Free energies for biological electron transfer from QM/MM calculation: method, application and critical assessment

Computer simulations of biological electron transfer reactions are reviewed with a focus on the calculation of reaction free energy (driving force) and reorganization free energy. Then a mixed quantum mechanical/molecular mechanical (QM/MM) approach is described which is designed for computation of these quantities for pure electron transfer reactions with large donor-acceptor separation distances. The method is applied to intra-protein electron transfer in Ru(bpy)(2)(im)His33 cytochrome c and the results compared to experimental data. Several modeling aspects which are important for successful calculation of free energies with QM/MM are discussed in detail.     

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.     

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 extensible interface for QM/MM molecular dynamics simulations with AMBER

We present an extensible interface between the AMBER molecular dynamics (MD) software package and electronic structure software packages for quantum mechanical (QM) and mixed QM and classical molecular mechanical (MM) MD simulations within both mechanical and electronic embedding schemes. With this interface, ab initio wave function theory and density functional theory methods, as available in the supported electronic structure software packages, become available for QM/MM MD simulations with AMBER. The interface has been written in a modular fashion that allows straight forward extensions to support additional QM software packages and can easily be ported to other MD software. Data exchange between the MD and QM software is implemented by means of files and system calls or the message passing interface standard. Based on extensive tests, default settings for the supported QM packages are provided such that energy is conserved for typical QM/MM MD simulations in the microcanonical ensemble. Results for the free energy of binding of calcium ions to aspartate in aqueous solution comparing semiempirical and density functional Hamiltonians are shown to demonstrate features of this interface. © 2013 Wiley Periodicals, Inc.     

QuanPol: A full spectrum and seamless QM/MM program

The quantum chemistry polarizable force field program (QuanPol) is implemented to perform combined quantum mechanical and molecular mechanical (QM/MM) calculations with induced dipole polarizable force fields and induced surface charge continuum solvation models. The QM methods include Hartree–Fock method, density functional theory method (DFT), generalized valence bond theory method, multiconfiguration self‐consistent field method, Møller–Plesset perturbation theory method, and time‐dependent DFT method. The induced dipoles of the MM atoms and the induced surface charges of the continuum solvation model are self‐consistently and variationally determined together with the QM wavefunction. The MM force field methods can be user specified, or a standard force field such as MMFF94, Chemistry at Harvard Molecular Mechanics (CHARMM), Assisted Model Building with Energy Refinement (AMBER), and Optimized Potentials for Liquid Simulations‐All Atom (OPLS‐AA). Analytic gradients for all of these methods are implemented so geometry optimization and molecular dynamics (MD) simulation can be performed. MD free energy perturbation and umbrella sampling methods are also implemented. © 2013 Wiley Periodicals, Inc.     

New QM/MM implementation of the DFTB3 method in the gromacs package

The approximate density‐functional tight‐binding theory method DFTB3 has been implemented in the quantum mechanics/molecular mechanics (QM/MM) framework of the Gromacs molecular simulation package. We show that the efficient smooth particle–mesh Ewald implementation of Gromacs extends to the calculation of QM/MM electrostatic interactions. Further, we make use of the various free‐energy functionalities provided by Gromacs and the PLUMED plugin. We exploit the versatility and performance of the current framework in three typical applications of QM/MM methods to solve biophysical problems: (i) ultrafast proton transfer in malonaldehyde, (ii) conformation of the alanine dipeptide, and (iii) electron‐induced repair of a DNA lesion. Also discussed is the further development of the framework, regarding mostly the options for parallelization. © 2015 Wiley Periodicals, Inc.     

A QM/MM study of a nucleophilic aromatic substitution reaction catalyzed by 4-chlorobenzoyl-CoA dehalogenase

Calculated using a QM/MM method, the free energy profile for the conversion of 4-chlorobenzoate to 4-hydroxybenzoate catalyzed by 4-chlorobenzoyl-CoA dehalogenase indicates the existence of a stable Meisenheimer complex.     

Geometry optimization with QM/MM,ONIOM, and other combined methods. I. Microiterations and constraints

Hybrid energy methods such as QM/MM and ONIOM, that combine different levels of theory into one calculation, have been very successful in describing large systems. Geometry optimization methods can take advantage of the partitioning of these calculations into a region treated at a quantum mechanical (QM) level of theory and the larger, remaining region treated by an inexpensive method such as molecular mechanics (MM). A series of microiterations can be employed to fully optimize the MM region for each optimization step in the QM region. Cartesian coordinates are used for the MM region and are chosen so that the internal coordinates of the QM region remain constant during the microiterations. The coordinates of the MM region are augmented to permit rigid body translation and rotation of the QM region. This is essential if any atoms in the MM region are constrained, but it also improves the efficiency of unconstrained optimizations. Because of the microiterations, special care is needed for the optimization step in the QM region so that the system remains in the same local valley during the course of the optimization. The optimization methodology with microiterations, constraints, and step-size control are illustrated by calculations on bacteriorhodopsin and other systems.     

QM/MM connection atoms for the multistate treatment of organic and biological molecules

We present an extension of our semiempirical floating occupation MO-CI approach for the determination of ground and excited state potential energy surfaces of interest in photochemistry. The QM/MM variant of the method, which allows for electrostatic and van der Waals interactions between the QM and MM subsystems, is supplemented with a treatment of covalent interactions based on Antes and Thiels connection atom approach. We concentrate on the correct treatment of electrostatic interactions concerning the connection atom, on the specific requirements for the representation of excited states, and on the transferability of the optimal parameters. We show the viability of the method with four examples of connection atoms: S in a thioether bridge, acylic C, aliphatic C, and N in a peptide. The results obtained with the QM/MM treatment compare well with all-QM results of the same level.     

pKa calculations in solution and proteins with QM/MM free energy perturbation simulations: a quantitative test of QM/MM protocols

The accuracy of biological simulations depends, in large part, on the treatment of electrostatics. Due to the availability of accurate experimental values, calculation of pKa provides stringent evaluation of computational methods. The generalized solvent boundary potential (GSBP) and Ewald summation electrostatic treatments were recently implemented for combined quantum mechanical and molecular mechanics (QM/MM) simulations by our group. These approaches were tested by calculating pKa shifts due to differences in electronic structure and electrostatic environment; the shifts were determined for a series of small molecules in solution, using various electrostatic treatments, and two residues (His 31, Lys 102) in the M102K T4-lysozyme mutant with large pKa shifts, using the GSBP approach. The calculations utilized a free energy perturbation scheme with the QM/MM potential function involving the self-consistent charge density functional tight binding (SCC-DFTB) and CHARMM as the QM and MM methods, respectively. The study of small molecules demonstrated that inconsistent electrostatic models produced results that were difficult to correct in a robust manner; by contrast, extended electrostatics, GSBP, and Ewald simulations produced consistent results once a bulk solvation contribution was carefully chosen. In addition to the electrostatic treatment, the pKa shifts were also sensitive to the level of the QM method and the scheme of treating QM/MM Coulombic interactions; however, simple perturbative corrections based on SCC-DFTB/CHARMM trajectories and higher level single point energy calculations were found to give satisfactory results. Combining all factors gave a root-mean-square difference of 0.7 pKa units for the relative pKa values of the small molecules compared to experiment. For the residues in the lysozyme, an accurate pKa shift was obtained for His 31 with multiple nanosecond simulations. For Lys 102, however, the pKa shift was estimated to be too large, even after more than 10 nanosecond simulations for each lambda window; the difficulty was due to the significant, but slow, reorganization of the protein and water structure when Lys 102 was protonated. The simulations support that Lys 102 is deprotonated in the X-ray structure and the protein is highly destabilized when this residue is protonated.     

Role of structural water molecule in HIV protease-inhibitor complexes: a QM/MM study

Structural water molecule 301 found at the interface of HIV protease-inhibitor complexes function as a hydrogen bond (H-bond) donor to carbonyl groups of the inhibitor as well as H-bond acceptor to amide/amine groups of the flap region of the protease. In this study, six systems of HIV protease-inhibitor complexes were analyzed, which have the presence of this "conserved" structural water molecule using a two-layer QM/MM ONIOM method. The combination of QM/MM and QM method enabled the calculation of strain energies of the bound ligands as well as the determination of their binding energies in the ligand-water and ligand-water-protease complexes. Although the ligand experiences considerable strain in the protein bound structure, the H-bond interactions through the structural water overcomes this strain effect to give a net stability in the range of 16-24 kcal/mol. For instance, in 1HIV system, the strain energy of the ligand was 12.2 kcal/mol, whereas the binding energy associated with the structural water molecule was 20.8 kcal/mol. In most of the cases, the calculated binding energy of structural water molecule showed the same trend as that of the experimental binding free energy values. Further, the classical MD simulations carried out on 1HVL system with and without structural water 301 showed that this conserved water molecule enhances the H-bond dynamics occurring at the Asp-bound active site region of the protease-inhibitor system, and therefore it will have a direct influence on the mechanism of drug action.     

A Mixed QM/MM Scoring Function to Predict Protein-Ligand Binding Affinity

Computational methods for predicting protein-ligand binding free energy continue to be popular as a potential cost-cutting method in the drug discovery process. However, accurate predictions are often difficult to make as estimates must be made for certain electronic and entropic terms in conventional force field based scoring functions. Mixed quantum mechanics/molecular mechanics (QM/MM) methods allow electronic effects for a small region of the protein to be calculated, treating the remaining atoms as a fixed charge background for the active site. Such a semi-empirical QM/MM scoring function has been implemented in AMBER using DivCon and tested on a set of 23 metalloprotein-ligand complexes, where QM/MM methods provide a particular advantage in the modeling of the metal ion. The binding affinity of this set of proteins can be calculated with an R(2) of 0.64 and a standard deviation of 1.88 kcal/mol without fitting and 0.71 and a standard deviation of 1.69 kcal/mol with fitted weighting of the individual scoring terms. In this study we explore using various methods to calculate terms in the binding free energy equation, including entropy estimates and minimization standards. From these studies we found that using the rotational bond estimate to ligand entropy results in a reasonable R(2) of 0.63 without fitting. We also found that using the ESCF energy of the proteins without minimization resulted in an R(2) of 0.57, when using the rotatable bond entropy estimate.     

Pseudobond ab initio QM/MM approach and its applications to enzyme reactions

This perspective article mainly focuses on the development and applications of a pseudobond ab initio QM/MM approach to study enzyme reactions. The following aspects of methodology development are discussed: the approaches for the QM/MM covalent boundary problem, an efficient iterative optimization procedure, the methods to determine enzyme reaction paths, and the approaches to calculate free energy change in enzyme reactions. Several applications are described to illustrate the capability of the methods. Finally, future directions are discussed.     

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