Crooks Fluctuation Theorem Research Articles

Free energy and kinetic rate calculation via non-equilibrium molecular simulation: application to biomolecules.

Non-equilibrium molecular dynamics (NEMD) simulation has been recognized as a powerful tool for examining biomolecules and provides fruitful insights into not only non-equilibrium but also equilibrium processes. We review recent advances in NEMD simulation and relevant, fundamental results of non-equilibrium statistical mechanics. We first introduce Crooks fluctuation theorem and Jarzynski equality that relate free energy difference to work done on a physical system during a non-equilibrium process. The theorems are beneficial for the analysis of NEMD trajectories. We then describe rate theory, a framework to calculate molecular kinetics from a non-equilibrium process; this theoretical framework enables us to calculate a reaction time-mean-first passage time-from NEMD trajectories. We, in turn, present recent NEMD techniques that apply an external force to a system to enhance molecular dissociation and introduce their application to biomolecules. Lastly, we show the current status of an appropriate selection of reaction coordinates for NEMD simulation.

Folding Free Energy Determination of an RNA Three-Way Junction Using Fluctuation Theorems.

Nonequilibrium work relations and fluctuation theorems permit us to extract equilibrium information from nonequilibrium measurements. They find application in single-molecule pulling experiments where molecular free energies can be determined from irreversible work measurements by using unidirectional (e.g., Jarzynski’s equality) and bidirectional (e.g., Crooks fluctuation theorem and Bennet’s acceptance ratio (BAR)) methods. However, irreversibility and the finite number of pulls limit their applicability: the higher the dissipation, the larger the number of pulls necessary to estimate G within a few . Here, we revisit pulling experiments on an RNA three-way junction (3WJ) that exhibits significant dissipation and work-distribution long tails upon mechanical unfolding. While bidirectional methods are more predictive, unidirectional methods are strongly biased. We also consider a cyclic protocol that combines the forward and reverse work values to increase the statistics of the measurements. For a fixed total experimental time, faster pulling rates permit us to efficiently sample rare events and reduce the bias, compensating for the increased dissipation. This analysis provides a more stringent test of the fluctuation theorem in the large irreversibility regime.

Crooks Fluctuation Theorem for Single Polymer Dynamics in Time-Dependent Flows: Understanding Viscoelastic Hysteresis.

Nonequilibrium work relations have fundamentally advanced our understanding of molecular processes. In recent years, fluctuation theorems have been extensively applied to understand transitions between equilibrium steady-states, commonly described by simple control parameters such as molecular extension of a protein or polymer chain stretched by an external force in a quiescent fluid. Despite recent progress, far less is understood regarding the application of fluctuation theorems to processes involving nonequilibrium steady-states such as those described by polymer stretching dynamics in nonequilibrium fluid flows. In this work, we apply the Crooks fluctuation theorem to understand the nonequilibrium thermodynamics of dilute polymer solutions in flow. We directly determine the nonequilibrium free energy for single polymer molecules in flow using a combination of single molecule experiments and Brownian dynamics simulations. We further develop a time-dependent extensional flow protocol that allows for probing viscoelastic hysteresis over a wide range of flow strengths. Using this framework, we define quantities that uniquely characterize the coil-stretch transition for polymer chains in flow. Overall, generalized fluctuation theorems provide a powerful framework to understand polymer dynamics under far-from-equilibrium conditions.

Implementing and Assessing an Alchemical Method for Calculating Protein-Protein Binding Free Energy.

Protein-protein binding is fundamental to most biological processes. It is important to be able to use computation to accurately estimate the change in protein-protein binding free energy due to mutations in order to answer biological questions that would be experimentally challenging, laborious, or time-consuming. Although nonrigorous free-energy methods are faster, rigorous alchemical molecular dynamics-based methods are considerably more accurate and are becoming more feasible with the advancement of computer hardware and molecular simulation software. Even with sufficient computational resources, there are still major challenges to using alchemical free-energy methods for protein-protein complexes, such as generating hybrid structures and topologies, maintaining a neutral net charge of the system when there is a charge-changing mutation, and setting up the simulation. In the current study, we have used the pmx package to generate hybrid structures and topologies, and a double-system/single-box approach to maintain the net charge of the system. To test the approach, we predicted relative binding affinities for two protein-protein complexes using a nonequilibrium alchemical method based on the Crooks fluctuation theorem and compared the results with experimental values. The method correctly identified stabilizing from destabilizing mutations for a small protein-protein complex, and a larger, more challenging antibody complex. Strong correlations were obtained between predicted and experimental relative binding affinities for both protein-protein systems.

Stochastic thermodynamics and fluctuation theorems for non-linear systems

We extend stochastic thermodynamics by relaxing the two assumptions that the Markovian dynamics must be linear and that the equilibrium distribution must be a Boltzmann distribution. We show that if we require the second law to hold when those assumptions are relaxed, then it cannot be formulated in terms of Shannon entropy. However, thermodynamic consistency is salvaged if we reformulate the second law in terms of generalized entropy; our first result is an equation relating the precise form of the non-linear master equation to the precise associated generalized entropy which results in thermodynamic consistency. We then build on this result to extend the usual trajectory-level definitions of thermodynamic quantities that are appropriate even when the two assumptions are relaxed. We end by using these trajectory-level definitions to derive extended versions of the Crooks fluctuation theorem and Jarzynski equality which apply when the two assumptions are relaxed.

Fluctuating relativistic hydrodynamics from Crooks theorem

We use the Crooks fluctuation theorem [1, 2] together with Zubarev hydro- dynamics [3] to develop a bottom-up theory of hydrodynamic fluctuations. We also use thermodynamic uncertainity relations to estimate bottom-up limits to dissipative transport coefficients.

Quantifying the Native Energetics Stabilizing Bacteriorhodopsin by Single-Molecule Force Spectroscopy.

We quantified the equilibrium (un)folding free energy ΔG_{0} of an eight-amino-acid region starting from the fully folded state of the model membrane-protein bacteriorhodopsin using single-molecule force spectroscopy. Analysis of equilibrium and nonequilibrium data yielded consistent, high-precision determinations of ΔG_{0} via multiple techniques (force-dependent kinetics, Crooks fluctuation theorem, and inverse Boltzmann analysis). We also deduced the full 1D projection of the free-energy landscape in this region. Importantly, ΔG_{0} was determined in bacteriorhodopsin's native bilayer, an advance over traditional results obtained by chemical denaturation in nonphysiological detergent micelles.

Recovery of Equilibrium Free Energy from Nonequilibrium Thermodynamics with Mechanosensitive Ion Channels in E.coli.

Insitu measurements of the free energy difference between the open and closed states of ion channels are challenging due to hysteresis effects and inactivation. Exploiting recent developments in statistical physics, we present a general formalism to extract the free energy difference ΔF between the closed and open states of mechanosensitive ion channels from nonequilibrium work distributions associated with the opening and closing of the channels (gating) in response to ramp stimulation protocols recorded in native patches. We show that the work distributions obtained from the gating of MscS channels in E.coli membrane satisfy the strong symmetry relation predicted by the Crooks fluctuation theorem. Our approach enables the determination of ΔF using patch-clamp experiments, which are often inherently restricted to the nonequilibrium regime.

Hot Wired

Two physicists engage in a courteous and exhilarating exchange. Jeremy England argues that descendants of the Crooks fluctuation theorem play, and should play, a role in explaining life’s origin. He is, in his optimism, a nightingale among pigeons. Brian Miller is equally persuaded that these theorems are of little relevance in explaining the origin of life. Pigeons remain well-satisfied.

Stochastic thermodynamics of nonharmonic oscillators in high vacuum.

We perform an analytic study on the stochastic thermodynamics of a small classical particle trapped in a time-dependent single-well potential in the highly underdamped limit. It is shown that the nonequilibrium probability density function for the system's energy is a Maxwell-Boltzmann distribution (as in equilibrium) with a closed form time-dependent effective temperature and fractional degrees of freedom. We also find that the solvable model satisfies the Crooks fluctuation theorem, as expected. Moreover, we compute the average work in this isothermal process and characterize analytically the optimal protocol for minimum work. The optimal protocol presents an initial and a final jump which correspond to adiabatic processes linked by a smooth exponential time-dependent part for all kinds of single-well potentials. Furthermore, we argue that this result connects two distinct relevant experimental setups for trapped nanoparticles, the levitated particle in a harmonic trap, and the free particle in a box, as they are limiting cases of the general single-well potential and display the time-dependent optimal protocols. Finally, we highlight the connection between our system and an equivalent model of a gas of Brownian particles.

Conservation laws and work fluctuation relations in chemical reaction networks.

We formulate a nonequilibrium thermodynamic description for open chemical reaction networks (CRNs) described by a chemical master equation. The topological properties of the CRN and its conservation laws are shown to play a crucial role. They are used to decompose the entropy production into a potential change and two work contributions, the first due to time dependent changes in the externally controlled chemostats concentrations and the second due to flows maintained across the system by nonconservative forces. These two works jointly satisfy a Jarzynski and Crooks fluctuation theorem. In the absence of work, the potential is minimized by the dynamics as the system relaxes to equilibrium and its equilibrium value coincides with the maximum entropy principle. A generalized Landauer's principle also holds: the minimal work needed to create a nonequilibrium state is the relative entropy of that state to its equilibrium value reached in the absence of any work.

Flaw in Crooks Fluctuation Theorem

The existence of Crooks fluctuation theorem (even at microscopic level, in a very short time period) is a direct threat to the second law of thermodynamics. In this paper, we will underline the flaw that exists in Crooks fluctuation theorem assumptions, and thus, we will confirm the validity of the second law of thermodynamics at any temperature, pressure, and at any scale (time, and length-scale) in nature. We will validate the Loschmidts paradox, and will show that no physical directional-process can be perfectly-reversible at any non-zero, finite temperature (T>0 K) and pressure (P>0) in nature.

Optical tweezers as a mathematically driven spatio-temporal potential generator.

The ability to create and manipulate spatio-temporal potentials is essential in the diverse fields of science and technology. Here, we introduce an optical feedback trap system based on high precision position detection and ultrafast feedback control of a Brownian particle in the optical tweezers to generate spatio-temporal virtual potentials of the desired shape in a controlled manner. As an application, we study the nonequilibrium fluctuation dynamics of the particle in a time-varying virtual harmonic potential and validate the Crooks fluctuation theorem in the highly nonequilibrium condition.

Experimental test of ensemble inequivalence and the fluctuation theorem in the force ensemble in DNA pulling experiments

We experimentally test the validity of the Crooks fluctuation theorem (CFT) in the force ensemble by pulling DNA hairpins, first with magnetic tweezers, next with optical tweezers using force feedback. The CFT holds when using the definition of work ${W}_{f}=\ensuremath{-}\ensuremath{\int}xdf$, where $x$ is the molecular extension and $f$ is the force. In contrast, it does not hold when using the usual definition, appropriate for the constant extension ensemble, ${W}_{x}=\ensuremath{\int}fdx$, showing the importance of the contribution of boundary terms to the full entropy production in a clear example of statistical ensemble inequivalence in small systems. We also evaluate the differences in the average dissipated work in the force ensemble as compared to the extension ensemble, highlighting ensemble inequivalence also at the level of molecular kinetics.

Fully Quantum Fluctuation Theorems

Systems that are driven out of thermal equilibrium typically dissipate random quantities of energy on microscopic scales. Crooks fluctuation theorem relates the distribution of these random work costs with the corresponding distribution for the reverse process. By an analysis that explicitly incorporates the energy reservoir that donates the energy, and the control system that implements the dynamic, we here obtain a quantum generalization of Crooks theorem that not only includes the energy changes in the reservoir, but the full description of its evolution, including coherences. This approach moreover opens up for generalizations of the concept of fluctuation relations. Here we introduce `conditional' fluctuation relations that are applicable to non-equilibrium systems, as well as approximate fluctuation relations that allow for the analysis of autonomous evolution generated by global time-independent Hamiltonians. We furthermore extend these notions to Markovian master equations, implicitly modeling the influence of the heat bath.

Asymptotics of work distributions in a stochastically driven system

We determine the asymptotic forms of work distributions at arbitrary times T, in a class of driven stochastic systems using a theory developed by Nickelsen and Engel (EN theory) [D. Nickelsen and A. Engel, Eur. Phys. J. B 82, 207 (2011)], which is based on the contraction principle of large deviation theory. In this paper, we extend the theory, previously applied in the context of deterministically driven systems, to a model in which the driving is stochastic. The models we study are described by overdamped Langevin equations and the work distributions in path integral form, are characterised by having quadratic augmented actions. We first illustrate EN theory, for a deterministically driven system – the breathing parabola model, and show that within its framework, the Crooks fluctuation theorem manifests itself as a reflection symmetry property of a certain characteristic polynomial, which also determines the exact moment-generating-function at arbitrary times. We then extend our analysis to a stochastically driven system, studied in references [S. Sabhapandit, EPL 89, 60003 (2010); A. Pal, S. Sabhapandit, Phys. Rev. E 87, 022138 (2013); G. Verley, C. Van den Broeck, M. Esposito, New J. Phys. 16, 095001 (2014)], for both equilibrium and non-equilibrium steady state initial distributions. In both cases we obtain new analytic solutions for the asymptotic forms of (dissipated) work distributions at arbitrary T. For dissipated work in the steady state, we compare the large T asymptotic behaviour of our solution to the functional form obtained in reference [New J. Phys. 16, 095001 (2014)]. In all cases, special emphasis is placed on the computation of the pre-exponential factor and the results show excellent agreement with numerical simulations. Our solutions are exact in the low noise (β → ∞) limit.

Crooks fluctuation theorem in -symmetric quantum mechanics

Following the recent work by Deffner and Saxena (2015 Phys. Rev. Lett. 114 150601), where the Jarzynski equality is generalised to non-Hermitian quantum mechanics, we prove in this work a stronger form of Jarzynski equality, the Crooks fluctuation theorem, also in the non-Hermitian formalism when the system is in the unbroken -symmetric phase.

Fluctuations When Driving Between Nonequilibrium Steady States

Maintained by environmental fluxes, biological systems are thermodynamic processes that operate far from equilibrium without detailed-balance dynamics. Yet, they often exhibit well defined nonequilibrium steady states (NESSs). More importantly, critical thermodynamic functionality arises directly from transitions among their NESSs, driven by environmental switching. Here, we identify constraints on excess thermodynamic quantities that ride above the NESS housekeeping background. We do this by extending the Crooks fluctuation theorem to transitions among NESSs, without invoking an unphysical dual dynamics. This and corresponding integral fluctuation theorems determine how much work must be expended when controlling systems maintained far from equilibrium. This generalizes feedback control theory, showing that Maxwellian Demons can leverage mesoscopic-state information to take advantage of the excess energetics in NESS transitions. Altogether, these point to universal thermodynamic laws that are immediately applicable to the accessible degrees of freedom within the effective dynamic at any emergent level of hierarchical organization. By way of illustration, this readily allows analyzing a voltage-gated sodium ion channel whose molecular conformational dynamics play a critical functional role in propagating action potentials in mammalian neuronal membranes.

Quantum fluctuation relations

We present here a set of lecture notes on exact fluctuation relations. We prove the Jarzynski equality and the Crooks fluctuation theorem, two paradigmatic examples of classical fluctuation relations. Finally, we consider their quantum versions, and analyze analogies and differences with the classical case.

Non-Boltzmann stationary distributions and nonequilibrium relations in active baths.

Most natural and engineered processes, such as biomolecular reactions, protein folding, and population dynamics, occur far from equilibrium and therefore cannot be treated within the framework of classical equilibrium thermodynamics. Here we experimentally study how some fundamental thermodynamic quantities and relations are affected by the presence of the nonequilibrium fluctuations associated with an active bath. We show in particular that, as the confinement of the particle increases, the stationary probability distribution of a Brownian particle confined within a harmonic potential becomes non-Boltzmann, featuring a transition from a Gaussian distribution to a heavy-tailed distribution. Because of this, nonequilibrium relations (e.g., the Jarzynski equality and Crooks fluctuation theorem) cannot be applied. We show that these relations can be restored by using the effective potential associated with the stationary probability distribution. We corroborate our experimental findings with theoretical arguments.