A finite‐time variational method for determining optimal paths and obtaining bounds on free energy changes from computer simulations

Probabilistic and maximum entropy modeling of chemical reaction systems: Characteristics and comparisons to mass action kinetic models

We demonstrate and characterize a first-principles approach to modeling the mass action dynamics of metabolism. Starting from a basic definition of entropy expressed as a multinomial probability density using Boltzmann probabilities with standard chemical potentials, we derive and compare the free energy dissipation and the entropy production rates. We express the relation between entropy production and the chemical master equation for modeling metabolism, which unifies chemical kinetics and chemical thermodynamics. Because prediction uncertainty with respect to parameter variability is frequently a concern with mass action models utilizing rate constants, we compare and contrast the maximum entropy model, which has its own set of rate parameters, to a population of standard mass action models in which the rate constants are randomly chosen. We show that a maximum entropy model is characterized by a high probability of free energy dissipation rate and likewise entropy production rate, relative to other models. We then characterize the variability of the maximum entropy model predictions with respect to uncertainties in parameters (standard free energies of formation) and with respect to ionic strengths typically found in a cell.

Exploring Routes to Enhance the Calculation of Free Energy Differences via Non-Equilibrium Work SQM/MM Switching Simulations Using Hybrid Charge Intermediates between MM and SQM Levels of Theory or Non-Linear Switching Schemes

Non-equilibrium work switching simulations and Jarzynski's equation are a reliable method for computing free energy differences, ΔAlow→high, between two levels of theory, such as a pure molecular mechanical (MM) and a quantum mechanical/molecular mechanical (QM/MM) description of a system of interest. Despite the inherent parallelism, the computational cost of this approach can quickly become very high. This is particularly true for systems where the core region, the part of the system to be described at different levels of theory, is embedded in an environment such as explicit solvent water. We find that even for relatively simple solute-water systems, switching lengths of at least 5 ps are necessary to compute ΔAlow→high reliably. In this study, we investigate two approaches towards an affordable protocol, with an emphasis on keeping the switching length well below 5 ps. Inserting a hybrid charge intermediate state with modified partial charges, which resembles the charge distribution of the desired high level, makes it possible to obtain reliable calculations with 2 ps switches. Attempts using step-wise linear switching paths, on the other hand, did not lead to improvement, i.e., a faster convergence for all systems. To understand these findings, we analyzed the solutes' properties as a function of the partial charges used and the number of water molecules in direct contact with the solute, and studied the time needed for water molecules to reorient themselves upon a change in the solute's charge distribution.

Skewed thermodynamic geometry and optimal free energy estimation

Free energy differences are a central quantity of interest in physics, chemistry, and biology. We develop design principles that improve the precision and accuracy of free energy estimators, which have potential applications to screening for targeted drug discovery. Specifically, by exploiting the connection between the work statistics of time-reversed protocol pairs, we develop near-equilibrium approximations for moments of the excess work and analyze the dominant contributions to the precision and accuracy of standard nonequilibrium free-energy estimators. Within linear response, minimum-dissipation protocols follow the geodesics of the Riemannian metric induced by the Stokes friction tensor. We find that the next-order contribution arises from the rank-3 supra-Stokes tensor that skews the geometric structure such that minimum-dissipation protocols follow the geodesics of a generalized cubic Finsler metric. Thus, near equilibrium, the supra-Stokes tensor determines the leading-order contribution to the bias of bidirectional free-energy estimators.

Extensive Assessment of Various Computational Methods for Aspartate's pKa Shift

A series of computational methods for pK_a shift prediction are extensively tested on a set of benchmark protein systems, aiming at identifying pitfalls and evaluating their performance on high variants. Including 19 ASP residues in 10 protein systems, the benchmark set consists of both residues with highly shifted pK_a values as well as those varying little from the reference value, with an experimental RMS free energy differences of 2.49 kcal/mol with respect to blocked amino acid, namely the RMS pK_a shift being 1.82 pK_a units. The constant pH molecular dynamics (MD), alchemical methods, PROPKA3.1, and multiconformation continuum electrostatics give RMSDs of 1.52, 2.58, 1.37, and 3.52 pK_a units, respectively, on the benchmark set. The empirical scoring method is the most accurate one with extremely low computational cost, and the pH-dependent model is also able to provide accurate results, while the accuracy of MD sampling incorporating alchemical free energy simulation is prohibited by convergence achievement and the performance of conformational search incorporating multiconformation continuum electrostatics is bad. Former research works did not define statistical uncertainty with care and yielded the questionable conclusion that alchemical methods perform well in most benchmarks. In this work the traditional alchemical methods are thoroughly tested for high variants. We also performed the first application of nonequilibrium alchemical methods to the pK_a cases.

BAR-based optimum adaptive sampling regime for variance minimization in alchemical transformation

The efficiency of alchemical free energy simulations with a staging strategy is improved by adaptively manipulating the significance of each ensemble followed by importance sampling.

Stochastic Thermodynamics of a Particle in a Box

The piston system (particles in a box) is the simplest paradigmatic model in traditional thermodynamics. However, the recently established framework of stochastic thermodynamics (ST) fails to apply to this model system due to the embedded singularity in the potential. In this Letter, we study the ST of a particle in a box by adopting a novel coordinate transformation technique. Through comparing with the exact solution of a breathing harmonic oscillator, we obtain analytical results of work distribution for an arbitrary protocol in the linear response regime and verify various predictions of the fluctuation-dissipation relation. When applying to the Brownian Szilard engine model, we obtain the optimal protocol λ_{t}=λ_{0}2^{t/τ} for a given sufficiently long total time τ. Our study not only establishes a paradigm for studying ST of a particle in a box but also bridges the long-standing gap in the development of ST.

Optimal control in nonequilibrium systems: Dynamic Riemannian geometry of the Ising model

A general understanding of optimal control in nonequilibrium systems would illuminate the operational principles of biological and artificial nanoscale machines. Recent work has shown that a system driven out of equilibrium by a linear response protocol is endowed with a Riemannian metric related to generalized susceptibilities, and that geodesics on this manifold are the nonequilibrium control protocols with the lowest achievable dissipation. While this elegant mathematical framework has inspired numerous studies of exactly solvable systems, no description of the thermodynamic geometry yet exists when the metric cannot be derived analytically. Herein, we numerically construct the dynamic metric of the two-dimensional Ising model in order to study optimal protocols for reversing the net magnetization.

Degenerate optimal paths in thermally isolated systems

We present an analysis of the work performed on a system of interest that is kept thermally isolated during the switching of a control parameter. We show that there exists, for a certain class of systems, a finite-time family of switching protocols for which the work is equal to the quasistatic value. These optimal paths are obtained within linear response for systems initially prepared in a canonical distribution. According to our approach, such protocols are composed of a linear part plus a function which is odd with respect to time reversal. For systems with one degree of freedom, we claim that these optimal paths may also lead to the conservation of the corresponding adiabatic invariant. This points to an interesting connection between work and the conservation of the volume enclosed by the energy shell. To illustrate our findings, we solve analytically the harmonic oscillator and present numerical results for certain anharmonic examples.

Current trends in finite-time thermodynamics

The cornerstone of finite-time thermodynamics is all about the price of haste and how to minimize it. Reversible processes may be ultimately efficient, but they are unrealistically slow. In all situations-chemical, mechanical, economical-we pay extra to get the job done quickly. Finite-time thermodynamics can be used to develop methods to limit that extra expenditure, be it in energy, entropy production, money, or something entirely different. Finite-time thermodynamics also includes methods to calculate the optimal path or mode of operation to achieve this minimal expenditure. The concept is to place the system of interest in contact with a time-varying environment which will coax the system along the desired path, much like guiding a horse along by waving a carrot in front of it.

Optimal estimators and asymptotic variances for nonequilibrium path-ensemble averages

Existing optimal estimators of nonequilibrium path-ensemble averages are shown to fall within the framework of extended bridge sampling. Using this framework, we derive a general minimal-variance estimator that can combine nonequilibrium trajectory data sampled from multiple path-ensembles to estimate arbitrary functions of nonequilibrium expectations. The framework is also applied to obtain asymptotic variance estimates, which are a useful measure of statistical uncertainty. In particular, we develop asymptotic variance estimates pertaining to Jarzynski's equality for free energies and the Hummer-Szabo expressions for the potential of mean force, calculated from uni- or bidirectional path samples. These estimators are demonstrated on a model single-molecule pulling experiment. In these simulations, the asymptotic variance expression is found to accurately characterize the confidence intervals around estimators when the bias is small. Hence, the confidence intervals are inaccurately described for unidirectional estimates with large bias, but for this model it largely reflects the true error in a bidirectional estimator derived by Minh and Adib.

Density-dependent analysis of nonequilibrium paths improves free energy estimates

When a system is driven out of equilibrium by a time-dependent protocol that modifies the Hamiltonian, it follows a nonequilibrium path. Samples of these paths can be used in nonequilibrium work theorems to estimate equilibrium quantities such as free energy differences. Here, we consider analyzing paths generated with one protocol using another one. It is posited that analysis protocols which minimize the lag, the difference between the nonequilibrium and the instantaneous equilibrium densities, will reduce the dissipation of reprocessed trajectories and lead to better free energy estimates. Indeed, when minimal lag analysis protocols based on exactly soluble propagators or relative entropies are applied to several test cases, substantial gains in the accuracy and precision of estimated free energy differences are observed.

Optimizing the driving function for nonequilibrium free-energy calculations in the linear regime: a variational approach

We consider the issue of optimizing linear-regime nonequilibrium simulations to estimate free-energy differences. In particular, we focus on the problem of finding the best-possible driving function lambda(t) that, for a given thermodynamic path, simulation algorithm, and amount of computational effort, minimizes dissipation. From the fluctuation-dissipation theorem it follows that, in the linear-response regime, the dissipation is controlled by the magnitude and characteristic correlation time of the equilibrium fluctuations in the driving force. As a result, the problem of finding the optimal switching scheme involves the solution of a standard problem in variational calculus: the minimization of a functional with respect to the switching function. In practice, the minimization involves solving the associated Euler-Lagrange equation subject to a set of boundary conditions. As a demonstration we apply the approach to the simple, yet illustrative problem of computing the free-energy difference between two classical harmonic oscillators with very different characteristic frequencies.

Histogram analysis as a method for determining the line tension of a three-phase contact region by Monte Carlo simulations

A method is proposed for determining the line tension, which is the main physical characteristic of a three-phase contact region, by Monte Carlo (MC) simulations. The key idea of the proposed method is that if a three-phase equilibrium involves a three-phase contact region, the probability distribution of states of a system as a function of two order parameters depends not only on the surface tension, but also on the line tension. This probability distribution can be obtained as a normalized histogram by appropriate MC simulations, so one can use the combination of histogram analysis and finite-size scaling to study the properties of a three phase contact region. Every histogram and results extracted therefrom will depend on the size of the simulated system. Carrying out MC simulations for a series of system sizes and extrapolating the results, obtained from the corresponding series of histograms, to infinite size, one can determine the line tension of the three phase contact region and the interfacial tensions of all three interfaces (and hence the contact angles) in an infinite system. To illustrate the proposed method, it is applied to the three-dimensional ternary fluid mixture, in which molecular pairs of like species do not interact whereas those of unlike species interact as hard spheres. The simulated results are in agreement with expectations.

Test-area simulation method for the direct determination of the interfacial tension of systems with continuous or discontinuous potentials

A novel test-area (TA) technique for the direct simulation of the interfacial tension of systems interacting through arbitrary intermolecular potentials is presented in this paper. The most commonly used method invokes the mechanical relation for the interfacial tension in terms of the tangential and normal components of the pressure tensor relative to the interface (the relation of Kirkwood and Buff [J. Chem. Phys. 17, 338 (1949)]). For particles interacting through discontinuous intermolecular potentials (e.g., hard-core fluids) this involves the determination of delta functions which are impractical to evaluate, particularly in the case of nonspherical molecules. By contrast we employ a thermodynamic route to determine the surface tension from a free-energy perturbation due to a test change in the surface area. There are important distinctions between our test-area approach and the computation of a free-energy difference of two (or more) systems with different interfacial areas (the method of Bennett [J. Comput. Phys. 22, 245 (1976)]), which can also be used to determine the surface tension. In order to demonstrate the adequacy of the method, the surface tension computed from test-area Monte Carlo (TAMC) simulations are compared with the data obtained with other techniques (e.g., mechanical and free-energy differences) for the vapor-liquid interface of Lennard-Jones and square-well fluids; the latter corresponds to a discontinuous potential which is difficult to treat with standard methods. Our thermodynamic test-area approach offers advantages over existing techniques of computational efficiency, ease of implementation, and generality. The TA method can easily be implemented within either Monte Carlo (TAMC) or molecular-dynamics (TAMD) algorithms for different types of interfaces (vapor-liquid, liquid-liquid, fluid-solid, etc.) of pure systems and mixtures consisting of complex polyatomic molecules.

Fcc-bcc transition for Yukawa interactions determined by applied strain deformation

Calculations of the work required to transform between bcc and fcc phases yield a high-precision bcc-fcc transition line for monodisperse point Yukawa (screened-Coulomb) systems. Our results agree qualitatively but not quantitatively with recently published simulations and phenomenological criteria for the bcc-fcc transition. In particular, the bcc-fcc-fluid triple point lies at a higher inverse screening length than previously reported.

Efficient use of nonequilibrium measurement to estimate free energy differences for molecular systems

A promising method for calculating free energy differences DeltaF is to generate nonequilibrium data via "fast-growth" simulations or by experiments--and then use Jarzynski's equality. However, a difficulty with using Jarzynski's equality is that DeltaF estimates converge very slowly and unreliably due to the nonlinear nature of the calculation--thus requiring large, costly data sets. The purpose of the work presented here is to determine the best estimate for DeltaF given a (finite) set of work values previously generated by simulation or experiment. Exploiting statistical properties of Jarzynski's equality, we present two fully automated analyses of nonequilibrium data from a toy model, and various simulated molecular systems. Both schemes remove at least several k(B)T of bias from DeltaF estimates, compared to direct application of Jarzynski's equality, for modest sized data sets (100 work values), in all tested systems. Results from one of the new methods suggest that good estimates of DeltaF can be obtained using 5-40-fold less data than was previously possible. Extending previous work, the new results exploit the systematic behavior of bias due to finite sample size. A key innovation is better use of the more statistically reliable information available from the raw data.

Variational formula for the free energy based on incomplete sampling in a molecular simulation

Finite sampling in free-energy perturbation (FEP) calculations by molecular simulation leads to reproducible systematic errors, with sign shown to depend (in a known way) only on which system governs sampling in the simulation. Thus the result of a FEP calculation can be used as a bound on the true free energy. This inequality is of a wholly different nature from established forms such as the Gibbs-Bogoliubov inequality or the second law, in that its origins relate to the performance of a molecular simulation. If one can identify a suitable reference system having a free energy known as a function of some defining parameter, variational schemes based on the finite-sampling inequalities can be implemented. This idea is demonstrated through calculation of the free energy of a hard-sphere solid by perturbing from harmonic references and of a hard-sphere fluid by perturbing from infinitely polydisperse references. The tightness of the bounds increases with the amount of sampling in the simulation and correlates with the entropy difference between the target and reference systems. The bounds are tightest near the point where the entropy difference is least.