Analysis of Coupling Schemes in Free Energy Simulations: A Unified Description of Nonbonded Contributions to Solvation Free Energies

Multifaceted aspects of charge transfer

Charge transfer and charge transport are by far among the most important processes for sustaining life on Earth and for making our modern ways of living possible. Involving multiple electron-transfer steps, photosynthesis and cellular respiration have been principally responsible for managing the energy flow in the biosphere of our planet since the Great Oxygen Event. It is impossible to imagine living organisms without charge transport mediated by ion channels, or electron and proton transfer mediated by redox enzymes. Concurrently, transfer and transport of electrons and holes drive the functionalities of electronic and photonic devices that are intricate for our lives. While fueling advances in engineering, charge-transfer science has established itself as an important independent field, originating from physical chemistry and chemical physics, focusing on paradigms from biology, and gaining momentum from solar-energy research. Here, we review the fundamental concepts of charge transfer, and outline its core role in a broad range of unrelated fields, such as medicine, environmental science, catalysis, electronics and photonics. The ubiquitous nature of dipoles, for example, sets demands on deepening the understanding of how localized electric fields affect charge transfer. Charge-transfer electrets, thus, prove important for advancing the field and for interfacing fundamental science with engineering. Synergy between the vastly different aspects of charge-transfer science sets the stage for the broad global impacts that the advances in this field have.

Dipole-induced effects on charge transfer and charge transport. Why do molecular electrets matter?

Charge transfer (CT) and charge transport (CTr) are at the core of life-sustaining biological processes and of processes that govern the performance of electronic and energy-conversion devices. Electric fields are invaluable for guiding charge movement. Therefore, as electrostatic analogues of magnets, electrets have unexplored potential for generating local electric fields for accelerating desired CT processes and suppressing undesired ones. The notion about dipole-generated local fields affecting CT has evolved since the middle of the 20th century. In the 1990s, the first reports demonstrating the dipole effects on the kinetics of long-range electron transfer appeared. Concurrently, the development of molecular-level designs of electric junctions has led the exploration of dipole effects on CTr. Biomimetic molecular electrets such as polypeptide helices are often the dipole sources in CT systems. Conversely, surface-charge electrets and self-assembled monolayers of small polar conjugates are the preferred sources for modifying interfacial electric fields for controlling CTr. The multifaceted complexity of such effects on CT and CTr testifies for the challenges and the wealth of this field that still remains largely unexplored. This review outlines the basic concepts about dipole effects on CT and CTr, discusses their evolution, and provides accounts for their future developments and impacts.

Multistate λ-local-elevation umbrella-sampling (MS-λ-LEUS): method and application to the complexation of cations by crown ethers

An extension of the λ-local-elevation umbrella-sampling (λ-LEUS) scheme [ Bieler et al. J. Chem. Theory Comput. 2014 , 10 , 3006 ] is proposed to handle the multistate (MS) situation, i.e. the calculation of the relative free energies of multiple physical states based on a single simulation. The key element of the MS-λ-LEUS approach is to use a single coupling variable Λ controlling successive pairwise mutations between the states of interest in a cyclic fashion. The Λ variable is propagated dynamically as an extended-system variable, using a coordinate transformation with plateaus and a memory-based biasing potential as in λ-LEUS. Compared to other available MS schemes (one-step perturbation, enveloping distribution sampling and conventional λ-dynamics) the proposed method presents a number of important advantages, namely: (i) the physical states are visited explicitly and over finite time periods; (ii) the extent of unphysical space required to ensure transitions is kept minimal and, in particular, one-dimensional; (iii) the setup protocol solely requires the topologies of the physical states; and (iv) the method only requires limited modifications in a simulation code capable of handling two-state mutations. As an initial application, the absolute binding free energies of five alkali cations to three crown ethers in three different solvents are calculated. The results are found to reproduce qualitatively the main experimental trends and, in particular, the experimental selectivity of 18C6 for K(+) in water and methanol, which is interpreted in terms of opposing trends along the cation series between the solvation free energy of the cation and the direct electrostatic interactions within the complex.

Problems of robustness in Poisson-Boltzmann binding free energies

Although models based on the Poisson–Boltzmann (PB) equation have been fairly successful at predicting some experimental quantities, such as solvation free energies (ΔG), these models have not been consistently successful at predicting binding free energies (ΔΔG). Here we found that ranking a set of protein–protein complexes by the electrostatic component (ΔΔGel) of ΔΔG was more difficult than ranking the same molecules by the electrostatic component (ΔGel) of ΔG. This finding was unexpected because ΔΔGel can be calculated by combining estimates of ΔGel for the complex and its components with estimates of the ΔΔGel in vacuum. One might therefore expect that if a theory gave reliable estimates of ΔGel, then its estimates of ΔΔGel would be reliable. However, ΔΔGel for these complexes were orders of magnitude smaller than ΔGel, so although estimates of ΔGel obtained with different force fields and surface definitions were highly correlated, similar estimates of ΔΔGel were often not correlated.

Importance of van der Waals Interactions in QM/MM Simulations

The importance of accurately treating van der Waals interactions between the quantum mechanical (QM) and molecular mechanical (MM) atoms in hybrid QM/MM simulations has been investigated systematically. First, a set of van der Waals (vdW) parameters was optimized for an approximate density functional method, the self-consistent charge-tight binding density functional (SCC-DFTB) approach, based on small hydrogen-bonding clusters. The sensitivity of condensed phase observables to the SCC-DFTB vdW parameters was then quantitatively investigated by SCC-DFTB/MM simulations of several model systems using the optimized set and two sets of extreme vdW parameters selected from the CHARMM22 forcefield. The model systems include a model FAD molecule in solution and a solvated enediolate, and the properties studied include the radial distribution functions of water molecules around the solute (model FAD and enediolate), the reduction potential of the model FAD and the potential of mean force for an intramolecular proton transfer in the enediolate. Although there are noticeable differences between parameter sets for gas-phase clusters and solvent structures around the solute, thermodynamic quantities in the condensed phase (e.g., reduction potential and potential of mean force) were found to be less sensitive to the numerical values of vdW parameters. The differences between SCC-DFTB/MM results with the three vdW parameter sets for SCC-DFTB atoms were explained in terms of the effects of the parameter set on solvation. The current study has made it clear that efforts in improving the reliability of QM/MM methods for energetical properties in the condensed phase should focus on components other than van der Waals interactions between QM and MM atoms.

Towards a universal method for calculating hydration free energies: a 3D reference interaction site model with partial molar volume correction

We report a simple universal method to systematically improve the accuracy of hydration free energies calculated using an integral equation theory of molecular liquids, the 3D reference interaction site model. A strong linear correlation is observed between the difference of the experimental and (uncorrected) calculated hydration free energies and the calculated partial molar volume for a data set of 185 neutral organic molecules from different chemical classes. By using the partial molar volume as a linear empirical correction to the calculated hydration free energy, we obtain predictions of hydration free energies in excellent agreement with experiment (R = 0.94, σ = 0.99 kcal mol (- 1) for a test set of 120 organic molecules).

Dielectric relaxation of cytochrome c oxidase: Comparison of the microscopic and continuum models

We have studied a charge-insertion process that models the deprotonation of a histidine side chain in the active site of cytochrome c oxidase (CcO) using both the continuum electrostatic calculations and the microscopic simulations. The group of interest is a ligand to Cu(B) center of CcO, which has been previously suggested to play the role of the proton pumping element in the enzyme; the group is located near a large internal water cavity in the protein. Using the nonpolarizable Amber-99 force field in molecular dynamics (MD) simulations, we have calculated the nuclear part of the reaction-field energy of charging of the His group and combined it with the electronic part, which we estimated in terms of the electronic continuum (EC) model, to obtain the total reaction-field energy of charging. The total free energy obtained in this MDEC approach was then compared with that calculated using pure continuum electrostatic model with variable dielectric parameters. The dielectric constant for the "dry" protein and that of the internal water cavity of CcO were determined as those parameters that provide best agreement between the continuum and microscopic MDEC model. The nuclear (MD) polarization alone (without electronic part) of a dry protein was found to correspond to an unphysically low dielectric constant of only about 1.3, whereas the inclusion of electronic polarizability increases the protein dielectric constant to 2.6-2.8. A detailed analysis is presented as to how the protein structure should be selected for the continuum calculations, as well as which probe and atomic radii should be used for cavity definition. The dielectric constant of the internal water cavity was found to be 80 or even higher using "standard" parameters of water probe radius, 1.4 A, and protein atomic radii from the MD force field for cavity description; such high values are ascribed to the fact that the standard procedure produces unphysically small cavities. Using x-ray data for internal water in CcO, we have explored optimization of the parameters and the algorithm of cavity description. For Amber radii, the optimal probe size was found to be 1.25 A; the dielectric of water cavity in this case is in the range of 10-16. The most satisfactory cavity description, however, was achieved with ProtOr atomic radii, while keeping the probe radius to be standard 1.4 A. In this case, the value of cavity dielectric constant was found to be in the range of 3-6. The obtained results are discussed in the context of recent calculations and experimental measurements of dielectric properties of proteins.

Species with negative electron affinity and standard DFT methods. Finding the valence anions

Recently, we have shown that traditional bound-electron DFT models are reliable enough to reproduce negative electron affinities (EA) within a few meV, as long as the valence anion state is found, but they seem to fail in predicting the lowest EA when the ground anion state obtained is non-valence, which holds the extra electron in a diffuse orbital around the molecule; here we propose an alternative approach for finding the valence anion state, based on the stabilization exerted by a polar solvent; the methodology yields correct EA values (i.e. beyond the Koopman's theorem approximation) by gradually decreasing the dielectric constant of the medium.

Short-range and long-range solvent effects on charge-transfer-to-solvent transitions of I- and K+I- contact ion pair dissolved in supercritical ammonia

Vertical excitation and electron detachment energies associated with the optical absorption of iodide ions dissolved in supercritical ammonia at 420 K have been calculated in two limiting scenarios: as a solvated free I- ion and forming a K+I- contact ion pair (CIP). The evolution of the transition energies as a result of the gradual building up of the solvation structure was studied for each absorbing species as the solvent's density increased, i.e., changing the NH3 supercritical thermodynamic state. In both cases, if the solvent density is sufficiently high, photon absorption produces a spatially extended electron charge beyond the volume occupied by the solvated solute core; this excited state resembles a typical charge-transfer-to-solvent (CTTS) state. A combination of classical molecular dynamics simulations followed by quantum mechanical calculations for the ground, first-excited, and electron-detached electronic states have been carried out for the system consisting of one donor species (free I- ion or K+I- CIP) surrounded by ammonia molecules. Vertical excitation and electron detachment energies were obtained by averaging 100 randomly chosen microconfigurations along the molecular dynamics trajectory computed for each thermodynamic condition (fluid density). Short- and long-range contributions of the solvent-donor interaction upon the CTTS states of I- and K+I- were identified by performing additional electronic structure calculations where only the solvent interaction due to the first neighbor molecules was taken into account. These computations, together with previous experimental evidence that we collected for the system, have been used to analyze the solvent effects on the CTTS transition. In this paper we have established the following: (i) the CTTS electron of free I- ion or K+I- CIP presents similar features, and it gradually localizes in close proximity of the iodine parent atom when the ammonia density is increased; (ii) for the free I- ion, the short-range solvent interaction contributes to the stabilization of the ground state more than it does for the CTTS excited state, which is evidenced experimentally as a blueshift in the maximum absorption of the CTTS transition when the density is increased; (iii) this effect is less noticeable for the K+I- ion pair, because in this case a tight solvation structure, formed by four NH3 molecules wedged between the ions, appears at very low density and is very little affected by changes in the density; (iv) the long-range contribution to the solvent stabilization can be neglected for the K+I- CIP, since the main features of its electronic transition can be explained on the basis of the vicinity of the cation; (v) however, the long-range solvent field contribution is essential for the free I- ion to become an efficient CTTS donor upon photoexcitation, and this establishes a difference in the CTTS behavior of I- in bulk and in clusters.

Absolute free energy calculations by thermodynamic integration in four spatial dimensions

An optimized technique for calculating the excess chemical potential of small molecules in dense liquids and the binding affinity of molecular ligands to biomolecules is reported. In this method, a molecular species is coupled to the system of interest via a nonphysical fourth spatial dimension w through which insertion or extraction can be carried out [R. Pomes, E. Eisenmesser, C. B. Post et al., J. Chem. Phys. 111, 3387 (1999)]. Molecular simulations are used to compute the potential of mean force (PMF) acting on the solute molecule in the fourth dimension. The excess chemical potential of that molecule is obtained as the difference in the PMF between fully coupled and fully decoupled systems. The simplicity, efficiency, and generality of the method are demonstrated for the calculation of the hydration free energies of water and methanol as well as sodium, cesium, and chloride ions. A significant advantage over other methods is that the 4D-PMF approach provides a single effective and general route for decoupling all nonbonded interactions (i.e., both Lennard-Jones and Coulombic) at once for both neutral and charged solutes. Direct calculation of the mean force from thermodynamic integration is shown to be more computationally efficient than calculating the PMF from umbrella sampling. Statistical error analysis suggests a simple strategy for optimizing sampling. The detailed analysis of systematic errors arising from the truncation of Coulombic interactions in a solvent droplet of finite size leads to straightforward corrections to ionic hydration free energies.

Phase-space overlap measures. I. Fail-safe bias detection in free energies calculated by molecular simulation

We consider ways to quantify the overlap of the parts of phase space important to two systems, labeled A and B. Of interest is how much of the A-important phase space lies in that important to B, and how much of B lies in A. Two measures are proposed. The first considers four total-energy distributions, formed from all combinations made by tabulating either the A-system or the B-system energy when sampling either the A or B system. Measures for A in B and B in A are given by two overlap integrals defined on pairs of these distributions. The second measure is based on information theory, and defines two relative entropies which are conveniently expressed in terms of the dissipated work for free-energy perturbation (FEP) calculations in the A-->B and B-->A directions, respectively. Phase-space overlap is an important consideration in the performance of free-energy calculations. To demonstrate this connection, we examine bias in FEP calculations applied to a system of independent particles in a harmonic potential. Systems are selected to represent a range of overlap situations, including extreme subset, subset, partial overlap, and nonoverlap. The magnitude and symmetry of the bias (A-->B vs B-->A) are shown to correlate well with the overlap, and consequently with the overlap measures. The relative entropies are used to scale the amount of sampling to obtain a universal bias curve. This result leads to develop a simple heuristic that can be applied to determine whether a work-based free-energy measurement is free of bias. The heuristic is based in part on the measured free energy, but we argue that it is fail-safe inasmuch as any bias in the measurement will not promote a false indication of accuracy.

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.

Solvation free energies of amino acid side chain analogs for common molecular mechanics water models

Quantitative free energy computation involves both using a model that is sufficiently faithful to the experimental system under study (accuracy) and establishing statistically meaningful measures of the uncertainties resulting from finite sampling (precision). In order to examine the accuracy of a range of common water models used for protein simulation for their solute/solvent properties, we calculate the free energy of hydration of 15 amino acid side chain analogs derived from the OPLS-AA parameter set with the TIP3P, TIP4P, SPC, SPC/E, TIP3P-MOD, and TIP4P-Ew water models. We achieve a high degree of statistical precision in our simulations, obtaining uncertainties for the free energy of hydration of 0.02-0.06 kcal/mol, equivalent to that obtained in experimental hydration free energy measurements of the same molecules. We find that TIP3P-MOD, a model designed to give improved free energy of hydration for methane, gives uniformly the closest match to experiment; we also find that the ability to accurately model pure water properties does not necessarily predict ability to predict solute/solvent behavior. We also evaluate the free energies of a number of novel modifications of TIP3P designed as a proof of concept that it is possible to obtain much better solute/solvent free energetic behavior without substantially negatively affecting pure water properties. We decrease the average error to zero while reducing the root mean square error below that of any of the published water models, with measured liquid water properties remaining almost constant with respect to our perturbations. This demonstrates there is still both room for improvement within current fixed-charge biomolecular force fields and significant parameter flexibility to make these improvements. Recent research in computational efficiency of free energy methods allows us to perform simulations on a local cluster that previously required large scale distributed computing, performing four times as much computational work in approximately a tenth of the computer time as a similar study a year ago.

Hybrid integral equation/simulation model for enhancing free energy computations

Integral equation theory is used for extrapolating free energy data from molecular simulations of a reference state with respect to a modification of the interaction potential. The methodology is applied to the correction of artefacts arising from potential shifting and truncation. Corrective contributions for the hydration free energy with respect to the full potential are analysed for the case that both the solute-solvent as well as the solvent-solvent potentials are truncated and modified by a shifted-force term, reaching beyond the range of the dielectric continuum approximation and simple long-range correction expressions. The model systems argon in water and pure water are used as examples for apolar and polar solutes, revealing significant correction contributions even for the short-ranged dispersive interactions and the magnitude of solute-solvent and solvent-solvent components. In comparison with simulation-based extrapolation techniques the integral equation method is shown to be capable of quantitatively predicting truncation artefacts at negligible computational overhead.

Ensemble variance in free energy calculations by thermodynamic integration: theory, optimal "Alchemical" path, and practical solutions

Thermodynamic integration is a widely used method to calculate and analyze the effect of a chemical modification on the free energy of a chemical or biochemical process, for example, the impact of an amino acid substitution on protein association. Numerical fluctuations can introduce large uncertainties, limiting the domain of application of the method. The parametric energy function describing the chemical modification in the thermodynamic integration, the "Alchemical path," determines the amplitudes of the fluctuations. In the present work, I propose a measure of the fluctuations in the thermodynamic integration and an approach to search for a parametric energy path minimizing that measure. The optimal path derived with this approach is very close to the theoretical minimum of the measure, but produces nonergodic sampling. Nevertheless, this path is used to guide the design of a practical and efficient path producing correct sampling. The convergence with this practical path is evaluated on test cases, and compares favorably with that of other methods such as power or polynomial path, soft-core van der Waals, and some other approaches presented in the literature.

Solvation effects are responsible for the reduced inhibitor affinity of some HIV-1 PR mutants

The formulation of HIV-1 PR inhibitors as anti-viral drugs has been hindered by the appearance of protease strains that present drug resistance to these compounds. The mechanism by which the HIV-1 PR mutants lower their affinity for the inhibitor is not yet fully understood. We have applied a modified Poisson-Boltzmann method to the evaluation of the molecular interactions that contribute to the lowering of the inhibitor affinity to some polar mutants at position 82. These strains present drug resistance behavior and hence are ideally suited for these studies. Our results indicate that the reduction in binding affinity is due to the solvation effects that penalize the binding to the more polar mutants. The inhibitor binding ranking of the different mutants can be explained from the analysis of the different components of our free energy scoring function.