A charge-scaling implementation of the variational electrostatic projection method

Two new charge-scaling methods for efficient modeling of the solvated macromolecular environment in hybrid QM/MM calculations of biological reactions are presented. The methods are extensions of the variational electrostatic projection (VEP) method, and allows a subset of atomic charges in the external environment to be adjusted to mimic, in the active dynamical region, the electrostatic potential and field due to the large surrounding macromolecule and solvent. The method has the advantages that it offers improved accuracy, does not require the use of a three-dimensional grid or auxiliary set of fitting points, and requires only minor molecular simulation code modifications. The VEP-cs and VEP-RVM+cs methods are able to attain very high accuracy (relative force errors of 10(-7) or better with appropriate choice of control parameters), and take advantage of a recently introduced set of high-order discretization schemes and Gaussian exponents for boundary element solvation and VEP methods. The methods developed here serve as potentially powerful tools in the arsenal of computational techniques used in multiscale computational modeling problems.     

Interactions between phosphate and water in solution: a natural bond orbital based analysis in a QM/MM framework

The natural bond orbital (NBO) and natural energy decomposition analysis (NEDA) calculations are used to analyze the interaction between mono-methyl phosphate-ester (MMP) and its solvation environment in a combined quantum mechanical/molecular mechanical (QM/MM) framework. The solute-solvent configurations are generated using a specific parametrization of the self-consistent-charge density functional tight-binding (SCC-DFTB) model for the MMP and TIP3P for water. The NBO and NEDA calculations are done with several QM/MM partitioning schemes with HF/6-31+G** as the QM level. Regardless of the size of the QM region, a notable amount of charge transfer is observed between MMP and the neighboring water molecules and the charge-transfer interactions are, in the NEDA framework, as important as the electric (electrostatic and polarization) components. This work illustrates that NBO based analyses are effective tools for probing intermolecular interactions in condensed phase systems.     

Toward a new approach for determination of solute's charge distribution to analyze interatomic electrostatic interactions in quantum mechanical/molecular mechanical simulations

We present an alternative approach to determine "density-dependent property"-derived charges for molecules in the condensed phase. In the case of a solution, it is essential to take into consideration the electron polarization of molecules in the active site of this system. The solute and solvent molecules in this site have to be described by a quantum mechanical technique and the others are allowed to be treated by a molecular mechanical method (QM/MM scheme). For calculations based on this scheme, using the forces and interaction energy as density-dependent property our charges from interaction energy and forces (CHIEF) approach can provide the atom-centered charges on the solute atoms. These charges reproduce well the electrostatic potentials around the solvent molecules and present properly the picture of the electron density of the QM subsystem in the solution system. Thus, the CHIEF charges can be considered as the atomic charges under the conditions of the QM/MM simulation, and then enable one to analyze electrostatic interactions between atoms in the QM and MM regions. This approach would give a view of the QM nuclei and electrons different from the conventional methods.     

High-order discretization schemes for biochemical applications of boundary element solvation and variational electrostatic projection methods

A series of high-order surface element discretization schemes for variational boundary element methods are introduced. The surface elements are chosen in accord with angular quadrature rules for integration of spherical harmonics. Surface element interactions are modeled by Coulomb integrals between spherical Gaussian functions with exponents chosen to reproduce the exact variational energy and Gauss's law for a point charge in a spherical cavity. The present work allows high-order surface element expansions to be made for variational methods such as the conductorlike screening model for solvation and the variational electrostatic projection method for generalized solvent boundary potentials in molecular simulations.     

Density embedded VB/MM: a hybrid ab initio VB/MM with electrostatic embedding

A hybrid QM/MM method that combines ab initio valence-bond (VB) with molecular mechanics (MM) is presented. The method utilizes the ab initio VB approach to describe the reactive fragments and MM to describe the environment thus allows VB calculations of reactions in large biological systems. The method, termed density embedded VB/MM (DE-VB/MM), is an extension of the recently developed VB/MM method. It involves calculation of the electrostatic interaction between the reactive fragments and their environment using the electrostatic embedding scheme. Namely, the electrostatic interactions are represented as one-electron integrals in the ab initio VB Hamiltonian, hence taking into account the wave function polarization of the reactive fragments due to the environment. Moreover, the assumptions that were utilized in an earlier version of the method, VB/MM, to formulate the electrostatic interactions effect on the off-diagonal matrix elements are no longer required in the DE-VB/MM methodology. Using DE-VB/MM, one can calculate, in addition to the adiabatic ground state reaction profile, the energy of the diabatic VB configurations as well as the VB state correlation diagram for the reaction. The abilities of the method are exemplified on the identity SN2 reaction of a chloride anion with methyl chloride in aqueous solution. Both the VB configurations diagram and the state correlation diagram are presented. The results are shown to be in very good agreement with both experimental and other computational data, suggesting that DE-VB/MM is a proper method for application to different reactivity problems in biological systems.     

Prediction of the mechanisms of enzyme-catalysed reactions using hybrid quantum mechanical/molecular mechanical methods

The use of hybrid methods, involving both quantum mechanics and molecular mechanics, to model the mechanism of enzyme-catalysed reactions, is discussed. Two alternative approaches to treating the electrostatic interactions between the quantum mechanical and molecular mechanical regions are studied, involving either the inclusion of this term in the electronic Hamiltonian (QM/MM), or evaluating it purely classically (MO + MM). In the latter scheme, possible problems of using force fields that are standard for macromolecular modelling are identified. The use of QM/MM schemes to investigate the mechanism of the enzymes thymidine phosphorylase (ThdPase) and protein tyrosine phosphatase (PTP) is described. For both systems, transition states have been identified using a PM3 Hamiltonian. For ThdPase, concerted motion of the enzyme during the course of the reaction is suggested and, for PTP, a two-step dephosphorylation reaction is indicated, both with quite low barriers.     

Fragment-based quantum mechanical methods for periodic systems with Ewald summation and mean image charge convention for long-range electrostatic interactions

We describe an Ewald-summation method to incorporate long-range electrostatic interactions into fragment-based electronic structure methods for periodic systems. The present method is an extension of the particle-mesh Ewald technique for combined quantum mechanical and molecular mechanical (QM/MM) calculations, and it has been implemented into the explicit polarization (X-Pol) potential to illustrate the computational details. As in the QM/MM-Ewald method, the X-Pol-Ewald approach is a linear-scaling electrostatic method, in which the short-range electrostatic interactions are determined explicitly in real space and the long-range Ewald pair potential is incorporated into the Fock matrix as a correction. To avoid the time-consuming Fock matrix update during the self-consistent field procedure, a mean image charge (MIC) approximation is introduced, in which the running average with a user-chosen correlation time is used to represent the long-range electrostatic correction as an average effect. Test simulations on liquid water show that the present X-Pol-Ewald method takes about 25% more CPU time than the usual X-Pol method using spherical cutoff, whereas the use of the MIC approximation reduces the extra costs for long-range electrostatic interactions by 15%. The present X-Pol-Ewald method provides a general procedure for incorporating long-range electrostatic effects into fragment-based electronic structure methods for treating biomolecular and condensed-phase systems under periodic boundary conditions.     

QM/MM-PBSA method to estimate free energies for reactions in proteins

We have developed a method to estimate free energies of reactions in proteins, called QM/MM-PBSA. It estimates the internal energy of the reactive site by quantum mechanical (QM) calculations, whereas bonded, electrostatic, and van der Waals interactions with the surrounding protein are calculated at the molecular mechanics (MM) level. The electrostatic part of the solvation energy of the reactant and the product is estimated by solving the Poisson-Boltzmann (PB) equation, and the nonpolar part of the solvation energy is estimated from the change in solvent-accessible surface area (SA). Finally, the change in entropy is estimated from the vibrational frequencies. We test this method for five proton-transfer reactions in the active sites of [Ni,Fe] hydrogenase and copper nitrite reductase. We show that QM/MM-PBSA reproduces the results of a strict QM/MM free-energy perturbation method with a mean absolute deviation (MAD) of 8-10 kJ/mol if snapshots from molecular dynamics simulations are used and 4-14 kJ/mol if a single QM/MM structure is used. This is appreciably better than the original QM/MM results or if the QM energies are supplemented with a point-charge model, a self-consistent reaction field, or a PB model of the protein and the solvent, which give MADs of 22-36 kJ/mol for the same test set.     

Reliable treatment of electrostatics in combined QM/MM simulation of macromolecules

A robust approach for dealing with electrostatic interactions for spherical boundary conditions has been implemented in the QM/MM framework. The development was based on the generalized solvent boundary potential (GSBP) method proposed by Im et al. [J. Chem. Phys. 114, 2924 (2001)], and the specific implementation was applied to the self-consistent-charge density-functional tight-binding approach as the quantum mechanics (QM) level, although extension to other QM methods is straightforward. Compared to the popular stochastic boundary-condition scheme, the new protocol offers a balanced treatment between quantum mechanics/molecular mechanics (QM/MM) and MM/MM interactions; it also includes the effect of the bulk solvent and macromolecule atoms outside of the microscopic region at the Poisson-Boltzmann level. The new method was illustrated with application to the enzyme human carbonic anhydrase II and compared to stochastic boundary-condition simulations using different electrostatic treatments. The GSBP-based QM/MM simulations were most consistent with available experimental data, while conventional stochastic boundary simulations yielded various artifacts depending on different electrostatic models. The results highlight the importance of carefully treating electrostatics in QM/MM simulations of biomolecules and suggest that the commonly used truncation schemes should be avoided in QM/MM simulations, especially in simulations that involve extensive conformational samplings. The development of the GSBP-based QM/MM protocol has opened up the exciting possibility of studying chemical events in very complex biomolecular systems in a multiscale framework.     

Assessment of weak intermolecular interactions across QM/MM noncovalent boundaries

An assessment of a number of quantum mechanical/molecular mechanical (QM/MM) combinations was performed for weak intermolecular interactions across noncovalent QM/MM 'boundaries'. The popular S22 data set, comprising of a number of weak hydrogen-bonded, dispersion-bound and complexes with mixed interactions was used for the assessment. A range of QM methods was combined with a number of popular MM force fields. The single-point interaction energies, at reference geometries, are presented as deviations to accurate CCSD(T)/CBS reference values. This investigation employed both additive and subtractive QM/MM schemes. The density functional has only a negligible effect; the choice of basis set was also negligible in terms of accuracy. The importance of selecting the most appropriate MM force field for accurately describing interactions at the noncovalent 'boundary' region has a dramatic effect on the accuracy.     

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.     

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.     

The Jahn-Teller effect of the TiIII ion in aqueous solution: extended ab initio QM/MM molecular dynamics simulations.

Combined ab initio quantum mechanical/molecular mechanical (QM/MM) molecular dynamics simulations, including only the first and the first and second hydration shells in the QM region, were performed for TiIII in aqueous solution. The hydration structure of TiIII is discussed in terms of radial distribution functions, coordination-number distributions and several angle distributions. Dynamical properties, such as librational and vibrational motions and TiIII-O vibrations, were evaluated. A fast dynamical Jahn-Teller effect of TiIII(aq) was observed in the QM/MM simulations, in particular when the second hydration shell was included into the QM region. The results justify the computational effort required for the inclusion of the second hydration shell into the QM region and show the importance of this effort for obtaining accurate hydration-shell geometries, dynamical properties, and details of the Jahn-Teller effect.     

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.     

Effect of polarization on the opsin shift in rhodopsins. 1. A combined QM/QM/MM model for bacteriorhodopsin and pharaonis sensory rhodopsin II

The optical and IR-spectroscopic properties of the protonated Schiff base of retinal are highly sensitive to the electrostatic environment. This feature makes retinal a useful probe to study structural differences and changes in rhodopsins. It also raises an interest to theoretically predict the spectroscopic response to mutation and structural evolution. Computational models appropriate for this purpose usually combine sophisticated quantum mechanical (QM) methods with molecular mechanics (MM) force fields. In an effort to test and improve the accuracy of these QM/MM models, we consider in this article the effects of polarization and inter-residual charge transfer within the binding pocket of bacteriorhodopsin (bR) and pharaonis sensory rhodopsin II (psRII, also called pharaonis phoborhodopsin, ppR) on the excitation energy using an ab initio QM/QM/MM approach. The results will serve as reference for assessing empirical polarization models in a consecutive article.     

Contributions of pauli repulsions to the energetics and physical properties computed in QM/MM methods

Conventional combined quantum mechanical/molecular mechanical (QM/MM) methods lack explicit treatment of Pauli repulsions between the quantum‐mechanical and molecular‐mechanical subsystems. Instead, classical Lennard‐Jones (LJ) potentials between QM and MM nuclei are used to model electronic Pauli repulsion and long‐range London dispersion, despite the fact that the latter two are inherently of quantum nature. Use of the simple LJ potential in QM/MM methods can reproduce minimal geometries and energies of many molecular clusters reasonably well, as compared to full QM calculations. However, we show here that the LJ potential cannot correctly describe subtle details of the electron density of the QM subsystem because of the neglect of Pauli repulsions between the QM and MM subsystems. The inaccurate electron density subsequently affects the calculation of electronic and magnetic properties of the QM subsystem. To explicitly consider Pauli interactions with QM/MM methods, we propose a method to use empirical effective potentials on the MM atoms. The test case of the binding energy and magnetic properties of a water dimer shows promising results for the general application of effective potentials to mimic Pauli repulsions in QM/MM calculations. © 2013 Wiley Periodicals, Inc.     

Experimental and theoretical investigation of the first overtone spectrum of 1,3,5-trinitrotoluene

Vibrational overtone spectroscopy is a powerful tool for studying intramolecular and intermolecular interactions. We report on a combined experimental and modeling study of the C-H stretch first overtone of bulk 1,3,5-trinitrotoluene (TNT) and TNT on fumed-silica powder. We recorded the overtone spectra by laser photoacoustic spectroscopy and compared them with those predicted with the harmonically coupled anharmonic oscillator model in the 5600-6600 cm(-1) region. The model systems included single molecules and hybrid quantum and molecular mechanical (QM:MM) clusters to account for the effects of intermolecular interactions on the observed spectra. We performed the hybrid QM:MM calculations at the HF/6-31+G(d,p), B3LYP/6-31+G(d,p), and MP2/6-31+G(d,p) levels of theory and with the universal force field (UFF) to account for van der Waals and electrostatic effects from surrounding molecules. Overtone spectra calculated from the MP2 level of theory, using a HF/3-21+G* calculation to assign molecular charges in the MM layer, and the Merz-Singh-Kollman population analysis for assigning partial charge in the QM layer and determining the transition dipole moment agreed best with the experimental data.     

A regularized and renormalized electrostatic coupling Hamiltonian for hybrid quantum-mechanical-molecular-mechanical calculations

We describe a regularized and renormalized electrostatic coupling Hamiltonian for hybrid quantum-mechanical (QM)-molecular-mechanical (MM) calculations. To remedy the nonphysical QM/MM Coulomb interaction at short distances arising from a point electrostatic potential (ESP) charge of the MM atom and also to accommodate the effect of polarized MM atom in the coupling Hamiltonian, we propose a partial-wave expansion of the ESP charge and describe the effect of a s-wave expansion, extended over the covalent radius r(c), of the MM atom. The resulting potential describes that, at short distances, large scale cancellation of Coulomb interaction arises intrinsically from the localized expansion of the MM point charge and the potential self-consistently reduces to 1r(c) at zero distance providing a renormalization to the Coulomb energy near interatomic separations. Employing this renormalized Hamiltonian, we developed an interface between the Car-Parrinello molecular-dynamics program and the classical molecular-dynamics simulation program Groningen machine for chemical simulations. With this hybrid code we performed QM/MM calculations on water dimer, imidazole carbon monoxide (CO) complex, and imidazole-heme-CO complex with CO interacting with another imidazole. The QM/MM results are in excellent agreement with experimental data for the geometry of these complexes and other computational data found in literature.     

Hybrid density functional theory/molecular mechanics calculations of two-photon absorption of dimethylamino nitro stilbene in solution

The dimethylamino nitro stilbene (DANS) molecule is studied for exploring solvent effects on two-photon absorption using the quantum mechanical/molecular mechanical (QM/MM) response theory approach, where the quantum part is represented by density functional theory. We have explored the role of geometrical change of the chromophore in solution, the importance of taking a dynamical average over the sampled structures and the role of a granular representation of the polarization and electrostatic interactions with the classically described medium. The line shape function was simulated by the QM/MM technique thereby allowing for non-empirical prediction of the absolute two-photon cross section. We report a maximum in the TPA cross section for a medium of intermediate solvent polarity (i.e. in chloroform) and provide the grounds for an explanation of this effect which recently has been experimentally observed for a series of charge transfer species in solvents of different polarity. The calculations of absorption energies reproduce well the positive solvatochromic behavior of DANS and are in good agreement with experimental spectra available for the chloroform and DMSO solvents. In line with recent development of the QM/MM response technique for color modeling, we find this methodology to offer a versatile tool to predict and analyze two-photon absorption phenomena taking place within a medium.