Computational methods for the study of enzymic reaction mechanisms III: a perturbation plus QM/MM approach for calculating relative free energies of protonation

We describe a coupling parameter, that is, perturbation, approach to effectively create and annihilate atoms in the quantum mechanical Hamiltonian within the closed shell restricted Hartree-Fock formalism. This perturbed quantum mechanical atom (PQA) method is combined with molecular mechanics (MM) methods (PQA/MM) within a molecular dynamics simulation, to model the protein environment (MM region) effects that also make a contribution to the overall free energy change. Using the semiempirical PM3 method to model the QM region, the application of this PQA/MM method is illustrated by calculation of the relative protonation free energy of the conserved OD2 (Asp27) and the N5 (dihydrofolate) proton acceptor sites in the active site of Escherichia coli dihydrofolate reductase (DHFR) with the bound nicotinamide adenine dinucleotide phosphate (NADPH) cofactor. For a number of choices for the QM region, the relative protonation free energy was calculated as the sum of contributions from the QM region and the interaction between the QM and MM regions via the thermodynamic integration (TI) method. The results demonstrate the importance of including the whole substrate molecule in the QM region, and the overall protein (MM) environment in determining the relative stabilities of protonation sites in the enzyme active site. The PQA/MM free energies obtained by TI were also compared with those estimated by a less computationally demanding nonperturbative method based on the linear response approximation (LRA). For some choices of QM region, the total free energies calculated using the LRA method were in very close agreement with the PQA/MM values. However, the QM and QM/MM component free energies were found to differ significantly between the two methods.     

Use of quantum mechanics/molecular mechanics-based FEP method for calculating relative binding affinities of FBPase inhibitors for type-2 diabetes

A quantum mechanics (QM)/molecular mechanics (MM)-based free energy perturbation (FEP) method, developed recently, provides most accurate estimation of binding affinities. The validity of the method was evaluated for a large set of diverse inhibitors for fructose 1,6-bisphosphatase (FBPase), a target enzyme for type-II diabetes mellitus. The validation set comprises of 22 important structurally different mutations. The calculated relative binding free energies using the QM/MM-based FEP method reproduce the experimental values with exceptional precision of less than ±0.5 kcal/mol. The CPU requirements for QM/MM-based FEP are about fivefold greater than conventional FEP methods, but it is superior in accuracy of predictions. In addition, the QM/MM-based FEP method eliminates the need for time-consuming development of MM force field parameters, which are frequently required for novel inhibitors described by MM. Future automation of the method and parallelization of the code for 128/256/512 cluster computers is expected to enhance the speed and increase its use for drug design and lead optimization. The present application of QM/MM-based FEP method for structurally diverse set of analogs serves to enhance the scope of FEP method and demonstrate the utility of QM/MM-based FEP method for its potential in drug discovery.     

Comparison of linear-scaling semiempirical methods and combined quantum mechanical/molecular mechanical methods for enzymic reactions. II. An energy decomposition analysis

QM/MM methods have been developed as a computationally feasible solution to QM simulation of chemical processes, such as enzyme-catalyzed reactions, within a more approximate MM representation of the condensed-phase environment. However, there has been no independent method for checking the quality of this representation, especially for highly nonisotropic protein environments such as those surrounding enzyme active sites. Hence, the validity of QM/MM methods is largely untested. Here we use the possibility of performing all-QM calculations at the semiempirical PM3 level with a linear-scaling method (MOZYME) to assess the performance of a QM/MM method (PM3/AMBER94 force field). Using two model pathways for the hydride-ion transfer reaction of the enzyme dihydrofolate reductase studied previously (Titmuss et al., Chem Phys Lett 2000, 320, 169-176), we have analyzed the reaction energy contributions (QM, QM/MM, and MM) from the QM/MM results and compared them with analogous-region components calculated via an energy partitioning scheme implemented into MOZYME. This analysis further divided the MOZYME components into Coulomb, resonance and exchange energy terms. For the model in which the MM coordinates are kept fixed during the reaction, we find that the MOZYME and QM/MM total energy profiles agree very well, but that there are significant differences in the energy components. Most significantly there is a large change (approximately 16 kcal/mol) in the MOZYME MM component due to polarization of the MM region surrounding the active site, and which arises mostly from MM atoms close to (<10 A) the active-site QM region, which is not modelled explicitly by our QM/MM method. However, for the model where the MM coordinates are allowed to vary during the reaction, we find large differences in the MOZYME and QM/MM total energy profiles, with a discrepancy of 52 kcal/mol between the relative reaction (product-reactant) energies. This is largely due to a difference in the MM energies of 58 kcal/mol, of which we can attribute approximately 40 kcal/mol to geometry effects in the MM region and the remainder, as before, to MM region polarization. Contrary to the fixed-geometry model, there is no correlation of the MM energy changes with distance from the QM region, nor are they contributed by only a few residues. Overall, the results suggest that merely extending the size of the QM region in the QM/MM calculation is not a universal solution to the MOZYME- and QM/MM-method differences. They also suggest that attaching physical significance to MOZYME Coulomb, resonance and exchange components is problematic. Although we conclude that it would be possible to reparameterize the QM/MM force field to reproduce MOZYME energies, a better way to account for both the effects of the protein environment and known deficiencies in semiempirical methods would be to parameterize the force field based on data from DFT or ab initio QM linear-scaling calculations. Such a force field could be used efficiently in MD simulations to calculate free energies.     

Quantum mechanical/molecular mechanical study on the mechanism of the enzymatic Baeyer-Villiger reaction

We report a combined quantum mechanical/molecular mechanical (QM/MM) study on the mechanism of the enzymatic Baeyer-Villiger reaction catalyzed by cyclohexanone monooxygenase (CHMO). In QM/MM geometry optimizations and reaction path calculations, density functional theory (B3LYP/TZVP) is used to describe the QM region consisting of the substrate (cyclohexanone), the isoalloxazine ring of C4a-peroxyflavin, the side chain of Arg-329, and the nicotinamide ring and the adjacent ribose of NADP(+), while the remainder of the enzyme is represented by the CHARMM force field. QM/MM molecular dynamics simulations and free energy calculations at the semiempirical OM3/CHARMM level employ the same QM/MM partitioning. According to the QM/MM calculations, the enzyme-reactant complex contains an anionic deprotonated C4a-peroxyflavin that is stabilized by strong hydrogen bonds with the Arg-329 residue and the NADP(+) cofactor. The CHMO-catalyzed reaction proceeds via a Criegee intermediate having pronounced anionic character. The initial addition reaction has to overcome an energy barrier of about 9 kcal/mol. The formed Criegee intermediate occupies a shallow minimum on the QM/MM potential energy surface and can undergo fragmentation to the lactone product by surmounting a second energy barrier of about 7 kcal/mol. The transition state for the latter migration step is the highest point on the QM/MM energy profile. Gas-phase reoptimizations of the QM region lead to higher barriers and confirm the crucial role of the Arg-329 residue and the NADP(+) cofactor for the catalytic efficiency of CHMO. QM/MM calculations for the CHMO-catalyzed oxidation of 4-methylcyclohexanone reproduce and rationalize the experimentally observed (S)-enantioselectivity for this substrate, which is governed by the conformational preferences of the corresponding Criegee intermediate and the subsequent transition state for the migration step.     

Convergence in the QM‐only and QM/MM modeling of enzymatic reactions: A case study for acetylene hydratase

We report systematic quantum mechanics‐only (QM‐only) and QM/molecular mechanics (MM) calculations on an enzyme‐catalyzed reaction to assess the convergence behavior of QM‐only and QM/MM energies with respect to the size of the chosen QM region. The QM and MM parts are described by density functional theory (typically B3LYP/def2‐SVP) and the CHARMM force field, respectively. Extending our previous work on acetylene hydratase with QM regions up to 157 atoms (Liao and Thiel, J. Chem. Theory Comput. 2012, 8, 3793), we performed QM/MM geometry optimizations with a QM region M4 composed of 408 atoms, as well as further QM/MM single‐point calculations with even larger QM regions up to 657 atoms. A charge deletion analysis was conducted for the previously used QM/MM model ( M3a , with a QM region of 157 atoms) to identify all MM residues with strong electrostatic contributions to the reaction energetics (typically more than 2 kcal/mol), which were then included in M4 . QM/MM calculations with this large QM region M4 lead to the same overall mechanism as the previous QM/MM calculations with M3a , but there are some variations in the relative energies of the stationary points, with a mean absolute deviation (MAD) of 2.7 kcal/mol. The energies of the two relevant transition states are close to each other at all levels applied (typically within 2 kcal/mol), with the first (second) one being rate‐limiting in the QM/MM calculations with M3a ( M4 ). QM‐only gas‐phase calculations give a very similar energy profile for QM region M4 (MAD of 1.7 kcal/mol), contrary to the situation for M3a where we had previously found significant discrepancies between the QM‐only and QM/MM results (MAD of 7.9 kcal/mol). Extension of the QM region beyond M4 up to M7 (657 atoms) leads to only rather small variations in the relative energies from single‐point QM‐only and QM/MM calculations (MAD typically about 1–2 kcal/mol). In the case of acetylene hydratase, a model with 408 QM atoms thus seems sufficient to achieve convergence in the computed relative energies to within 1–2 kcal/mol.Copyright © 2013 Wiley Periodicals, Inc.     

Binding free energies in the SAMPL6 octa-acid host–guest challenge calculated with MM and QM methods

We have estimated free energies for the binding of eight carboxylate ligands to two variants of the octa-acid deep-cavity host in the SAMPL6 blind-test challenge (with or without endo methyl groups on the four upper-rim benzoate groups, OAM and OAH, respectively). We employed free-energy perturbation (FEP) for relative binding energies at the molecular mechanics (MM) and the combined quantum mechanical (QM) and MM (QM/MM) levels, the latter obtained with the reference-potential approach with QM/MM sampling for the MM → QM/MM FEP. The semiempirical QM method PM6-DH+ was employed for the ligand in the latter calculations. Moreover, binding free energies were also estimated from QM/MM optimised structures, combined with COSMO-RS estimates of the solvation energy and thermostatistical corrections from MM frequencies. They were performed at the PM6-DH+ level of theory with the full host and guest molecule in the QM system (and also four water molecules in the geometry optimisations) for 10–20 snapshots from molecular dynamics simulations of the complex. Finally, the structure with the lowest free energy was recalculated using the dispersion-corrected density-functional theory method TPSS-D3, for both the structure and the energy. The two FEP approaches gave similar results (PM6-DH+/MM slightly better for OAM), which were among the five submissions with the best performance in the challenge and gave the best results without any fit to data from the SAMPL5 challenge, with mean absolute deviations (MAD) of 2.4–5.2 kJ/mol and a correlation coefficient (R²) of 0.77–0.93. This is the first time QM/MM approaches give binding free energies that are competitive to those obtained with MM for the octa-acid host. The QM/MM-optimised structures gave somewhat worse performance (MAD?=?3–8 kJ/mol and R²?=?0.1–0.9), but the results were improved compared to previous studies of this system with similar methods.     

Ab initio quantum mechanics-based free energy perturbation method for calculating relative solvation free energies

A free energy perturbation (FEP) method was developed that uses ab initio quantum mechanics (QM) for treating the solute molecules and molecular mechanics (MM) for treating the surroundings. Like our earlier results using AM1 semi empirical QMs, the ab initio QM/MM-based FEP method was shown to accurately calculate relative solvation free energies for a diverse set of small molecules that differ significantly in structure, aromaticity, hydrogen bonding potential, and electron density. Accuracy was similar to or better than conventional FEP methods. The QM/MM-based methods eliminate the need for time-consuming development of MM force field parameters, which are frequently required for drug-like molecules containing structural motifs not adequately described by MM. Future automation of the method and parallelization of the code for Linux 128/256/512 clusters is expected to enhance the speed and increase its use for drug design and lead optimization.     

Use of a QM/MM-based FEP method to evaluate the anomalous hydration behavior of simple alkyl amines and amides: application to the design of FBPase inhibitors for the treatment of type-2 diabetes

Standard molecular mechanics (MM) force fields predict a nearly linear decrease in hydration free energy with each successive addition of a methyl group to ammonia or acetamide, whereas a nonadditive relationship is observed experimentally. In contrast, the non-additive hydration behavior is reproduced directly using a quantum mechanics (QM)/MM-based free-energy perturbation (FEP) method wherein the solute partial atomic charges are updated at every window. Decomposing the free energies into electrostatic and van der Waals contributions and comparing the results with the corresponding free energies obtained using a conventional FEP method and a QM/MM method wherein the charges are not updated suggests that inaccuracies in the electrostatic free energies are the primary reason for the inability of the conventional FEP method to predict the experimental findings. The QM/MM-based FEP method was subsequently used to evaluate inhibitors of the diabetes drug target fructose-1,6-bisphosphatase adenosine 5'-monophosphate and 6-methylamino purine riboside 5'-monophosphate. The predicted relative binding free energy was consistent with the experimental findings, whereas the relative binding free energy predicted using the conventional FEP method differed from the experimental finding by an amount consistent with the overestimated relative solvation free energies calculated for alkylamines. Accordingly, the QM/MM-based FEP method offers potential advantages over conventional FEP methods, including greater accuracy and reduced user input. Moreover, since drug candidates often contain either functionality that is inadequately treated by MM (e.g., simple alkylamines and alkylamides) or new molecular scaffolds that require time-consuming development of MM parameters, these advantages could enable future automation of FEP calculations as well as greatly increase the use and impact of FEP calculations in drug discovery.     

Predicting hydration free energies with a hybrid QM/MM approach: an evaluation of implicit and explicit solvation models in SAMPL4

The correct representation of solute-water interactions is essential for the accurate simulation of most biological phenomena. Several highly accurate quantum methods are available to deal with solvation by using both implicit and explicit solvents. So far, however, most evaluations of those methods were based on a single conformation, which neglects solute entropy. Here, we present the first test of a novel approach to determine hydration free energies that uses molecular mechanics (MM) to sample phase space and quantum mechanics (QM) to evaluate the potential energies. Free energies are determined by using re-weighting with the Non-Boltzmann Bennett (NBB) method. In this context, the method is referred to as QM-NBB. Based on snapshots from MM sampling and accounting for their correct Boltzmann weight, it is possible to obtain hydration free energies that incorporate the effect of solute entropy. We evaluate the performance of several QM implicit solvent models, as well as explicit solvent QM/MM for the blind subset of the SAMPL4 hydration free energy challenge. While classical free energy simulations with molecular dynamics give root mean square deviations (RMSD) of 2.8 and 2.3 kcal/mol, the hybrid approach yields an improved RMSD of 1.6 kcal/mol. By selecting an appropriate functional and basis set, the RMSD can be reduced to 1 kcal/mol for calculations based on a single conformation. Results for a selected set of challenging molecules imply that this RMSD can be further reduced by using NBB to reweight MM trajectories with the SMD implicit solvent model.     

A novel quantum mechanical/molecular mechanical approach to the free energy calculation for isomerization of glycine in aqueous solution

The free energy change associated with the isomerization reaction of glycine in water solution has been studied by a hybrid quantum mechanical/molecular mechanical (QM/MM) approach combined with the theory of energy representation (QM/MM-ER) recently developed. The solvation free energies for both neutral and zwitterionic form of glycine have been determined by means of the QM/MM-ER simulation. The contributions of the electronic polarization and the fluctuation of the QM solute to the solvation free energy have been investigated. It has been found that the contribution of the density fluctuation of the zwitterionic solute is estimated as -4.2 kcal/mol in the total solvation free energy of -46.1 kcal/mol, while that of the neutral form is computed as -3.0 kcal/mol in the solvation free energy of -15.6 kcal/mol. The resultant free energy change associated with the isomerization of glycine in water has been obtained as -7.8 kcal/mol, in excellent agreement with the experimental data of -7.3 or -7.7 kcal/mol, implying the accuracy of the QM/MM-ER approach. The results have also been compared with those computed by other methodologies such as the polarizable continuum model and the classical molecular simulation. The efficiency and advantage of the QM/MM-ER method has been discussed.     

Peptide hydrolysis catalyzed by matrix metalloproteinase 2: a computational study

The MMP-2 reaction mechanism is investigated by using different computational methodologies. First, quantum mechanical (QM) calculations are carried out on a cluster model of the active site bound to an Ace-Gly approximately Ile-Nme peptide. Along the QM reaction path, a Zn-bound water molecule attacks the Gly carbonyl group to give a tetrahedral intermediate. The breaking of the C-N bond is completed thanks to the Glu 404 residue that shuttles a proton from the water molecule to Ile-N atom. The gas-phase QM energy barrier is quite low ( approximately 14 kcal/mol), thus suggesting that the essential catalytic machinery is included in the cluster model. A similar reaction path occurs in the MMP-2 catalytic domain bound to an octapeptide substrate according to hybrid QM and molecular mechanical (QM/MM) geometry optimizations. However, the rupture of the Gly( P 1) approximately Ile( P 1') amide bond is destabilized in the static QM/MM calculations, owing to the positioning of the Ile( P 1') side chain inside the MMP-2 S 1' pocket and to the inability of simple energy miminization methodologies to properly relax complex systems. Molecular dynamics simulations show that these steric limitations are overcome easily through structural fluctuations. The energetic effect of structural fluctuations is taken into account by combining QM energies with average MM Poisson-Boltzmann free energies, resulting in a total free energy barrier of 14.8 kcal/mol in good agreement with experimental data. The rate-determining event in the MMP-2 mechanism corresponds to a H-bond rearrangement involving the Glu 404 residue and/or the Glu 404-COOH --> N-Ile( P 1') proton transfer. Overall, the present computational results and previous experimental data complement each other well in order to provide a detailed view of the MMPs catalytic mechanism.     

Mechanism of proteolysis in matrix metalloproteinase‐2 revealed by QM/MM modeling

The mechanism of enzymatic peptide hydrolysis in matrix metalloproteinase‐2 (MMP‐2) was studied at atomic resolution through quantum mechanics/molecular mechanics (QM/MM) simulations. An all‐atom three‐dimensional molecular model was constructed on the basis of a crystal structure from the Protein Data Bank (ID: 1QIB), and the oligopeptide Ace‐Gln‐Gly～Ile‐Ala‐Gly‐Nme was considered as the substrate. Two QM/MM software packages and several computational protocols were employed to calculate QM/MM energy profiles for a four‐step mechanism involving an initial nucleophilic attack followed by hydrogen bond rearrangement, proton transfer, and C? N bond cleavage. These QM/MM calculations consistently yield rather low overall barriers for the chemical steps, in the range of 5–10 kcal/mol, for diverse QM treatments (PBE0, B3LYP, and BB1K density functionals as well as local coupled cluster treatments) and two MM force fields (CHARMM and AMBER). It, thus, seems likely that product release is the rate‐limiting step in MMP‐2 catalysis. This is supported by an exploration of various release channels through QM/MM reaction path calculations and steered molecular dynamics simulations. © 2015 Wiley Periodicals, Inc.     

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.     

Force matching as a stepping stone to QM/MM CB[8] host/guest binding free energies: a SAMPL6 cautionary tale

Use of quantum mechanical/molecular mechanical (QM/MM) methods in binding free energy calculations, particularly in the SAMPL challenge, often fail to achieve improvement over standard additive (MM) force fields. Frequently, the implementation is through use of reference potentials, or the so-called “indirect approach”, and inherently relies on sufficient overlap existing between MM and QM/MM configurational spaces. This overlap is generally poor, particularly for the use of free energy perturbation to perform the MM to QM/MM free energy correction at the end states of interest (e.g., bound and unbound states). However, by utilizing MM parameters that best reproduce forces obtained at the desired QM level of theory, it is possible to lessen the configurational disparity between MM and QM/MM. To this end, we sought to use force matching to generate MM parameters for the SAMPL6 CB[8] host–guest binding challenge, classically compute binding free energies, and apply energetic end state corrections to obtain QM/MM binding free energy differences. For the standard set of 11 molecules and the bonus set (including three additional challenge molecules), error statistics, such as the root mean square deviation (RMSE) were moderately poor (5.5 and 5.4 kcal/mol). Correlation statistics, however, were in the top two for both standard and bonus set submissions (\(R^{2}\) of 0.42 and 0.26, \(\tau\) of 0.64 and 0.47 respectively). High RMSE and moderate correlation strongly indicated the presence of systematic error. Identifiable issues were ameliorated for two of the guest molecules, resulting in a reduction of error and pointing to strong prospects for the future use of this methodology.     

Protein environment facilitates O2 binding in non-heme iron enzyme. An insight from ONIOM calculations on isopenicillin N synthase (IPNS)

Binding of dioxygen to a non-heme enzyme has been modeled using the ONIOM combined quantum mechanical/molecular mechanical (QM/MM) method. For the present system, isopenicillin N synthase (IPNS), binding of dioxygen is stabilized by 8-10 kcal/mol for a QM:MM (B3LYP:Amber) protein model compared to a quantum mechanical model of the active site only. In the protein system, the free energy change of O2 binding is close to zero. Two major factors consistently stabilize O2 binding. The first effect, evaluated at the QM level, originates from a change in coordination geometry of the iron center. The active-site model artificially favors the deoxy state (O2 not bound) because it allows too-large rearrangements of the five-coordinate iron site. This error is corrected when the protein is included. The corresponding effect on binding energies is 3-6 kcal/mol, depending on the coordination mode of O2 (side-on or end-on). The second major factor that stabilizes O2 binding is van der Waals interactions between dioxygen and the surrounding enzyme. These interactions, 3-4 kcal/mol at the MM level, are neglected in models that include only the active site. Polarization of the active site by surrounding amino acids does not have a significant effect on the binding energy in the present system.     

Computational study of the phosphoryl transfer catalyzed by a cyclin-dependent kinase

A cyclin-dependent kinase, Cdk2, catalyzes the transfer of the gamma-phosphate from ATP to a threonine or serine residue of its polypeptide substrates. Here, we investigate aspects of the reaction mechanism of Cdk2 by gas-phase density functional calculations, classical molecular dynamics, and Car-Parrinello QM/MM simulations. We focus on the role of the conserved Asp127 and on the nature of the phosphoryl transfer reaction mechanism catalyzed by Cdk2. Our findings suggest that Asp127 is active in its deprotonated form by assisting the formation of the near-attack orientation of the substrate serine or threonine. Therefore, the residue does not act as a general base during the catalysis. The mechanism for the phosphoryl transfer is a single SN2-like concerted step, which shows a phosphorane-like transition state geometry. Although the resulting reaction mechanism is in agreement with a previous density functional study of the same catalytic reaction mechanism (Cavalli et al., Chem. Comm. 2003, 1308-1309), the reaction barrier is considerably lower when QM/MM calculations are performed, as in this study ( approximately 42 kcal mol(-1) QM vs. approximately 24 kcal mol(-1) QM/MM); this indicates that important roles for the catalysis are played by the protein environment and solvent waters. Because of the high amino acid sequence conservation among the whole family of cyclin-dependent kinases (CDKs), these results could be general for the CDK family.     

Comparison of 6-31G*-based MST/SCRF and FEP evaluations of the free energies of hydration for small neutral molecules

A comparison between Miertus–Scrocco–Tomasi (MST) SCRF and free energy perturbation (FEP) estimates of the free energy of hydration of eight small neutral molecules is presented. In both cases, the 6-31G* molecular electrostatic potential is used to describe the electrostatic properties of the molecules. The results demonstrate the ability of both methodologies to provide useful theoretical estimates of the total free energy of hydration; the average errors are only 1.5 kcal/mol (FEP) and 0.8 kcal/mol (MST/SCRF). The largest errors in the FEP and MST/SCRF results are less than 1.5 kcal/mol for all molecules except acetic acid, where the FEP method overestimates the free energy of hydration by 3.3 kcal/mol. © John Wiley & Sons, Inc.