Classical Free Energy Research Articles

Incorporating Electrolyte Correlation Effects into Variational Models of Electrochemical Interfaces.

We propose a way for obtaining a classical free energy density functional for electrolytes based on a first-principle many-body partition function. Via a one-loop expansion, we include coulombic correlations beyond the conventional mean-field approximation. To examine electrochemical interfaces, we integrate the electrolyte free energy functional into a hybrid quantum-classical model. This scheme self-consistently couples electronic, ionic, and solvent degrees of freedom and incorporates electrolyte correlation effects. The derived free energy functional causes a correlation-induced enhancement in interfacial counterion density and leads to an overall increase in capacitance. This effect is partially compensated by a reduction of the dielectric permittivity of interfacial water. At larger surface charge densities, ion crowding at the interface stifles these correlation effects. While scientifically intriguing already at planar interfaces, we anticipate these correlation effects to play an essential role for electrolytes in nanoconfinement.

Engines for predictive work extraction from memoryful quantum stochastic processes

Quantum information-processing techniques enable work extraction from a system's inherently quantum features, in addition to the classical free energy it contains. Meanwhile, the science of computational mechanics affords tools for the predictive modeling of non-Markovian classical and quantum stochastic processes. We combine tools from these two sciences to develop a technique for predictive work extraction from non-Markovian stochastic processes with quantum outputs. We demonstrate that this technique can extract more work than non-predictive quantum work extraction protocols, on the one hand, and predictive work extraction without quantum information processing, on the other. We discover a phase transition in the efficacy of memory for work extraction from quantum processes, which is without classical precedent. Our work opens up the prospect of machines that harness environmental free energy in an essentially quantum, essentially time-varying form.

Leading quantum correction to the classical free energy

The quantum free energy of a system governed by a standard Hamiltonian is larger than its classical counterpart. The lowest-order correction, first calculated by Wigner, is proportional to ℏ2 and involves the sum of the mean squared forces. We present an elementary derivation of this result by drawing upon the Zassenhaus formula, an operator-generalization for the main functional relation of the exponential map. Our approach highlights the central role of non-commutativity between kinetic and potential energy and is more direct than Wigner's original calculation, or even streamlined variations thereof found in modern textbooks. We illustrate the quality of the correction for the simple harmonic oscillator (analytically) and the purely quartic oscillator (numerically) in the limit of high temperature. We also demonstrate that the Wigner correction fails in situations with sufficiently rapidly changing potentials, for instance, the particle in a box.

Estimation of frequency factors for the calculation of kinetic isotope effects from classical and path integral free energy simulations.

We use the modified Bigeleisen-Mayer equation to compute kinetic isotope effect values for non-enzymatic phosphoryl transfer reactions from classical and path integral molecular dynamics umbrella sampling. The modified form of the Bigeleisen-Mayer equation consists of a ratio of imaginary mode vibrational frequencies and a contribution arising from the isotopic substitution's effect on the activation free energy, which can be computed from path integral simulation. In the present study, we describe a practical method for estimating the frequency ratio correction directly from umbrella sampling in a manner that does not require normal mode analysis of many geometry optimized structures. Instead, the method relates the frequency ratio to the change in the mass weighted coordinate representation of the minimum free energy path at the transition state induced by isotopic substitution. The method is applied to the calculation of 16/18O and 32/34S primary kinetic isotope effect values for six non-enzymatic phosphoryl transfer reactions. We demonstrate that the results are consistent with the analysis of geometry optimized transition state ensembles using the traditional Bigeleisen-Mayer equation. The method thus presents a new practical tool to enable facile calculation of kinetic isotope effect values for complex chemical reactions in the condensed phase.

Nucleation-mediated reshaping of facetted metallic nanocrystals: Breakdown of the classical free energy picture.

Shape stability is key to avoiding degradation of performance for metallic nanocrystals synthesized with facetted non-equilibrium shapes to optimize properties for catalysis, plasmonics, and so on. Reshaping of facetted nanocrystals is controlled by the surface diffusion-mediated nucleation and growth of new outer layers of atoms. Kinetic Monte Carlo (KMC) simulation of a realistic stochastic atomistic-level model is applied to precisely track the reshaping of Pd octahedra and nanocubes. Unexpectedly, separate constrained equilibrium Monte Carlo analysis of the free energy profile during reshaping reveals a fundamental failure of the classical nucleation theory (CNT) prediction for the reshaping barrier and rate. Why? Nucleation barriers can be relatively low for these processes, so the system is not locally equilibrated before crossing the barrier, as assumed in CNT. This claim is supported by an analysis of a first-passage problem for reshaping within a master equation framework for the model that reasonably captures the behavior in KMC simulations.

Collective Proton Transfers in Cyclic Water–Ammonia Tetramers: A Path Integral Machine-Learning Study

We present results from machine-learning-based path integral molecular dynamics simulations that describe isomerization paths articulated via collective proton transfers along mixed, cyclic tetramers combining water and ammonia at cryogenic conditions. The net result of such isomerizations is a reverse of the chirality of the global hydrogen-bonding architecture along the different cyclic moieties. In monocomponent tetramers, the classical free energy profiles associated with these isomerizations present the usual symmetric double-well characteristics whereas the reactive paths exhibit full concertedness among the different intermolecular transfer processes. Contrastingly, in mixed water/ammonia tetramers, the incorporation of a second component introduces imbalances in the strengths of the different hydrogen bonds leading to a partial loss of concertedness, most notably at the vicinity of the transition state. As such, the highest and lowest degrees of progression are registered along OH···N and O···HN coordinations, respectively. These characteristics lead to polarized transition state scenarios akin to solvent-separated ion-pair configurations. The explicit incorporation of nuclear quantum effects promotes drastic depletions in the activation free energies and modifications in the overall shape of the profiles which include central plateau-like stages, indicating the prevalence of deep tunneling regimes. On the other hand, the quantum treatment of the nuclei partially restores the degree of concertedness among the evolutions of the individual transfers.

Statistical Mechanical Design Principles for Coarse-Grained Interactions across Different Conformational Free Energy Surfaces.

Systematic bottom-up coarse-graining (CG) of molecular systems provides a means to explore different coupled length and time scales while treating the molecular-scale physics at a reduced level. However, the configuration dependence of CG interactions often results in CG models with limited applicability for exploring the parametrized configurations. We propose a statistical mechanical theory to design CG interactions across different configurations and conditions. In order to span wide ranges of conformational space, distinct classical CG free energy surfaces for characteristic configurations are identified using molecular collective variables. The coupling interaction between different CG free energy surfaces can then be systematically determined by analogy to quantum mechanical approaches describing coupled states. The present theory can accurately capture the underlying many-body potentials of mean force in the CG variables for various order parameters applied to liquids, interfaces, and in principle proteins, uncovering the complex nature underlying the coupling interaction and imparting a new protocol for the design of predictive multiscale models.

On the Permeation of Polychlorinated Dibenzodioxins and Dibenzofurans through Lipid Membranes: Classical MD and Hybrid QM/MM-EDA Analysis.

The permeation of dioxin-like pollutants, namely, chlorinated dibenzodioxins and dibenzofurans, through lipid membranes has been simulated using classic molecular dynamics (CMD) combined with the umbrella sampling approach. The most toxic forms of chlorinated dibenzodioxin and dibenzofuran, 2,3,7,8-tetrachloro-p-dibenzodioxin (TCDD) and 2,3,7,8-tetrachlorodibenzofuran (TCDF), and a dioleyl-phosphatidylcholine (DOPC) lipid membrane of 50 Å wide have been chosen for our study. The free energy profile shows the penetration process is largely favoured thermodynamically (ΔG ≈ -12 kcal/mol), with a progressively decrease of the free energy until reaching the energy minima at distances of 8 Å and 9.5 Å from the centre of the membrane for, respectively, TCDD and TCDF. At the centre of the membrane, both molecules display subtle local maxima with free energy differences of 0.5 and 1 kcal/mol with respect to the energy minima for TCDD and TCDF, respectively. Furthermore, the intermolecular interactions between the molecules and the lipid membrane have been characterized at the minima and the local maxima using hybrid quantum mechanics/molecular mechanics energy decomposition analysis (QM/MM-EDA). Total interaction energies of -17.5 and -16.5 kcal/mol have been found at the energy minima for TCDD and TCDF, respectively. In both cases, the dispersion forces govern the molecule-membrane interactions, no significant changes have been found at the local maxima, in agreement with the classical free energy profile. The small differences found in the results obtained for TCDD and TCDF point out that the adsorption and diffusion processes through the cell membrane are not related to the different toxicity shown by these pollutants.

Quantum Free Energy Profiles for Molecular Proton Transfers.

Although many molecular dynamics simulations treat the atomic nuclei as classical particles, an adequate description of nuclear quantum effects (NQEs) is indispensable when studying proton transfer reactions. Herein, quantum free energy profiles are constructed for three typical proton transfers, which properly take NQEs into account using the path integral formalism. The computational cost of the simulations is kept tractable by deriving machine learning potentials. It is shown that the classical and quasi-classical centroid free energy profiles of the proton transfers deviate substantially from the exact quantum free energy profile.

Linear-Time Algorithms for RNA Structure Prediction.

RNA secondary structure prediction is widely used to understand RNA function. Existing dynamic programming-based algorithms, both the classical minimum free energy (MFE) methods and partition function methods, suffer from a major limitation: their runtimes scale cubically with the RNA length, and this slowness limits their use in genome-wide applications. Inspired by incremental parsing for context-free grammars in computational linguistics, we designed linear-time heuristic algorithms, LinearFold and LinearPartition, to approximate the MFE structure, partition function and base pairing probabilities. These programs are orders of magnitude faster than Vienna RNAfold and CONTRAfold on long sequences. More interestingly, LinearFold and LinearPartition lead to more accurate predictions on the longest sequence families for which the structures are well established (16S and 23S Ribosomal RNAs), as well as improved accuracies for long-range base pairs (500 + nucleotides apart). This chapter provides protocols for using LinearFold and LinearPartition for secondary structure prediction.

On the Wigner-Kirkwood Expansion of the Free Energy and the Evaluation of the Quantum Correction

The Wigner-Kirkwood expansion of the quantum correction to the classical free energy is generally said to be in powers of ℏ2 and only its first few terms are presented. In this work, we use the Bloch differential equation to obtain a general description of all terms in a dimensionless form. The first corrective term turns out to be proportional to the product of λ2/a2, where λ is the thermal de Broglie wavelength and a3 is the volume per particle, by an effective coupling constant. This dimensionless parameter can be used to assess the magnitude of the quantum correction. Using the one-component plasma as an illustration we highlight the importance of the magnitude of the potential on the quantum correction. The results presented are not formally new; the emphasis is placed on a simple and didactic presentation.

A trace inequality of Ando, Hiai, and Okubo and a monotonicity property of the Golden–Thompson inequality

The Golden–Thompson trace inequality, which states that Tr eH+K ≤ Tr eHeK, has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here, we make this G–T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H = Δ or H=−−Δ+m and K = potential, Tr eH+(1−u)KeuK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr XsYtX1−sY1−t ≤ Tr XY for positive operators X, Y and for 12≤s,t≤1, and s+t≤32. The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the 32,1 range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem.

Mechanistic insights into the Japanese encephalitis virus RNA dependent RNA polymerase protein inhibition by bioflavonoids from Azadirachta indica

Japanese encephalitis (JE) virus is a flavivirus causing encephalitis causing neurological damage. RNA-dependent-RNA-polymerase (RdRp) is responsible for genome replication making it excellent anti-viral target. In this study, the crystal structure of JE RdRp (jRdRp) and bioflavonoids reported in Azadirachta indica were retrieved from specific databases. Structure-based virtual screening was employed using MTiOpenScreen server and top four compounds selected with the most negative docking scores. Conformations were redocked using AutoDock Vina; these complexes showed mechanistic interactions with Arg474, Gly605, Asp668, and Trp800 residues in the active site of jRdRp, i.e., guanosine-5′-triphosphate. Furthermore, 100 ns classical molecular dynamics simulation and binding free energy calculation showed stability of docked bioflavonoids in the active jRdRp pocket and significant contribution of van-der-Waals interactions for docked complex stability during simulation. Therefore, this study predicted the anti-viral activity of Gedunin, Nimbolide, Ohchinin acetate, and Kulactone against jRdRp and can be considered for further antiviral drug development.

Classical Casimir free energy for two Drude spheres of arbitrary radii: A plane-wave approach

We derive an exact analytic expression for the high-temperature limit of the Casimir interaction between two Drude spheres of arbitrary radii. Specifically, we determine the Casimir free energy by using the scattering approach in the plane-wave basis. Within a round-trip expansion, we are led to consider the combinatorics of certain partitions of the round trips. The relation between the Casimir free energy and the capacitance matrix of two spheres is discussed. Previously known results for the special cases of a sphere-plane geometry as well as two spheres of equal radii are recovered. An asymptotic expansion for small distances between the two spheres is determined and analytical expressions for the coefficients are given.

Generating Excess Protons in Microsolvated Acid Clusters under Ambient Conditions: An Issue of Configurational Entropy versus Internal Energy.

Acid dissociation, and thus liberation of excess protons in small water droplets, impacts on diverse fields such as interstellar, atmospheric or environmental chemistry. At cryogenic temperatures below 1 K, it is now well established that as few as four water molecules suffice to dissociate the generic strong acid HCl, yet temperature‐driven recombination sets in simply upon heating that cluster. Here, the fundamental question is posed of how many more water molecules are required to stabilize a hydrated excess proton at room temperature. Ab initio path integral simulations disclose that not five, but six water molecules are needed at 300 K to allow for HCl dissociation independently from nuclear quantum effects. In order to provide the molecular underpinnings of these observations, the classical and quantum free energy profiles were decomposed along the dissociation coordinate in terms of the corresponding internal energy and entropy profiles. What decides in the end about acid dissociation, and thus ion pair formation, in a specific microsolvated water cluster at room temperature is found to be a fierce competition between classical configurational entropy and internal energy, where the former stabilizes the undissociated state whereas the latter favors dissociation. It is expected that these are generic findings with broad implications on acid–base chemistry depending on temperature in small water assemblies.

A Simple Method for Including Polarization Effects in Solvation Free Energy Calculations When Using Fixed-Charge Force Fields: Alchemically Polarized Charges.

The incorporation of polarizability in classical force-field molecular simulations is an ongoing area of research. We focus here on its application to hydration free energy simulations of organic molecules. In contrast to computationally complex approaches involving the development of explicitly polarizable force fields, we present herein a simple methodology for incorporating polarization into such simulations using standard fixed-charge force fields, which we call the alchemically polarized charges (APolQ) method. APolQ employs a standard classical alchemical free energy change simulation to calculate the free energy difference between a fully polarized solute particle in a condensed phase and its unpolarized state in a vacuum. APolQ can in principle be applied to any microscopically homogeneous system (e.g., pure or mixed solvents). We applied APolQ to hydration free energy data for a test set of 45 neutral solute molecules in the FreeSolv database and compared results obtained using three different water models (SPC/E, TIP3P, and OPC3) and using minimal basis iterative Stockholder (MBIS) and restrained electrostatic potential (RESP) partial charge methodologies. In comparison with AM1-BCC, we found that APolQ outperforms it for the test set. Despite our method using default GAFF parameters, the MBIS partial charges yield absolute average deviations 1.5–1.9 kJ mol–1 lower than using AM1 bond charge correction (AM1-BCC). We conjecture that this method can be further improved by fitting the Lennard-Jones and torsional parameters to partial charges derived using MBIS or RESP methodologies.

LinearPartition: linear-time approximation of RNA folding partition function and base-pairing probabilities.

MotivationRNA secondary structure prediction is widely used to understand RNA function. Recently, there has been a shift away from the classical minimum free energy methods to partition function-based methods that account for folding ensembles and can therefore estimate structure and base pair probabilities. However, the classical partition function algorithm scales cubically with sequence length, and is therefore prohibitively slow for long sequences. This slowness is even more severe than cubic-time free energy minimization due to a substantially larger constant factor in runtime.ResultsInspired by the success of our recent LinearFold algorithm that predicts the approximate minimum free energy structure in linear time, we design a similar linear-time heuristic algorithm, LinearPartition, to approximate the partition function and base-pairing probabilities, which is shown to be orders of magnitude faster than Vienna RNAfold and CONTRAfold (e.g. 2.5 days versus 1.3 min on a sequence with length 32 753 nt). More interestingly, the resulting base-pairing probabilities are even better correlated with the ground-truth structures. LinearPartition also leads to a small accuracy improvement when used for downstream structure prediction on families with the longest length sequences (16S and 23S rRNAs), as well as a substantial improvement on long-distance base pairs (500+ nt apart).Availability and implementationCode: http://github.com/LinearFold/LinearPartition; Server: http://linearfold.org/partition.Supplementary information Supplementary data are available at Bioinformatics online.

Combined Effects from Solvation and Nuclear Quantum Fluctuations on Autoionization Mechanisms in Aqueous Clusters.

Using path-integral molecular dynamics simulations, we examine isomerization paths involving collective proton transfers in [H2O]5 and [H2O]8 clusters under cryogenic conditions. We focused attention on combined effects derived from solvation and nuclear quantum fluctuations on the characteristics of free energy barriers and relative stabilities of reactants and products. In particular, we analyzed two different processes: the first one involves the exchange of donor-acceptor hydrogen bond roles along cyclic moieties, whereas the second one corresponds to charge separation leading to stable [H3O]+[OH]- ion pairs. In the first case, the explicit incorporation of quantum tunneling introduces important modifications in the classical free energy profile. The resulting quantum profile presents two main contributions, one corresponding to compressions of O-O distances and a second one ascribed to nuclear tunneling of the light protons. Solvation effects promote a moderate polarization of the cyclic structures and a partial loss of concertedness in the collective modes, most notably, at the onset of tunneling. Still, the latter effects are also sufficiently strong to promote the stabilization of ion pairs along the classical trajectories. In contrast, the explicit incorporation of nuclear quantum fluctuations leads to charge separated configurations that are marginally stable. As such, the latter states could also be regarded as short-lived intermediate states along the reactive exchange path.

Analytical classical density functionals from an equation learning network.

We explore the feasibility of using machine learning methods to obtain an analytic form of the classical free energy functional for two model fluids, hard rods and Lennard-Jones, in one dimension. The equation learning network proposed by Martius and Lampert [e-print arXiv:1610.02995 (2016)] is suitably modified to construct free energy densities which are functions of a set of weighted densities and which are built from a small number of basis functions with flexible combination rules. This setup considerably enlarges the functional space used in the machine learning optimization as compared to the previous work [S.-C. Lin and M. Oettel, SciPost Phys. 6, 025 (2019)] where the functional is limited to a simple polynomial form. As a result, we find a good approximation for the exact hard rod functional and its direct correlation function. For the Lennard-Jones fluid, we let the network learn (i) the full excess free energy functional and (ii) the excess free energy functional related to interparticle attractions. Both functionals show a good agreement with simulated density profiles for thermodynamic parameters inside and outside the training region.

Simulation of Calcium Phosphate Prenucleation Clusters in Aqueous Solution: Association beyond Ion Pairing.

Classical molecular dynamics simulations and free energy methods have been used to obtain a better understanding of the molecular processes occurring prior to the first nucleation event for calcium phosphate biominerals. The association constants for the formation of negatively charged complexes containing calcium and phosphate ions in aqueous solution have been computed, and these results suggest that the previously proposed calcium phosphate building unit, [Ca(HPO4)3]4–, should only be present in small amounts under normal experimental conditions. However, the presence of an activation barrier for the removal of an HPO42– ion from this complex indicates that this species could be kinetically trapped. Aggregation pathways involving CaHPO4, [Ca(HPO4)2]2–, and [Ca(HPO4)3]4– complexes have been explored with the finding that dimerization is favorable up to a Ca/HPO4 ratio of 1:2.