Quantum mechanics/molecular mechanics electrostatic embedding with continuous and discrete functions

A quantum mechanics/molecular mechanics (QM/MM) implementation that uses the Gaussian electrostatic model (GEM) as the MM force field is presented. GEM relies on the reproduction of electronic density by using auxiliary basis sets to calculate each component of the intermolecular interaction. This hybrid method has been used, along with a conventional QM/MM (point charges) method, to determine the polarization on the QM subsystem by the MM environment in QM/MM calculations on 10 individual H(2)O dimers and a Mg(2+)-H(2)O dimer. We observe that GEM gives the correct polarization response in cases when the MM fragment has a small charge, while the point charges produce significant over-polarization of the QM subsystem and in several cases present an opposite sign for the polarization contribution. In the case when a large charge is located in the MM subsystem, for example, the Mg(2+) ion, the opposite is observed at small distances. However, this is overcome by the use of a damped Hermite charge, which provides the correct polarization response.     

Performance assessment of semiempirical molecular orbital methods in describing halogen bonding: quantum mechanical and quantum mechanical/molecular mechanical-molecular dynamics study

The performance of semiempirical molecular-orbital methods--MNDO, MNDO-d, AM1, RM1, PM3 and PM6--in describing halogen bonding was evaluated, and the results were compared with molecular mechanical (MM) and quantum mechanical (QM) data. Three types of performance were assessed: (1) geometrical optimizations and binding energy calculations for 27 halogen-containing molecules complexed with various Lewis bases (Two of the tested methods, AM1 and RM1, gave results that agree with the QM data.); (2) charge distribution calculations for halobenzene molecules, determined by calculating the solvation free energies of the molecules relative to benzene in explicit and implicit generalized Born (GB) solvents (None of the methods gave results that agree with the experimental data.); and (3) appropriateness of the semiempirical methods in the hybrid quantum-mechanical/molecular-mechanical (QM/MM) scheme, investigated by studying the molecular inhibition of CK2 protein by eight halobenzimidazole and -benzotriazole derivatives using hybrid QM/MM molecular-dynamics (MD) simulations with the inhibitor described at the QM level by the AM1 method and the rest of the system described at the MM level. The pure MM approach with inclusion of an extra point of positive charge on the halogen atom approach gave better results than the hybrid QM/MM approach involving the AM1 method. Also, in comparison with the pure MM-GBSA (generalized Born surface area) binding energies and experimental data, the calculated QM/MM-GBSA binding energies of the inhibitors were improved by replacing the G(GB,QM/MM) solvation term with the corresponding G(GB,MM) term.     

Effective multipoles and Yukawa electrostatics in dressed molecule theory

In this paper we derive the multipolar expansion of the screened Coulomb potential in electrolyte solutions with molecular solvent. The solute and solvent molecules can have arbitrary sizes, shapes, and internal charge distributions. We use the exact statistical mechanical definition of renormalized charge distributions coming from "dressed molecule theory" to determine the effective multipoles of a molecule immersed in an electrolyte. The effects of many-body correlations are fully included in our formally exact theory. We restrict ourselves to sufficiently dilute solutions so the screened Coulomb potential decays for large distances like a Yukawa function, exp(-kappa r)/r, where r is the distance and 1/kappa is the decay length (it is normally different from the Debye length). The resulting "Yukawa electrostatics" differ in many respects from ordinary, unscreened electrostatics. The "Yukawa charge" of a molecule (the lowest order moment in the multipolar expansion) is in general not equal to its Coulombic charge and it is not the integral of the renormalized charge distribution of the molecule. Moreover, as shown in this paper, the multipolar expansion of the Yukawa potential does not correspond, contrary to the case of the Coulomb potential, to its asymptotic expansion for large r. As a consequence, the charge term in the multipolar expansion is not the leading term in the asymptotic expansion. Instead, for large r values, multipoles of all orders contribute to the leading asymptotic term. Thus, the electrostatic potential from, for example, an electroneutral solvent molecule in an electrolyte solution has generally the same range as that from an ion. The proper asymptotic expansion for electrostatic interactions in electrolytes is derived. It is briefly shown how the multipole expansion formalism can also be applied in the Poisson-Boltzmann approximation for primitive model electrolytes.     

Clarification of the role of protein in carbonmonoxy myoglobin by investigating electronic states

This article reports the proton tautomerization effects of distal histidine residues in carbonmonoxy myoglobin according to the density functional calculations of the whole protein. The electron eigenstates and electrostatic potential (ESP) distributed around heme and its pocket vary significantly depending on the protonation positions of the distal histidine residues. To investigate the range over which the electronic structures are affected by the proton tautomerization, the quantum mechanics/molecular mechanics (QM/MM) method is applied to probe the QM size to reproduce the atomic partial charges and ESP around the active center. Consequently, we show that these properties converged for the 300 pm QM/MM system in this study. During the analysis, we also find that amino residues such as Phe43, Val68, and Phe138 interact strongly with heme through orbital mixing, indicating that the protein is a medium not only interacting with the reaction center, but also buffering on electrons. © 2013 Wiley Periodicals, Inc.     

CPMD/GULP QM/MM interface for modeling periodic solids: Implementation and its application in the study of Y‐zeolite supported Rhn clusters

We report here the development of hybrid quantum mechanics/molecular mechanics (QM/MM) interface between the plane‐wave density functional theory based CPMD code and the empirical force‐field based GULP code for modeling periodic solids and surfaces. The hybrid QM/MM interface is based on the electrostatic coupling between QM and MM regions. The interface is designed for carrying out full relaxation of all the QM and MM atoms during geometry optimizations and molecular dynamics simulations, including the boundary atoms. Both Born–Oppenheimer and Car–Parrinello molecular dynamics schemes are enabled for the QM part during the QM/MM calculations. This interface has the advantage of parallelization of both the programs such that the QM and MM force evaluations can be carried out in parallel to model large systems. The interface program is first validated for total energy conservation and parallel scaling performance is benchmarked. Oxygen vacancy in α‐cristobalite is then studied in detail and the results are compared with a fully QM calculation and experimental data. Subsequently, we use our implementation to investigate the structure of rhodium cluster (Rh_n; n = 2 to 6) formed from Rh(C₂H₄)₂ complex adsorbed within a cavity of Y‐zeolite in a reducible atmosphere of H₂ gas. © 2016 Wiley Periodicals, Inc.     

Reaction path potential for complex systems derived from combined ab initio quantum mechanical and molecular mechanical calculations

Combined ab initio quantum mechanical and molecular mechanical calculations have been widely used for modeling chemical reactions in complex systems such as enzymes, with most applications being based on the determination of a minimum energy path connecting the reactant through the transition state to the product in the enzyme environment. However, statistical mechanics sampling and reaction dynamics calculations with a combined ab initio quantum mechanical (QM) and molecular mechanical (MM) potential are still not feasible because of the computational costs associated mainly with the ab initio quantum mechanical calculations for the QM subsystem. To address this issue, a reaction path potential energy surface is developed here for statistical mechanics and dynamics simulation of chemical reactions in enzymes and other complex systems. The reaction path potential follows the ideas from the reaction path Hamiltonian of Miller, Handy and Adams for gas phase chemical reactions but is designed specifically for large systems that are described with combined ab initio quantum mechanical and molecular mechanical methods. The reaction path potential is an analytical energy expression of the combined quantum mechanical and molecular mechanical potential energy along the minimum energy path. An expansion around the minimum energy path is made in both the nuclear and the electronic degrees of freedom for the QM subsystem internal energy, while the energy of the subsystem described with MM remains unchanged from that in the combined quantum mechanical and molecular mechanical expression and the electrostatic interaction between the QM and MM subsystems is described as the interaction of the MM charges with the QM charges. The QM charges are polarizable in response to the changes in both the MM and the QM degrees of freedom through a new response kernel developed in the present work. The input data for constructing the reaction path potential are energies, vibrational frequencies, and electron density response properties of the QM subsystem along the minimum energy path, all of which can be obtained from the combined quantum mechanical and molecular mechanical calculations. Once constructed, it costs much less for its evaluation. Thus, the reaction path potential provides a potential energy surface for rigorous statistical mechanics and reaction dynamics calculations of complex systems. As an example, the method is applied to the statistical mechanical calculations for the potential of mean force of the chemical reaction in triosephosphate isomerase.     

A critical evaluation of different QM/MM frontier treatments with SCC-DFTB as the QM method

The performance of different link atom based frontier treatments in QM/MM simulations was evaluated critically with SCC-DFTB as the QM method. In addition to the analysis of gas-phase molecules as in previous studies, an important element of the present work is that chemical reactions in realistic enzyme systems were also examined. The schemes tested include all options available in the program CHARMM for SCC-DFTB/MM simulation, which treat electrostatic interactions due to the MM atoms close to the QM/MM boundary in different ways. In addition, a new approach, the divided frontier charge (DIV), has been implemented in which the partial charge associated with the frontier MM atom ("link host") is evenly distributed to the other MM atoms in the same group. The performance of these schemes was evaluated based on properties including proton affinities, deprotonation energies, dipole moments, and energetics of proton transfer reactions. Similar to previous work, it was found that calculated proton affinities and deprotonation energies of alcohols, carbonic acids, amino acids, and model DNA bases are very sensitive to the link atom scheme; the commonly used single link atom approach often gives error on the order of 15 to 20 kcal/mol. Other schemes give better and, on average, mutually comparable results. For proton transfer reactions, encouragingly, both activation barriers and reaction energies are fairly insensitive (within a typical range of 2-4 kcal/mol) to the link atom scheme due to error cancellation, and this was observed for both gas-phase and enzyme systems. Therefore, the effect of using different link atom schemes in QM/MM simulations is rather small for chemical reactions that conserve the total charge. Although the current study used an approximate DFT method as the QM level, the observed trends are expected to be applicable to QM/MM methods with use of other QM approaches. This observation does not mean to encourage QM/MM simulations without careful benchmark in the study of specific systems, rather it emphasizes that other technical details, such as the treatment of long-range electrostatics, tend to play a more important role and need to be handled carefully.     

Geometry optimization using tuned and balanced redistributed charge schemes for combined quantum mechanical and molecular mechanical calculations

We performed geometry optimizations using the tuned and balanced redistributed charge algorithms to treat the QM-MM boundary in combined quantum mechanical and molecular mechanical (QM/MM) methods. In the tuned and balanced redistributed charge (TBRC) scheme, the QM boundary atom is terminated by a tuned F link atom, and the charge of the MM boundary atom is properly adjusted to conserve the total charge of the entire QM/MM system; then the adjusted MM boundary charge is moved evenly to the midpoints of the bonds between the MM boundary atom and its neighboring MM atoms. In the tuned and balanced redistributed charge-2 (TBRC2) scheme, the adjusted MM boundary charge is moved evenly to all MM atoms that are attached to the MM boundary atom. A new option, namely charge smearing, has been added to the TBRC scheme, yielding the tuned and balanced smeared redistributed charge (TBSRC) scheme. In the new scheme, the redistributed charges near the QM-MM boundary are smeared to make the electrostatic interactions between the QM region and the redistributed charges more realistic. The TBRC2 scheme and new TBSRC scheme have been tested for various kinds of bonds at a QM-MM boundary, including C-C, C-N, C-O, O-C, N-C, C-S, S-S, S-C, C-Si, and O-N bonds. Charge smearing is necessary if the redistributed charges are close to the QM region, as in the TBSRC scheme, but not if the redistributed charge is farther from the QM region, as in the TBRC2 scheme. We found that QM/MM results using either the TBRC2 scheme or the TBSRC scheme agree well with full QM results; the mean unsigned error (MUE) of the QM/MM deprotonation energy is 1.6 kcal/mol in both cases, and the MUE of QM/MM optimized bond lengths over the three bonds closest to the QM-MM boundary, with errors averaged over the protonated forms and unprotonated forms, is 0.015 ? for TBRC2 and 0.021 ? for TBSRC. The improvements in the new scheme are essential for QM-MM boundaries that pass through a polar bond, but even for boundaries that pass through C-C bonds, the improvement can be quite significant.     

A numerically stable restrained electrostatic potential charge fitting method

Inspired by the idea of charge decomposition in calculation of the dipole preserving and polarization consistent charges (Zhang et al., J. Comput. Chem. 2011, 32, 2127), we have proposed a numerically stable restrained electrostatic potential (ESP)‐based charge fitting method for protein. The atomic charge is composed of two parts. The dominant part is fixed to a predefined value (e.g., AMBER charge), and the residual part is to be determined by restrained fitting to residual ESP on grid points around the molecule. Nonuniform weighting factors as a function of the dominant charge are assigned to the atoms. Because the residual part is several folds to several orders smaller than the dominant part, the impact of ill‐conditioning is alleviated. This charge fitting method can be used in quantum mechanical/molecular mechanical (QM/MM) simulations and similar studies, where QM calculated electronic properties are frequently mapped to partial atomic charges. © 2012 Wiley Periodicals, Inc.     

Communication: Quantum polarized fluctuating charge model: a practical method to include ligand polarizability in biomolecular simulations

We present a simple and practical method to include ligand electronic polarization in molecular dynamics (MD) simulation of biomolecular systems. The method involves periodically spawning quantum mechanical (QM) electrostatic potential (ESP) calculations on an extra set of computer processors using molecular coordinate snapshots from a running parallel MD simulation. The QM ESPs are evaluated for the small-molecule ligand in the presence of the electric field induced by the protein, solvent, and ion charges within the MD snapshot. Partial charges on ligand atom centers are fit through the multi-conformer restrained electrostatic potential (RESP) fit method on several successive ESPs. The RESP method was selected since it produces charges consistent with the AMBER/GAFF force-field used in the simulations. The updated charges are introduced back into the running simulation when the next snapshot is saved. The result is a simulation whose ligand partial charges continuously respond in real-time to the short-term mean electrostatic field of the evolving environment without incurring additional wall-clock time. We show that (1) by incorporating the cost of polarization back into the potential energy of the MD simulation, the algorithm conserves energy when run in the microcanonical ensemble and (2) the mean solvation free energies for 15 neutral amino acid side chains calculated with the quantum polarized fluctuating charge method and thermodynamic integration agree better with experiment relative to the Amber fixed charge force-field.     

Toward a consistent treatment of polarization in model QM/MM calculations

The concept of model chemistries within hybrid QM/MM calculations has been addressed through analysis of the polarization energy determined by two distinct approaches based on (i) induced charges and (ii) induced dipoles. The quantum mechanical polarization energy for four configurations of the water dimer has been determined for a range of basis sets using Morokuma energy decomposition analysis. This benchmark value has been compared to the fully classical polarization energy determined using the induced dipole approach, and the molecular mechanics polarization energy calculated using induced charges within the MM region of hybrid QM/MM calculations. From the water dimer calculations, it is concluded that the induced charge approach is consistent with medium sized basis set calculations whereas the induced dipole approach is consistent with large basis set calculations. This result is highly relevant to the concept of QM/MM model chemistries.     

A Molecular Mechanics Energy Partitioning Software for Biomolecular Systems

The partitioning of the molecular mechanics (MM) energy in calculations involving biomolecular systems is important to identify the source of major stabilizing interactions, e.g., in ligand–protein interactions, or to identify residues with considerable contributions in hybrid multiscale calculations, i.e., quantum mechanics/molecular mechanics (QM/MM). Here, we describe Energy Split, a software program to calculate MM energy partitioning considering the AMBER Hamiltonian and parameters. Energy Split includes a graphical interface plugin for VMD to facilitate the selection of atoms and molecules belonging to each part of the system. Energy Split is freely available at or can be easily installed through the VMD Store.     

Intraprotein electrostatics derived from first principles: divide-and-conquer approaches for QM/MM calculations

Two divide-and-conquer (DAQ) approaches for building multipole-based molecular electrostatic potentials of proteins are presented and evaluated for use in QM/MM calculations. One approach is a further development of the neutralization method of Bellido and Rullmann (J Comput Chem 1989, 10, 479-487) while the other is based on removing part of the electron density before performing the multipole expansion. Both methods create systems with integer charges without using charge renormalization. To determine their performance in terms of location of cuts and distance to QM region, the new DAQ approaches are tested in calculations of the proton affinity of N(zeta) of Lys55 in the inhibitor turkey ovomucoid third domain. Finally, the two methods are used to build a variety of MM regions, applied to calculations of the pK(a) of Lys55, and compared to other computational methodologies in which force field charges are employed.     

The large quadrupole of water molecules

Many quantum mechanical calculations indicate water molecules in the gas and liquid phase have much larger quadrupole moments than any of the common site models of water for computer simulations. Here, comparisons of multipoles from quantum mechanical∕molecular mechanical (QM∕MM) calculations at the MP2∕aug-cc-pVQZ level on a B3LYP∕aug-cc-pVQZ level geometry of a waterlike cluster and from various site models show that the increased square planar quadrupole can be attributed to the p-orbital character perpendicular to the molecular plane of the highest occupied molecular orbital as well as a slight shift of negative charge toward the hydrogens. The common site models do not account for the p-orbital type electron density and fitting partial charges of TIP4P- or TIP5P-type models to the QM∕MM dipole and quadrupole give unreasonable higher moments. Furthermore, six partial charge sites are necessary to account reasonably for the large quadrupole, and polarizable site models will not remedy the problem unless they account for the p-orbital in the gas phase since the QM calculations show it is present there too. On the other hand, multipole models by definition can use the correct multipoles and the electrostatic potential from the QM∕MM multipoles is much closer than that from the site models to the potential from the QM∕MM electron density. Finally, Monte Carlo simulations show that increasing the quadrupole in the soft-sticky dipole-quadrupole-octupole multipole model gives radial distribution functions that are in good agreement with experiment.     

Conformational dependence of charges in protein simulations

We have studied the conformational dependence of molecular mechanics atomic charges for proteins by calculating the charges fitted to the quantum mechanical (QM) electrostatic potential (ESP) for all atoms in complexes between avidin and seven biotin analogues for 20 snapshots from molecular dynamics simulations. We have studied how various other charge sets reproduce those charges. The QM charges, even if averaged over all snapshots or all residues, in general have a larger magnitude than standard Amber charges, indicating that the restraint toward zero in the restrained ESP method is too strong. This has a significant influence on the electrostatic conformational energies and the interaction energy between the biotin ligand and the protein, giving a difference between the QM and Amber charges of 43 and 8 kJ/mol for the negatively charged and neutral biotin analogues, respectively (3-4%). However, this energy difference is strongly reduced if the solvation energy (calculated by the Poisson-Boltzmann or Generalized Born methods) is added, viz., to 7 kJ/mol for charged and 3 kJ/mol for uncharged ligand. In fact, charges need to be recalculated with a QM method only for residues within 7 or 4 A of the ligand, if the error should be less than 4 kJ/mol. Unfortunately, the QM charges do not give significantly better MM/PBSA estimates of ligand-binding affinities than standard Amber charges.     

Theoretical modeling of the heme group with a hybrid QM/MM method

The quality of the results obtained in calculations with the hybrid QM/MM method IMOMM on systems where the heme group is partitioned in QM and MM regions is evaluated through the performance of calculations on the 4‐coordinate [Fe(P)] (P = porphyrin), the 5‐coordinate [Fe(P)(1−(Me)Im)] (Im = imidazole) and the 6‐coordinate [Fe(P)(1−(Me)Im)(O₂)] systems. The results are compared with those obtained from much more expensive pure quantum mechanics calculations on model systems. Three different properties are analyzed—namely, the optimized geometries, the binding energies of the axial ligands to the heme group, and the energy cost of the biochemically relevant out‐of‐plane displacement of the iron atom. Agreement is especially good in the case of optimized geometries and energy cost of out‐of‐plane displacements, with larger discrepancies in the case of binding energies. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 282–294, 2000