Tryptophan side chain electrostatic interactions determine edge-to-face vs parallel-displaced tryptophan side chain geometries in the designed beta-hairpin "trpzip2"

The interaction geometries of the four tryptophan (Trp) side chains in the 12-residue designed beta-hairpin trpzip2 are investigated using all-atom explicit-solvent molecular dynamics simulations. The experimentally observed edge-to-face (EtF) pairwise interaction geometries are stable on a time scale of 10 ns. However, removing the electrostatic multipoles of the Trp side chains while retaining the dipoles of the side chains' NH moieties induces a conformational change to a geometry in which three of the four side chains interact in a parallel-displaced (PD) manner. Free energy simulations of the Etf to PD conformational change reveal that, with the side chain multipole moments intact (+MP), the EtF conformation is preferred by 5.79 kcal/mol. Conversely, with only the dipole moments of the side chain NH moieties intact (-MP), the PD conformation's free energy is more favorable by 1.71 kcal/mol. In contrast to energetic similarities for Trp side chain-water electrostatic and Trp side chain-Trp side chain and Trp side chain-water van der Waals, +MP Trp side chain-Trp side chain electrostatic interactions are more favorable by 4.21 kcal/mol in the EtF conformation, while in the -MP case the EtF and PD conformations' Trp side chain-Trp side chain electrostatic energies are nearly identical. The results highlight the importance of electrostatic multipole moments in determining aromatic-aromatic interaction geometries in aqueous biomolecular systems and argue for the inclusion of this physics in simplified models used for protein-ligand docking and protein structure prediction, possibly through a truncated Coulomb term between aromatic moieties.     

Progress in the understanding of drug-receptor interactions, part 2: experimental and theoretical electrostatic moments and interaction energies of an angiotensin II receptor antagonist (C30H30N6(O)3S)

A combined experimental and theoretical charge density study of an angiotensin II receptor antagonist (1) is presented focusing on electrostatic properties such as atomic charges, molecular electric moments up to the fourth rank and energies of the intermolecular interactions, to gain an insight into the physical nature of the drug-receptor interaction. Electrostatic properties were derived from both the experimental electron density (multipole refinement of X-ray data collected at T=17 K) and the ab initio wavefunction (single molecule and fully periodic calculations at the DFT level). The relevance of SO and SN intramolecular interactions on the activity of 1 is highlighted by using both the crystal and gas-phase geometries and their electrostatic nature is documented by means of QTAIM atomic charges. The derived electrostatic properties are consistent with a nearly spherical electron density distribution, characterised by an intermingling of electropositive and -negative zones rather than by a unique electrophilic region opposed to a nucleophilic area. This makes the first molecular moment scarcely significant and ill-determined, whereas the second moment is large, significant and highly reliable. A comparison between experimental and theoretical components of the third electric moment shows a few discrepancies, whereas the agreement for the fourth electric moment is excellent. The most favourable intermolecular bond is show to be an NHN hydrogen bond with an energy of about 50 kJ mol(-1). Key pharmacophoric features responsible for attractive electrostatic interactions include CHX hydrogen bonds. It is shown that methyl and methylene groups, known to be essential for the biological activity of the drug, provide a significant energetic contribution to the total binding energy. Dispersive interactions are important at the thiophene and at both the phenyl fragments. The experimental estimates of the electrostatic contribution to the intermolecular interaction energies of six molecular pairs, obtained by a new model proposed by Spackman, predict the correct relative electrostatic energies with no exceptions.     

Prediction of conformationally dependent atomic multipole moments in carbohydrates

The conformational flexibility of carbohydrates is challenging within the field of computational chemistry. This flexibility causes the electron density to change, which leads to fluctuating atomic multipole moments. Quantum Chemical Topology (QCT) allows for the partitioning of an “atom in a molecule,” thus localizing electron density to finite atomic domains, which permits the unambiguous evaluation of atomic multipole moments. By selecting an ensemble of physically realistic conformers of a chemical system, one evaluates the various multipole moments at defined points in configuration space. The subsequent implementation of the machine learning method kriging delivers the evaluation of an analytical function, which smoothly interpolates between these points. This allows for the prediction of atomic multipole moments at new points in conformational space, not trained for but within prediction range. In this work, we demonstrate that the carbohydrates erythrose and threose are amenable to the above methodology. We investigate how kriging models respond when the training ensemble incorporating multiple energy minima and their environment in conformational space. Additionally, we evaluate the gains in predictive capacity of our models as the size of the training ensemble increases. We believe this approach to be entirely novel within the field of carbohydrates. For a modest training set size of 600, more than 90% of the external test configurations have an error in the total (predicted) electrostatic energy (relative to ab initio) of maximum 1 kJ mol^?1 for open chains and just over 90% an error of maximum 4 kJ mol^?1 for rings. © 2015 Wiley Periodicals, Inc.     

Accuracy and tractability of a kriging model of intramolecular polarizable multipolar electrostatics and its application to histidine

We propose a generic method to model polarization in the context of high‐rank multipolar electrostatics. This method involves the machine learning technique kriging, here used to capture the response of an atomic multipole moment of a given atom to a change in the positions of the atoms surrounding this atom. The atoms are malleable boxes with sharp boundaries, they do not overlap and exhaust space. The method is applied to histidine where it is able to predict atomic multipole moments (up to hexadecapole) for unseen configurations, after training on 600 geometries distorted using normal modes of each of its 24 local energy minima at B3LYP/apc‐1 level. The quality of the predictions is assessed by calculating the Coulomb energy between an atom for which the moments have been predicted and the surrounding atoms (having exact moments). Only interactions between atoms separated by three or more bonds (“1, 4 and higher” interactions) are included in this energy error. This energy is compared with that of a central atom with exact multipole moments interacting with the same environment. The resulting energy discrepancies are summed for 328 atom–atom interactions, for each of the 29 atoms of histidine being a central atom in turn. For 80% of the 539 test configurations (outside the training set), this summed energy deviates by less than 1 kcal mol^?1. © 2013 Wiley Periodicals, Inc.     

Physical nature of ethidium and proflavine interactions with nucleic acid bases in the intercalation plane

On the basis of the crystallographic structures of three nucleic acid intercalation complexes involving ethidium and proflavine, we have analyzed the interaction energies between intercalator chromophores and their four nearest bases, using a hybrid variation-perturbation method at the second-order M?ller-Plesset theory level (MP2) with a 6-31G(d,p) basis set. A total MP2 interaction energy minimum precisely reproduces the crystallographic position of the ethidium chromophore in the intercalation plane between UA/AU bases. The electrostatic component constitutes the same fraction of the total energy for all three studied structures. The multipole electrostatic interaction energy, calculated from cumulative atomic multipole moments (CAMMs), was found to converge only after including components above the fifth order. CAMM interaction surfaces, calculated on grids in the intercalation planes of these structures, reasonably reproduce the alignment of intercalators in crystal structures; they exhibit additional minima in the direction of the DNA grooves, however, which also need to be examined at higher theory levels if no crystallographic data are given.     

Transferability of quantum topological atoms in terms of electrostatic interaction energy

Understanding atomic transferability is important to guide the design of a force field. Atoms in molecules are defined and computed according to the theory of quantum chemical topology (QCT). The electron density associated with such topological atoms is conveniently described by high-rank multipole moments. Here, we assess the transferability of atoms by means of their electrostatic interaction energy, using a convergent multipole expansion. The test systems are (H2O)3 and serine...(H2O)5. The effect of a varying electron density cutoff (i.e., truncating the atoms) is discussed and the effect of polarization is quantified.     

The effects of atomic multipole moments obtained by the potential-derived method on hydrogen bonding

A potential-derived atomic multipole method called the cumulative potential-derived atomic multipole method is developed, with which electrostatic atomic multipole moments are derived by fitting the molecular electric potential in a cumulative way. It is applied to the hydrides of N , O , F , S , Cl , and methanol and the hydrogen-bonded dimers formed between them. The relationship between atomic multipole moments and molecular charge distributions is found. The structures calculated with Buckingham's electrostatic model are in good agreement with experiments. The phenomena of nonlinear structures of most H -bonded complexes—the deviations of symmetry axes of electron donors from H bonds—and correct distinguishing between two alternative structures are attributed to atomic dipole and quadrupole moments. Compared with other methods, this method has a quantitative and qualitative advantage and simple algorithm. The main conclusion is that the atomic multipole moments play a substantial role, although a potential-derived charge model was deemed sufficient previously. © 1993 John Wiley & Sons, Inc.     

The elusive atomic rationale for DNA base pair stability

A systematic analysis of the electrostatic interaction between 27 natural DNA base pairs was carried out, based on ab initio correlated wave functions and the topology of the electron density. Using high rank multipole moments we show that the atomic partitioning of the interaction energy contains many substantial contributions between distant atoms. Profiles of cumulative energy versus internuclear distance show large fluctuations and provide an electrostatic fingerprint of the partitioning of interaction energy in a complex. A quantified comparison between each pair of energy profiles, one for each base pair, makes clear that there is no correlation between the total base pair interaction energy and the shape of the profile. In other words, base pairs with similar interaction energy are not stable for the same reasons in terms of atomic partitioning. In summary, simple rules to rationalize the pattern of energetic stability of naturally occurring base pairs in terms of subsets of atoms are elusive. Our work cautions against inappropriate use of Jorgensen's secondary interaction hypothesis.     

Calculation of electrostatic interaction energies in molecular dimers from atomic multipole moments obtained by different methods of electron density partitioning

Accurate and fast evaluation of electrostatic interactions in molecular systems is still one of the most challenging tasks in the rapidly advancing field of macromolecular chemistry, including molecular recognition, protein modeling and drug design. One of the most convenient and accurate approaches is based on a Buckingham-type approximation that uses the multipole moment expansion of molecular/atomic charge distributions. In the mid-1980s it was shown that the pseudoatom model commonly used in experimental X-ray charge density studies can be easily combined with the Buckingham-type approach for calculation of electrostatic interactions, plus atom-atom potentials for evaluation of the total interaction energies in molecular systems. While many such studies have been reported, little attention has been paid to the accuracy of evaluation of the purely electrostatic interactions as errors may be absorbed in the semiempirical atom-atom potentials that have to be used to account for exchange repulsion and dispersion forces. This study is aimed at the evaluation of the accuracy of the calculation of electrostatic interaction energies with the Buckingham approach. To eliminate experimental uncertainties, the atomic moments are based on theoretical single-molecule electron densities calculated at various levels of theory. The electrostatic interaction energies for a total of 11 dimers of alpha-glycine, N-acetylglycine and L-(+)-lactic acid structures calculated according to Buckingham with pseudoatom, stockholder and atoms-in-molecules moments are compared with those evaluated with the Morokuma-Ziegler energy decomposition scheme. For alpha-glycine a comparison with direct "pixel-by-pixel" integration method, recently developed Gavezzotti, is also made. It is found that the theoretical pseudoatom moments combined with the Buckingham model do predict the correct relative electrostatic interactions energies, although the absolute interaction energies are underestimated in some cases. The good agreement between electrostatic interaction energies computed with Morokuma-Ziegler partitioning, Gavezzotti's method, and the Buckingham approach with atoms-in-molecules moments demonstrates that reliable and accurate evaluation of electrostatic interactions in molecular systems of considerable complexity is now feasible.     

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.     

Cumulative potential-derived atomic multipoles

A method to derive the atomic multipole moments cumulatively up to quadrupole moments was developed. The multipole moments are obtained by least-square simulating the molecular electrostatic potentials. Only the components of the term of highest order in the atomic multipole expansion are optimized while the lower terms remain fixed. The calculations on HF, H₂O and NH₃ show that the cumulative method can give reasonable qualitative and fairly good quantitative results.     

Point atomic multipole moments for simulation of electrostatic potential and field in all-siliceous zeolites

Calibration method of atomic multipole moments (AMMs) is presented with respect to geometries of all-siliceous zeolite models obtained with X-ray diffraction (XRD) methods. Mulliken atomic charges and AMMs are calculated for all-siliceous types possessing small size elementary unit cells at the hybrid density functional theory (DFT) (B3LYP) and general gradient approximation (GGA) Perdew-Burke-Ernzerhof (PBE) levels and then used to fit the dependences versus geometry variables for the Mulliken charges and versus special coordinate for the AMMs. Fitted and exact charges and AMMs are used to compute electrostatic potential (EP) and electric field (EF) for all-siliceous zeolites with CRYSTAL. A possibility of application of the point AMMs to quantum mechanical/molecular mechanics computations or classic simulation of physical adsorption is evaluated. The considered models expand over wide range of structural parameters and could be applied even to amorphous all-siliceous systems.     

Base stacking in cytosine dimer. A comparison of correlated ab initio calculations with three empirical potential models and density functional theory calculations

Ab initio MP2/6-31G* interaction energies were calculated for more than 80 geometries of stacked cytosine dimer. Diffuse polarization functions were used to properly cover the dispersion energy. The results of ab initio calculations were compared with those obtained from three electrostatic empirical potential models, constructed as the sum of a Lennard-Jones potential (covering dispersion and repulsion contributions) and the electrostatic term. Point charges and point multipoles of the electrostatic term were also obtained at the MP2/6-31G* level of theory. The point charge MEP model (atomic charges derived from molecular electrostatic potential) satisfactorily reproduced the ab initio data. Addition of π-charges localized below and above the cytosine plane did not affect the calculated energies. The model employing the distributed multipole analysis gave worse agreement with the ab initio data than the MEP approach. The MP2 MEP charges were also derived using larger sets of atomic orbitals: cc-pVDZ, 6-311 + G(2d, p), and aug-cc-pVDZ. Differences between interaction energies calculated using these three sets of point charges and the MP2/6-31G* charges were smaller than 0.8 kcal/mol. The correlated ab initio calculations were also compared with the density functional theory (DFT) method. DFT calculations well reproduced the electrostatic part of interaction energy. They also covered some nonelectrostatic short-range effects which were not reproduced by the empirical potentials. The DFT method does not include the dispersion energy. This energy, approximated by an empirical term, was therefore added to the DFT interaction energy. The resulting interaction energy exhibited an artifact secondary minimum for a 3.9-4.0 vertical separation of bases. This defect is inherent in the DFT functionals, because it is not observed for the Hartree-Fock + dispersion interaction energy.© 1996 John Wiley & Sons, Inc.     

A convergent multipole expansion for 1,3 and 1,4 Coulomb interactions

Traditionally force fields express 1,3 and 1,4 interactions as bonded terms via potentials that involve valence and torsion angles, respectively. These interactions are not modeled by point charge terms, which are confined to electrostatic interactions between more distant atoms (1,n where n>4). Here we show that both 1,3 and 1,4 interactions can be described on the same footing as 1,n (n>4) interactions by a convergent multipole expansion of the Coulomb energy of the participating atom pairs. The atomic multipole moments are generated by the theory of quantum chemical topology. The procedure to make the multipole expansion convergent is based on a "shift procedure" described in earlier work [L. Joubert and P. L. A. Popelier, Molec. Phys. 100, 3357 (2002)].     

Halogen bonding: a theoretical study based on atomic multipoles derived from quantum theory of atoms in molecules

Atomic multipole moments derived from quantum theory of atoms in molecules are used to study halogen bonds in dihalogens (with general formula YX, in which X refers to the halogen directly interacted with the Lewis base) and some molecules containing C–X group. Multipole expansion is used to calculate the electrostatic potential in a vicinity of halogen atom (which is involved in halogen bonding) in terms of atomic monopole, dipole, and quadrupole moments. In all the cases, the zz component of atomic traceless quadrupole moments (where z axis taken along Y–X or C–X bonds) of the halogens plays a stabilizing role in halogen bond formation. The effects of atomic monopole and dipole moments on the formation of a halogen bond in YX molecules depend on Y and X atoms. In Br₂ and Cl₂, the monopole moment of halogens is zero and has no contribution in electrostatic potential and hence in halogen bonding, while in ClBr, FBr, and FCl it is positive and therefore stabilize the halogen bonds. On the other hand, the negative sign of dipole moment of X in all the YX molecules weakens the corresponding halogen bonds. In the C–X-containing molecules, monopole and dipole moments of X atom are negative and consequently destabilize the halogen bonds. So, in these molecules the quadrupole moment of X atom is the only electrostatic term which strengthens the halogen bonds. In addition, we found good linear correlations between halogen bonds strength and electrostatic potentials calculated from multipole expansion.     

Long range behavior of high-rank topological multipole moments

The construction of a high-rank multipolar force field (for peptides) is a complex task, leading to several intermediate questions in need of a clear answer. Here we focus on the convergence of the (electrostatic) multipolar expansion at medium and long range. Using molecular electron densities, quantum chemical topology (QCT) defines the atoms as finite volumes, each endowed with multipole moments. The terms in the multipole expansion are grouped according to powers of the internuclear distance, R(-L). Given two atom types at a given distance, we determine which rank (L) is necessary for the electrostatic energy to converge to the exact interaction energy within a certain error. With this information, the rank of the expansion for each interaction can be adapted to the required accuracy and the available computing power.     

Atomic forces for geometry‐dependent point multipole and Gaussian multipole models

In standard treatments of atomic multipole models, interaction energies, total molecular forces, and total molecular torques are given for multipolar interactions between rigid molecules. However, if the molecules are assumed to be flexible, two additional multipolar atomic forces arise because of (1) the transfer of torque between neighboring atoms and (2) the dependence of multipole moment on internal geometry (bond lengths, bond angles, etc.) for geometry‐dependent multipole models. In this study, atomic force expressions for geometry‐dependent multipoles are presented for use in simulations of flexible molecules. The atomic forces are derived by first proposing a new general expression for Wigner function derivatives . The force equations can be applied to electrostatic models based on atomic point multipoles or Gaussian multipole charge density. Hydrogen‐bonded dimers are used to test the intermolecular electrostatic energies and atomic forces calculated by geometry‐dependent multipoles fit to the ab initio electrostatic potential. The electrostatic energies and forces are compared with their reference ab initio values. It is shown that both static and geometry‐dependent multipole models are able to reproduce total molecular forces and torques with respect to ab initio, whereas geometry‐dependent multipoles are needed to reproduce ab initio atomic forces. The expressions for atomic force can be used in simulations of flexible molecules with atomic multipoles. In addition, the results presented in this work should lead to further development of next generation force fields composed of geometry‐dependent multipole models. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010