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1.  The active site of hen egg-white lysozyme: flexibility and chemical bonding 
Chemical bonding at the active site of lysozyme is analyzed on the basis of a multipole model employing transferable multipole parameters from a database. Large B factors at low temperatures reflect frozen-in disorder, but therefore prevent a meaningful free refinement of multipole parameters.
Chemical bonding at the active site of hen egg-white lysozyme (HEWL) is analyzed on the basis of Bader’s quantum theory of atoms in molecules [QTAIM; Bader (1994 ▶), Atoms in Molecules: A Quantum Theory. Oxford University Press] applied to electron-density maps derived from a multipole model. The observation is made that the atomic displacement parameters (ADPs) of HEWL at a temperature of 100 K are larger than ADPs in crystals of small biological molecules at 298 K. This feature shows that the ADPs in the cold crystals of HEWL reflect frozen-in disorder rather than thermal vibrations of the atoms. Directly generalizing the results of multipole studies on small-molecule crystals, the important consequence for electron-density analysis of protein crystals is that multipole parameters cannot be independently varied in a meaningful way in structure refinements. Instead, a multipole model for HEWL has been developed by refinement of atomic coordinates and ADPs against the X-ray diffraction data of Wang and coworkers [Wang et al. (2007), Acta Cryst. D63, 1254–1268], while multipole parameters were fixed to the values for transferable multipole parameters from the ELMAM2 database [Domagala et al. (2012), Acta Cryst. A68, 337–351] . Static and dynamic electron densities based on this multipole model are presented. Analysis of their topological properties according to the QTAIM shows that the covalent bonds possess similar properties to the covalent bonds of small molecules. Hydrogen bonds of intermediate strength are identified for the Glu35 and Asp52 residues, which are considered to be essential parts of the active site of HEWL. Furthermore, a series of weak C—H⋯O hydrogen bonds are identified by means of the existence of bond critical points (BCPs) in the multipole electron density. It is proposed that these weak interactions might be important for defining the tertiary structure and activity of HEWL. The deprotonated state of Glu35 prevents a distinction between the Phillips and Koshland mechanisms.
PMCID: PMC3975892  PMID: 24699657
hen egg-white lysozyme; multipole model; multipole parameters
2.  Topological Properties of Chemical Bonds from Static and Dynamic Electron Densities 
Dynamic and static electron densities (EDs) based on the independent spherical atom model (IAM) and multipole (MP) models of crambin were successfully computed, holding no series-termination effects. The densities are compared to EDs of small biological molecules at diverse temperatures. It is outlined that proteins exhibit an intrinsic flexibility, present as frozen disorder at 100 K, in contrast to small molecules. The flexibility of the proteins is reflected by atomic displacement parameters (B-factors), which are considerably larger than for small molecules at 298 K. Thus, an optimal deconvolution of deformation density and thermal motion is not guaranteed, which prevents a free refinement of MP parameters but allows an application of transferable, fixed MP parameters. The analysis of the topological properties, such as the density at bond critical points (BCPs) and the Laplacian, reveals systematic differences between static and dynamic EDs. Zero-point-vibrations, yet present in dynamic EDs at low temperature, affect but marginally the EDs of small molecules. The zero-point-vibrations cause a smearing of the ED, which becomes more pronounced with increasing temperature. Topological properties, primarily the Laplacian, of covalent bonds appear to be more sensitive to effects by temperature and the polarity of the bonds. However, dynamic EDs at ca. 20 K based on MP models provide a good characterization of chemical bonding. Both the density at BCPs and the Laplacian of hydrogen bonds constitute similar values from static and dynamic EDs for all studied temperatures. Deformation densities demonstrate the necessity of the employment of MP parameters in order to comprise the nature of covalent bonds. The character of hydrogen bonds can be roughly pictured by IAM, whereas MP parameters are recommended for a classification of hydrogen bonds beyond a solely interpretation of topological properties.
PMCID: PMC4431502  PMID: 25995522
Amino acids; Electron densities; Chemical bonding; Proteins; X-ray diffraction
3.  Atomic Forces for Geometry-Dependent Point Multipole and Gaussian Multipole Models 
Journal of computational chemistry  2010;31(15):2702-2713.
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 due to 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 the current 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 ∂Dlm′m/∂Ω. 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 inter-molecular electrostatic energies and atomic forces calculated by geometry-dependent multipoles fit to the ab initio electrostatic potential (ESP). The electrostatic energies and forces are compared to 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, while 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.
PMCID: PMC2941241  PMID: 20839297
Multipole; Gaussian Multipole; Force; Torque; Wigner Function
4.  High-Resolution Crystal Structures of Protein Helices Reconciled with Three-Centered Hydrogen Bonds and Multipole Electrostatics 
PLoS ONE  2015;10(4):e0123146.
Theoretical and experimental evidence for non-linear hydrogen bonds in protein helices is ubiquitous. In particular, amide three-centered hydrogen bonds are common features of helices in high-resolution crystal structures of proteins. These high-resolution structures (1.0 to 1.5 Å nominal crystallographic resolution) position backbone atoms without significant bias from modeling constraints and identify Φ = -62°, ψ = -43 as the consensus backbone torsional angles of protein helices. These torsional angles preserve the atomic positions of α-β carbons of the classic Pauling α-helix while allowing the amide carbonyls to form bifurcated hydrogen bonds as first suggested by Némethy et al. in 1967. Molecular dynamics simulations of a capped 12-residue oligoalanine in water with AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications), a second-generation force field that includes multipole electrostatics and polarizability, reproduces the experimentally observed high-resolution helical conformation and correctly reorients the amide-bond carbonyls into bifurcated hydrogen bonds. This simple modification of backbone torsional angles reconciles experimental and theoretical views to provide a unified view of amide three-centered hydrogen bonds as crucial components of protein helices. The reason why they have been overlooked by structural biologists depends on the small crankshaft-like changes in orientation of the amide bond that allows maintenance of the overall helical parameters (helix pitch (p) and residues per turn (n)). The Pauling 3.613 α-helix fits the high-resolution experimental data with the minor exception of the amide-carbonyl electron density, but the previously associated backbone torsional angles (Φ, Ψ) needed slight modification to be reconciled with three-atom centered H-bonds and multipole electrostatics. Thus, a new standard helix, the 3.613/10-, Némethy- or N-helix, is proposed. Due to the use of constraints from monopole force fields and assumed secondary structures used in low-resolution refinement of electron density of proteins, such structures in the PDB often show linear hydrogen bonding.
PMCID: PMC4403875  PMID: 25894612
5.  Gaussian Multipole Model (GMM) 
An electrostatic model based on charge density is proposed as a model for future force fields. The model is composed of a nucleus and a single Slater-type contracted Gaussian multipole charge density on each atom. The Gaussian multipoles are fit to the electrostatic potential (ESP) calculated at the B3LYP/6-31G* and HF/aug-cc-pVTZ levels of theory and tested by comparing electrostatic dimer energies, inter-molecular density overlap integrals, and permanent molecular multipole moments with their respective ab initio values. For the case of water, the atomic Gaussian multipole moments Qlm are shown to be a smooth function of internal geometry (bond length and bond angle), which can be approximated by a truncated linear Taylor series. In addition, results are given when the Gaussian multipole charge density is applied to a model for exchange-repulsion energy based on the inter-molecular density overlap.
PMCID: PMC2832208  PMID: 20209077
Gaussian multipoles; charge density; electrostatic model; multipole; overlap
6.  HPAM: Hirshfeld Partitioned Atomic Multipoles 
Computer physics communications  2012;183(2):390-397.
An implementation of the Hirshfeld (HD) and Hirshfeld-Iterated (HD-I) atomic charge density partitioning schemes is described. Atomic charges and atomic multipoles are calculated from the HD and HD-I atomic charge densities for arbitrary atomic multipole rank lmax on molecules of arbitrary shape and size. The HD and HD-I atomic charges/multipoles are tested by comparing molecular multipole moments and the electrostatic potential (ESP) surrounding a molecule with their reference ab initio values. In general, the HD-I atomic charges/multipoles are found to better reproduce ab initio electrostatic properties over HD atomic charges/multipoles. A systematic increase in precision for reproducing ab initio electrostatic properties is demonstrated by increasing the atomic multipole rank from lmax = 0 (atomic charges) to lmax = 4 (atomic hexadecapoles). Both HD and HD-I atomic multipoles up to rank lmax are shown to exactly reproduce ab initio molecular multipole moments of rank L for L ≤ lmax. In addition, molecular dipole moments calculated by HD, HD-I, and ChelpG atomic charges only (lmax = 0) are compared with reference ab initio values. Significant errors in reproducing ab initio molecular dipole moments are found if only HD or HD-I atomic charges used.
PMCID: PMC3225920  PMID: 22140274
Atomic multipoles; Hirshfeld charges; dipole; quadrupole
7.  Modulated anharmonic ADPs are intrinsic to aperiodic crystals: a case study on incommensurate Rb2ZnCl4  
The superspace maximum entropy method (MEM) density in combination with structure refinements has been used to uncover the modulation in incommensurate Rb2ZnCl4 close to the lock-in transition. Modulated atomic displacement parameters (ADPs) and modulated anharmonic ADPs are found to form an intrinsic part of the modulation. Refined values for the displacement modulation function depend on the presence or absence of modulated ADPs in the model.
A combination of structure refinements, analysis of the superspace MEM density and interpretation of difference-Fourier maps has been used to characterize the incommensurate modulation of rubidium tetrachlorozincate, Rb2ZnCl4, at a temperature of T = 196 K, close to the lock-in transition at T lock-in = 192 K. The modulation is found to consist of a combination of displacement modulation functions, modulated atomic displacement parameters (ADPs) and modulated third-order anharmonic ADPs. Up to fifth-order Fourier coefficients could be refined against diffraction data containing up to fifth-order satellite reflections. The center-of-charge of the atomic basins of the MEM density and the displacive modulation functions of the structure model provide equivalent descriptions of the displacive modulation. Modulations of the ADPs and anharmonic ADPs are visible in the MEM density, but extracting quantitative information about these modulations appears to be difficult. In the structure refinements the modulation parameters of the ADPs form a dependent set, and ad hoc restrictions had to be introduced in the refinements. It is suggested that modulated harmonic ADPs and modulated third-order anharmonic ADPs form an intrinsic part, however small, of incommensurately modulated structures in general. Refinements of alternate models with and without parameters for modulated ADPs lead to significant differences between the parameters of the displacement modulation in these two types of models, thus showing the modulation of ADPs to be important for a correct description of the displacive modulation. The resulting functions do not provide evidence for an interpretation of the modulation by a soliton model.
PMCID: PMC3098556  PMID: 21586828
aperiodic crystals; incommensurate modulated structures; MEM density; ADPs
8.  Directional Dependence of Hydrogen Bonds: a Density-based Energy Decomposition Analysis and Its Implications on Force Field Development 
One well-known shortcoming of widely-used biomolecular force fields is the description of the directional dependence of hydrogen bonding (HB). Here we aim to better understand the origin of this difficulty and thus provide some guidance for further force field development. Our theoretical approaches center on a novel density-based energy decomposition analysis (DEDA) method [J. Chem. Phys., 131, 164112 (2009)], in which the frozen density energy is variationally determined through constrained search. This unique and most significant feature of DEDA enables us to find that the frozen density interaction term is the key factor in determining the HB orientation, while the sum of polarization and charge-transfer components shows very little HB directional dependence. This new insight suggests that the difficulty for current non-polarizable force fields to describe the HB directional dependence is not due to the lack of explicit polarization or charge-transfer terms. Using the DEDA results as reference, we further demonstrate that the main failure coming from the atomic point charge model can be overcome largely by introducing extra charge sites or higher order multipole moments. Among all the electrostatic models explored, the smeared charge distributed multipole model (up to quadrupole), which also takes account of charge penetration effects, gives the best agreement with the corresponding DEDA results. Meanwhile, our results indicate that the van der Waals interaction term needs to be further improved to better model directional hydrogen bonding.
PMCID: PMC3259744  PMID: 22267958
9.  Modeling Organochlorine Compounds and the σ-Hole Effect Using a Polarizable Multipole Force Field 
The journal of physical chemistry. B  2014;118(24):6456-6465.
The charge distribution of halogen atoms on organochlorine compounds can be highly anisotropic and even display a so-called σ-hole, which leads to strong halogen bonds with electron donors. In this paper, we have systematically investigated a series of chloromethanes with one to four chloro substituents using a polarizable multipole-based molecular mechanics model. The atomic multipoles accurately reproduced the ab initio electrostatic potential around chloromethanes, including CCl4, which has a prominent σ-hole on the Cl atom. The van der Waals parameters for Cl were fitted to the experimental density and heat of vaporization. The calculated hydration free energy, solvent reaction fields, and interaction energies of several homo- and heterodimer of chloromethanes are in good agreement with experimental and ab initio data. This study suggests that sophisticated electrostatic models, such as polarizable atomic multipoles, are needed for accurate description of electrostatics in organochlorine compounds and halogen bonds, although further improvement is necessary for better transferability.
PMCID: PMC4065202  PMID: 24484473
molecular mechanics simulation; halogen bond; ab initio quantum mechanics; hydration free energy; parameter optimization
10.  Polarizable atomic multipole X-ray refinement: application to peptide crystals 
A method to accelerate the computation of structure factors from an electron density described by anisotropic and aspherical atomic form factors via fast Fourier transformation is described for the first time.
Recent advances in computational chemistry have produced force fields based on a polarizable atomic multipole description of biomolecular electrostatics. In this work, the Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) force field is applied to restrained refinement of molecular models against X-ray diffraction data from peptide crystals. A new formalism is also developed to compute anisotropic and aspherical structure factors using fast Fourier transformation (FFT) of Cartesian Gaussian multipoles. Relative to direct summation, the FFT approach can give a speedup of more than an order of magnitude for aspherical refinement of ultrahigh-resolution data sets. Use of a sublattice formalism makes the method highly parallelizable. Application of the Cartesian Gaussian multipole scattering model to a series of four peptide crystals using multipole coefficients from the AMOEBA force field demonstrates that AMOEBA systematically underestimates electron density at bond centers. For the trigonal and tetrahedral bonding geometries common in organic chemistry, an atomic multipole expansion through hexadecapole order is required to explain bond electron density. Alternatively, the addition of inter­atomic scattering (IAS) sites to the AMOEBA-based density captured bonding effects with fewer parameters. For a series of four peptide crystals, the AMOEBA–IAS model lowered R free by 20–40% relative to the original spherically symmetric scattering model.
PMCID: PMC2733883  PMID: 19690373
scattering factors; aspherical; anisotropic; force fields; multipole; polarization; AMOEBA; bond density; direct summation; FFT; SGFFT; Ewald; PME
11.  Solid-state tautomeric structure and invariom refinement of a novel and potent HIV integrase inhibitor 
The conformation and tautomeric structure of (Z)-4-[5-(2,6-difluoro­benzyl)-1-(2-fluoro­benzyl)-2-oxo-1,2-dihydro­pyridin-3-yl]-4-hy­droxy-2-oxo-N-(2-oxopyrrolidin-1-yl)but-3-enamide, C27H22F3N3O5, in the solid state has been resolved by single-crystal X-ray crystallography. The electron distribution in the mol­ecule was evaluated by refinements with invarioms, aspherical scattering factors by the method of Dittrich et al. [Acta Cryst. (2005), A61, 314–320] that are based on the Hansen–Coppens multipole model [Hansen & Coppens (1978 ▶). Acta Cryst. A34, 909–921]. The β-diketo portion of the mol­ecule exists in the enol form. The enol –OH hydrogen forms a strong asymmetric hydrogen bond with the carbonyl O atom on the β-C atom of the chain. Weak intra­molecular hydrogen bonds exist between the weakly acidic α-CH hydrogen of the keto–enol group and the pyridinone carbonyl O atom, and also between the hydrazine N—H group and the carbonyl group in the β-position from the hydrazine N—H group. The electrostatic properties of the mol­ecule were derived from the mol­ecular charge density. The mol­ecule is in a lengthened conformation and the rings of the two benzyl groups are nearly orthogonal. Results from a high-field 1H and 13C NMR correlation spectroscopy study confirm that the same tautomer exists in solution as in the solid state.
PMCID: PMC3589111  PMID: 23459357
12.  Anharmonicity and isomorphic phase transition: a multi-temperature X-ray single-crystal and powder diffraction study of 1-(2′-aminophenyl)-2-methyl-4-nitroimidazole 
Iucrj  2014;1(Pt 2):110-118.
Multi-temperature single-crystal and powder diffraction experiments on 1-(2′-aminophenyl)-2-methyl-4-nitroimidazole show that this crystal undergoes an isomorphic phase transition with the coexistence of two phase domains over a wide temperature range. The anharmonic approach was the only way to model the resulting disorder.
The harmonic model of atomic nuclear motions is usually enough for multipole modelling of high-resolution X-ray diffraction data; however, in some molecular crystals, such as 1-(2′-aminophenyl)-2-methyl-4-nitro-1H-imidazole [Paul, Kubicki, Jelsch et al. (2011 ▶). Acta Cryst. B67, 365–378], it may not be sufficient for a correct description of the charge-density distribution. Multipole refinement using harmonic atom vibrations does not lead to the best electron density model in this case and the so-called ‘shashlik-like’ pattern of positive and negative residual electron density peaks is observed in the vicinity of some atoms. This slight disorder, which cannot be modelled by split atoms, was solved using third-order anharmonic nuclear motion (ANM) parameters. Multipole refinement of the experimental high-resolution X-ray diffraction data of 1-(2′-aminophenyl)-2-methyl-4-nitro-1H-imidazole at three different temperatures (10, 35 and 70 K) and a series of powder diffraction experiments (20 ≤ T ≤ 300 K) were performed to relate this anharmonicity observed for several light atoms (N atoms of amino and nitro groups, and O atoms of nitro groups) to an isomorphic phase transition reflected by a change in the b cell parameter around 65 K. The observed disorder may result from the coexistence of domains of two phases over a large temperature range, as shown by low-temperature powder diffraction.
PMCID: PMC4062092  PMID: 25075327
anharmonicity; isomorphic phase transition; experimental charge density; X-ray closed-circuit helium cryostat; Hansen–Coppens model; multiple-temperature powder diffraction
13.  Polarizable Atomic Multipole-based Molecular Mechanics for Organic Molecules 
An empirical potential based on permanent atomic multipoles and atomic induced dipoles is reported for alkanes, alcohols, amines, sulfides, aldehydes, carboxylic acids, amides, aromatics and other small organic molecules. Permanent atomic multipole moments through quadrupole moments have been derived from gas phase ab initio molecular orbital calculations. The van der Waals parameters are obtained by fitting to gas phase homodimer QM energies and structures, as well as experimental densities and heats of vaporization of neat liquids. As a validation, the hydrogen bonding energies and structures of gas phase heterodimers with water are evaluated using the resulting potential. For 32 homo- and heterodimers, the association energy agrees with ab initio results to within 0.4 kcal/mol. The RMS deviation of hydrogen bond distance from QM optimized geometry is less than 0.06 Å. In addition, liquid self-diffusion and static dielectric constants computed from molecular dynamics simulation are consistent with experimental values. The force field is also used to compute the solvation free energy of 27 compounds not included in the parameterization process, with a RMS error of 0.69 kcal/mol. The results obtained in this study suggest the AMOEBA force field performs well across different environments and phases. The key algorithms involved in the electrostatic model and a protocol for developing parameters are detailed to facilitate extension to additional molecular systems.
PMCID: PMC3196664  PMID: 22022236
14.  A Transferable Coarse-Grained Model for Hydrogen Bonding Liquids 
We present here a recent development of a generalized coarse-grained model for use in molecular simulations. In this model, interactions between coarse-grained particles consist of both van der Waals and explicit electrostatic components. As a result, the coarse-grained model offers the transferability that is lacked by most current effectivepotential based approaches. The previous center-of-mass framework1 is generalized here to include arbitrary off-center interaction sites for both Gay-Berne and multipoles. The new model has been applied to molecular dynamic simulations of neat methanol liquid. By placing a single point multipole at the oxygen atom rather than at the center of mass of methanol, there is a significant improvement in the ability to capture hydrogen-bonding. The critical issue of transferability of the coarse-grained model is verified on methanol-water mixtures, using parameters derived from neat liquids without any modification. The mixture density and internal energy from coarse-grained molecular dynamics simulations show good agreement with experimental measurements, on a par with what has been obtained from more detailed atomic models. By mapping the dynamics trajectory from the coarse-grained simulation into the all-atom counterpart, we are able to investigate atomic .level structure and interaction. Atomic radial distribution functions of neat methanol, neat water and mixtures compare favorably to experimental measurements. Furthermore, hydrogen-bonded 6- and 7-molecule chains of water and methanol observed in the mixture are in agreement with previous atomic simulations.
PMCID: PMC2443098  PMID: 18688358
15.  Restrained electrostatic potential atomic partial charges for condensed-phase simulations of carbohydrates* 
Theochem  2000;527(1-3):149-156.
Charges derived from fitting a classical Coulomb model to quantum mechanical molecular electrostatic potentials (so called ESP-charges) are frequently used in simulations of macromolecules. Simulational methods that use ESP-charges generally reproduce the geometries of hydrogen bonded complexes, despite the fact that these charges are known to overestimate the strengths of these interactions. Through the use of a restraint function during the fitting of the partial charges to the electrostatic potentials the magnitudes of the charges may be attenuated (so called RESP-charges). For the AMBER force field RESP-charges have been proposed for proteins and nucleic acids. Here we examine a novel approach for determining the RESP-charges for carbohydrates based on molecular dynamics (MD) simulations of crystal structures. During a simulation, the crystallographic unit cell geometry is sensitive to both inter-molecular non-bonded forces and internal torsional rotations. However, for polar molecules, and specifically carbohydrates, the crystal geometries are particularly sensitive to the set of partial atomic charges employed in the simulation. Thus, given a force field in which the van der Waals and torsion terms are well parameterized, it is possible to assess the suitability of a set of partial charges by monitoring the properties of the crystal during an MD simulation. We have examined several charge sets for use with the GLYCAM parameters for carbohydrate and glycoprotein simulations and found that a restraint weight of 0.01 gives the best agreement with the neutron diffraction structure of α-d-glucopyranose. Unrestrained ESP-charges performed poorly as did the charges obtained from Mulliken and distributed multipole analyses of the quantum mechanical HF/6-31G* wavefunctions.
PMCID: PMC4191892  PMID: 25309012
AMBER; Carbohydrate; Electrostatic potential; GLYCAM; Molecular dynamics simulations; Restrained electrostatic charges
16.  A variational linear-scaling framework to build practical, efficient next-generation orbital-based quantum force fields 
We introduce a new hybrid molecular orbital/density-functional modified divide-and-conquer (mDC) approach that allows the linear-scaling calculation of very large quantum systems. The method provides a powerful framework from which linear-scaling force fields for molecular simulations can be developed. The method is variational in the energy, and has simple, analytic gradients and essentially no break-even point with respect to the corresponding full electronic structure calculation. Furthermore, the new approach allows intermolecular forces to be properly balanced such that non-bonded interactions can be treated, in some cases, to much higher accuracy than the full calculation. The approach is illustrated using the second-order self-consistent charge density-functional tight-binding model (DFTB2). Using this model as a base Hamiltonian, the new mDC approach is applied to a series of water systems, where results show that geometries and interaction energies between water molecules are greatly improved relative to full DFTB2. In order to achieve substantial improvement in the accuracy of intermolecular binding energies and hydrogen bonded cluster geometries, it was necessary to extend the DFTB2 model to higher-order atom-centered multipoles for the second-order self-consistent intermolecular electrostatic term. Using generalized, linear-scaling electrostatic methods, timings demonstrate that the method is able to calculate a water system of 3000 atoms in less than half of a second, and systems of up to one million atoms in only a few minutes using a conventional desktop workstation.
PMCID: PMC3694615  PMID: 23814506
17.  Point Charges Optimally Placed to Represent the Multipole Expansion of Charge Distributions 
PLoS ONE  2013;8(7):e67715.
We propose an approach for approximating electrostatic charge distributions with a small number of point charges to optimally represent the original charge distribution. By construction, the proposed optimal point charge approximation (OPCA) retains many of the useful properties of point multipole expansion, including the same far-field asymptotic behavior of the approximate potential. A general framework for numerically computing OPCA, for any given number of approximating charges, is described. We then derive a 2-charge practical point charge approximation, PPCA, which approximates the 2-charge OPCA via closed form analytical expressions, and test the PPCA on a set of charge distributions relevant to biomolecular modeling. We measure the accuracy of the new approximations as the RMS error in the electrostatic potential relative to that produced by the original charge distribution, at a distance the extent of the charge distribution–the mid-field. The error for the 2-charge PPCA is found to be on average 23% smaller than that of optimally placed point dipole approximation, and comparable to that of the point quadrupole approximation. The standard deviation in RMS error for the 2-charge PPCA is 53% lower than that of the optimal point dipole approximation, and comparable to that of the point quadrupole approximation. We also calculate the 3-charge OPCA for representing the gas phase quantum mechanical charge distribution of a water molecule. The electrostatic potential calculated by the 3-charge OPCA for water, in the mid-field (2.8 Å from the oxygen atom), is on average 33.3% more accurate than the potential due to the point multipole expansion up to the octupole order. Compared to a 3 point charge approximation in which the charges are placed on the atom centers, the 3-charge OPCA is seven times more accurate, by RMS error. The maximum error at the oxygen-Na distance (2.23 Å ) is half that of the point multipole expansion up to the octupole order.
PMCID: PMC3701554  PMID: 23861790
18.  Polymer Amide as an Early Topology 
PLoS ONE  2014;9(7):e103036.
Hydrophobic polymer amide (HPA) could have been one of the first normal density materials to accrete in space. We present ab initio calculations of the energetics of amino acid polymerization via gas phase collisions. The initial hydrogen-bonded di-peptide is sufficiently stable to proceed in many cases via a transition state into a di-peptide with an associated bound water molecule of condensation. The energetics of polymerization are only favorable when the water remains bound. Further polymerization leads to a hydrophobic surface that is phase-separated from, but hydrogen bonded to, a small bulk water complex. The kinetics of the collision and subsequent polymerization are discussed for the low-density conditions of a molecular cloud. This polymer in the gas phase has the properties to make a topology, viz. hydrophobicity allowing phase separation from bulk water, capability to withstand large temperature ranges, versatility of form and charge separation. Its flexible tetrahedral carbon atoms that alternate with more rigid amide groups allow it to deform and reform in hazardous conditions and its density of hydrogen bonds provides adhesion that would support accretion to it of silicon and metal elements to form a stellar dust material.
PMCID: PMC4105422  PMID: 25048204
19.  Theoretical description of halogen bonding – an insight based on the natural orbitals for chemical valence combined with the extended-transition-state method (ETS-NOCV) 
Journal of Molecular Modeling  2012;19(11):4681-4688.
In the present study we have characterized the halogen bonding in selected molecules H3N–ICF3 (1-NH3), (PH3)2C–ICF3 (1-CPH3), C3H7Br–(IN2H2C3)2C6H4 (2-Br), H2–(IN2H2C3)2C6H4 (2-H2) and Cl–(IC6F5)2C7H10N2O5 (3-Cl), containing from one halogen bond (1-NH3, 1-CPH3) up to four connections in 3-Cl (the two Cl–HN and two Cl–I), based on recently proposed ETS-NOCV analysis. It was found based on the NOCV-deformation density components that the halogen bonding C–X…B (X-halogen atom, B-Lewis base), contains a large degree of covalent contribution (the charge transfer to X…B inter-atomic region) supported further by the electron donation from base atom B to the empty σ*(C–X) orbital. Such charge transfers can be of similar importance compared to the electrostatic stabilization. Further, the covalent part of halogen bonding is due to the presence of σ-hole at outer part of halogen atom (X). ETS-NOCV approach allowed to visualize formation of the σ-hole at iodine atom of CF3I molecule. It has also been demonstrated that strongly electrophilic halogen bond donor, [C6H4(C3H2N2I)2][OTf]2, can activate chemically inert isopropyl bromide (2-Br) moiety via formation of Br–I bonding and bind the hydrogen molecule (2-H2). Finally, ETS-NOCV analysis performed for 3-Cl leads to the conclusion that, in terms of the orbital-interaction component, the strength of halogen (Cl–I) bond is roughly three times more important than the hydrogen bonding (Cl–HN).
FigureETS-NOCV reprezentation of σ-hole at iodine together with the molecular electrostatic potential picture
PMCID: PMC3825488  PMID: 22669533
Covalency; ETS-NOCV; Halogen bonding
20.  Experimental Electron Density and Neutron Diffraction Studies on the Polymorphs of Sulfathiazole 
Crystal Growth & Design  2014;14(3):1227-1239.
High resolution X-ray diffraction data on forms I–IV of sulfathiazole and neutron diffraction data on forms II–IV have been collected at 100 K and analyzed using the Atoms in Molecules topological approach. The molecular thermal motion as judged by the anisotropic displacement parameters (adp’s) is very similar in all four forms. The adp of the thiazole sulfur atom had the greatest amplitude perpendicular to the five-membered ring, and analysis of the temperature dependence of the adps indicates that this is due to genuine thermal motion rather than a concealed disorder. A minor disorder (∼1–2%) is evident for forms I and II, but a statistical analysis reveals no deleterious effect on the derived multipole populations. The topological analysis reveals an intramolecular S–O···S interaction, which is consistently present in all experimental topologies. Analysis of the gas-phase conformation of the molecule indicates two low-energy theoretical conformers, one of which possesses the same intramolecular S–O···S interaction observed in the experimental studies and the other an S–O···H–N intermolecular interaction. These two interactions appear responsible for “locking” the molecular conformation. The lattice energies of the various polymorphs computed from the experimental multipole populations are highly dependent on the exact refinement model. They are similar in magnitude to theoretically derived lattice energies, but the relatively high estimated errors mean that this method is insufficiently accurate to allow a definitive stability order for the sulfathiazole polymorphs at 0 K to be determined.
High resolution X-ray diffraction data on sulfathiazole (forms I−IV) and neutron diffraction data have been used to analyze the polymorphic electron density using Quantum Theory of Atoms in Molecules. Two low-energy theoretical conformers are found in the gas phase, one of which possesses an S−O···S interaction (a) and the other an S−O···H−N (b) intermolecular interaction. These interactions appear responsible for “locking” the molecular conformation.
PMCID: PMC3963452  PMID: 24672285
21.  Ultrahigh-resolution crystallography and related electron density and electrostatic properties in proteins 
Journal of Synchrotron Radiation  2008;15(Pt 3):202-203.
Ultrahigh-resolution protein diffraction data allow valence electron density modelling and calculations of experimental electrostatic properties. Protein–ligand interaction energy may therefore be estimated.
With an increasing number of biological macromolecular crystal structures measured at ultrahigh resolution (1 Å or better), it is necessary to extend to large systems the experimental valence electron density modelling that is applied to small molecules. A database of average multipole populations has been built, describing the electron density of chemical groups in all 20 amino acids found in proteins. It allows calculation of atomic aspherical scattering factors, which are the starting point for refinement of the protein electron density, using the MoPro software. It is shown that the use of non-spherical scattering factors has a major impact on crystallographic statistics and results in a more accurate crystal structure, notably in terms of thermal displacement parameters and bond distances involving H atoms. It is also possible to obtain a realistic valence electron density model, which is used in the calculation of the electrostatic potential and energetic properties of proteins.
PMCID: PMC2394818  PMID: 18421138
electron density; protein refinement; high-resolution crystallography
22.  Ethynyl and Propynylpyrene Inhibitors of Cytochrome P450 
The single-crystal X-ray structures and in vivo activities of three aryl acetylenic inhibitors of cytochromes P450 1A1, 1A2, 2A6, and 2B1 have been determined and are reported herein. These are 1-ethynylpyrene, 1-propy-nylpyrene, and 4-propynylpyrene. To investigate electronic influences on the mechanism of enzyme inhibition, the experimental electron density distribution of 1-ethynylpy-rene has been determined using low-temperature X-ray diffraction measurements, and the resulting net atomic charges compared with various theoretical calculations. A total of 82,390 reflections were measured with Mo Kα radiation to a (sinθ/λ)max = 0.985 Å−1. Averaging symmetry equivalent reflections yielded 8,889 unique reflections. A least squares refinement procedure was used in which multipole parameters were added to describe the distortions of the atomic electron distributions from spherical symmetry. A map of the model electron density distribution of 1-ethynylpyrene was obtained. Net atomic charges calculated from refined monopole population parameters yielded charges that showed that the terminal acetylenic carbon atom (C18) is more negative than the internal carbon (C17). Net atomic charges calculated by ab initio, density functional theory, and semi-empirical methods are consistent with this trend suggesting that the terminal acetylenic carbon atom is more likely to be the site of oxidation. This is consistent with the inhibition mechanism pathway that results in the formation of a reactive ketene intermediate. This is also consistent with assay results that determined that 1-ethynylpyrene acts as a mechanism-based inhibitor of P450s 1A1 and 1A2 and as a reversible inhibitor of P450 2B1. Crystallographic data: 1-ethynylpyrene, C18H10, P21/c, a = 14.571(2) Å, b = 3.9094(5) Å, c = 20.242(3) Å, β = 105.042(2)°, V = 1,113.5(2) Å3; 1-propynylpyrene, C19H12, P21/n, a = 8.970(2) Å, b = 10.136(1) Å, c = 14.080(3) Å, β = 99.77(2)°, V = 1,261.5(4) Å3; 4-propynylpyrene, C19H12, Pbca, a = 9.904(1) Å, b = 13.174(2) Å, c = 19.401(1) Å, V = 2,531.4(5) Å3.
PMCID: PMC2869100  PMID: 20473363
X-ray structure; Mechanism-based inhibitor; Ethynylpyrene; Propynylpyrene; Electron density distribution; Theoretical calculations
23.  Theoretical description of hydrogen bonding in oxalic acid dimer and trimer based on the combined extended-transition-state energy decomposition analysis and natural orbitals for chemical valence (ETS-NOCV) 
Journal of Molecular Modeling  2010;16(11):1789-1795.
In the present study we have analyzed hydrogen bonding in dimer and trimer of oxalic acid, based on a recently proposed charge and energy decomposition scheme (ETS-NOCV). In the case of a dimer, two conformations, α and β, were considered. The deformation density contributions originating from NOCV’s revealed that the formation of hydrogen bonding is associated with the electronic charge deformation in both the σ—(Δρσ) and π-networks (Δρπ). It was demonstrated that σ-donation is realized by electron transfer from the lone pair of oxygen on one monomer into the empty \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \rho_{H - O}^* $$\end{document} orbital of the second oxalic acid fragment. In addition, a covalent contribution is observed by the density transfer from hydrogen of H-O group in one oxalic acid monomer to the oxygen atom of the second fragment. The resonance assisted component (Δρπ), is based on the transfer of electron density from the π—orbital localized on the oxygen of OH on one oxalic acid monomer to the oxygen atom of the other fragment. ETS-NOCV allowed to conclude that the σ(O---HO) component is roughly eight times as important as π (RAHB) contribution in terms of energetic estimation. The electrostatic factor (ΔEelstat) is equally as important as orbital interaction term (ΔEorb). Finally, comparing β-dimer of oxalic acid with trimer we found practically no difference concerning each of the O---HO bonds, neither qualitative nor quantitative.
FigureThe contours of deformation density σ- and π-contributions describing the hydrogen bonding between the monomers in the oxalic acid dimer, together with the corresponding ETS-NOCV-based orbital-interaction energies (in kcal/mol).
PMCID: PMC2949554  PMID: 20505966
ETS-NOCV; Extended transition state; Natural orbitals for chemical valence; Oxalic acid dimer and trimer; Resonance assisted hydrogen bonds
24.  Incorporating Phase-Dependent Polarizability in Non-Additive Electrostatic Models for Molecular Dynamics Simulations of the Aqueous Liquid-Vapor Interface 
We discuss a new classical water force field that explicitly accounts for differences in polarizability between liquid and vapor phases. The TIP4P-QDP (4-point transferable intermolecular potential with charge dependent-polarizability) force field is a modification of the original TIP4P-FQ fluctuating charge water force field of Rick et al.1 that self-consistently adjusts its atomic hardness parameters via a scaling function dependent on the M-site charge. The electronegativity (χ) parameters are also scaled in order to reproduce condensed-phase dipole moments of comparable magnitude to TIP4P-FQ. TIP4P-QDP is parameterized to reproduce experimental gas-phase and select condensed-phase properties. The TIP4P-QDP water model possesses a gas phase polarizability of 1.40 Å3 and gas-phase dipole moment of 1.85 Debye, in excellent agreement with experiment and high-level ab initio predictions. The liquid density of TIP4P-QDP is 0.9954(±0.0002) g/cm3 at 298 K and 1 atmosphere, and the enthalpy of vaporization is 10.55(±0.12) kcal/mol. Other condensed-phase properties such as the isobaric heat capacity, isothermal compressibility, and diffusion constant are also calculated within reasonable accuracy of experiment and consistent with predictions of other current state-of-the-art water force fields. The average molecular dipole moment of TIP4P-QDP in the condensed phase is 2.641(±0.001) Debye, approximately 0.02 Debye higher than TIP4P-FQ and within the range of values currently surmised for the bulk liquid. The dielectric constant, ε = 85.8 ± 1.0, is 10% higher than experiment. This is reasoned to be due to the increase in the condensed phase dipole moment over TIP4P-FQ, which estimates ε remarkably well. Radial distribution functions for TIP4P-QDP and TIP4P-FQ show similar features, with TIP4P-QDP showing slightly reduced peak heights and subtle shifts towards larger distance interactions. Since the greatest effects of the phase-dependent polarizability are anticipated in regions with both liquid and vapor character, interfacial simulations of TIP4P-QDP were performed and compared to TIP4P-FQ, a static polarizability analog. Despite similar features in density profiles such as the position of the GDS and interfacial width, enhanced dipole moments are observed for the TIP4P-QDP interface and onset of the vapor phase. Water orientational profiles show an increased preference (over TIP4P-FQ) in the orientation of the permanent dipole vector of the molecule within the interface; an enhanced z-induced dipole moment directly results from this preference. Hydrogen bond formation is lower, on average, in the bulk for TIP4P-QDP than TIP4P-FQ. However, the average number of hydrogen bonds formed by TIP4P-QDP in the interface exceeds that of TIP4P-FQ, and observed hydrogen bond networks extend further into the gaseous region. The TIP4P-QDP interfacial potential, calculated to be -11.98(±0.08) kcal/mol, is less favorable than that for TIP4P-FQ by approximately 2% as a result of a diminished quadrupole contribution. Surface tension is calculated within a 1.3% reduction from the experimental value. Results reported demonstrate TIP4P-QDP as a model comparable to the popular TIP4P-FQ while accounting for a physical effect previously neglected by other water models. Further refinements to this model, as well as future applications are discussed.
PMCID: PMC3488353  PMID: 23133341
Phase Dependent Polarizability; Molecular Dynamics; Charge Equilibration; Polarizable Force Field; Liquid-Vapor Interface; TIP4P-QDP
25.  ETS-NOCV description of σ-hole bonding 
Journal of Molecular Modeling  2012;19(7):2747-2758.
The ETS-NOCV analysis was applied to describe the σ-hole in a systematic way in a series of halogen compounds, CF3-X (X = I, Br, Cl, F), CH3I, and C(CH3)nH3-n-I (n = 1,2,3), as well as for the example germanium-based systems. GeXH3, X = F, Cl, H. Further, the ETS-NOCV analysis was used to characterize bonding with ammonia for these systems. The results show that the dominating contribution to the deformation density, Δρ1, exhibits the negative-value area with a minimum, corresponding to σ-hole. The “size” (spatial extension of negative value) and “depth” (minium value) of the σ-hole varies for different X in CF3-X, and is influenced by the carbon substituents (fluorine atoms, hydrogen atoms, methyl groups). The size and depth of σ-hole decreases in the order: I, Br, Cl, F in CF3-X. In CH3-I and C(CH3)nH3-n-I, compared to CF3-I, introduction of hydrogen atoms and their subsequent replacements by methyl groups lead to the systematic decrease in the σ-hole size and depth. The ETS-NOCV σ-hole picture is consistent with the existence the positive MEP area at the extension of σ-hole generating bond. Finally, the NOCV deformation density contours as well as by the ETS orbital-interaction energy indicate that the σ-hole-based bond with ammonia contains a degree of covalent contribution. In all analyzed systems, it was found that the electrostatic energy is approximately two times larger than the orbital-interaction term, confirming the indisputable role of the electrostatic stabilization in halogen bonding and σ-hole bonding.
FigureGraphical representation of the σ-hole on the halogen atom, based on the molecular electrostatic potential (upper row) and the NOCV deformation-density channel Δρ1 (lower row and the right-hand side plot)
PMCID: PMC3693432  PMID: 23053006
ETS-NOCV; Halogen bonding; Sigma hole bonding

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