Calculation of relative solvation free energy differences by thermodynamic perturbation method: Dependence of free energy results on simulation length

Molecular dynamics (MD) simulations in conjunction with the thermodynamic cycle perturbation approach has been used to calculate relative solvation free energies for acetone to acetaldehyde, acetone to pyruvic acid, acetone to 1,1,1-trifluoroacetone, acetone to 1,1,1-trichloroacetone, acetone to 2,3-butanedione, acetone to cyclopropanone, and formaldehyde hydrate to formaldehyde. To evaluate the dependence of relative solvation free energy convergence on MD simulation length and starting configuration two studies were performed. In the first study, each simulation started from the same well-equilibrated configuration and the length was varied from 153 to 1530 ps. In the second study, the relative solvation free energy differences were calculated starting from three different configurations and using 510 ps of MD simulation for each mutation. These results clearly indicate that, even for molecules with limited conformational flexibility, a simulation length of 510 ps or greater is required to obtain satisfactory convergence and, for the mutations of large structural changes between reactant and product, such as cyclopropanone to acetone, require much longer simulation lengths to achieve satisfactory convergence. These results also show that performing one long simulation is better than averaging results from three shortest simulations of the same length using different starting conformations. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1018–1027, 1999     

Calculation of relative binding affinities of fructose 1,6-bisphosphatase mutants with adenosine monophosphate using free energy perturbation method

The free energy perturbation (FEP) methodology is the most accurate means of estimating relative binding affinities between inhibitors and protein variants. In this article, the importance of hydrophobic and hydrophilic residues to the binding of adenosine monophosphate (AMP) to the fructose 1,6-bisphosphatase (FBPase), a target enzyme for type-II diabetes, was examined by FEP method. Five mutations were made to the FBPase enzyme with AMP inhibitor bound: 113Tyr --> 113Phe, 31Thr --> 31Ala, 31Thr --> 31Ser, 177Met --> 177Ala, and 30Leu --> 30Phe. These mutations test the strength of hydrogen bonds and van der Waals interactions between the ligand and enzyme. The calculated relative free energies indicated that: 113Tyr and 31Thr play an important role, each via two hydrogen bonds affecting the binding affinity of inhibitor AMP to FBPase, and any changes in these hydrogen bonds due to mutations on the protein will have significant effect on the binding affinity of AMP to FBPase, consistent to experimental results. Also, the free energy calculations clearly show that the hydrophilic interactions are more important than the hydrophobic interactions of the binding pocket of FBPase.     

Relative solvation free energies calculated using an ab initio QM/MM-based free energy perturbation method: dependence of results on simulation length

Molecular dynamics (MD) simulations in conjunction with thermodynamic perturbation approach was used to calculate relative solvation free energies of five pairs of small molecules, namely; (1) methanol to ethane, (2) acetone to acetamide, (3) phenol to benzene, (4) 1,1,1 trichloroethane to ethane, and (5) phenylalanine to isoleucine. Two studies were performed to evaluate the dependence of the convergence of these calculations on MD simulation length and starting configuration. In the first study, each transformation started from the same well-equilibrated configuration and the simulation length was varied from 230 to 2,540 ps. The results indicated that for transformations involving small structural changes, a simulation length of 860 ps is sufficient to obtain satisfactory convergence. In contrast, transformations involving relatively large structural changes, such as phenylalanine to isoleucine, require a significantly longer simulation length (>2,540 ps) to obtain satisfactory convergence. In the second study, the transformation was completed starting from three different configurations and using in each case 860 ps of MD simulation. The results from this study suggest that performing one long simulation may be better than averaging results from three different simulations using a shorter simulation length and three different starting configurations.     

Calculation of relative binding free energy differences for fructose 1,6-bisphosphatase inhibitors using the thermodynamic cycle perturbation approach.

An iterative, computer-assisted, drug design strategy that combines molecular design, molecular mechanics, molecular dynamics (MD), and free energy perturbation (FEP) calculations with compound synthesis, biochemical testing of inhibitors, and crystallographic structure determination of protein-inhibitor complexes was successfully used to predict the rank order of a series of nucleoside monophosphate analogues as fructose 1,6-bisphosphatase (FBPase) inhibitors. The X-ray structure of FBPase complexed with 5-aminoimidazole-4-carboxamide-1-beta-D-ribofuranosyl 5'-monophosphate (ZMP) provided structural information used for subsequent analogue design and free energy calculations. The FEP protocol was validated by calculating the free energy differences for the mutation of ZMP (1) to AMP (2). The calculated results showed a net gain of 1.7 kcal/mol, which agreed with the experimental result of 1.3 kcal/mol. FEP calculations were performed for 18 other AMP analogues. Inhibition constants were determined for over half of these analogues, usually after completion of the calculation, and were consistent with the predictions. Solvation free energy differences between AMP and various AMP analogues proved to be an important factor in binding free energies, suggesting that increased desolvation costs associated with the addition of polar groups to an inhibitor must be overcome by stronger ligand-protein interactions if the structural modification is to enhance inhibitor potency. The results indicate that FEP calculations predict relative binding affinities with high accuracy and provide valuable insight into the factors that influence inhibitor binding and therefore should greatly aid efforts to optimize initial lead compounds and reduce the time required for the discovery of new drug candidates.     

Converging ligand‐binding free energies obtained with free‐energy perturbations at the quantum mechanical level

In this article, the convergence of quantum mechanical (QM) free‐energy simulations based on molecular dynamics simulations at the molecular mechanics (MM) level has been investigated. We have estimated relative free energies for the binding of nine cyclic carboxylate ligands to the octa‐acid deep‐cavity host, including the host, the ligand, and all water molecules within 4.5 Å of the ligand in the QM calculations (158–224 atoms). We use single‐step exponential averaging (ssEA) and the non‐Boltzmann Bennett acceptance ratio (NBB) methods to estimate QM/MM free energy with the semi‐empirical PM6‐DH2X method, both based on interaction energies. We show that ssEA with cumulant expansion gives a better convergence and uses half as many QM calculations as NBB, although the two methods give consistent results. With 720,000 QM calculations per transformation, QM/MM free‐energy estimates with a precision of 1 kJ/mol can be obtained for all eight relative energies with ssEA, showing that this approach can be used to calculate converged QM/MM binding free energies for realistic systems and large QM partitions. © 2016 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Achieving fast convergence of ab initio free energy perturbation calculations with the adaptive force-matching method

This paper studies the possibility of improving the convergence of ab initio free energy perturbation (FEP) calculations by developing customized force fields with the adaptive force-matching (AFM) method. The ab initio FEP method relies on a molecular mechanics (MM) potential to sample configuration space. If the Boltzmann weight of the MM sampling is close to that of the ab initio method, the efficiency of ab initio FEP will be optimal. The difference in the Boltzmann weights can be quantified by the relative energy difference distribution (REDD). The force field developed through AFM significantly improves the REDD when compared with standard MM models, thus improving the convergence of the ab initio FEP calculation. The static dielectric constant ε_s of ice-Ih was studied with PW-91 through ab initio FEP. With a customized force field developed through AFM, we were able to converge ε_s to 80 ± 4 with 3,600 configurations. A similar ab initio FEP calculation with the TIP4P model would require 220 times more configurations to achieve the same accuracy. Our study indicates that the PW-91 functional underestimates ice-Ih ε_s by about 20%.     

Free energy difference calculations by thermodynamic integration: Difficulties in obtaining a precise value

Free energy difference calculations have been performed by the “slow growth” method of thermodynamic integration of the AMBER 3.0 molecular dynamics program for the mutation of a conformationally restricted threonine dipeptide, N-acetyl threonyl-N-methylamide, to the corresponding alanyl dipeptide. By varying the total simulation length, it has been determined that precise free energy values are obtained only for simulations of greater than 100 ps total simulation time length. By varying the starting configurations for simulations of the same length, it has been determined that averaging the free energies obtained from shorter simulations may not give precise answers. Possible reasons for this behavior are discussed.     

X‐Ray Crystallography and Free Energy Calculations Reveal the Binding Mechanism of A2A Adenosine Receptor Antagonists

We present a robust protocol based on iterations of free energy perturbation (FEP) calculations, chemical synthesis, biophysical mapping and X‐ray crystallography to reveal the binding mode of an antagonist series to the A_2A adenosine receptor (AR). Eight A_2AAR binding site mutations from biophysical mapping experiments were initially analyzed with sidechain FEP simulations, performed on alternate binding modes. The results distinctively supported one binding mode, which was subsequently used to design new chromone derivatives. Their affinities for the A_2AAR were experimentally determined and investigated through a cycle of ligand‐FEP calculations, validating the binding orientation of the different chemical substituents proposed. Subsequent X‐ray crystallography of the A_2AAR with a low and a high affinity chromone derivative confirmed the predicted binding orientation. The new molecules and structures here reported were driven by free energy calculations, and provide new insights on antagonist binding to the A_2AAR, an emerging target in immuno‐oncology.     

Improved precision and efficiency of free energy calculations for small systems using lambda-scaled atomic masses and separating conformational and transformational sampling

We present results showing the importance of appropriate treatment of atomic masses in molecular dynamics (MD)-based single topology free-energy perturbations (FEPs) on small molecule systems. The reversibility of gas phase simulations is significantly improved by scaling the atomic mass of mutated atoms with the lambda variable normally used for the scaling of energy terms. Because this effect is less pronounced for solvated systems, it will not cancel in estimates of the relative hydration free energy difference. The advantage of mass scaling is demonstrated by a null mutation of ethane to ethane and the calculation of the relative hydration free energy difference between ethane and n-propane. Furthermore, it is found that the simulation time necessary for converged MD/FEPs is prohibitively large for relative hydration free energy calculations on cyclic alkanes. Therefore, we explore an alternative free energy pathway including strongly constrained conformations to improve convergence in FEP simulations of flexible molecules.     

Molecular dynamics simulations and free energy calculations on the enzyme 4‐hydroxyphenylpyruvate dioxygenase

4‐Hydroxyphenylpyruvate dioxygenase is a relevant target in both pharmaceutical and agricultural research. We report on molecular dynamics simulations and free energy calculations on this enzyme, in complex with 12 inhibitors for which experimental affinities were determined. We applied the thermodynamic integration approach and the more efficient one‐step perturbation. Even though simulations seem well converged and both methods show excellent agreement between them, the correlation with the experimental values remains poor. We investigate the effect of slight modifications on the charge distribution of these highly conjugated systems and find that accurate models can be obtained when using improved force field parameters. This study gives insight into the applicability of free energy methods and current limitations in force field parameterization. © 2011 Wiley Periodicals, Inc. J Comput Chem 2011     

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.     

Calculation of a divergence‐free lamb shift

A divergence‐free theory of the Lamb shift is presented, and calculations are carried out for hydrogen‐like ions. The theory uses two Lorentz‐invariant equations of motion, one for the electron in the presence of the photon, which is Dirac's four‐space time‐dependent equation, and the other for the photon in the presence of the electron. The electron–photon interaction is calculated from the Dirac current. The distribution of photon frequencies is bounded by the distribution of electron frequencies intrinsic to the Dirac current, thereby guaranteeing a divergence‐free result for the Lamb shift. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Improving the efficiency and reliability of free energy perturbation calculations using overlap sampling methods

A challenge in free energy calculation for complex molecular systems by computer simulation is to obtain a reliable estimate within feasible computational time. In this study, we suggest an answer to this challenge by exploring a simple method, overlap sampling (OS), for producing reliable free-energy results in an efficient way. The formalism of the OS method is based on ensuring sampling of important overlapping phase space during perturbation calculations. This technique samples both forward and reverse free energy perturbation (FEP) to improve the free-energy calculation. It considers the asymmetry of the FEP calculation and features an ability to optimize both the precision and the accuracy of the measurement without affecting the simulation process itself. The OS method is tested at two optimization levels: no optimization (simple OS), and full optimization (equivalent to Bennett's method), and compared to conventional FEP techniques, including the widely used direct FEP averaging method, on three alchemical mutation systems: (a) an anion transformation in water solution, (b) mutation between methanol and ethane, and (c) alchemical change of an adenosine molecule. It is consistently shown that the reliability of free-energy estimates can be greatly improved using the OS techniques at both optimization levels, while the performance of Bennett's method is particularly striking. In addition, the efficiency of a calculation can be significantly improved because the method is able to (a) converge to the right answer quickly, and (b) work for large perturbations. The basic two-stage OS method can be extended to admit additional stages, if needed. We suggest that the OS method can be used as a general perturbation technique for computing free energy differences in molecular simulations.     

Efficient synthesis of 4‐heteroaryl‐substituted triphenylamine derivatives via a ligand‐free Suzuki reaction

This paper reports an efficient synthesis of 4‐heteroaryl‐substituted triphenylamine derivatives via the palladium‐catalyzed Suzuki reaction of heteroaryl halides with 4‐(diphenylamino)phenylboronic acid in ethylene glycol under ligand‐free and aerobic conditions. These derivatives are important structural motifs for use in dye‐sensitized solar cells and organic electroluminescence materials. Copyright © 2011 John Wiley & Sons, Ltd.     

Rapid,metal‐free room temperature vulcanization produces silicone elastomers

Silicone room temperature vulcanization elastomers are usually formed through either a platinum‐catalyzed hydrosilylation or tin‐catalyzed moisture cure. In this article, we show that it is possible to create robust, transparent silicone elastomers without the need for metal catalysts. Hydrogen‐terminated silicone polymers are crosslinked by tri‐ or tetraalkoxysilane crosslinkers in a condensation process catalyzed by the presence of trispentafluorophenylborane catalyst to give elastomers and alkane by‐products. This procedure allows for very fast cure times (< 30 s to a tack free state): the process is more conveniently controlled with the addition of a small amount of solvent. Physical and mechanical properties are readily modified by control of the chain length of the starting polymer, the functionality and nature of the alkoxy group on the crosslinker. Organofunctional groups, useful for further polymer modification, can optionally be incorporated by judicious choice of readily available starting materials. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2013     

Implementation of the diagonalization‐free algorithm in the self‐consistent field procedure within the four‐component relativistic scheme

A recently developed Thouless‐expansion‐based diagonalization‐free approach for improving the efficiency of self‐consistent field (SCF) methods (Noga and ?imunek, J. Chem. Theory Comput. 2010, 6, 2706) has been adapted to the four‐component relativistic scheme and implemented within the program package ReSpect. In addition to the implementation, the method has been thoroughly analyzed, particularly with respect to cases for which it is difficult or computationally expensive to find a good initial guess. Based on this analysis, several modifications of the original algorithm, refining its stability and efficiency, are proposed. To demonstrate the robustness and efficiency of the improved algorithm, we present the results of four‐component diagonalization‐free SCF calculations on several heavy‐metal complexes, the largest of which contains more than 80 atoms (about 6000 4‐spinor basis functions). The diagonalization‐free procedure is about twice as fast as the corresponding diagonalization. © 2014 Wiley Periodicals, Inc.     

Aminopropylated PEG as a novel,eco‐friendly and biodegradable basic catalyst for bis‐Michael addition to α,β‐unsaturated ketones under solvent‐free conditions

A solvent‐free and highly efficient protocol has been developed for the synthesis of novel bis‐Michael addition products ( 3a , 3b , 3c , 3d , 3e , 3f , 3g , 3h , 3i , 3j , 3k , 3l , 3m , 3n , 3o ) using aminopropylated PEG‐6000 (NH₂‐PEG) as a biodegradable and recyclable catalyst in excellent yields under solvent‐free conditions. Other remarkable features of this environmentally benign protocol are shorter reaction time, tolerance of a wide range of C―H‐activated acids, high yield of products, and simple experimental and work‐up procedure as compared to conventional methods. The NH₂‐PEG catalyst is characterized by using FT‐IR, powder XRD and scanning electron microscopy–energy dispersion X‐ray spectrometric analyses. The catalyst can be recycled several times without significant loss of its catalytic activity. Copyright © 2014 John Wiley & Sons, Ltd.