Near-equilibrium measurements of nonequilibrium free energy

A central endeavor of thermodynamics is the measurement of free energy changes. Regrettably, although we can measure the free energy of a system in thermodynamic equilibrium, typically all we can say about the free energy of a nonequilibrium ensemble is that it is larger than that of the same system at equilibrium. Herein, we derive a formally exact expression for the probability distribution of a driven system, which involves path ensemble averages of the work over trajectories of the time-reversed system. From this we find a simple near-equilibrium approximation for the free energy in terms of an excess mean time-reversed work, which can be experimentally measured on real systems. With analysis and computer simulation, we demonstrate the accuracy of our approximations for several simple models.     

Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences

There are only a very few known relations in statistical dynamics that are valid for systems driven arbitrarily far-from-equilibrium. One of these is the fluctuation theorem, which places conditions on the entropy production probability distribution of nonequilibrium systems. Another recently discovered far from equilibrium expression relates nonequilibrium measurements of the work done on a system to equilibrium free energy differences. In this paper, we derive a generalized version of the fluctuation theorem for stochastic, microscopically reversible dynamics. Invoking this generalized theorem provides a succinct proof of the nonequilibrium work relation.     

Free energy reconstruction from steered dynamics without post-processing

Various methods achieving importance sampling in ensembles of nonequilibrium trajectories enable one to estimate free energy differences and, by maximum-likelihood post-processing, to reconstruct free energy landscapes. Here, based on Bayes theorem, we propose a more direct method in which a posterior likelihood function is used both to construct the steered dynamics and to infer the contribution to equilibrium of all the sampled states. The method is implemented with two steering schedules. First, using non-autonomous steering, we calculate the migration barrier of the vacancy in Fe-α. Second, using an autonomous scheduling related to metadynamics and equivalent to temperature-accelerated molecular dynamics, we accurately reconstruct the two-dimensional free energy landscape of the 38-atom Lennard-Jones cluster as a function of an orientational bond-order parameter and energy, down to the solid–solid structural transition temperature of the cluster and without maximum-likelihood post-processing.     

平衡和非平衡方法计算相对结合自由能的精度和效率比较

Estimation of protein-ligand binding affinity within chemical accuracy is one of the grand challenges in structure-based rational drug design. With the efforts over three decades, free energy methods based on equilibrium molecular dynamics (MD) simulations have become mature and are nowadays routinely applied in the community of computational chemistry. On the contrary, nonequilibrium MD simulation methods have attracted less attention, despite their underlying rigor in mathematics and potential advantage in efficiency. In this work, the equilibrium and nonequilibrium simulation methods are compared in terms of accuracy and convergence rate in the calculations of relative binding free energies. The proteins studied are T4-lysozyme mutant L99A and COX-2. For each protein, two ligands are studied. The results show that the nonequilibrium simulation method can be competitively as accurate as the equilibrium method, and the former is more efficient than the latter by considering the convergence rate with respect to the cost of wall clock time. In addition, Bennett acceptance ratio, which is a bidirectional post-processing method, converges faster than the unidirectional Jarzynski equality for the nonequilibrium simulations.     

Nonequilibrium Thermodynamics of Polymeric Liquids via Atomistic Simulation

The challenge of calculating nonequilibrium entropy in polymeric liquids undergoing flow was addressed from the perspective of extending equilibrium thermodynamics to include internal variables that quantify the internal microstructure of chain-like macromolecules and then applying these principles to nonequilibrium conditions under the presumption of an evolution of quasie equilibrium states in which the requisite internal variables relax on different time scales. The nonequilibrium entropy can be determined at various levels of coarse-graining of the polymer chains by statistical expressions involving nonequilibrium distribution functions that depend on the type of flow and the flow strength. Using nonequilibrium molecular dynamics simulations of a linear, monodisperse, entangled C₁₀₀₀H₂₀₀₂ polyethylene melt, nonequilibrium entropy was calculated directly from the nonequilibrium distribution functions, as well as from their second moments, and also using the radial distribution function at various levels of coarse-graining of the constituent macromolecular chains. Surprisingly, all these different methods of calculating the nonequilibrium entropy provide consistent values under both planar Couette and planar elongational flows. Combining the nonequilibrium entropy with the internal energy allows determination of the Helmholtz free energy, which is used as a generating function of flow dynamics in nonequilibrium thermodynamic theory.     

Thermodynamically guided nonequilibrium Monte Carlo method for generating realistic shear flows in polymeric systems

A thermodynamically guided atomistic Monte Carlo methodology is presented for simulating systems beyond equilibrium by expanding the statistical ensemble to include a tensorial variable accounting for the overall structure of the system subjected to flow. For a given shear rate, the corresponding tensorial conjugate field is determined iteratively through independent nonequilibrium molecular dynamics simulations. Test simulations for the effect of flow on the conformation of a C50H102 polyethylene liquid show that the two methods (expanded Monte Carlo and nonequilibrium molecular dynamics) provide identical results.     

Free energy functional for nonequilibrium systems: an exactly solvable case

We consider the steady state of an open system in which there is a flux of matter between two reservoirs at different chemical potentials. For a large system of size N, the probability of any macroscopic density profile rho(x) is exp[-NF([rho])]; F thus generalizes to nonequilibrium systems the notion of free energy density for equilibrium systems. Our exact expression for F is a nonlocal functional of rho, which yields the macroscopically long range correlations in the nonequilibrium steady state previously predicted by fluctuating hydrodynamics and observed experimentally.     

Probing molecular free energy landscapes by periodic loading

Single molecule pulling experiments provide information about interactions in biomolecules that cannot be obtained by any other method. However, the reconstruction of the molecule's free energy profile from the experimental data is still a challenge, in particular, for the unstable barrier regions. We propose a new method for obtaining the full profile by introducing a periodic ramp and using Jarzynski's relation for obtaining equilibrium quantities from nonequilibrium data. Our simulated experiments show that this method delivers significant more accurate data than previous methods, under the constraint of equal experimental effort.     

Reconstructing the free-energy landscape of a mechanically unfolded model protein

The equilibrium free-energy landscape of an off-lattice model protein as a function of an internal (reaction) coordinate is reconstructed from out-of-equilibrium mechanical unfolding manipulations. This task is accomplished via two independent methods: by employing an extended version of the Jarzynski equality (EJE) and the protein inherent structures (ISs). In a range of temperatures around the "folding transition" we find a good quantitative agreement between the free energies obtained via EJE and IS approaches. This indicates that the two methodologies are consistent and able to reproduce equilibrium properties of the examined system. Moreover, for the studied model the structural transitions induced by pulling can be related to thermodynamical aspects of folding.     

Protein folding kinetics and thermodynamics from atomistic simulations

Determining protein folding kinetics and thermodynamics from all-atom molecular dynamics (MD) simulations without using experimental data represents a formidable scientific challenge because simulations can easily get trapped in local minima on rough free energy landscapes. This necessitates the computation of multiple simulation trajectories, which can be independent from each other or coupled in some manner, as, for example, in the replica exchange MD method. Here we present results obtained with a new analysis tool that allows the deduction of faithful kinetics data from a heterogeneous ensemble of simulation trajectories. The method is demonstrated on the decapeptide Chignolin for which we predict folding and unfolding time constants of 1.0 +/- 0.3 and 2.6 +/- 0.4 micros, respectively. We also derive the energetics of folding, and calculate a realistic melting curve for Chignolin.     

Equilibrium folding and unfolding dynamics to reveal detailed free energy landscape of src SH3 protein by magnetic tweezers

Src SH3 protein domain is a typical two-state protein which has been confirmed by research of denaturant-induced unfolding dynamics. Force spectroscopy experiments by optical tweezers and atomic force microscopy have measured the force-dependent unfolding rates with different kinds of pulling geometry. However, the equilibrium folding and unfolding dynamics at constant forces has not been reported. Here, using stable magnetic tweezers, we performed equilibrium folding and unfolding dynamic measurement and force-jump measurement of src SH3 domain with tethering points at its N-and C-termini. From the obtained force-dependent transition rates, a detailed two-state free energy landscape of src SH3 protein is constructed with quantitative information of folding free energy, transition state barrier height and position,which exemplifies the capability of magnetic tweezers to study protein folding and unfolding dynamics.     

Derivation of the nonlinear hydrodynamic equations using multi-mode techniques

A general mode-mode coupling theory is developed for the microscopic mass, energy and momentum densities of a simple classical fluid. A projection operator method is employed to derive a generalized Langevin equation that contains nonlinearities of all orders with both convective and dissipative terms. A general nonequilibrium ensemble average, which contains local equilibrium as a special case, is employed to derive nonlinear transport equations that are nonlocal in both space and time.The nonlinear Euler and Navier-Stokes equations are recovered using a factorization procedure based on an inverse system size approximation. We show that in the context of mode-mode coupling theory, nonlinearities of all orders must be retained to derive the full nonlinear transport equations. We also slow that the space and time dependent nonequilibrium pressure and transport coefficients are functions of the nonequilibrium mass and internal energy densities. The thermodynamic closure relationships follow as a natural consequence of mode-mode coupling theory. For a system linearly displaced from equilibrium we demonstrate the role of the corrections to our factorization approximation in renormalizing the transport coefficients.     

Conformational temperature characterizing the folding of a protein

The time sequences of the molecular dynamics simulation for the folding process of a protein is analyzed with the inherent structure landscape which focuses on the configurational dynamics of the system. Time-dependent energy and entropy for inherent structures are introduced, and from these quantities a conformational temperature is defined. The conformational temperature follows the time evolution of a slow relaxation process and reaches the bath temperature when the system is equilibrated. We show that the nonequilibrium system is described by two temperatures, one for fast vibration and the other for slow configurational relaxation, while the equilibrium system is described by one temperature. The proposed formalism is applicable widely for systems with many metastable states.     

Nonequilibrium work relations: foundations and applications

When a macroscopic system in contact with a heat reservoir is driven away from equilibrium, the second law of thermodynamics places a strict bound on the amount of work performed on the system. With a microscopic system the situation is more subtle, as thermal fluctuations give rise to a statistical distribution of work values. In recent years it has been realized that such distributions encode surprisingly more information than one might expect from traditional thermodynamic arguments. I will discuss a number of exact results that relate equilibrium properties of the system, in particular free energy differences, to the fluctuations in the work performed during such a nonequilibrium process. I will describe the theoretical foundations of these relations, connections with irreversibility and the second law of thermodynamics, and potential experimental and computational applications.     

Hydrodynamic and Thermodynamic Nonequilibrium Effects around Shock Waves: Based on a Discrete Boltzmann Method

A shock wave that is characterized by sharp physical gradients always draws the medium out of equilibrium. In this work, both hydrodynamic and thermodynamic nonequilibrium effects around the shock wave are investigated using a discrete Boltzmann model. Via Chapman–Enskog analysis, the local equilibrium and nonequilibrium velocity distribution functions in one-, two-, and three-dimensional velocity space are recovered across the shock wave. Besides, the absolute and relative deviation degrees are defined in order to describe the departure of the fluid system from the equilibrium state. The local and global nonequilibrium effects, nonorganized energy, and nonorganized energy flux are also investigated. Moreover, the impacts of the relaxation frequency, Mach number, thermal conductivity, viscosity, and the specific heat ratio on the nonequilibrium behaviours around shock waves are studied. This work is helpful for a deeper understanding of the fine structures of shock wave and nonequilibrium statistical mechanics.     

Nonequilibrium thermodynamics and nonlinear kinetics in a cellular signaling switch

We develop a rigorous nonequilibrium thermodynamics for an open system of nonlinear biochemical reactions responsible for cell signal processing. We show that the quality of the biological switch consisting of a phosphorylation-dephosphorylation cycle, such as those in protein kinase cascade, is controlled by the available intracellular free energy from the adenosine triphosphate (ATP) hydrolysis in vivo: DeltaG=k(B)Tln(([ATP]/K(eq)[ADP]), where K(eq) is the equilibrium constant. The model reveals the correlation between the performance of the switch and the level of DeltaG. The result demonstrates the importance of nonequilibrium thermodynamics in analyzing biological information processing, provides its energetic cost, establishes an interplay between signal transduction and energy metabolism in cells, and suggests a biological function for phosphoenergetics in the ubiquitous phosphorylation signaling.     

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.     

Extremal properties of distribution function and statistical operator in equilibrium and nonequilibrium thermodynamics

It is shown that the distribution function and the statistical operator, in the case that the considered system is close to the equilibrium state, can be received by the method relying upon minimizing the information gain, which is connected with the transition of the system from a nonequilibrium state to the equilibrium state. For the systems far from equilibrium the nonequilibrium distribution function or the nonequilibrium statistical operator can be derived using a variational principle based on Jaynes' maximum entropy formalism including memory effects.