Comparison of two simulation methods to compute solvation free energies and partition coefficients

The thermodynamic integration (TI) and expanded ensemble (EE) methods are used here to calculate the hydration free energy in water, the solvation free energy in 1‐octanol, and the octanol‐water partition coefficient for a six compounds of varying functionality using the optimized potentials for liquid simulations (OPLS) all‐atom (AA) force field parameters and atomic charges. Both methods use the molecular dynamics algorithm as a primary component of the simulation protocol, and both have found wide applications in fields such as the calculation of activity coefficients, phase behavior, and partition coefficients. Both methods result in solvation free energies and 1‐octanol/water partition coefficients with average absolute deviations (AAD) from experimental data to within 4 kJ/mol and 0.5 log units, respectively. Here, we find that in simulations the OPLS‐AA force field parameters (with fixed charges) can reproduce solvation free energies of solutes in 1‐octanol with AAD of about half that for the solute hydration free energies using a extended simple point charge (SPC/E) model of water. The computational efficiency of the two simulation methods are compared based on the time (in nanoseconds) required to obtain similar standard deviations in the solvation free energies and 1‐octanol/water partition coefficients. By this analysis, the EE method is found to be a factor of nine more efficient than the TI algorithm. For both methods, solvation free energy calculations in 1‐octanol consume roughly an order of magnitude more CPU hours than the hydration free energy calculations. © 2012 Wiley Periodicals, Inc.     

Free energy perturbation study of water dimer dissociation kinetics

An efficient approach is described for using accurate ab initio calculations to determine the rates of elementary condensation and evaporation processes that lead to nucleation of aqueous aerosols. The feasibility of the method is demonstrated in an application to evaporation rates of water dimer at 230 K. The method, known as ABC-FEP (ab initio/classical free energy perturbation), begins with a calculation of the potential of mean force for the dissociation (evaporation) of small water clusters using a molecular dynamics (MD) simulation with a model potential. The free energy perturbation is used to calculate how changing from the model potential to a potential calculated from ab initio methods would alter the potential of mean force. The difference in free energy is the Boltzmann-weighted average of the difference between the ab initio and classical potential energies, with the average taken over a sample of configurations from the MD simulation. In principle, the method does not require a highly accurate model potential, though more accurate potentials require fewer configurations to achieve a small sampling error in the free energy perturbation step. To test the feasibility of obtaining accurate potentials of mean force from ab initio calculations at a modest number of configurations, the free energy perturbation method has been used to correct the errors when some standard models for bulk water (SPC, TIP4P, and TIP4PFQ) are applied to water dimer. To allow a thorough exploration of sampling issues, a highly accurate fit to results of accurate ab initio calculations, known as SAPT-5s, as been used a proxy for the ab initio calculations. It is shown that accurate values for a point on the potential of mean force can be obtained from any of the water models using ab initio calculations at only 50 configurations. Thus, this method allows accurate simulations of small clusters without the need to develop water models specifically for clusters.     

Identifying low variance pathways for free energy calculations of molecular transformations in solution phase

Improving the efficiency of free energy calculations is important for many biological and materials design applications, such as protein-ligand binding affinities in drug design, partitioning between immiscible liquids, and determining molecular association in soft materials. We show that for any pair potential, moderately accurate estimation of the radial distribution function for a solute molecule is sufficient to accurately estimate the statistical variance of a sampling along a free energy pathway. This allows inexpensive analytical identification of low statistical error free energy pathways. We employ a variety of methods to estimate the radial distribution function (RDF) and find that the computationally cheap two-body "dilute gas" limit performs as well or better than 3D-RISM theory and other approximations for identifying low variance free energy pathways. With a RDF estimate in hand, we can search for pairwise interaction potentials that produce low variance. We give an example of a search minimizing statistical variance of solvation free energy over the entire parameter space of a generalized "soft core" potential. The free energy pathway arising from this optimization procedure has lower curvature in the variance and reduces the total variance by at least 50% compared to the traditional soft core solvation pathway. We also demonstrate that this optimized pathway allows free energies to be estimated with fewer intermediate states due to its low curvature. This free energy variance optimization technique is generalizable to solvation in any homogeneous fluid and for any type of pairwise potential and can be performed in minutes to hours, depending on the method used to estimate g(r).     

Corresponding states principle for the alkaline earth dimers and the van der Waals potential of Ba2

A simple method based on the corresponding states principle and the Tang-Toennies potential model is used for the determination of the ground state van der Waals potential of barium dimer. The potential energy curve calculated from the theoretical dispersion coefficients and the experimental vibrational frequency is comparable to other quantum chemical calculations. To assess its validity, the same method is used to calculate the potential of the strontium dimer, for which there are many theoretical and experimental determinations. A comparison with experimental measurements suggests that the present method is highly accurate. The potentials of five alkaline earth dimers are compared. The potentials of Mg(2), Ca(2), Sr(2), and Ba(2) are shown to be nearly conformal with each other. The potential of Be(2) has a different shape.     

Calibration of force-field dependency in free energy landscapes of peptide conformations by quantum chemical calculations

The free energy landscapes of peptide conformations were calibrated by ab initio quantum chemical calculations, after the enhanced conformational diversity search using the multicanonical molecular dynamics simulations. Three different potentials of mean force for an isolated dipeptide were individually obtained by the multicanonical molecular dynamics simulations using the conventional force fields, AMBER parm94, AMBER parm96, and CHARMm22. Each potential of mean force was then calibrated based upon the umbrella sampling algorithm from the adiabatic energy map that was calculated separately by the ab initio molecular orbital method, and all of the calibrated potentials of mean force coincided well. The calibration method was also applied to the simulations of a peptide dimer in explicit water models, and it was shown that the calibrated free energy landscapes did not depend on the force field used in the classical simulations, as far as the conformational space was sampled well. The current calibration method fuses the classical free energy calculation with the quantum chemical calculation, and it should generally make simulations for biomolecular systems much more reliable when combining with enhanced conformational sampling.     

Calculation of free-energy differences and potentials of mean force by a multi-energy gap method

A method is proposed to significantly accelerate the convergence of free-energy calculations. It introduces a bias factor in Monte Carlo simulations or, equivalently, a bias force in molecular dynamics simulations. The bias factor targets the energy gap, i.e., the difference in energy function between two states, and is therefore specifically designed for calculating free-energy differences. The goal is to make the probability density of the energy gap as uniform as possible, thus allowing for its accurate determination. An iterative procedure, based on simulations at higher temperatures, is devised to obtain the bias factor. The same method naturally extends to the calculation of potentials of mean force. The generalized coordinate, for which the potential of mean force is to be calculated, now plays the role of the energy gap. Applications to model systems confirm the expected increase in accuracy of calculated free-energy differences and potentials of mean force.     

Dispersion self-free energies and interaction free energies of finite-sized ions in salt solutions

The role for many-body dipolar (dispersion) potentials in ion-solvent and ion-solvent-interface interactions is explored. Such many-body potentials, accessible in principle from measured dielectric data, are necessary in accounting for Hofmeister specific ion effects. Dispersion self-energy is the quantum electrodynamic analogue of the Born electrostatic self-energy of an ion. We here describe calculations of dispersion self-free energies of four different anions (OH-, Cl-, Br-, and I-) that take finite ion size into account. Three different examples of self-free energy calculations are presented. These are the self-free energy of transfer of an ion to bulk solution, which influences solubility; the dispersion potential acting between one ion and an air-water interface (important for surface tension calculations); and the dispersion potential acting between two ions (relevant to activity coefficient calculations). To illustrate the importance of dispersion self-free energies, we compare the Born and dispersion contributions to the free energy of ion transfer from water to air (oil). We have also calculated the change in interfacial tension with added salt for air (oil)-water interfaces. A new model is used that includes dispersion potentials acting on the ions near the interface, image potentials, and ions of finite size that are allowed to spill over the solution-air interface. It is shown that interfacial free energies require a knowledge of solvent profiles at the interface.     

Embedded density functional theory for covalently bonded and strongly interacting subsystems

Embedded density functional theory (e-DFT) is used to describe the electronic structure of strongly interacting molecular subsystems. We present a general implementation of the Exact Embedding (EE) method [J. Chem. Phys. 133, 084103 (2010)] to calculate the large contributions of the nonadditive kinetic potential (NAKP) in such applications. Potential energy curves are computed for the dissociation of Li(+)-Be, CH(3)-CF(3), and hydrogen-bonded water clusters, and e-DFT results obtained using the EE method are compared with those obtained using approximate kinetic energy functionals. In all cases, the EE method preserves excellent agreement with reference Kohn-Sham calculations, whereas the approximate functionals lead to qualitative failures in the calculated energies and equilibrium structures. We also demonstrate an accurate pairwise approximation to the NAKP that allows for efficient parallelization of the EE method in large systems; benchmark calculations on molecular crystals reveal ideal, size-independent scaling of wall-clock time with increasing system size.     

Thermodynamics of electron flow in the bacterial deca-heme cytochrome MtrF

Electron-transporting multi-heme cytochromes are essential to the metabolism of microbes that inhabit soils and carry out important biogeochemical processes. Recently the first crystal structure of a prototype bacterial deca-heme cytochrome (MtrF) has been resolved and its electrochemistry characterized. However, the molecular details of electron transport along heme chains in the cytochrome are difficult to access via experiment due to the nearly identical chemical nature of the heme cofactors. Here we employ large-scale molecular dynamics simulations to compute the redox potentials of the 10 hemes of MtrF in aqueous solution. We find that as a whole they fall within a range of ~0.3 V, in agreement with experiment. Individual redox potentials give rise to a free energy profile for electron transport that is approximately symmetric with respect to the center of the protein. Our calculations indicate that there is no significant potential bias along the orthogonal octa- and tetra-heme chains, suggesting that under aqueous conditions MtrF is a nearly reversible two-dimensional conductor.     

On the possibility of using model potentials for collision integral calculations of interest for planetary atmospheres

The interaction energy in systems (atom–atom, atom–ion and atom–molecule) involving open-shell species, predicted by a phenomenological method, is used for collision integral calculations. The results are compared with those obtained by different authors by using the complete set of quantum mechanical interaction potentials arizing from the electronic configurations of separate partners. A satisfactory agreement is achieved, implying that the effect of deep potential wells, present in some of the chemical potentials, is cancelled by the effect of strong repulsive potentials.     

Theoretical determination of standard oxidation and reduction potentials of chlorophyll-a in acetonitrile

QM/MM calculations were performed on ethyl chlorophyllide-a and its radical cation and anion, by using the density functional (DF) B3LYP method to determine the molecular characteristics, and a molecular mechanics (MM) method to simulate the solvating medium. The presence of the solvent was accounted for during the optimization of the geometry of the 85-atom chlorophyll-a system by using an ONIOM methodology. A total of 24 solvent molecules were explicitly considered during the optimization process, and these were treated by the universal force field (UFF) method. Initially, the split-valence 3-21G basis set was used for optimizing the geometry of the 85-atom species, neutral, cation and anion. Electronic energies were then determined for the optimized species by making use of the polarized 6-31G(d) basis set. The ionization energy calculated (6.0 eV) is in very good agreement with the observed one (6.1 eV). The MM+ force field was used to investigate the dynamics of the acetonitrile molecules around the neutral species as well as the radical ions of chlorophyll. The required atomic charges on all the atoms were obtained from calculations on all involved molecules at the DFT/6-31G(d) level. Randomly sampled configurations were used to determine the first solvation layer contribution to the free energy of solvation of various species. A truncated 46-atom model of ethyl chlorophyllide-a was used to evaluate the thermal energies of neutral chlorophyll molecule relative to its two radical ions in the gas phase. Born energy, Onsager energy, and the Debye-Huckel energy of the chlorophyll-solvent aggregate were added as perturbative corrections to the free energy of solvation that was initially obtained through molecular dynamics method for the same complex. These calculations yield the oxidation potential as 0.75 +/- 0.32 V and the reduction potential -1.18 +/- 0.31 V at 298.15 K. The calculated values are in good agreement with the experimental midpoint potentials of +0.76 and -1.04 V, respectively.     

Sampling free energy surfaces as slices by combining umbrella sampling and metadynamics

Metadynamics (MTD) is a very powerful technique to sample high‐dimensional free energy landscapes, and due to its self‐guiding property, the method has been successful in studying complex reactions and conformational changes. MTD sampling is based on filling the free energy basins by biasing potentials and thus for cases with flat, broad, and unbound free energy wells, the computational time to sample them becomes very large. To alleviate this problem, we combine the standard Umbrella Sampling (US) technique with MTD to sample orthogonal collective variables (CVs) in a simultaneous way. Within this scheme, we construct the equilibrium distribution of CVs from biased distributions obtained from independent MTD simulations with umbrella potentials. Reweighting is carried out by a procedure that combines US reweighting and Tiwary–Parrinello MTD reweighting within the Weighted Histogram Analysis Method (WHAM). The approach is ideal for a controlled sampling of a CV in a MTD simulation, making it computationally efficient in sampling flat, broad, and unbound free energy surfaces. This technique also allows for a distributed sampling of a high‐dimensional free energy surface, further increasing the computational efficiency in sampling. We demonstrate the application of this technique in sampling high‐dimensional surface for various chemical reactions using ab initio and QM/MM hybrid molecular dynamics simulations. Further, to carry out MTD bias reweighting for computing forward reaction barriers in ab initio or QM/MM simulations, we propose a computationally affordable approach that does not require recrossing trajectories. © 2016 Wiley Periodicals, Inc.     

A comparison of different methods for the calculation of the chemical potentials of polymer systems

In an earlier publication we had presented a Monte Carlo method to calculate the incremental chemical potential between a chain of length v and one of length v+1 at any density. (Ref.9) In this paper we begin by formally proving that the chemical potential of a whole chain of length v can be obtained by sequentially summing the incremental chemical potentials of all chains whose lengths are smaller than v under the same conditions. We then compare the numerical value of chain chemical potentials computed in this fashion to those obtained by inserting whole chains into the simulation cell following the configurational bias procedure proposed by Smit and Frenkel. (Ref.5) It is found, for the two cases considered here, that the results obtained from the two different calculations agree within simulation uncertainty, thus verifying the validity of the proposed arguments.     

A bias-exchange approach to protein folding

By suitably extending a recent approach [Bussi, G.; et al. J. Am. Chem. Soc. 2006, 128, 13435] we introduce a powerful methodology that allows the parallel reconstruction of the free energy of a system in a virtually unlimited number of variables. Multiple metadynamics simulations of the same system at the same temperature are performed, biasing each replica with a time-dependent potential constructed in a different set of collective variables. Exchanges between the bias potentials in the different variables are periodically allowed according to a replica exchange scheme. Due to the efficaciously multidimensional nature of the bias the method allows exploring complex free energy landscapes with high efficiency. The usefulness of the method is demonstrated by performing an atomistic simulation in explicit solvent of the folding of a Triptophane cage miniprotein. It is shown that the folding free energy landscape can be fully characterized starting from an extended conformation with use of only 40 ns of simulation on 8 replicas.     

Efficient free energy calculation of water across lipid membranes

An efficient free energy (FE) calculation of a water molecule to go across lipid membranes is presented. Both overlapping distribution and cavity insertion Widom methods are complementarily used. The former is useful for a dense region where water molecules are found, i.e., from the interfacial to bulk water region, while the latter works well in the low density region, i.e., the hydrocarbon region. Since both methods evaluate the excess chemical potential of water, the obtained FE profile is free from the fitting problem usually arisen when a FE difference method is used. A diphytanyl phosphatidylcholine bilayer is used for our test calculations. An excellent and fast convergence of the chemical potential is obtained when each method is applied for the appropriate region. The estimated FE barrier using the Ewald method for the electrostatic interaction is approximately 7.2 kcal/mol, which is higher than that using the interaction cutoff of 20 A by about 0.9 kcal/mol.     

Calculation of chemical potentials of chain molecules by the incremental gauge cell method

The gauge cell Monte Carlo method is extended to calculations of the incremental chemical potentials and free energies of linear chain molecules. The method was applied to chains of Lennard-Jones beads with stiff harmonic bonds up to 500 monomers in length. We show that the suggested method quantitatively reproduces the modified Widom particle insertion method of Kumar et al. [S. K. Kumar, I. Szleifer, and A. Z. Panagiotopoulos, Phys. Rev. Lett. 66(22), 2935 (1991)], and is by an order of magnitude more efficient for long chains in terms of the computational time required for the same accuracy of chemical potential calculations. The chain increment ansatz, which suggests that the incremental chemical potential is independent of the chain length, was tested at different temperatures. We confirmed that the ansatz holds only for coils above the θ temperature. Special attention is paid to the effects of the magnitude of adsorption potential and temperature on the behavior of single chains in confinements that are comparable in size with the free chain radius of gyration. At sufficiently low temperatures, the dependence of the incremental chemical potential on the chain length in wetting pores is superficially similar to a capillary condensation isotherm, reflecting monolayer formation following by pore volume filling, as the chain length increases. We find that the incremental gauge cell method is an accurate and efficient technique for calculations of the free energies of chain molecules in bulk systems and nanoconfinements alike. The suggested method may find practical applications, such as modeling polymer partitioning on porous substrates and dynamics of chain translocation into nanopores.     

Efficient and precise solvation free energies via alchemical adiabatic molecular dynamics

A new molecular dynamics method for calculating free energies associated with transformations of the thermodynamic state or chemical composition of a system (also known as alchemical transformations) is presented. The new method extends the adiabatic dynamics approach recently introduced by Rosso et al. [J. Chem. Phys. 116, 4389 (2002)] and is based on the use of an additional degree of freedom, lambda, that is used as a switching parameter between the potential energy functions that characterize the two states. In the new method, the coupling parameter lambda is introduced as a fictitious dynamical variable in the Hamiltonian, and a system of switching functions is employed that leads to a barrier in the lambda free energy profile between the relevant thermodynamic end points. The presence of such a barrier, therefore, enhances sampling in the end point (lambda = 0 and lambda = 1) regions which are most important for computing relevant free energy differences. In order to ensure efficient barrier crossing, a high temperature T(lambda) is assigned to lambda and a fictitious mass m(lambda) is introduced as a means of creating an adiabatic separation between lambda and the rest of the system. Under these conditions, it is shown that the lambda free energy profile can be directly computed from the adiabatic probability distribution function of lambda without any postprocessing or unbiasing of the output data. The new method is illustrated on two model problems and in the calculation of the solvation free energy of amino acid side-chain analogs in TIP3P water. Comparisons to previous work using thermodynamic integration and free energy perturbation show that the new lambda adiabatic free energy dynamics method results in very precise free energy calculations using significantly shorter trajectories.     

Calculation of hydrophobic interactions from molecular dynamics,surface areas,and experimental hydrocarbon solubilities

Using experimental solubilities and partial pressures for hydrocarbon solution in water and molecular dynamics calculations of hydrocarbon water interaction energies, hydrocarbon–water cavity potentials are obtained and then plotted vs. accessible surface area. The data used is mainly for aliphatic hydrocarbons, but benzene is included. Molecular dynamics calculations of pairs of hydrocarbon molecules together with the cavity potential curve are then used to obtain hydrophobic interaction free energies between the hydrocarbon pairs. While the cavity potential change is related to a change in surface area for hydrocarbon systems, the hydrocarbon–water interaction energy is not, so that the hydrophobic binding energy is not. The results are in agreement with previous results by a different method (R.B. Hermann, In Seventh Jerusalem Symposium on Quantum Pharmacology, E. Bergman and B. Pullman, Eds., D. Reidel, Dordrecht, 1974, p. 441) in that there is little or no solvent-induced binding free energy between small hydrocarbon molecules in a dilute aqueous solution. It is proposed that the cavity potential vs. accessible surface area curve obtained here can be used together with OPLS parameters to calculate both hydrocarbon–water solvation free energies and hydrophobic interactions. © 1993 John Wiley & Sons, Inc.