Theoretical study of the amino acid interactions in the binding region of the trypsin–pancreatic trypsin inhibitor complex

The meaningful interactions in the contact region of the trypsin–pancreatic trypsin inhibitor complex have been evaluated using free energy simulation methods and appropriate thermodynamic cycles. Consequently, mutations on a few selected residues were performed to destroy the specificity of these interactions preserving the three-dimensional structure of the original species; therefore, the original and mutated residues involved had to be similar from a topological point of view but with opposite chemical properties. The thermodynamic perturbation, conventional thermodynamic integration, thermodynamic integration with end-point perturbational correction and thermodynamic integration conjugated with the extrapolation method of Brooks formalisms have been used in these calculations. The results obtained, with these four alternative approaches, are in reasonable, mutual agreement and reveal that the strength of a specific interaction increases with the charge separation between the amino acid residues involved, and with the hydrophilicity of the surrounding environment.     

Calculation of the relative free energy of oxidation of azurin at pH 5 and pH 9

Free energy calculations are described for the small copper‐containing redox protein Azurin from Pseudomonas aeruginosa. A thermodynamic cycle connecting the reduced and oxidized states at pH 5 and pH 9 is considered, allowing for an assessment of convergence in terms of hysteresis and cycle closure. Previously published thermodynamic integration (TI) data is compared to Hamiltonian replica exchange TI (RE‐TI) simulations using different simulation setups. The effects of varying simulation length, initial structure, position restraints on particular atoms, and the strength of temperature coupling are studied. Although the overall simulation times are too short to observe an experimentally described peptide plane rotation, it is found that RE‐TI simulations do stimulate the distribution of conformational changes over the relevant values of the TI coupling parameter λ. This results in significantly improved values for hysteresis and cycle closure when compared to regular TI. © 2012 Wiley Periodicals, Inc.     

Rapid approximate calculation of water binding free energies in the whole hydration domain of (bio)macromolecules

The evaluation of water binding free energies around solute molecules is important for the thermodynamic characterization of hydration or association processes. Here, a rapid approximate method to estimate water binding free energies around (bio)macromolecules from a single molecular dynamics simulation is presented. The basic idea is that endpoint free‐energy calculation methods are applied and the endpoint quantities are monitored on a three‐dimensional grid around the solute. Thus, a gridded map of water binding free energies around the solute is obtained, that is, from a single short simulation, a map of favorable and unfavorable water binding sites can be constructed. Among the employed free‐energy calculation methods, approaches involving endpoint information pertaining to actual thermodynamic integration calculations or endpoint information as exploited in the linear interaction energy method were examined. The accuracy of the approximate approaches was evaluated on the hydration of a cage‐like molecule representing either a nonpolar, polar, or charged water binding site and on α‐ and β‐cyclodextrin molecules. Among the tested approaches, the linear interaction energy method is considered the most viable approach. Applying the linear interaction energy method on the grid around the solute, a semi‐quantitative thermodynamic characterization of hydration around the whole solute is obtained. Disadvantages are the approximate nature of the method and a limited flexibility of the solute. © 2016 Wiley Periodicals, Inc.     

Comparison of Accuracy and Convergence Rate between Equilibrium and Nonequilibrium Alchemical Transformations for Calculation of Relative Binding Free Energy

Estimation of protein-ligand binding affinity within chemical accuracy is one of the grand challenges in structure-based rational drug design. With the efforts over three decades, free energy methods based on equilibrium molecular dynamics (MD) simulations have become mature and are nowadays routinely applied in the community of computational chemistry. On the contrary, nonequilibrium MD simulation methods have attracted less attention, despite their underlying rigor in mathematics and potential advantage in efficiency. In this work, the equilibrium and nonequilibrium simulation methods are compared in terms of accuracy and convergence rate in the calculations of relative binding free energies. The proteins studied are T4-lysozyme mutant L99A and COX-2. For each protein, two ligands are studied. The results show that the nonequilibrium simulation method can be competitively as accurate as the equilibrium method, and the former is more efficient than the latter by considering the convergence rate with respect to the cost of wall clock time. In addition, Bennett acceptance ratio, which is a bidirectional post-processing method, converges faster than the unidirectional Jarzynski equality for the nonequilibrium simulations.     

Molecular simulation of the hydration Gibbs energy of barbiturates

In the present work, molecular dynamics calculations of the Gibbs energy of hydration of 10 different substituted barbiturates in SPC/E water were performed using thermodynamic integration. Given that experimental determination of the Gibbs hydration energy for this class of compounds is currently unfeasible, computer simulations appear as the only alternative for the estimation of this important quantity. Several simulation parameters are discussed and optimized based on calculations for barbituric acid. It is concluded that accounting for electrostatic interactions with the Reaction-Field method can be up to two times faster than with Particle-Mesh-Ewald method, without loss of accuracy. Different number of solvent molecules and simulation lengths were also tested. Lennard-Jones and electrostatic contributions were scaled down to zero in an independent way. It is shown that the electrostatic contribution is dominant (representing approximately 90% of the total Gibbs energy of hydration) and that barbiturate intra-molecular interactions cannot be neglected. The importance of the electrostatic contribution is attributed to the formation of hydrogen bonds between the barbiturates and water, which play an important role in the solvation process. The influence of the different substituents and their contribution to the Gibbs energy of hydration was assessed. Finally, the Lennard-Jones contributions and the total hydration Gibbs energy can both be correlated against molecular weight or partition coefficient data for mono- and di-substituted barbiturates.     

Precision of free energies calculated by molecular dynamics simulations of peptides in solution

Errors in free energies for molecular replacement and for conformation change of a small model peptide have been determined empirically by repeated simulations from different starting points. All calculations have been done using thermodynamic integration, in which the system's potential energy is coupled to a parameter λ, that is increased or decreased by a small amount at each step of the simulation. The effects of several factors that may alter the precision are evaluated. These factors include: the length of the simulation, the dependence of the potential energy on λ, the use of conformational restraints, and their magnitude and form. The methods used for restraint and conformational forcing are described in detail. The free energy change, calculated as the mean from several successive simulations with alternately increasing and decreasing λ, is found to be independent of the length of the simulations. As expected, longer simulations produce more precise results. The variation of the calculated free energies is found to consist of two parts, a random error and a systematic hysteresis, i.e., a dependence on the direction in which λ changes. The hysteresis varies as the inverse of the length of the simulation and the random error as the inverse square root The advantage of the use of a different (nonlinear) dependence of the attractive and repulsive parts of the nonbonded potential energy on the coupling parameter when “creating” particles in solution is found to be very large. This nonlinear coupling was found to be superior to the use of linear coupling and a nonlinear change of the coupling parameter with the simulation time. The hysteresis in conformational free energy calculations is found to increase markedly if too weak a forcing restraint is chosen. It is shown how to deconvolute the contribution of a torsional restraint from the dependence of the free energy on a torsion angle.     

Free energy perturbation and MM/PBSA studies on inclusion complexes of some structurally related compounds with β-cyclodextrin

Molecular dynamics (MD) simulations have been conducted to explore time-resolved guest–host interactions involving inclusion complex formation between β-cyclodextrin and organic molecules bearing two peripheral benzene rings in aqueous solution. Moreover, free energy perturbation (FEP) and thermodynamic integration (TI) methods at different simulation times have been employed to estimate the relative free energy of complexation. Also, the less computer-time demanding molecular mechanics/Poisson–Boltzmann surface area (MM/PBSA) method was used to estimate the free energy of complexation based on only 1-ns MD simulation. Results showed that both FEP and TI methods were able to reasonably reproduce the experimental thermodynamic quantities. However, long simulation times (e.g. 15 ns) were needed for benzoin mutating to benzanilide (BAN), while moderately shorter times were sufficient for BAN mutating to phenyl benzoate and for benzilic acid mutating to diphenylacetic acid. The results have been discussed in the light of the differences in the chemical structural and conformational features of the guest molecules. In general, it was apparent that the TI method requires less time for convergence of results than the FEP method. However, the less expensive MM/PBSA method proved capable of producing results that are in agreement with those of the more expensive TI and FEP methods.     

Theoretical calculations: Can Gibbs free energy for intermolecular complexes be predicted efficiently and accurately?

The theoretical study has been performed to refine the procedure for calculations of Gibbs free energy with a relative accuracy of less than 1 kcal/mol. Three benchmark intermolecular complexes are examined via several quantum-chemical methods, including the second-order Moller-Plesset perturbation (MP2), coupled cluster (CCSD(T)), and density functional (BLYP, B3LYP) theories augmented by Dunnings correlation-consistent basis sets. The effects of electron correlation, basis set size, and anharmonicity are systematically analyzed, and the results are compared with available experimental data. The results of the calculations suggest that experimental accuracy can be reached only by extrapolation of MP2 and CCSD(T) total energies to the complete basis set. The contribution of anharmonicity to the zero point energy and TDeltaSint values is fairly small. The new, economic way to reach chemical accuracy in the calculations of the thermodynamic parameters of intermolecular interactions is proposed. In addition, interaction energy (De) and free energy change (DeltaA) for considered species have been evaluated by Carr-Parrinello molecular dynamics (CPMD) simulations and static BLYP-plane wave calculations. The free energy change along the reaction paths were determined by the thermodynamic integration/"Blue Moon Ensemble" technique. Comparison between obtained values, and available experimental and conventional ab initio results has been made. We found that the accuracy of CPMD simulations is affected by several factors, including statistical uncertainty and convergence of constrained forces (TD integration), and the nature of DFT (density functional theory) functional. The results show that CPMD technique is capable of reproducing interaction and free energy with an accuracy of 1 kcal/mol and 2-3 kcal/mol respectively.     

FEW: A workflow tool for free energy calculations of ligand binding

In the later stages of drug design projects, accurately predicting relative binding affinities of chemically similar compounds to a biomolecular target is of utmost importance for making decisions based on the ranking of such compounds. So far, the extensive application of binding free energy approaches has been hampered by the complex and time‐consuming setup of such calculations. We introduce the free energy workflow (FEW) tool that facilitates setup and execution of binding free energy calculations with the AMBER suite for multiple ligands. FEW allows performing free energy calculations according to the implicit solvent molecular mechanics (MM‐PB(GB)SA), the linear interaction energy, and the thermodynamic integration approaches. We describe the tool's architecture and functionality and demonstrate in a show case study on Factor Xa inhibitors that the time needed for the preparation and analysis of free energy calculations is considerably reduced with FEW compared to a fully manual procedure. © 2013 Wiley Periodicals, Inc.     

Conformational free energy surface of the linear DPDPE peptide: cost of pre-organization for disulfide bond formation

We have calculated the free energy differences between four conformers of the linear form of the opioid pentapeptide DPDPE in aqueous solution. The conformers are Cyc, representing the structure adopted by the linear peptide prior to disulfide bond formation, β _C and β _E , two slightly different β-turns previously identified in unconstrained molecular dynamics simulations, and Ext, an extended structure. Our simulations indicate that β _E is the most stable of the studied conformers of linear DPDPE in aqueous solution, with β _C , Cyc and Ext having free energies higher by 2.3, 6.3, and 28.2 kcal/mol, respectively. The free energy differences of 4.0 kcal/mol between β _C and Cyc, and 6.3 kcal/mol between β _E and Cyc, reflect the cost of pre-organizing the linear peptide into a conformation conducive for disulfide bond formation. Such a conformational change is a pre-requisite for the chemical reaction of S–S bond formation to proceed. The relatively low population of the cyclic-like structure agrees qualitatively with observed lower potency and different receptor specificity of the linear form relative to the cyclic peptide, and with previous unconstrained simulation results. Free energy component analysis indicates that the moderate stability difference of 4.0–6.3 kcal/mol between the β-turns and the cyclic-like structure results from cancellation of two large opposing effects. In accord with intuition, the relaxed β-turns have conformational strain 43–45 kcal/mol lower than the Cyc structure. However, the cyclic-like conformer interacts with water about 39 kcal/mol strongly than the open β-turns. Our simulations are the first application of the recently developed multidimensional conformational free energy thermodynamic integration (CFTI) protocol to a solvated system, with fast convergence of the free energy obtained by fixing all flexible dihedrals. Additionally, the availability of the CFTI multidimensional free energy gradient leads to a new decomposition scheme, giving the contribution of each fixed dihedral to the overall free energy change and providing additional insight into the microscopic mechanisms of the studied processes. Received: 20 April 1998 / Accepted: 9 September 1998 / Published online: 7 December 1998     

Automated Search of Minimum Free‐Energy Path by Umbrella Integration

The free‐energy landscape is an important factor for understanding the conformational equilibria of chemical reactions, and many techniques have been developed to calculate the potential of the mean force. Unfortunately, these methods require a previous knowledge of the system for calculations because the results depend on the reaction coordinates. In this study, we combine the scaled hypersphere search method with the umbrella integration method to obtain the transition states on free‐energy landscapes and minimum‐free‐energy paths (MFEPs). With this approach, the MFEP connections between known and unknown equilibrium points are constructed without the prior knowledge of the free‐energy landscape. The problem of reaction coordinates can be solved by using a multidimensional, fully automated interrogation of MFEPs for acquiring the potential of mean force. The efficiency of the proposed method is demonstrated by applying it to alanine dipeptide and alanine tripeptide. © 2018 Wiley Periodicals, Inc.     

Partition coefficients of organic molecules in squalane and water/ethanol mixtures by molecular dynamics simulations

Accurate partition coefficient data of migrants between a polymer and a solvent are of paramount importance for estimating the migration of the migrant over time, including the concentration of the migrant at infinite time in the two solvents. In this article it is shown how this partition coefficient can be estimated for both a small hydrophilic and a hydrophobic organic molecules between squalane (used here to mimic low density poly ethylene) and water/ethanol solutes using thermodynamic integration to calculate the free energy of solvation. Molecular dynamics simulations are performed, using the GROMACS software, by slowly decoupling of firstly the electrostatic and then the Lennard-Jones interactions between molecules in the simulation box. These calculations depend very much on the choice of force field. Two force fields have been tested in this work, the TraPPE-UA (united-atom) and the OPLS-AA (all-atom). The computational cheaper TraPPE-UA force field showed to be more accurate over the whole range of systems compared to the OPLS-AA force field. Moreover, some of the calculations were done with five different water models to investigate the influence of the specific water model on the calculations. It was found that the combination of the TraPPE-UA force field and the TIP4p water model gave the best results. Based on the methodology proposed in this article, it is possible to obtain good partition coefficients only knowing the chemical structure of the molecules in the system.     

On‐the‐fly reconstruction of free‐energy profiles using logarithmic mean‐force dynamics

Mean‐force dynamics (MFD), which is a fictitious dynamics for a set of collective variables on a potential of mean‐force, is a powerful algorithm to efficiently explore free‐energy landscapes. Recently, we have introduced logarithmic MFD (LogMFD) (Morishita et al., Phys. Rev. E 2012, 85, 066702) which overcomes difficulties encounterd in free‐energy calculations using standard approaches such as thermodynamic integration. Here, we present a guide to implementing LogMFD calculations paying attention to the practical issues in choosing the parameters in LogMFD. A primary focus is given to the effect of the parameters on the accuracy of the reconstructed free‐energy profiles. A recipe for reducing the errors due to energy dissipation is presented. We also demonstrate that multidimensional free‐energy landscapes can be reconstructed on‐the‐fly using LogMFD, which cannot be accomplished using any other free‐energy calculation techniques. © 2013 Wiley Periodicals, Inc.     

Free energy derivatives: A new method for probing the convergence problem in free energy calculations

The convergence behavior of free energy calculations has been explored in more detail than in any previously reported work, using a model system of two neon atoms in a periodic box of water. We find that for thermodynamic integration-type free energy calculations as much as a nanosecond or more molecular dynamics sampling is required to obtain a fully converged value for a single λ point of the integrand. The concept of “free energy derivatives” with respect to the individual parameters of the force field is introduced. This formalism allows the total convergence of the simulation to be deconvoluted into components. A determination of the statistical “sampling ratio” from these simulations indicates that for window-type free energy calculations carried out in a periodic waterbox of typical size at least 0.6 ps of sampling should be performed at each window (0.7 ps if constraint contributions to the free energy are being determined). General methods to estimate and reduce the error in thermodynamic integration and free energy perturbation calculations are discussed. We show that the difficulty in applying such methods is determining a reliable estimate of the correlation length from a short series of data. © 1994 by John Wiley & Sons, Inc.     

Free Energy Calculations using a Swarm‐Enhanced Sampling Molecular Dynamics Approach

Free energy simulations are an established computational tool in modelling chemical change in the condensed phase. However, sampling of kinetically distinct substates remains a challenge to these approaches. As a route to addressing this, we link the methods of thermodynamic integration (TI) and swarm‐enhanced sampling molecular dynamics (sesMD), where simulation replicas interact cooperatively to aid transitions over energy barriers. We illustrate the approach by using alchemical alkane transformations in solution, comparing them with the multiple independent trajectory TI (IT‐TI) method. Free energy changes for transitions computed by using IT‐TI grew increasingly inaccurate as the intramolecular barrier was heightened. By contrast, swarm‐enhanced sampling TI (sesTI) calculations showed clear improvements in sampling efficiency, leading to more accurate computed free energy differences, even in the case of the highest barrier height. The sesTI approach, therefore, has potential in addressing chemical change in systems where conformations exist in slow exchange.     

Pathway dependence of the efficiency of calculating free energy and entropy of solute–solute association in water

In this study we investigate two alternative pathways to compute the free energy and the entropy of small molecule association (ΔF_assoc and ΔS_assoc) in water. The first route (direct pathway) uses thermodynamic integration as function of the distance R between the solutes. The mean force and the mean covariance of the force with the energy in solution are calculated from molecular dynamics simulation followed by integration of these quantities with respect to the reaction coordinate R. The alternative approach examined (solvation pathway) would first remove the solutes from the solution using thermodynamic integration as function of a solvation coupling parameter λ, change the solute–solute distance in vacuo and then solvate back the solute pair at the new separation distance. The system studied was a pair of CH₄ molecules in water. We investigate the influence of the CH₄–water interaction strength on the obtained ΔF_assoc and ΔS_assoc values by changing van der Waals and Coulomb interaction and evaluated the accuracy and efficiency for the two pathways. We find that the direct route seems more suitable for the calculation of free energies of hydrophobic solutes while the solvation pathway performs better when calculating entropy changes for solutes that have a stronger interaction with the solvent.     

Molecular dynamics simulation reveals structural and thermodynamic features of kinase activation by cancer mutations within the epidermal growth factor receptor

The epidermal growth factor receptor (EGFR) is a major target for drugs in treating lung carcinoma as it promotes cell growth and tumor progression. Structural studies have demonstrated that EGFR exists in an equilibrium between catalytically active and inactive forms, and dramatic conformational transitions occur during its activation. It is known that EGFR mutations promote such conformational changes that affect its activation and drug efficacy. The most common point mutation in lung cancer patients is a leucine to arginine substitution at amino acid 834 (L834R). In a recent article, we have studied changes in drug binding affinities due to cancer mutations of EGFR using ensemble molecular dynamics (MD) simulations. Here, we address an enhanced activation mechanism thought to be associated with this mutation. Using extended timescale MD simulations, the structural and energetic properties are studied for both active and inactive conformations of EGFR. The thermodynamic stabilities of these two conformations are characterized by free energy landscapes estimated from molecular mechanics/Poisson-Boltzmann solvent area calculations. Our study reveals that the L834R mutation introduces conformational changes in both states, adjusting the relative stabilities of active and inactive conformations and hence the activation of the EGFR kinase.     

Calculation of the free energy of solvation for neutral analogs of amino acid side chains

The ability of the GROMOS96 force field to reproduce partition constants between water and two less polar solvents (cyclohexane and chloroform) for analogs of 18 of the 20 naturally occurring amino acids has been investigated. The estimations of the solvation free energies in water, in cyclohexane solution, and chloroform solution are based on thermodynamic integration free energy calculations using molecular dynamics simulations. The calculations show that while the force field reproduces the experimental solvation free energies of nonpolar analogs with reasonable accuracy the solvation free energies of polar analogs in water are systematically overestimated (too positive). The dependence of the calculated free energies on the atomic partial charges was also studied.     

Free Energy Change of Micelle Formation for Sodium Dodecyl Sulfate from a Dispersed State in Solution to Complete Micelles along Its Aggregation Pathways Evaluated by Chemical Species Model Combined with Molecular Dynamics Calculations

Surfactant molecules, when dispersed in solution, have been shown to spontaneously form aggregates. Our previous studies on molecular dynamics(MD) calculations have shown that ionic sodium dodecyl sulfate molecules quickly aggregated even when the aggregation number is small. The aggregation rate, however, decreased for larger aggregation numbers. In addition, studies have shown that micelle formation was not completed even after a 100 ns-long MD run(Chem. Phys. Lett. 2016, 646, 36). Herein, we analyze the free energy change of micelle formation based on chemical species model combined with molecular dynamics calculations. First, the free energy landscape of the aggregation, ?G_(i+j)~+, where two aggregates with sizes i and j associate to form the(i + j)-mer, was investigated using the free energy of micelle formation of the i-mer, G_i~+, which was obtained through MD calculations. The calculated ?G_(i+j)~+ was negative for all the aggregations where the sum of DS ions in the two aggregates was 60 or less. From the viewpoint of chemical equilibrium, aggregation to the stable micelle is desired. Further, the free energy profile along possible aggregation pathways was investigated, starting from small aggregates and ending with the complete thermodynamically stable micelles in solution. The free energy profiles, G(l, k), of the aggregates at l-th aggregation path and k-th state were evaluated by the formation free energy ∑_in_i( l,k)G_i~+ and the free energy of mixing ∑_in_i( l,k)k_BTln( n_i( l,k)/n( l,k)), where ni(l, k) is the number of i-mer in the system at the l-th i aggregation path and k-th state, with n(l,k)= ∑_n_i( l,k). All the aggregation pathways were obtained from the initial i state of 12 pentamers to the stable micelle with i = 60. All the calculated G(l, k) values monotonically decreased with increasing k. This indicates that there are no free energy barriers along the pathways. Hence, the slowdown is not due to the thermodynamic stability of the aggregates, but rather the kinetics that inhibit the association of the fragments. The time required for a collision between aggregates, one of the kinetic factors, was evaluated using the fast passage time, t_(FPT). The calculated t_(FPT) was about 20 ns for the aggregates with N = 31. Therefore, if aggregation is a diffusion-controlled process, it should be completed within the 100 ns-simulation. However, aggregation does not occur due to the free energy barrier between the aggregates, that is, the repulsive force acting on them. This may be caused by electrostatic repulsions produced by the overlap of the electric double layers, which are formed by the negative charge of the hydrophilic groups and counter sodium ions on the surface of the aggregates.