Python implementation of the restrained electrostatic potential charge model

The restrained electrostatic potential (RESP) charge model is widely used in molecular dynamics simulations, especially for the AMBER and GAFF force fields. We have implemented the RESP scheme using the accessible and widely used Python language and the NumPy numerical library. This article provides a programming-oriented introduction to the RESP scheme and highlights some of the features of NumPy that are useful in scientific computing.     

A new method to determine electrostatic potential around a macromolecule in solution from molecular wave functions

The three-dimensional reference interaction site model integral equation theory (3D-RISM) combined with the ab initio molecular orbital method (3D-RISM-SCF) is applied to a solvated macromolecular system. The solvation structure around a solute molecule is obtained from the 3D-RISM integral equation under the electrostatic potential of the solute molecule, calculated by the ab initio molecular orbital theory. The electrostatic potential should be calculated on each grid point in the three-dimensional real space. Therefore, the calculation of the electrostatic potential is the most time consuming part in this method. In this article, we propose a new procedure to save the computational cost for calculating the electrostatic potential and the solvated fock matrix. The strategy of this procedure is to evaluate the electrostatic potential and the solvated fock matrix in different ways, depending on the distance between solute and solvent. Inside the repulsive cores of solute atoms, it is possible to avoid the calculation of electrostatic potential and solvated Fock matrix by assuming the potential to be infinity. In the region sufficiently far from solute, they are evaluated classically by putting the effective point charge on each atom. In the intermediate region, the electrostatic potential is evaluated directly by integrating the molecular orbitals of the solute molecule. The electronic structure and the energy gradient of Methionine-Enkephalin and solvation structure are estimated by using this procedure in aqueous solution, and are compared with the results from other procedures. The results are compared also with those from the continuum model.     

Ab initio quantum chemical calculation of electron density,electrostatic potential,and electric field of biomolecule based on fragment molecular orbital method

Efficient quantum chemical calculations of electrostatic properties, namely, the electron density (EDN), electrostatic potential (ESP), and electric field (EFL), were performed using the fragment molecular orbital (FMO) method. The numerical errors associated with the FMO scheme were examined at the HF, MP2, and RI‐MP2 levels of theory using 4 small peptides. As a result, the FMO errors in the EDN, ESP, and EFL were significantly smaller than the magnitude of the electron correlation effects, which indicated that the FMO method provides sufficiently accurate values of electrostatic properties. In addition, an attempt to reduce the computational effort was proposed by combining the FMO scheme and a point charge approximation. The error due to this approximation was examined using 2 proteins, prion protein and human immunodeficiency virus type 1 protease. As illustrative examples, the ESP values at the molecular surface of these proteins were calculated at the MP2 level of theory.     

Nonexistence of local maxima in molecular electrostatic potential maps

It has been rigorously established by means of classical electrostatic arguments, that molecular electrostatic potential maps are devoid of local maxima. This forms a generalization of the earlier works of Politzer and co-workers which were restricted to the case of atoms.     

The molecular electrostatic potential of some simple molecules

The calculation of the molecular electrostatic potential from simplified models of the electron density is considered. Results are shown for water, hydrogen fluoride and ammonia. Little loss of accuracy is evident when the density is represented by a linear sum of well-chosen Gaussians. When these are further simplified into sets of point charges the inner parts of the molecule are poorly represented. More elaborate point moments make the representation worse. On the other hand a mixed representation with point charges and one diffuse Gaussian gives all the essential features of the potential of these molecules.     

Protein‐specific force field derived from the fragment molecular orbital method can improve protein–ligand binding interactions

Accurate computational estimate of the protein–ligand binding affinity is of central importance in rational drug design. To improve accuracy of the molecular mechanics (MM) force field (FF) for protein–ligand simulations, we use a protein‐specific FF derived by the fragment molecular orbital (FMO) method and by the restrained electrostatic potential (RESP) method. Applying this FMO‐RESP method to two proteins, dodecin, and lysozyme, we found that protein‐specific partial charges tend to differ more significantly from the standard AMBER charges for isolated charged atoms. We did not see the dependence of partial charges on the secondary structure. Computing the binding affinities of dodecin with five ligands by MM PBSA protocol with the FMO‐RESP charge set as well as with the standard AMBER charges, we found that the former gives better correlation with experimental affinities than the latter. While, for lysozyme with five ligands, both charge sets gave similar and relatively accurate estimates of binding affinities. © 2013 Wiley Periodicals, Inc.     

A potential with low point charges for pure siliceous zeolites

A modified CHARMM force‐field (ZHB potential) with low point charges for silica was previously proposed by Zimmerman et al. (J. Chem. Theory Comput. 2011, 7, 1695). The ZHB potential is advantageous for quantum mechanics/molecular mechanics simulations as it minimizes the electron spill‐out problems. In the same spirit, here we propose a modified ZHB potential (MZHB) by reformulating its bonding potential, while retaining the nonbonding potential as in the ZHB force‐field. We show that several structural and dynamic properties of silica, like the IR spectrum, distribution functions, mechanical properties, and negative thermal expansion computed using the MZHB potential agree well with experimental data. Further, transferability of MZHB is also tested for reproducing the crystallographic structures of several polymorphs of silica. © 2015 Wiley Periodicals, Inc.     

Site charge models for molecular electrostatic potentials of cycloalkanes and tetrahedrane

The molecular structures of cycloalkanes (from cyclopropane to cyclodecane) and tetrahedrane were optimized at the Hartree–Fock/6–31G** level and their molecular electric potentials (MEPs) were calculated using a geodesic grid. The MEPs were fitted using net atomic charges and several site charge models. The net atomic charge model gave very poor fits to the MEPs in every case. A model with additional methylene bisector charge sites, similar to one successfully used previously for linear alkanes, greatly improved the fits to the MEPs of these cycloalkanes. The MEPs of the highly strained molecules cyclopropane and tetrahedrane were further studied using ring center and displaced bond charge sites. The fitting of the MEP of cyclopropane was consistent with a banana bond model with asymmetrically displaced electron density in the C C bonds. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 579–585, 1999     

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.     

A combined electronegativity equalization and electrostatic potential fit method for the determination of atomic point charges

We report an approach for the determination of atomic monopoles of macromolecular systems using connectivity and geometry parameters alone. The method is appropriate also for the calculation of charge distributions based on the quantum mechanically determined wave function and does not suffer from the mathematical instability of other electrostatic potential fit methods.     

Accurate molecular electrostatic potentials based on modified PRDDO/M wave functions. I. Electrostatic potential derived atomic charges

A new approach for the calculation of electrostatic potential derived atomic charges is presented. Based on molecular orbital calculations in the PRDDO/M approximation, the new parametrized electrostatic potential (PESP) method is parametrized against ab initio MP2/6-31G** calculations. For a data set of 820 atoms in 145 molecules containing H, C, N. O, F, P, S, Cl, and Br (including hypervalent species), the PESP method achieves a mean absolute error of 0.037 e⁻ with a correlation coefficient of 0.990. Unlike other approximate approaches, no scaling factor is required to improve the agreement between PESP charges and the underlying ab initio results. PESP calculations are an order of magnitude faster than the simplest ab initio calculation (STO-3G) on large molecules while achieving a level of accuracy that rivals much more elaborate ab initio methods. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 955–969, 1997     

Near ab initio quality molecular electrostatic potential maps using hybridization displacement charges at PM3 level and effects of geometrical changes in amino groups

Charge distributions, dipole moments, and molecular electrostatic potentials (MEP) around several molecules consisting of carbon, nitrogen, oxygen, fluorine, sulfur, and chlorine atoms were studied using the PM3 semiempirical method and the results compared with those obtained using ab initio calculations at the RHF/6‐31G** level. Thus it is shown that relative MEP values near different atoms can be obtained using hybridization displacement charges (HDC) obtained by employing PM3 density matrices that usually agree quite satisfactorily with the ab initio ones. Further, positions of ab initio MEP minima are correctly located and the corresponding relative MEP values usually correctly predicted using the PM3(HDC) charges distributed continuously in three dimensions according to the forms of squares of valence s atomic orbitals. The necessary parameters for HDC calculations using the PM3 method were optimized. It is shown how within the frameworks of both PM3 and AM1 methods the π electrons or lone pairs associated with amino group nitrogen atoms and ring atoms can be satisfactorily treated in different situations. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 299–312, 2001     

Predicting reduction potentials of 1,3,6‐triphenyl fulvenes using molecular electrostatic potential analysis of substituent effects

The influence of mono‐ and multiple substituent effect on the reduction potential (E⁰) of 1,3,6‐triphenyl fulvenes is investigated using B3LYP‐SMD/6‐311+G(d,p) level density functional theory. The molecular electrostatic potential (MESP) minimum at the fulvene π‐system (V_min) and the change in MESP at any of the fulvene carbon atoms (ΔV_C) for both neutral and reduced forms are used as excellent measures of substituent effect from the para and meta positions of the 1,3 and 6‐phenyl moieties. Substitution at 6‐phenyl para position has led to significant change in E⁰ than any other positions. By applying the additivity rule of substituent effects, an equation in ΔV_C is derived to predict E⁰ for multiply substituted fulvenes. Further, E⁰ is predicted for a set of 2000 hexa‐substituted fulvene derivatives where the substituents and their positions in the system are chosen in a random way. The calculated E⁰ agreed very well with the experimental E⁰ reported by Godman et al. Predicting E⁰ solely by substituent effect offers a simple and powerful way to select suitable combinations of substituents on fulvene system for light harvesting applications. © 2018 Wiley Periodicals, Inc.     

New charged microheterogeneous system — A microemulsion with droplets of variable electrostatic potential

A method has been proposed for varying the electrostatic potential of droplets in emulsions of the oil/water type by introducing nonsurface-active hydrophilic—hydrophobic electrolytes into the aqueous phase. Consideration has been given to the simplest two-phase model using standard Galvani potentials for interphase transfer of the individual ions to describe the properties of the electrolyte. From calculations based on this model relationships have been obtained between the electrostatic potential of the microphase and the electrolyte properties: concentration, droplet dimensions of the dispersed phase, and its mole fraction. Limiting conditions have been discovered in terms of the droplet dimensions and mole fraction of the dispersed phase. Calculations have demonstrated the possibility of controlling the electrostatic potential of the emulsion droplets in the range ± 300 mV.N. N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, 117977 Moscow. Translated from Izvestiya Akademii Nauk, Seriya Khimicheskaya, No. 10, pp. 2245–2252, October, 1992.     

Principal component analysis of the effects of wavefunction modification on the electrostatic potential of indole

The molecular electrostatic potential (MEP) of the indole molecule was calculated in a three‐dimensional grid in which the molecule was centered at the origin. To evaluate the dependence of MEP on the type of calculation, semiempirical, ab initio, and density functional theory methods with different basis sets were employed. The data matrix generated by these calculations was analyzed by principal component analysis (PCA). The appearance of outliers and the effect of wavefunction modifications such as the introduction of electron correlations and diffuse functions were highlighted by the use of PCA. The spatial localization of such effects around the molecule was possible from the loadings values associated with the graphical analysis of the grid points. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Analysis of the transferability of atomic multipoles for amino acids in modeling macromolecular charge distribution from fragments

Cumulative Atomic Multipole Moments were calculated for all natural amino acids and symmetric cyclic hexapeptides within Self‐Consisted Field (SCF) and Density Functional Theory (DFT) approaches using a standard 6‐31G(d,p) basis set. These data were used to analyze in detail the quality and the conformational and the intermolecular transferability of molecular charge distributions expressed in the atomic multipole form. Intermolecular interaction energies were reproduced reasonably by CAMM transformed from other conformations. Good transferability of CAMM based model was also achieved between similar molecular environments, which opens a route to modeling electrostatic effects in highly symmetric (e.g., crystalline) systems. Transferability deficiencies of various charge distribution models were analyzed and attributed to different levels of multipole expansion. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1082–1097, 2001     

Quantum mechanical/molecular mechanical self-consistent Madelung potential method for treatment of polar molecular crystals

The self-consistent Madelung potential (SCMP) approach for calculating molecular wave functions for a subunit embedded in a symmetrical environment constituted by the copies of the subunit is implemented with semiempirical NDDO model Hamiltonians and supplemented with empirically parameterized dispersion–repulsion interaction potentials. This model yields sublimation enthalpies in good agreement with available experimental data for a series of molecular crystals, including imidazol, benzimidazole, urea, urethane, dicyaneamide, formamide, uracil, cytosine, maleic anhydride, succinic anhydride, and 1,3,5-triamino-2,4,6-trinitro-benzene. The SCMP-NDDO method, which avoids difficulties concerning the parametrization of charges in the molecular mechanics force fields, is proposed mainly for the treatment of molecular crystals with large unit cells. It might be particularly useful where important charge reorganization is expected under the effect of the crystal field. Charge distributions, obtained by the SCMP and the simple dielectric cavity self-consistent reaction field models, are compared and analyzed. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 38–50, 1998     

Atomic multipoles: Electrostatic potential fit,local reference axis systems,and conformational dependence

Currently, all standard force fields for biomolecular simulations use point charges to model intermolecular electrostatic interactions. This is a fast and simple approach but has deficiencies when the electrostatic potential (ESP) is compared to that from ab initio methods. Here, we show how atomic multipoles can be rigorously implemented into common biomolecular force fields. For this, a comprehensive set of local reference axis systems is introduced, which represents a universal solution for treating atom‐centered multipoles for all small organic molecules and proteins. Furthermore, we introduce a new method for fitting atomic multipole moments to the quantum mechanically derived ESP. This methods yields a 50–90% error reduction compared to both point charges fit to the ESP and multipoles directly calculated from the ab initio electron density. It is shown that it is necessary to directly fit the multipole moments of conformational ensembles to the ESP. Ignoring the conformational dependence or averaging over parameters from different conformations dramatically deteriorates the results obtained with atomic multipole moments, rendering multipoles worse than partial charges. © 2012 Wiley Periodicals, Inc.