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.     

Simple and exact approach to the electronic polarization effect on the solvation free energy: formulation for quantum-mechanical/molecular-mechanical system and its applications to aqueous solutions

We present a simple and exact numerical approach to compute the free energy contribution δμ in solvation due to the electron density polarization and fluctuation of a quantum-mechanical solute in the quantum-mechanical/molecular-mechanical (QM/MM) simulation combined with the theory of the energy representation (QM/MM-ER). Since the electron density fluctuation is responsible for the many-body QM-MM interactions, the standard version of the energy representation method cannot be applied directly. Instead of decomposing the QM-MM polarization energy into the pairwise additive and non-additive contributions, we take sum of the polarization energies in the QM-MM interaction and adopt it as a new energy coordinate for the method of energy representation. Then, it is demonstrated that the free energy δμ can be exactly formulated in terms of the energy distribution functions for the solution and reference systems with respect to this energy coordinate. The benchmark tests were performed to examine the numerical efficiency of the method with respect to the changes in the individual properties of the solvent and the solute. Explicitly, we computed the solvation free energy of a QM water molecule in ambient and supercritical water, and also the free-energy change associated with the isomerization reaction of glycine from neutral to zwitterionic structure in aqueous solution. In all the systems examined, it was demonstrated that the computed free energy δμ agrees with the experimental value, irrespective of the choice of the reference electron density of the QM solute. The present method was also applied to a prototype reaction of adenosine 5'-triphosphate hydrolysis where the effect of the electron density fluctuation is substantial due to the excess charge. It was demonstrated that the experimental free energy of the reaction has been accurately reproduced with the present approach.     

An application of the novel quantum mechanical/molecular mechanical method combined with the theory of energy representation: An ionic dissociation of a water molecule in the supercritical water

A novel quantum chemical approach recently developed has been applied to an ionic dissociation of a water molecule (2H(2)O-->H(3)O(+)+OH(-)) in ambient and supercritical water. The method is based on the quantum mechanical/molecular mechanical (QM/MM) simulations combined with the theory of energy representation (QM/MM-ER), where the energy distribution function of MM solvent molecules around a QM solute serves as a fundamental variable to determine the hydration free energy of the solute according to the rigorous framework of the theory of energy representation. The density dependence of the dissociation free energy in the supercritical water has been investigated for the density range from 0.1 to 0.6 g/cm(3) with the temperature fixed at a constant. It has been found that the product ionic species significantly stabilizes in the high density region as compared with the low density. Consequently, the dissociation free energy decreases monotonically as the density increases. The decomposition of the hydration free energy has revealed that the entropic term (-TDeltaS) strongly depends on the density of the solution and dominates the behavior of the dissociation free energy with respect to the variation of the density. The increase in the entropic term in the low density region can be attributed to the decrease in the translational degrees of freedom brought about by the aggregation of solvent water molecules around the ionic solute.     

Novel quantum mechanical/molecular mechanical method combined with the theory of energy representation: free energy calculation for the Beckmann rearrangement promoted by proton transfers in the supercritical water

The Beckmann rearrangement of acetone oxime promoted by proton transfers in the supercritical water has been investigated by means of the hybrid quantum mechanical/molecular mechanical approach combined with the theory of energy representation (QM/MM-ER) recently developed. The transition state (TS) structures have been explored by ab initio calculations for the reaction of hydrated acetone oxime on the assumption that the reaction is catalyzed by proton transfers along the hydrogen bonds connecting the solute and the solvent water molecules. Up to two water molecules have been considered as reactants that take part in the proton transfers. As a result of the density functional theory calculations with B3LYP functional and aug-cc-pVDZ basis set, it has been found that participation of two water molecules in the reaction reduces the activation free energy by -12.3 kcal/mol. Furthermore, the QM/MM-ER simulations have revealed that the TS is more stabilized than the reactant state in the supercritical water by 2.7 kcal/mol when two water molecules are involved in the reaction. Solvation free energies of the reactant and the TS have been decomposed into terms due to the electronic polarization of the solute, electron density fluctuation, and others to elucidate the origin of the stabilization of the TS as compared with the reactant. It has been revealed that the promotion of the chemical reaction due to the hydration mainly originates from the interaction between the nonpolarized solute and the solvent water molecules at the supercritical state.     

Hydration effects on the reaction with an open-shell transition state: QM/MM-ER study for the dehydration reaction of alcohol in hot water

The reaction mechanism for the dehydration of 1,4-butanediol in hot water has been investigated by means of the hybrid quantum mechanical/molecular mechanical approach combined with the theory of energy representation (QM/MM-ER). We have assumed that the proton transfers along the hydrogen bonds of the water molecules catalyze the reaction, where the transition state (TS) forms a singlet biradical electronic structure. It has been revealed by the simulation that the biradical electronic state at the TS changes to zwitterionic structure in solution due to the hydration of the polar solvent. Such the electronic structure change gives rise to the substantial stabilization of the TS in hot water. As a result, the water-catalytic path becomes more favorable in aqueous solution than another possible path that proceeds without proton transfers as opposed to the reaction mechanism in the gas phase. Furthermore, the activation free energy computed by the present method is in excellent agreement with the experimental result.     

Investigation of the dominant hydration structures among the ionic species in aqueous solution: novel quantum mechanics/molecular mechanics simulations combined with the theory of energy representation

In the present work, we have performed quantum chemical calculations to determine preferable species among the ionic complexes that are present in ambient water due to the autodissociation of water molecule. First, we have formulated the relative population of the hydrated complexes with respect to the bare ion (H(3)O(+) or OH(-)) in terms of the solvation free energies of the relevant molecules. The solvation free energies for various ionic species (H(3)O(+), H(5)O(2) (+), H(7)O(3) (+), H(9)O(4) (+) or OH(-), H(3)O(2) (-), H(5)O(3) (-), H(7)O(4) (-), H(9)O(5) (-)), categorized as proton or hydroxide ion in solution, have been computed by employing the QM/MM-ER method recently developed by combining the quantum mechanical/molecular mechanical (QM/MM) approach with the theory of energy representation (ER). Then, the computed solvation free energies have been used to evaluate the ratio of the populations of the ionic complexes to that of the bare ion (H(3)O(+) or OH(-)). Our results suggest that the Zundel form, i.e., H(5)O(2) (+), is the most preferable in the solution among the cationic species listed above though the Eigen form (H(9)O(4) (+)) is very close to the Zundel complex in the free energy, while the anionic fragment from water molecules mostly takes the form of OH(-). It has also been found that the loss of the translational entropy of water molecules associated with the formation of the complex plays a role in determining the preferable size of the cluster.     

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.     

An efficient method for the calculation of quantum mechanics/molecular mechanics free energies

The combination of quantum mechanics (QM) with molecular mechanics (MM) offers a route to improved accuracy in the study of biological systems, and there is now significant research effort being spent to develop QM/MM methods that can be applied to the calculation of relative free energies. Currently, the computational expense of the QM part of the calculation means that there is no single method that achieves both efficiency and rigor; either the QM/MM free energy method is rigorous and computationally expensive, or the method introduces efficiency-led assumptions that can lead to errors in the result, or a lack of generality of application. In this paper we demonstrate a combined approach to form a single, efficient, and, in principle, exact QM/MM free energy method. We demonstrate the application of this method by using it to explore the difference in hydration of water and methane. We demonstrate that it is possible to calculate highly converged QM/MM relative free energies at the MP2/aug-cc-pVDZ/OPLS level within just two days of computation, using commodity processors, and show how the method allows consistent, high-quality sampling of complex solvent configurational change, both when perturbing hydrophilic water into hydrophobic methane, and also when moving from a MM Hamiltonian to a QM/MM Hamiltonian. The results demonstrate the validity and power of this methodology, and raise important questions regarding the compatibility of MM and QM/MM forcefields, and offer a potential route to improved compatibility.     

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.     

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.     

Importance of van der Waals Interactions in QM/MM Simulations

The importance of accurately treating van der Waals interactions between the quantum mechanical (QM) and molecular mechanical (MM) atoms in hybrid QM/MM simulations has been investigated systematically. First, a set of van der Waals (vdW) parameters was optimized for an approximate density functional method, the self-consistent charge-tight binding density functional (SCC-DFTB) approach, based on small hydrogen-bonding clusters. The sensitivity of condensed phase observables to the SCC-DFTB vdW parameters was then quantitatively investigated by SCC-DFTB/MM simulations of several model systems using the optimized set and two sets of extreme vdW parameters selected from the CHARMM22 forcefield. The model systems include a model FAD molecule in solution and a solvated enediolate, and the properties studied include the radial distribution functions of water molecules around the solute (model FAD and enediolate), the reduction potential of the model FAD and the potential of mean force for an intramolecular proton transfer in the enediolate. Although there are noticeable differences between parameter sets for gas-phase clusters and solvent structures around the solute, thermodynamic quantities in the condensed phase (e.g., reduction potential and potential of mean force) were found to be less sensitive to the numerical values of vdW parameters. The differences between SCC-DFTB/MM results with the three vdW parameter sets for SCC-DFTB atoms were explained in terms of the effects of the parameter set on solvation. The current study has made it clear that efforts in improving the reliability of QM/MM methods for energetical properties in the condensed phase should focus on components other than van der Waals interactions between QM and MM atoms.     

Increasing the time step with mass scaling in Born‐Oppenheimer ab initio QM/MM molecular dynamics simulations

Born‐Oppenheimer ab initio QM/MM molecular dynamics simulation with umbrella sampling is a state‐of‐the‐art approach to calculate free energy profiles of chemical reactions in complex systems. To further improve its computational efficiency, a mass‐scaling method with the increased time step in MD simulations has been explored and tested. It is found that by increasing the hydrogen mass to 10 amu, a time step of 3 fs can be employed in ab initio QM/MM MD simulations. In all our three test cases, including two solution reactions and one enzyme reaction, the resulted reaction free energy profiles with 3 fs time step and mass scaling are found to be in excellent agreement with the corresponding simulation results using 1 fs time step and the normal mass. These results indicate that for Born‐Oppenheimer ab initio QM/MM molecular dynamics simulations with umbrella sampling, the mass‐scaling method can significantly reduce its computational cost while has little effect on the calculated free energy profiles. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Molecular dynamics and free energy perturbation study of hydride-ion transfer step in dihydrofolate reductase using combined quantum and molecular mechanical model

We used molecular dynamics simulation and free energy perturbation (FEP) methods to investigate the hydride-ion transfer step in the mechanism for the nicotinamide adenine dinucleotide phosphate (NADPH)-dependent reduction of a novel substrate by the enzyme dihydrofolate reductase (DHFR). The system is represented by a coupled quantum mechanical and molecular mechanical (QM/MM) model based on the AM1 semiempirical molecular orbital method for the reacting substrate and NADPH cofactor fragments, the AMBER force field for DHFR, and the TIP3P model for solvent water. The FEP calculations were performed for a number of choices for the QM system. The substrate, 8-methylpterin, was treated quantum mechanically in all the calculations, while the larger cofactor molecule was partitioned into various QM and MM regions with the addition of “link” atoms (F, CH₃, and H). Calculations were also carried out with the entire NADPH molecule treated by QM. The free energies of reaction and the net charges on the NADPH fragments were used to determine the most appropriate QM/MM model. The hydride-ion transfer was also carried out over several FEP pathways, and the QM and QM/MM component free energies thus calculated were found to be state functions (i.e., independent of pathway). A ca. 10 kcal/mol increase in free energy for the hydride-ion transfer with an activation barrier of ca. 30 kcal/mol was calculated. The increase in free energy on the hydride-ion transfer arose largely from the QM/MM component. Analysis of the QM/MM energy components suggests that, although a number of charged residues may contribute to the free energy change through long-range electrostatic interactions, the only interaction that can account for the 10 kcal/mol increase in free energy is the hydrogen bond between the carboxylate side chain of Glu30 (avian DHFR) and the activated (protonated) substrate. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 977–988, 1998     

Frozen density functional free energy simulations of redox proteins: computational studies of the reduction potential of plastocyanin and rusticyanin

The evaluation of reduction potentials of proteins by ab initio approaches presents a major challenge for computational chemistry. This is addressed in the present investigation by reporting detailed calculations of the reduction potentials of the blue copper proteins plastocyanin and rusticyanin using the QM/MM all-atom frozen density functional theory, FDFT, method. The relevant ab initio free energies are evaluated by using a classical reference potential. This approach appears to provide a general consistent and effective way for reproducing the configurational ensemble needed for consistent ab initio free energy calculations. The FDFT formulation allows us to treat a large part of the protein quantum mechanically by a consistently coupled QM/QM/MM embedding method while still retaining a proper configurational sampling. To establish the importance of proper configurational sampling and the need for a complete representation of the protein+solvent environment, we also consider several classical approaches. These include the semi-macroscopic PDLD/S-LRA method and classical all-atom simulations with and without a polarizable force field. The difference between the reduction potentials of the two blue copper proteins is reproduced in a reasonable way, and its origin is deduced from the different calculations. It is found that the protein permanent dipole tunes down the reduction potential for plastocyanin compared to the active site in regular water solvent, whereas in rusticyanin it is instead tuned up. This electrostatic environment, which is the major effect determining the reduction potential, is a property of the entire protein and solvent system and cannot be ascribed to any particular single interaction.