Computing Relative Free Energies of Solvation using Single Reference Thermodynamic Integration Augmented with Hamiltonian Replica Exchange

Computing Relative Binding Affinity of Ligands to Receptor: An Effective Hybrid Single-Dual-Topology Free-Energy Perturbation Approach in NAMD

An effective hybrid single-dual-topology protocol is designed for the calculation of relative binding affinities of small ligands to a receptor. The protocol was developed as an extension of the NAMD molecular dynamics program, which exclusively supports a dual-topology framework for relative alchemical free-energy perturbation (FEP) calculations. In this protocol, the alchemical end states are represented as two separate molecules sharing a common substructure identified through maximum structural mapping. Within the substructure, an atom-to-atom correspondence is established, and each pair of corresponding atoms is holonomically constrained to share identical coordinates at all time throughout the simulation. The forces are projected and combined at each step for propagation. Following this formulation, a set of illustrative calculations of reliable experiment/simulation data, including relative solvation free energies of small molecules and relative binding affinities of drug compounds to proteins, are presented. To enhance sampling of the dual-topology region, the FEP calculations were carried out within a replica-exchange MD scheme supported by the multiple-copy algorithm module of NAMD, with periodically attempted swapping of the thermodynamic coupling parameter λ between neighboring states. The results are consistent with experiments and benchmarks reported in the literature, lending support to the validity of the current protocol. In summary, this hybrid single-dual-topology approach combines the conceptual simplicity of the dual-topology paradigm with the advantageous sampling efficiency of the single-topology approach, making it an ideal strategy for high-throughput in silico drug design.

Exploring Gatekeeper Mutations in EGFR through Computer Simulations

The emergence of resistance against drugs that inhibit a particular protein is a major problem in targeted therapy. There is a clear need for rigorous methods to predict the likelihood of specific drug-resistance mutations arising in response to the binding of a drug. In this work we attempt to develop a robust computational protocol for predicting drug resistant mutations at the gatekeeper position (T790) in EGFR. We explore how mutations at this site affects interactions with ATP and three drugs that are currently used in clinics. We found, as expected, that certain mutations are not tolerated structurally, while some other mutations interfere with the natural substrate and hence are unlikely to be selected for. However, we found five possible mutations that are well tolerated structurally and energetically. Two of these mutations were predicted to have increased affinity for the drugs over ATP, as has been reported earlier. By reproducing the trends in the experimental binding affinities of the data, the methods chosen here are able to correctly predict the effects of these mutations on the binding affinities of the drugs. However, the increased affinity does not always translate into increased efficacy, because the efficacy is affected by several other factors such as binding kinetics, competition with ATP, and residence times. The computational methods used in the current study are able to reproduce or predict the effects of mutations on the binding affinities. However, a different set of methods is required to predict the kinetics of drug binding.

Alchemical Free-Energy Calculations by Multiple-Replica λ-Dynamics: The Conveyor Belt Thermodynamic Integration Scheme

A new method is proposed to calculate alchemical free-energy differences based on molecular dynamics (MD) simulations, called the conveyor belt thermodynamic integration (CBTI) scheme. As in thermodynamic integration (TI), K replicas of the system are simulated at different values of the alchemical coupling parameter λ. The number K is taken to be even, and the replicas are equally spaced on a forward-turn-backward-turn path, akin to a conveyor belt (CB) between the two physical end-states; and as in λ-dynamics (λD), the λ-values associated with the individual systems evolve in time along the simulation. However, they do so in a concerted fashion, determined by the evolution of a single dynamical variable Λ of period 2π controlling the advance of the entire CB. Thus, a change of Λ is always associated with K/2 equispaced replicas moving forward and K/2 equispaced replicas moving backward along λ. As a result, the effective free-energy profile of the replica system along Λ is periodic of period 2 πK^-1, and the magnitude of its variations decreases rapidly upon increasing K, at least as K^-1 in the limit of large K. When a sufficient number of replicas is used, these variations become small, which enables a complete and quasi-homogeneous coverage of the λ-range by the replica system, without application of any biasing potential. If desired, a memory-based biasing potential can still be added to further homogenize the sampling, the preoptimization of which is computationally inexpensive. The final free-energy profile along λ is calculated similarly to TI, by binning of the Hamiltonian λ-derivative as a function of λ considering all replicas simultaneously, followed by quadrature integration. The associated quadrature error can be kept very low owing to the continuous and quasi-homogeneous λ-sampling. The CBTI scheme can be viewed as a continuous/deterministic/dynamical analog of the Hamiltonian replica-exchange/permutation (HRE/HRP) schemes or as a correlated multiple-replica analog of the λD or λ-local elevation umbrella sampling (λ-LEUS) schemes. Compared to TI, it shares the advantage of the latter schemes in terms of enhanced orthogonal sampling, i.e. the availability of variable-λ paths to circumvent conformational barriers present at specific λ-values. Compared to HRE/HRP, it permits a deterministic and continuous sampling of the λ-range, is expected to be less sensitive to possible artifacts of the thermo- and barostating schemes, and bypasses the need to carefully preselect a λ-ladder and a swapping-attempt frequency. Compared to λ-LEUS, it eliminates (or drastically reduces) the dead time associated with the preoptimization of a biasing potential. The goal of this article is to provide the mathematical/physical formulation of the proposed CBTI scheme, along with an initial application of the method to the calculation of the hydration free energy of methanol.

How Does Solvation Affect the Binding of Hydrophilic Amino Saccharides to Cucurbit[7]uril with Exceptional Anomeric Selectivity?

Cucurbit[7]uril (CB[7]) is known to bind strongly to hydrophilic amino saccharide guests with exceptional α-anomer selectivities under aqueous conditions. Single-crystal X-ray crystallography and computational methods were used to elucidate the reason behind this interesting phenomenon. The crystal structures of protonated galactosamine (GalN) and glucosamine (GluN) complexes confirm the inclusion of α anomers inside CB[7] and disclose the details of the host-guest binding. Whereas computed gas-phase structures agree with these crystal structures, gas-phase binding free energies show preferences for the β-anomer complexes over their α counterparts, in striking contrast to the experimental results under aqueous conditions. However, when the solvation effect is considered, the binding structures drastically change and the preference for the α anomers is recovered. The α anomers also tend to bind more tightly and leave less space in the CB[7] cavity toward inclusion of only one water molecule, whereas loosely bound β anomers leave more space toward accommodating two water molecules, with markedly different hydrogen-bonding natures. Surprisingly, entropy seems to contribute significantly to both anomeric discrimination and binding. This suggests that of all the driving factors for the strong complexation of the hydrophilic amino saccharide guests, water mediation plays a crucial role in the anomer discrimination.

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.

Intermediate Thermodynamic States Contribute Equally to Free Energy Convergence: A Demonstration with Replica Exchange

We investigate the relationship between the number of intermediate thermodynamic states along a pathway and the precision of free energy estimates. With a sufficient number of states, the asymptotic variance collapses as a function of the total sample size. Our analytical result is corroborated by replica exchange molecular dynamics simulations of model systems in which the neighbor exchange rate exceeds 35%. Precision collapse is also observed in heat capacity estimates based on the multistate Bennett acceptance ratio. In contrast to the relaxation and mean first-passage times, the autocorrelation time of state indices is found to be relevant to free energy convergence.

Prediction of Solvation Free Energies with Thermodynamic Integration Using the General Amber Force Field

Computer-aided drug design (CADD) techniques can be very effective in reducing costs and speeding up drug discovery. The determination of binding and solvation free energies is pivotal for this process and is, therefore, the subject of many studies. In this work, the solvation free energy change (ΔΔGsolv) for a total of 92 transformations in small molecules was predicted using Thermodynamic Integration (TI). It was our aim to compare experimental and calculated solvation free energies for typical and prime additions considered in drug optimizations, analyzing trends, and optimizing a TI protocol. The results showed a good agreement between experimental and predicted values, with an overestimation of the predicted values for CH3, halogens, and NH2, as well as an underestimation for CONH2, but all fall within ±1 kcal/mol. NO2 addition showed a larger and systematic underestimation of the predicted ΔΔGsolv, indicating the need for special attention in these cases. For small molecules, if no experimental data is available, using TI as a theoretical strategy thus appears to be a suitable choice in CADD. It provides a good compromise between time and accuracy.

Calculating binding free energies for protein-carbohydrate complexes

A variety of computational techniques may be applied to compute theoretical binding free energies for protein-carbohydrate complexes. Elucidation of the intermolecular interactions, as well as the thermodynamic effects, that contribute to the relative strength of receptor binding can shed light on biomolecular recognition, and the resulting initiation or inhibition of a biological process. Three types of free energy methods are discussed here, including MM-PB/GBSA, thermodynamic integration, and a non-equilibrium alternative utilizing SMD. Throughout this chapter, the well-known concanavalin A lectin is employed as a model system to demonstrate the application of these methods to the special case of carbohydrate binding.

Practical Aspects of Free-Energy Calculations: A Review

Free-energy calculations in the framework of classical molecular dynamics simulations are nowadays used in a wide range of research areas including solvation thermodynamics, molecular recognition, and protein folding. The basic components of a free-energy calculation, that is, a suitable model Hamiltonian, a sampling protocol, and an estimator for the free energy, are independent of the specific application. However, the attention that one has to pay to these components depends considerably on the specific application. Here, we review six different areas of application and discuss the relative importance of the three main components to provide the reader with an organigram and to make nonexperts aware of the many pitfalls present in free energy calculations.

Efficient Combination of Environment Change and Alchemical Perturbation within the Enveloping Distribution Sampling (EDS) Scheme: Twin-System EDS and Application to the Determination of Octanol-Water Partition Coefficients

The methodology of Enveloping Distribution Sampling (EDS) is extended to probe a single-simulation alternative to the thermodynamic cycle that is standardly used for measuring the effect of a modification of a chemical compound, e.g. from a given species to a chemical derivative for a ligand or solute molecule, on the free-enthalpy change associated with a change in environment, e.g. from the unbound state to the bound state for a protein-ligand system or from one solvent to another one for a solute molecule. This alternative approach relies on the coupled simulation of two systems (computational boxes) 1 and 2, and the method is therefore referred to as twin-system EDS. Systems 1 and 2 account for the two choices of environment. The end states of the alchemical perturbation for the twin-system associate the two alternative forms X and Y of the molecule to systems 1 and 2 or 2 and 1, respectively. In this way, the processes of transforming one molecule into the other are carried out simultaneously in opposite directions in the two environments, leading to a change in free enthalpy that is smaller than for the two individual processes and to an energy-difference distribution that is more symmetric. As an illustration, the method is applied to the calculation of octanol-water partition coefficients for C4 to C8 alkanes, 1-hexanol and 1,2-dimethoxyethane. It is shown in particular that the consideration of the residual hydration of octanol leads to calculated partition coefficients that are in better agreement with reported experimental numbers.

Fungal bis-Naphthopyrones as Inhibitors of Botulinum Neurotoxin Serotype A

An in silico screen of the NIH Molecular Library Small Molecule Repository (MLSMR) of ∼350000 compounds and confirmatory bioassays led to identification of chaetochromin A (1) as an inhibitor of botulinum neurotoxin serotype A (BoNT A). Subsequent acquisition and testing of analogues of 1 uncovered two compounds, talaroderxines A (2) and B (3), with improved activity. These are the first fungal metabolites reported to exhibit BoNT/A inhibitory activity.

Improved Binding Free Energy Predictions from Single-Reference Thermodynamic Integration Augmented with Hamiltonian Replica Exchange

Reliable predictions of relative binding free energies are essential in drug discovery, where chemists modify promising compounds with the aim of increasing binding affinity. Conventional Thermodynamic Integration (TI) approaches can estimate corresponding changes in binding free energies, but suffer from inadequate sampling due to ruggedness of the molecular energy surfaces. Here, we present an improved TI strategy for computing relative binding free energies of congeneric ligands. This strategy employs a specific, unphysical single-reference (SR) state and Hamiltonian replica exchange (HREX) to locally enhance sampling. We then apply this strategy to compute relative binding free energies of twelve ligands in the L99A mutant of T4 Lysozyme. Besides the ligands, our approach enhances hindered rotations of the important V111, as well as V87 and L118 sidechains. Concurrently, we devise practical strategies to monitor and improve HREX-SRTI efficiency. Overall, the HREX-SRTI results agree well (R(2) = 0.76, RMSE = 0.3 kcal/mol) with available experimental data. When optimized for efficiency, the HREX-SRTI precision matches that of experimental measurements.

Small Molecule Solvation Free Energy: Enhanced Conformational Sampling Using Expanded Ensemble Molecular Dynamics Simulation

We present an efficient expanded ensemble molecular dynamics method to calculate the solvation free energy (or residual chemical potential) of small molecules with complex topologies. The methodology is validated by computing the solvation free energy of ibuprofen in water, methanol, and ethanol at 300 K and 1 bar and comparing to reference simulation results using Bennett's acceptance ratio method. Difficulties with ibuprofen using conventional molecular dynamics methods stem from an inadequate sampling of the carboxylic acid functional group, which, for the present study, is subject to free energy barriers of rotation of 14-20 kBT. While several advances have been made to overcome such weaknesses, we demonstrate how this shortcoming is easily overcome by using an expanded ensemble methodology to facilitate conformational sampling. Not only does the method enhance conformational sampling but it also boosts the rate of exploration of the configurational phase space and requires only a single simulation to calculate the solvation free energy. Agreement between the expanded ensemble and the reference calculations is good for all three solvents, with the reported uncertainties of the expanded ensemble being comparable to the uncertainties of the reference calculations, while requiring less simulation time; the reduced simulation time demonstrates the improved performance gained from the expanded ensemble method.

Separation of Betti Reaction Product Enantiomers: Absolute Configuration and Inhibition of Botulinum Neurotoxin A

The racemic product of the Betti reaction of 5-chloro-8-hydroxyquinoline, benzaldehyde and 2-aminopyridine was separated by chiral HPLC to determine which enantiomer inhibited botulinum neurotoxin serotype A. When the enantiomers unexpectedly proved to have comparable activity, the absolute structures of (+)-(R)-1 and (-)-(S)-1 were determined by comparison of calculated and observed circular dichroism spectra. Molecular modeling studies were undertaken in an effort to understand the observed bioactivity and revealed different ensembles of binding modes, with roughly equal binding energies, for the two enantiomers.

Recent theoretical and computational advances for modeling protein-ligand binding affinities

We review recent theoretical and algorithmic advances for the modeling of protein ligand binding free energies. We first describe a statistical mechanics theory of noncovalent association, with particular focus on deriving the fundamental formulas on which computational methods are based. The second part reviews the main computational models and algorithms in current use or development, pointing out the relations with each other and with the theory developed in the first part. Particular emphasis is given to the modeling of conformational reorganization and entropic effect. The methods reviewed are free energy perturbation, double decoupling, the Binding Energy Distribution Analysis Method, the potential of mean force method, mining minima and MM/PBSA. These models have different features and limitations, and their ranges of applicability vary correspondingly. Yet their origins can all be traced back to a single fundamental theory.