Quantitative comparison and optimization of methods for evaluating the chemical potential by molecular simulation

Alchemical Enhanced Sampling with Optimized Phase Space Overlap

An alchemical enhanced sampling (ACES) method has recently been introduced to facilitate importance sampling in free energy simulations. The method achieves enhanced sampling from Hamiltonian replica exchange within a dual topology framework while utilizing new smoothstep softcore potentials. A common sampling problem encountered in lead optimization is the functionalization of aromatic rings that exhibit distinct conformational preferences when interacting with the protein. It is difficult to converge the distribution of ring conformations due to the long time scale of ring flipping events; however, the ACES method addresses this issue by modeling the syn and anti ring conformations within a dual topology. ACES thereby samples the conformer distributions by alchemically tunneling between states, as opposed to traversing a physical pathway with a high rotational barrier. We demonstrate the use of ACES to overcome conformational sampling issues involving ring flipping in ML300-derived noncovalent inhibitors of SARS-CoV-2 Main Protease (Mpro). The demonstrations explore how the use of replica exchange and the choice of softcore selection affects the convergence of the ring conformation distributions. Furthermore, we examine how the accuracy of the calculated free energies is affected by the degree of phase space overlap (PSO) between adjacent states (i.e., between neighboring λ-windows) and the Hamiltonian replica exchange acceptance ratios. Both of these factors are sensitive to the spacing between the intermediate states. We introduce a new method for choosing a schedule of λ values. The method analyzes short "burn-in" simulations to construct a 2D map of the nonlocal PSO. The schedule is obtained by optimizing an alchemical pathway on the 2D map that equalizes the PSO between the λ intervals. The optimized phase space overlap λ-spacing method (Opt-PSO) leads to more numerous end-to-end single passes and round trips due to the correlation between PSO and Hamiltonian replica exchange acceptance ratios. The improved exchange statistics enhance the efficiency of ACES method. The method has been implemented into the FE-ToolKit software package, which is freely available.

Modern Alchemical Free Energy Methods for Drug Discovery Explained

This Perspective provides a contextual explanation of the current state-of-the-art alchemical free energy methods and their role in drug discovery as well as highlights select emerging technologies. The narrative attempts to answer basic questions about what goes on "under the hood" in free energy simulations and provide general guidelines for how to run simulations and analyze the results. It is the hope that this work will provide a valuable introduction to students and scientists in the field.

Measuring many-body distribution functions in fluids using test-particle insertion

We derive a hierarchy of equations, which allow a general n-body distribution function to be measured by test-particle insertion of between 1 and n particles. We apply it to measure the pair and three-body distribution functions in a simple fluid using snapshots from Monte Carlo simulations in the grand canonical ensemble. The resulting distribution functions obtained from insertion methods are compared with the conventional distance-histogram method: the insertion approach is shown to overcome the drawbacks of the histogram method, offering enhanced structural resolution and a more straightforward normalization. At high particle densities, the insertion method starts breaking down, which can be delayed by utilizing the underlying hierarchical structure of the insertion method. Our method will be especially useful in characterizing the structure of inhomogeneous fluids and investigating closure approximations in liquid state theory.

Accessible Molecular System Creator: Building Molecular Configurations Based on the Inaccessible Molecular Volume and Accessible Molecular Surface via Static Monte Carlo Sampling

Monte Carlo (MC) stochastic sampling is a powerful tool in classical molecular simulations that directly connects the observable macroscopic properties of matter and the underlying atomistic interactions. This connection operates within the framework of the statistical mechanics proposed by Gibbs. Most MC simulations are "dynamic," creating statistical ensembles of microstates via a Markovian chain, where each microstate in the ensemble depends only on its previous microstate. Herein, we re-examine an alternative form of MC that generates ensemble members through a "static" approach, building molecular systems stepwise. The basic theory for such an approach traces back to Rosenbluth and Rosenbluth, who proposed "static" stepwise sampling of a polymeric chain. It is almost as old as the Metropolis importance sampling approach used in dynamic MC, although the latter has been considerably more popular than the former. Herein, we address the main obstacle in static MC that has hindered the widespread adoption of Rosenbluth-based approaches in atomistic simulations. The obstacle lies in mapping the molecular accessible volume for adding a molecule in a Rosenbluth-like static sampling of atomistic configurations. We demonstrate a breakthrough by leveraging the ability to analytically map the inaccessible molecular volume and the accessible molecular surface owing to interatomically excluded volume interactions. This advance substantially enhances the ability to create molecular samples using a Rosenbluth-like static building process. The proposed approach can be used as a tool for creating initial configurations in MC or molecular dynamics simulations─a field where Rosenbluth-like static building has been applied. Additionally, this approach can be used as the first step in a perturbation scheme that accurately estimates free energy differences by estimating the chemical work related to molecule addition, removal, or reinsertion within the context of free energy perturbation schemes employed in molecular simulations.

The Contribution of the Ion-Ion and Ion-Solvent Interactions in a Molecular Thermodynamic Treatment of Electrolyte Solutions

Developing molecular equations of state to treat electrolyte solutions is challenging due to the long-range nature of the Coulombic interactions. Seminal approaches commonly used are the mean spherical approximation (MSA) and the Debye-Hückel (DH) theory to account for ion-ion interactions and, often, the Born theory of solvation for ion-solvent interactions. We investigate the accuracy of the MSA and DH approaches using each to calculate the contribution of the ion-ion interactions to the chemical potential of NaCl in water, comparing these with newly computer-generated simulation data; the ion-ion contribution is isolated by selecting an appropriate primitive model with a Lennard-Jones force field to describe the solvent. A study of mixtures with different concentrations and ionic strengths reveals that the calculations from both MSA and DH theories are of similar accuracy, with the MSA approach resulting in marginally better agreement with the simulation data. We also demonstrate that the Born theory provides a good qualitative description of the contribution of the ion-solvent interactions; we employ an explicitly polar water model in these simulations. Quantitative agreement up to moderate salt concentrations and across the relevant range of temperature is achieved by adjusting the Born radius using simulation data of the free energy of solvation. We compute the radial and orientational distribution functions of the systems, thereby providing further insight on the differences observed between the theory and simulation. We thus provide rigorous benchmarks for use of the MSA, DH, and Born theories as perturbation approaches, which will be of value for improving existing models of electrolyte solutions, especially in the context of equations of state.

Predicting octanol/water partition coefficients using molecular simulation for the SAMPL7 challenge: comparing the use of neat and water saturated 1-octanol

Blind predictions of octanol/water partition coefficients at 298.15 K for 22 drug-like compounds were made for the SAMPL7 challenge. The octanol/water partition coefficients were predicted using solvation free energies computed using molecular dynamics simulations, wherein we considered the use of both pure and water-saturated 1-octanol to model the octanol-rich phase. Water and 1-octanol were modeled using TIP4P and TrAPPE-UA, respectively, which have been shown to well reproduce the experimental mutual solubility, and the solutes were modeled using GAFF. After the close of the SAMPL7 challenge, we additionally made predictions using TIP4P/2005 water. We found that the predictions were sensitive to the choice of water force field. However, the effect of water in the octanol-rich phase was found to be even more significant and non-negligible. The effect of inclusion of water was additionally sensitive to the chemical structure of the solute.

Development of a Robust Indirect Approach for MM → QM Free Energy Calculations That Combines Force-Matched Reference Potential and Bennett's Acceptance Ratio Methods

We use the PBE0/6-31G* density functional method to perform ab initio quantum mechanical/molecular mechanical (QM/MM) molecular dynamics (MD) simulations under periodic boundary conditions with rigorous electrostatics using the ambient potential composite Ewald method in order to test the convergence of MM → QM/MM free energy corrections for the prediction of 17 small-molecule solvation free energies and eight ligand binding free energies to T4 lysozyme. The "indirect" thermodynamic cycle for calculating free energies is used to explore whether a series of reference potentials improve the statistical quality of the predictions. Specifically, we construct a series of reference potentials that optimize a molecular mechanical (MM) force field's parameters to reproduce the ab initio QM/MM forces from a QM/MM simulation. The optimizations form a systematic progression of successively expanded parameters that include bond, angle, dihedral, and charge parameters. For each reference potential, we calculate benchmark quality reference values for the MM → QM/MM correction by performing the mixed MM and QM/MM Hamiltonians at 11 intermediate states, each for 200 ps. We then compare forward and reverse application of Zwanzig's relation, thermodynamic integration (TI), and Bennett's acceptance ratio (BAR) methods as a function of reference potential, simulation time, and the number of simulated intermediate states. We find that Zwanzig's equation is inadequate unless a large number of intermediate states are explicitly simulated. The TI and BAR mean signed errors are very small even when only the end-state simulations are considered, and the standard deviations of the TI and BAR errors are decreased by choosing a reference potential that optimizes the bond and angle parameters. We find a robust approach for the data sets of fairly rigid molecules considered here is to use bond + angle reference potential together with the end-state-only BAR analysis. This requires QM/MM simulations to be performed in order to generate reference data to parametrize the bond + angle reference potential, and then this same simulation serves a dual purpose as the full QM/MM end state. The convergence of the results with respect to time suggests that computational resources may be used more efficiently by running multiple simulations for no more than 50 ps, rather than running one long simulation.

Model-Free Measurement of the Pair Potential in Colloidal Fluids Using Optical Microscopy

We report a straightforward, model-free approach for measuring pair potentials from particle-coordinate data, based on enforcing consistency between the pair distribution function measured separately by the distance-histogram and test-particle insertion routes. We demonstrate the method's accuracy and versatility in simulations of simple fluids, before applying it to an experimental system composed of superparamagnetic colloidal particles. The method will enable experimental investigations into many-body interactions and allow for effective coarse graining of interactions from simulations.

Association of Nucleobases in Hydrated Ionic Liquid from Biased Molecular Dynamics Simulations

We employed metadynamics-based classical molecular dynamics simulations to methylated adenine-thymine (mA-mT) and guanine-cytosine (mG-mC) base pairs to see favorable conformations in various concentrations of hydrated 1-ethyl, 3-methyl imidazolium acetate. We investigated various stacked and hydrogen-bonded conformations of association of base pairs through appropriately chosen collective variables. Stacked conformations more favored in water for both base pairs, whereas Watson-Crick (WC) hydrogen-bonding conformations are favored in pure and hydrated ionic liquids (ILs) except for 0.75 mol fraction IL. We observe that EMIm cations surround the base pairs in WC conformations creating a kind of hydrophobic cavity and protect the hydrogen bonds between base pairs. However, the five-membered heteroaromatic rings of cations stack with the nucleobases in the cation-base-cation (π-π-π) model, which resembles the base-base-base stacking in a DNA duplex. Interestingly, from additional simulations of 0.5 mol fraction hydrated choline dihydrogen phosphate IL, we observe that the stacked conformations become more favored than the WC conformation due to the absence of π-bonds in cations. The calculated values of relative solubility of base pairs in pure and hydrated ionic liquids compared to those in pure water correlate well with the free energy values of WC and stacked conformations.

Chemical potential calculations in non-homogeneous liquids

The numerical computation of chemical potential in dense non-homogeneous fluids is a key problem in the study of confined fluid thermodynamics. To this day, several methods have been proposed; however, there is still need for a robust technique, capable of obtaining accurate estimates at large average densities. A widely established technique is the Widom insertion method, which computes the chemical potential by sampling the energy of insertion of a test particle. Non-homogeneity is accounted for by assigning a density dependent weight to the insertion points. However, in dense systems, the poor sampling of the insertion energy is a source of inefficiency, hampering a reliable convergence. We have recently presented a new technique for the chemical potential calculation in homogeneous fluids. This novel method enhances the sampling of the insertion energy via well-tempered metadynamics, reaching accurate estimates at very large densities. In this paper, we extend the technique to the case of non-homogeneous fluids. The method is successfully tested on a confined Lennard-Jones fluid. In particular, we show that, thanks to the improved sampling, our technique does not suffer from a systematic error that affects the classic Widom method for non-homogeneous fluids, providing a precise and accurate result.

Computational methodology for solubility prediction: Application to sparingly soluble organic/inorganic materials

The solubility of a crystalline material can be estimated from the absolute free energy of the solid and the excess solvation free energy. In the earlier work, we presented a general-purpose molecular-dynamics-based methodology enabling solubility predictions of crystalline compounds, yielding accurate estimates of the aqueous solubilities of naphthalene at various pressures and temperatures. In the present work, we investigate a number of prototypical complex materials, including phenanthrene, calcite, and aragonite over a range of temperatures and pressures. Our results provide stronger evidence for the power of the methodology for universal solubility predictions.

Computational methodology for solubility prediction: Application to the sparingly soluble solutes

The solubility of a crystalline substance in the solution can be estimated from its absolute solid free energy and excess solvation free energy. Here, we present a numerical method, which enables convenient solubility estimation of general molecular crystals at arbitrary thermodynamic conditions where solid and solution can coexist. The methodology is based on standard alchemical free energy methods, such as thermodynamic integration and free energy perturbation, and consists of two parts: (1) systematic extension of the Einstein crystal method to calculate the absolute solid free energies of molecular crystals at arbitrary temperatures and pressures and (2) a flexible cavity method that can yield accurate estimates of the excess solvation free energies. As an illustration, via classical Molecular Dynamic simulations, we show that our approach can predict the solubility of OPLS-AA-based (Optimized Potentials for Liquid Simulations All Atomic) naphthalene in SPC (Simple Point Charge) water in good agreement with experimental data at various temperatures and pressures. Because the procedure is simple and general and only makes use of readily available open-source software, the methodology should provide a powerful tool for universal solubility prediction.

Understanding the large solubility of lidocaine in 1-n-butyl-3-methylimidazolium based ionic liquids using molecular simulation

Room temperature ionic liquids have been proposed as replacement solvents in a wide range of industrial separation processes. Here, we focus on the use of ionic liquids as solvents for the pharmaceutical compound lidocaine. We show that the solubility of lidocaine in seven common 1-n-butyl-3-methylimidazolium based ionic liquids is greatly enhanced relative to water. The predicted solubility is greatest in [BMIM](+)[CH3CO2](-), which we find results from favorable hydrogen bonding between the lidocaine amine hydrogen and the [CH3CO2](-) oxygen, favorable electrostatic interactions between the lidocaine amide oxygen with the [BMIM](+) aromatic ring hydrogens, while lidocaine does not interfere with the association of [BMIM](+) with [CH3CO2](-). Additionally, by removing functional groups from the lidocaine scaffold while maintaining the important amide group, we found that as the van der Waals volume increases, solubility in [BMIM](+)[CH3CO2](-) relative to water increases.

Understanding the Solubility of Acetaminophen in 1-n-Alkyl-3-methylimidazolium-Based Ionic Liquids Using Molecular Simulation

During the manufacturing of pharmaceutical compounds, solvent mixtures are commonly used, where the addition of a cosolvent allows for the tuning of the intermolecular interactions present in the system. Here we demonstrate how a similar effect can be accomplished using a room temperature ionic liquid. The pharmaceutical compound acetaminophen is studied in 21 common ionic liquids composed of a 1-n-alkyl-3-methylimidazolium cation with 1 of 7 anions. Using the acetate anion, we predict a large enhancement in solubility of acetaminophen relative to water. We show how this is caused by a synergistic effect of favorable interactions between the ionic liquid and the phenyl, hydroxyl and amide groups of acetaminophen, demonstrating how the ionic liquid cation and anion may be chosen to preferentially solvate different functional groups of complex pharmaceutical compounds. Additionally, while the use of charge scaling in ionic liquid force fields has previously been found to have a minute effect on ionic liquid structural properties, we find it appreciably affects the computed solvation free energy of acetaminophen, which in turn affects the predicted solubility.

Predicting the excess solubility of acetanilide, acetaminophen, phenacetin, benzocaine, and caffeine in binary water/ethanol mixtures via molecular simulation

We present a general framework to predict the excess solubility of small molecular solids (such as pharmaceutical solids) in binary solvents via molecular simulation free energy calculations at infinite dilution with conventional molecular models. The present study used molecular dynamics with the General AMBER Force Field to predict the excess solubility of acetanilide, acetaminophen, phenacetin, benzocaine, and caffeine in binary water/ethanol solvents. The simulations are able to predict the existence of solubility enhancement and the results are in good agreement with available experimental data. The accuracy of the predictions in addition to the generality of the method suggests that molecular simulations may be a valuable design tool for solvent selection in drug development processes.

Fifth to eleventh virial coefficients of hard spheres

Virial coefficients B(n) of three-dimensional hard spheres are reported for n=5 to 11, with precision exceeding that presently available in the literature. Calculations are performed using the recursive method due to Wheatley, and a binning approach is proposed to allow more flexibility in where computational effort is directed in the calculations. We highlight the difficulty as a general measure that quantifies performance of an algorithm that computes a stochastic average and show how it can be used as the basis for optimizing such calculations.

Free energy calculations, enhanced by a Gaussian ansatz, for the "chemical work" distribution

The evaluation of the free energy is essential in molecular simulation because it is intimately related with the existence of multiphase equilibrium. Recently, it was demonstrated that it is possible to evaluate the Helmholtz free energy using a single statistical ensemble along an entire isotherm by accounting for the "chemical work" of transforming each molecule, from an interacting one, to an ideal gas. In this work, we show that it is possible to perform such a free energy perturbation over a liquid vapor phase transition. Furthermore, we investigate the link between a general free energy perturbation scheme and the novel nonequilibrium theories of Crook's and Jarzinsky. We find that for finite systems away from the thermodynamic limit the second law of thermodynamics will always be an inequality for isothermal free energy perturbations, resulting always to a dissipated work that may tend to zero only in the thermodynamic limit. The work, the heat, and the entropy produced during a thermodynamic free energy perturbation can be viewed in the context of the Crooks and Jarzinsky formalism, revealing that for a given value of the ensemble average of the "irreversible" work, the minimum entropy production corresponded to a Gaussian distribution for the histogram of the work. We propose the evaluation of the free energy difference in any free energy perturbation based scheme on the average irreversible "chemical work" minus the dissipated work that can be calculated from the variance of the distribution of the logarithm of the work histogram, within the Gaussian approximation. As a consequence, using the Gaussian ansatz for the distribution of the "chemical work," accurate estimates for the chemical potential and the free energy of the system can be performed using much shorter simulations and avoiding the necessity of sampling the computational costly tails of the "chemical work." For a more general free energy perturbation scheme that the Gaussian ansatz may not be valid, the free energy calculation can be expressed in terms of the moment generating function of the "chemical work" distribution.

A Wang-Landau study of a lattice model for lipid bilayer self-assembly

The Wang-Landau (WL) Monte Carlo method has been applied to simulate the self-assembly of a lipid bilayer on a 3D lattice. The WL method differs from conventional Monte Carlo methods in that a complete density of states is obtained directly for the system, from which properties, such as the free energy, can be derived. Furthermore, from a single WL simulation, continuous curves of the average energy and heat capacity can be determined, which provide a complete picture of the phase behavior. The lipid model studied consists of 3 or 5 coarse-grained segments on lattices of varying sizes, with the empty lattice sites representing water. A bilayer structure is found to form at low temperatures, with phase transitions to clusters as temperature increases. For 3-segment chains, varying lattice sizes were studied, with the observation that the ratio of chain number to lattice area (i.e., area per lipid) affects the phase transition temperature. At small ratios, only one phase transition occurs between the bilayer and cluster phases, while at high lipid ratios the phase transition occurs in a two-step process with a stable intermediate phase. This second phase transition was not observed in conventional Metropolis Monte Carlo simulations on the same model, demonstrating the advantage of being able to perform a complete scan of the whole temperature range with the WL method. For longer 5-segment chains similar phase transitions are also observed with changes in temperature. In the WL method, due to the extensive nature of the energy, the number of energy bins required to represent the density of states increases as the system size increases and so limits its practical application to larger systems. To improve this, an extension of the WL algorithm, the statistical-temperature Monte Carlo method that allows simulations with larger energy bin sizes, has recently been proposed and is implemented in this work for the 3-segment lattice model. The results obtained are in good agreement with the original WL method and appear to be independent of the energy bin size used.

Direct first-principles chemical potential calculations of liquids

We propose a scheme that drastically improves the efficiency of Widom's particle insertion method by efficiently sampling cavities while calculating the integrals providing the chemical potentials of a physical system. This idea enables us to calculate chemical potentials of liquids directly from first-principles without the help of any reference system, which is necessary in the commonly used thermodynamic integration method. As an example, we apply our scheme, combined with the density functional formalism, to the calculation of the chemical potential of liquid copper. The calculated chemical potential is further used to locate the melting temperature. The calculated results closely agree with experiments.

Microscopic origins of the anomalous melting behavior of sodium under high pressure

X-ray diffraction experiments have shown that sodium exhibits a dramatic pressure-induced drop in melting temperature, which extends from 1000 K at ~30 GPa to as low as room temperature at ~120 GPa. Despite significant theoretical effort to understand the anomalous melting, its origins are still debated. In this work, we reconstruct the sodium phase diagram by using an ab initio quality neural-network potential. Furthermore, we demonstrate that the reentrant behavior results from the screening of interionic interactions by conduction electrons, which at high pressure induces a softening in the short-range repulsion.