Efficient free energy calculations on small molecule host-guest systems - a combined linear interaction energy/one-step perturbation approach

Guidelines for Free-Energy Calculations Involving Charge Changes

The Coulomb interactions in molecular simulations are inherently approximated due to the finite size of the molecular box sizes amenable to current-day compute power. Several methods exist for treating long-range electrostatic interactions, yet these approaches are subject to various finite-size-related artifacts. Lattice-sum methods are frequently used to approximate long-range interactions; however, these approaches also suffer from artifacts which become particularly pronounced for free-energy calculations that involve charge changes. The artifacts, however, also affect the sampling when plain simulations are performed, leading to a biased ensemble. Here, we investigate two previously described model systems to determine if artifacts continue to play a role when overall neutral boxes are considered, in the context of both free-energy calculations and sampling. We find that ensuring that no net-charge changes take place, while maintaining a neutral simulation box, may be sufficient provided that the simulation boxes are large enough. Addition of salt to the solution (when appropriate) can further alleviate the remaining artifacts in the sampling or the calculated free-energy differences. We provide practical guidelines to avoid finite-size artifacts.

Charge-Changing Perturbations and Path Sampling via Classical Molecular Dynamic Simulations of Simple Guest-Host Systems

Currently, two different methods dominate the field of biomolecular free-energy calculations for the prediction of binding affinities. Pathway methods are frequently used for large ligands that bind on the surface of a host, such as protein-protein complexes. Alchemical methods, on the other hand, are preferably applied for small ligands that bind to deeply buried binding sites. The latter methods are also widely known to be heavily artifacted by the representation of electrostatic energies in periodic simulation boxes, in particular, when net-charge changes are involved. Different methods have been described to deal with these artifacts, including postsimulation correction schemes and instantaneous correction schemes (e.g., co-alchemical perturbation of ions). Here, we use very simple test systems to show that instantaneous correction schemes with no change in the system net charge lower the artifacts but do not eliminate them. Furthermore, we show that free energies from pathway methods suffer from the same artifacts.

Experimental characterization of the association of β-cyclodextrin and eight novel cyclodextrin derivatives with two guest compounds

We investigate the binding of native β-cyclodextrin (β-CD) and eight novel β-CD derivatives with two different guest compounds, using isothermal calorimetry and 2D NOESY NMR. In all cases, the stoichiometry is 1:1 and binding is exothermic. Overall, modifications at the 3' position of β-CD, which is at the secondary face, weaken binding by several kJ/mol relative to native β-CD, while modifications at the 6' position (primary face) maintain or somewhat reduce the binding affinity. The variations in binding enthalpy are larger than the variations in binding free energy, so entropy-enthalpy compensation is observed. Characterization of the bound conformations with NOESY NMR shows that the polar groups of the guests may be situated at either face, depending on the host molecule, and, in some cases, both orientations are populated. The present results were used in the SAMPL7 blinded prediction challenge whose results are detailed in the same special issue of JCAMD.

Toward Automated Free Energy Calculation with Accelerated Enveloping Distribution Sampling (A-EDS)

Free-energy perturbation (FEP) methods are commonly used in drug design to calculate relative binding free energies of different ligands to a common host protein. Alchemical ligand transformations are usually performed in multiple steps which need to be chosen carefully to ensure sufficient phase-space overlap between neighboring states. With one-step or single-step FEP techniques, a single reference state is designed that samples phase-space not only representative of a full transformation but also ideally resembles multiple ligand end states and hence allows for efficient multistate perturbations. Enveloping distribution sampling (EDS) is one example for such a method in which the reference state is created by a mathematical combination of the different ligand end states based on solid statistical mechanics. We have recently proposed a novel approach to EDS which enables efficient barrier crossing between the different end states, termed accelerated EDS (A-EDS). In this work, we further simplify the parametrization of the A-EDS reference state and demonstrate the automated calculation of multiple free-energy differences between different ligands from a single simulation in three different well-described drug design model systems.

Accelerating equilibrium isotope effect calculations. II. Stochastic implementation of direct estimators

Path integral calculations of equilibrium isotope effects and isotopic fractionation are expensive due to the presence of path integral discretization errors, statistical errors, and thermodynamic integration errors. Whereas the discretization errors can be reduced by high-order factorization of the path integral and statistical errors by using centroid virial estimators, two recent papers proposed alternative ways to completely remove the thermodynamic integration errors: Cheng and Ceriotti [J. Chem. Phys. 141, 244112 (2015)] employed a variant of free-energy perturbation called "direct estimators," while Karandashev and Vaníček [J. Chem. Phys. 143, 194104 (2017)] combined the thermodynamic integration with a stochastic change of mass and piecewise-linear umbrella biasing potential. Here, we combine the former approach with the stochastic change in mass in order to decrease its statistical errors when applied to larger isotope effects and perform a thorough comparison of different methods by computing isotope effects first on a harmonic model and then on methane and methanium, where we evaluate all isotope effects of the form CH_4-xD_x/CH₄ and CH_5-xD_x ⁺/CH₅ ⁺, respectively. We discuss the reasons for a surprising behavior of the original method of direct estimators, which performed well for a much larger range of isotope effects than what had been expected previously, as well as some implications of our work for the more general problem of free energy difference calculations.

Saturation Mutagenesis by Efficient Free-Energy Calculation

Single-point mutations in proteins can greatly influence protein stability, binding affinity, protein function or its expression per se. Here, we present accurate and efficient predictions of the free energy of mutation of amino acids. We divided the complete mutational free energy into an uncharging step, which we approximate by a third-power fitting (TPF) approach, and an annihilation step, which we approximate using the one-step perturbation (OSP) method. As a diverse set of test systems, we computed the solvation free energy of all amino acid side chain analogues and obtained an excellent agreement with thermodynamic integration (TI) data. Moreover, we calculated mutational free energies in model tripeptides and established an efficient protocol involving a single reference state. Again, the approximate methods agreed excellently with the TI references, with a root-mean-square error of only 3.6 kJ/mol over 17 mutations. Our combined TPF+OSP approach does show not only a very good agreement but also a 2-fold higher efficiency than full blown TI calculations.

Alchemical screening of ionic crystals

We introduce alchemical perturbations as a rapid and accurate tool to estimate fundamental structural and energetic properties in pure and mixed ionic crystals.

Molecular recognition of metabotropic glutamate receptor type 1 (mGluR1): synergistic understanding with free energy perturbation and linear response modeling

Metabotropic glutamate receptors (mGluRs) constitute an important family of the G-protein coupled receptors. Due to their widespread distribution in the central nervous system (CNS), these receptors are attractive candidates for understanding the molecular basis of various cognitive processes as well as for designing inhibitors for relevant psychiatric and neurological disorders. Despite many studies on drugs targeting the mGluR receptors to date, the molecular level details on the ligand binding dynamics still remain unclear. In this study, we performed in silico experiments for mGluR1 with 29 different ligands including known synthetic agonists and antagonists as well as natural amino acids. The ligand-receptor binding affinities were estimated by the use of atomistic simulations combined with the mathematically rigorous, Free Energy Perturbation (FEP) method, which successfully recognized the native agonist l-glutamate among the highly favorable binders, and also accurately distinguished antagonists from agonists. Comparative contact analysis also revealed the binding mode differences between natural and non-natural amino acid-based ligands. Several factors potentially affecting the ligand binding affinity and specificity were identified including net charges, dipole moments, and the presence of aromatic rings. On the basis of these findings, linear response models (LRMs) were built for different sets of ligands that showed high correlations (R(2) > 0.95) to the corresponding FEP binding affinities. These results identify some key factors that determine ligand-mGluR1 binding and could be used for future inhibitor designs and support a role for in silico modeling for understanding receptor ligand interactions.

Net charge changes in the calculation of relative ligand-binding free energies via classical atomistic molecular dynamics simulation

The calculation of binding free energies of charged species to a target molecule is a frequently encountered problem in molecular dynamics studies of (bio-)chemical thermodynamics. Many important endogenous receptor-binding molecules, enzyme substrates, or drug molecules have a nonzero net charge. Absolute binding free energies, as well as binding free energies relative to another molecule with a different net charge will be affected by artifacts due to the used effective electrostatic interaction function and associated parameters (e.g., size of the computational box). In the present study, charging contributions to binding free energies of small oligoatomic ions to a series of model host cavities functionalized with different chemical groups are calculated with classical atomistic molecular dynamics simulation. Electrostatic interactions are treated using a lattice-summation scheme or a cutoff-truncation scheme with Barker-Watts reaction-field correction, and the simulations are conducted in boxes of different edge lengths. It is illustrated that the charging free energies of the guest molecules in water and in the host strongly depend on the applied methodology and that neglect of correction terms for the artifacts introduced by the finite size of the simulated system and the use of an effective electrostatic interaction function considerably impairs the thermodynamic interpretation of guest-host interactions. Application of correction terms for the various artifacts yields consistent results for the charging contribution to binding free energies and is thus a prerequisite for the valid interpretation or prediction of experimental data via molecular dynamics simulation. Analysis and correction of electrostatic artifacts according to the scheme proposed in the present study should therefore be considered an integral part of careful free-energy calculation studies if changes in the net charge are involved.

Large scale affinity calculations of cyclodextrin host-guest complexes: Understanding the role of reorganization in the molecular recognition process

Host-guest inclusion complexes are useful models for understanding the structural and energetic aspects of molecular recognition. Due to their small size relative to much larger protein-ligand complexes, converged results can be obtained rapidly for these systems thus offering the opportunity to more reliably study fundamental aspects of the thermodynamics of binding. In this work, we have performed a large scale binding affinity survey of 57 β-cyclodextrin (CD) host guest systems using the binding energy distribution analysis method (BEDAM) with implicit solvation (OPLS-AA/AGBNP2). Converged estimates of the standard binding free energies are obtained for these systems by employing techniques such as parallel Hamitionian replica exchange molecular dynamics, conformational reservoirs and multistate free energy estimators. Good agreement with experimental measurements is obtained in terms of both numerical accuracy and affinity rankings. Overall, average effective binding energies reproduce affinity rank ordering better than the calculated binding affinities, even though calculated binding free energies, which account for effects such as conformational strain and entropy loss upon binding, provide lower root mean square errors when compared to measurements. Interestingly, we find that binding free energies are superior rank order predictors for a large subset containing the most flexible guests. The results indicate that, while challenging, accurate modeling of reorganization effects can lead to ligand design models of superior predictive power for rank ordering relative to models based only on ligand-receptor interaction energies.

Catalytic effects of mutations of distant protein residues in human DNA polymerase β: theory and experiment

We carried out free-energy calculations and transient kinetic experiments for the insertion of the right (dC) and wrong (dA) nucleotides by wild-type (WT) and six mutant variants of human DNA polymerase β (Pol β). Since the mutated residues in the point mutants, I174S, I260Q, M282L, H285D, E288K, and K289M, were not located in the Pol β catalytic site, we assumed that the WT and its point mutants share the same dianionic phosphorane transition-state structure of the triphosphate moiety of deoxyribonucleotide 5'-triphosphate (dNTP) substrate. On the basis of this assumption, we have formulated a thermodynamic cycle for calculating relative dNTP insertion efficiencies, Ω = (k(pol)/K(D))(mut)/(k(pol)/K(D))(WT) using free-energy perturbation (FEP) and linear interaction energy (LIE) methods. Kinetic studies on five of the mutants have been published previously using different experimental conditions, e.g., primer-template sequences. We have performed a presteady kinetic analysis for the six mutants for comparison with wild-type Pol β using the same conditions, including the same primer/template DNA sequence proximal to the dNTP insertion site used for X-ray crystallographic studies. This consistent set of kinetic and structural data allowed us to eliminate the DNA sequence from the list of factors that can adversely affect calculated Ω values. The calculations using the FEP free energies scaled by 0.5 yielded 0.9 and 1.1 standard deviations from the experimental log Ω values for the insertion of the right and wrong dNTP, respectively. We examined a hybrid FEP/LIE method in which the FEP van der Waals term for the interaction of the mutated amino acid residue with its surrounding environment was replaced by the corresponding van der Waals term calculated using the LIE method, resulting in improved 0.4 and 1.0 standard deviations from the experimental log Ω values. These scaled FEP and FEP/LIE methods were also used to predict log Ω for R283A and R283L Pol β mutants.

Efficient and Accurate Free Energy Calculations on Trypsin Inhibitors

Several combinations of free energy calculation methods have been applied to determine the relative free energies of binding between eight para-substituted benzamidines in complex with the serine protease trypsin. With the aim to improve efficiency and maintain accuracy, the linear response approximation (LRA), linear interaction energy (LIE) and third power fitting (TPF) are combined with the one-step perturbation (OSP) to determine the polar and apolar contributions to the free energy, respectively. It is shown that the combination TPF/OSP gives the most accurate results and is 4.5 times more efficient than the rigorous thermodynamic integration (TI). By projecting the electrostatic preorganization energy from the OSP simulations, an increase in efficiency of a factor 7.5 can even be achieved. Loss of accuracy with respect to the TI data is limited to 3.9 and 5.6 kJ/mol, respectively, making it an attractive approach for lead optimization programs in drug research.

Free energy calculations from one-step perturbations

The one-step perturbation approach offers an efficient means to estimate free energy differences. It may be applied to estimate solvation free energies, conformational preferences or relative free energies of binding of series of compounds to a common receptor. Applicability of the method depends on the possibility to define a proper reference state which may in itself be an unphysical molecule. Here, we describe practical considerations and explicit guidelines to define a proper reference state, and to efficiently calculate relative free energies. The strengths and limitations of the method are highlighted and special considerations are noted. The method may be applied using many different simulation programs. Here, analyses are exemplified at the hand of the GROMOS simulation package.

Path Integral Computation of Quantum Free Energy Differences Due to Alchemical Transformations Involving Mass and Potential

Thermodynamic integration, perturbation theory, and λ-dynamics methods were applied to path integral molecular dynamics calculations to investigate free energy differences due to "alchemical" transformations. Several estimators were formulated to compute free energy differences in solvable model systems undergoing changes in mass and/or potential. Linear and nonlinear alchemical interpolations were used for the thermodynamic integration. We find improved convergence for the virial estimators, as well as for the thermodynamic integration over nonlinear interpolation paths. Numerical results for the perturbative treatment of changes in mass and electric field strength in model systems are presented. We used thermodynamic integration in ab initio path integral molecular dynamics to compute the quantum free energy difference of the isotope transformation in the Zundel cation. The performance of different free energy methods is discussed.

Free energy calculations of protein-ligand interactions

In the calculation of free energies of binding for protein-ligand complexes, we distinguish endpoint methods, methods involving alchemical modifications and methods that physically displace the ligand from the protein. Most methodological advances seem to come from a clever combination of multiple existing methods to enhance the sampling or to utilize specific advantages of various approaches. The coupling parameters common in thermodynamic integration and in Hamiltonian replica exchange are for instance combined to yield replica exchange thermodynamic integration. As new methods mostly aim to improve efficiency or to attain more complete sampling, there are good prospects to understand and tackle the sampling problem better and to shift the focus towards the scoring problem in the context of more robust and accurate force fields.