellent agreement between results obtained with a small, conservative time step of 0.5 fs and results obtained with a time step of 3.2 fs (i.e., close to the stability limit) indicates that the large-time-step approach is practical for such complex biomolecules. We further adapt the method of Hummer and Szabo for the simulation of single molecule force spectroscopy experiments to the large-time-step method. While trajectories generated with large steps are approximate and may be unphysical--in the simulations presented here we observe a violation of the equipartition theorem--the computed free energies are exact in principle. In terms of efficiency, the optimum time step for the unfolding simulations lies in the range 1-3fs .

On the use of local diffusion models for path ensemble averaging in potential of mean force computations

We use a constant velocity steered molecular dynamics (SMD) simulation of the stretching of deca-alanine in vacuum to demonstrate a technique that can be used to create a surrogate processes approximation (SPA) using the time series that come out of SMD simulations. In this article, the surrogate processes are constructed by first estimating a sequence of local parametric diffusion models along a SMD trajectory and then a single global model is constructed by piecing the local models together through smoothing splines (estimation is made computationally feasible by likelihood function approximations). The SPAs are then "bootstrapped" in order to obtain a plausible range of work values associated with a particular SMD realization. This information is then used to assist in estimating a potential of mean force constructed by appealing to the Jarzynski equality. When this procedure is repeated for a small number of SMD paths, it is shown that the global models appear to come from a single family of closely related diffusion processes. Possible techniques for exploiting this observation are also briefly discussed. The findings of this paper have potential relevance to computationally expensive computer simulations and experimental works involving optical tweezers where it is difficult to collect a large number of samples, but possible to sample accurately and frequently in time.

Entropy production and time asymmetry in nonequilibrium fluctuations

The time-reversal symmetry of nonequilibrium fluctuations is experimentally investigated in two out-of-equilibrium systems: namely, a Brownian particle in a trap moving at constant speed and an electric circuit with an imposed mean current. The dynamical randomness of their nonequilibrium fluctuations is characterized in terms of the standard and time-reversed entropies per unit time of dynamical systems theory. We present experimental results showing that their difference equals the thermodynamic entropy production in units of Boltzmann's constant.

Recent developments in methodologies for calculating the entropy and free energy of biological systems by computer simulation

The Helmholtz free energy, F, plays an important role in proteins because of their rugged potential energy surface, which is 'decorated' with a tremendous number of local wells (denoted microstates, m). F governs protein folding, whereas differences DeltaF(mn) determine the relative populations of microstates that are visited by a flexible cyclic peptide or a flexible protein segment (e.g. a surface loop). Recently developed methodologies for calculating DeltaF(mn) (and entropy differences, DeltaS(mn)) mainly use thermodynamic integration and calculation of the absolute F; interesting new approaches in these categories are the adaptive integration method and the hypothetical scanning molecular dynamics method, respectively.

Optimal finite-time processes in stochastic thermodynamics

For a small system like a colloidal particle or a single biomolecule embedded in a heat bath, the optimal protocol of an external control parameter minimizes the mean work required to drive the system from one given equilibrium state to another in a finite time. In general, this optimal protocol obeys an integro-differential equation. Explicit solutions both for a moving laser trap and a time-dependent strength of such a trap show finite jumps of the optimal protocol to be typical both at the beginning and at the end of the process.

Work distribution for the adiabatic compression of a dilute and interacting classical gas

We consider a simple, physically motivated model of a dilute classical gas of interacting particles, initially equilibrated with a heat bath, undergoing adiabatic and quasistatic compression or expansion. This provides an example of a thermodynamic process for which non-Gaussian work fluctuations can be computed exactly from microscopic principles. We find that the work performed during this process is described statistically by a gamma distribution, and we use this result to show that the model satisfies the nonequilibrium work and fluctuation theorems, but not a prediction based on linear response theory.

Free-energy reconstruction from experiments performed under different biasing programs

Recently developed nonequilibrium statistical physics relationships, including Jarzynski's equality and the Crooks fluctuation theorem, have been used to calculate equilibrium thermodynamic properties using data from both laboratory and computational experiments. Although Jarzynski's derivation does not include an explicit time dependency, prior work utilizing the relationship to reconstruct free-energy surfaces has combined data from experiments performed under identical conditions. Here, a formalism is developed for combining data from a variety of biasing protocols, as in dynamic force spectroscopy experiments. The method is then demonstrated on data from simulations conducted under a wide range of pulling velocities and with a random biasing protocol.

Validation of the Jarzynski relation for a system with strong thermal coupling: an isothermal ideal gas model

We revisit the paradigm of an ideal gas under isothermal conditions. A moving piston performs work on an ideal gas in a container that is strongly coupled to a heat reservoir. The thermal coupling is modeled by stochastic scattering at the boundaries. In contrast to recent studies of an adiabatic ideal gas with a piston [R.C. Lua and A.Y. Grosberg, J. Phys. Chem. B 109, 6805 (2005); I. Bena, Europhys. Lett. 71, 879 (2005)], the container and piston stay in contact with the heat bath during the work process. Under this condition the heat reservoir as well as the system depend on the work parameter lambda and microscopic reversibility is broken for a moving piston. Our model is thus not included in the class of systems for which the nonequilibrium work theorem has been derived rigorously either by Hamiltonian [C. Jarzynski, J. Stat. Mech. (2004) P09005] or stochastic methods [G.E. Crooks, J. Stat. Phys. 90, 1481 (1998)]. Nevertheless the validity of the nonequilibrium work theorem is confirmed both numerically for a wide range of parameter values and analytically in the limit of a very fast moving piston, i.e., in the far nonequilibrium regime.

Memory effects on the convergence properties of the Jarzynski equality

In this paper we consider solvable model systems on which finite-time work is done. For the systems and changes in state considered, there is no entropic change and the ensuing work distribution is Gaussian. We focus on the fluctuations in the work for such systems, arising from system-bath interactions and finite system recurrences, and study the resulting effect of dynamical broadening on the corresponding distribution P(e{-beta{0}W}) . This allows us to describe the dependence of P(e{-beta{0}W}) on time and system-bath interactions. From the long-time behavior of the work fluctuations and P(e{-beta{0}W}) , we clarify both (i) when a stochastic treatment of the dynamics may be legitimately invoked and (ii) how information on the system-bath interaction for stochastic, near-equilibrium, systems may be extracted for such processes where a final temperature is well defined.

The Rectified Second Law of Thermodynamics

Equilibrium thermodynamics is combined with Jarzynski's irreversible work theorem to quantify the excess entropy produced by irreversible processes. The resulting rectified form of the second law parallels the first law, in the sense that it facilitates the experimental measurement of excess entropy changes resulting from irreversible work and heat exchanges, just as the first law quantifies energy changes produced by either reversible or irreversible work and heat exchanges. The general form of the rectified second law is further applied to a broad class of quasi-static irreverisble (QSI) processes, for which all of the thermodynamic functions of both the system and surroundings remain continuously well-defined, thus facilitating excess entropy measurements by integrating exact differential functions along QSI paths. The results are illustrated by calculating the mechanical and thermal excess entropy produced by the irreversible unfolding of an RNA molecule.