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.
Multipole; Gaussian Multipole; Force; Torque; Wigner Function
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.
Gaussian multipoles; charge density; electrostatic model; multipole; overlap
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.
Atomic multipoles; Hirshfeld charges; dipole; quadrupole
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.
aperiodic crystals; incommensurate modulated structures; MEM density; ADPs
We present a simple damping scheme for point-charge electrostatics that could be used directly in classical force fields. The approach acts at the charge (or monopole) level only and allows the inclusion of short-range electrostatic penetration effects at a very low cost. Results are compared with density functional theory Coulomb intermolecular interaction energies and with several other methods such as distributed multipoles, damped distributed multipoles, and transferable Hermite-Gaussian densities. Realistic trends in the interactions are observed for atom-centered Mertz-Kollman corrected point-charge distributions. The approach allows increasing the selectivity of parameters in the case of metal complexes. In addition, two QM/MM calculations are presented where the damping function is employed to include the MM atoms located at the QM/MM boundary. The first calculation corresponds to the gas-phase proton transfer of aspartic acid through water and the second is the first step of the reaction catalyzed by the 4-oxalocrotonate tautomerase (4OT) enzyme. First, improved agreement is observed when using the damping approach compared with the conventional excluded charge method or when including all charges in the calculation. Second, in the case of 4OT, the damped charge approach is in agreement with previous calculations, whereas including all charges gives a significantly higher energy barrier. In both cases, no reparameterization of the van der Waals part of the force field was performed.
electrostatics; force fields; QM/MM; embedding; penetration energy; reaction path
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.
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.
electron density; protein refinement; high-resolution crystallography
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.
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.
scattering factors; aspherical; anisotropic; force fields; multipole; polarization; AMOEBA; bond density; direct summation; FFT; SGFFT; Ewald; PME
The effects of water multipole moments on the aqueous solvation of ions were determined in Monte Carlo simulations using soft-sticky dipole-quadrupole-octupole (SSDQO) water. Water molecules formed linear hydrogen bonds to Cl− using the new SSDQO1 parameters, similar to multi-site models. However, the dipole vector was tilted rather than parallel to the oxygen-Na+ internuclear vector as in most multi-site model, while experiment and ab initio molecular dynamics simulations generally indicate a range of values between tilted and parallel. By varying the multipoles in SSDQO, the octupole was found to determine the orientation around Na+. Moreover, analysis of the multipoles of more conventional models is predictive of their performance as solvents.
Water structure around sugars modeled by partial charges is compared for soft-sticky dipole-quadrupole-octupole (SSDQO), a fast single-site multipole model, and commonly used multi-site models in Monte Carlo simulations. Radial distribution functions and coordination numbers of all the models indicate similar hydration by hydrogen-bond donor and acceptor waters. However, the new optimized SSDQO1 parameters as well as TIP4P-Ew and TIP5P predict a “lone-pair” orientation for the water accepting the sugar hydroxyl hydrogen bond that is more consistent with the limited experimental data than the “dipole” orientation in SPC/E, which has important implications for studies of the cryoprotectant properties of sugars.
The equations defining the variational explicit polarization (X-Pol) potential introduced in earlier work are modified in the present work so that multipole point charge distributions are used instead of Mulliken charges to polarize the monomers that comprise the system. In addition, when computing the electrostatic interaction between a monomer whose molecular orbitals are being optimized and a monomer whose electron density is being used to polarize the first monomer, the electron densities of both monomers are represented by atom-centered multipole point charge distributions. In the original formulation of the variational X-Pol potential, the continuous electron density of the monomer being optimized interacts with external Mulliken charges, but this corresponds to the monopole truncation in a multipole expansion scheme in the computation of the Fock matrix elements of the given monomer. The formulation of the variational X-Pol potential introduced in this work (which we are calling the “multipole variational X-Pol potential”) represents the electron density of the monomer whose wave function is being variationally optimized in the same way that it represents the electron densities of external monomers when computing the Coulomb interactions between them.
Explicit polarization (X-Pol); Polarizable force field; Fragment-based molecular orbital method; Atom-based multipole moments
Due to their small size, low weight, low cost and low energy consumption, MEMS accelerometers have achieved great commercial success in recent decades. The aim of this research work is to identify a MEMS accelerometer structure for human body dynamics measurements. Photogrammetry was used in order to measure possible maximum accelerations of human body parts and the bandwidth of the digital acceleration signal. As the primary structure the capacitive accelerometer configuration is chosen in such a way that sensing part measures on all three axes as it is 3D accelerometer and sensitivity on each axis is equal. Hill climbing optimization was used to find the structure parameters. Proof-mass displacements were simulated for all the acceleration range that was given by the optimization problem constraints. The final model was constructed in Comsol Multiphysics. Eigenfrequencies were calculated and model's response was found, when vibration stand displacement data was fed into the model as the base excitation law. Model output comparison with experimental data was conducted for all excitation frequencies used during the experiments.
capacitive; accelerometer; MEMS; optimization; proof-mass; L-shaped beam
MoleCoolQt is a molecule viewer designed for experimental charge density studies with a user-friendly graphical user interface.
MoleCoolQt is a molecule viewer for charge-density research. Features include the visualization of local atomic coordinate systems in multipole refinements based on the Hansen and Coppens formalism as implemented, for example, in the XD suite. Residual peaks and holes from XDfft are translated so that they appear close to the nearest atom of the asymmetric unit. Critical points from a topological analysis of the charge density can also be visualized. As in the program MolIso, color-mapped isosurfaces can be generated with a simple interface. Apart from its visualization features the program interactively helps in assigning local atomic coordinate systems and local symmetry, which can be automatically detected and altered. Dummy atoms – as sometimes required for local atomic coordinate systems – are calculated on demand; XD system files are updated after changes. When using the invariom database, potential scattering factor assignment problems can be resolved by the use of an interactive dialog. The following file formats are supported: XD, MoPro, SHELX, GAUSSIAN (com, FChk, cube), CIF and PDB. MoleCoolQt is written in C++ using the Qt4 library, has a user-friendly graphical user interface, and is available for several flavors of Linux, Windows and MacOS.
MoleCoolQt; charge density; visualization software
The structure of the title compound, C17H18FN3O3·6H2O, has been redetermined at 120 K. An invariom refinement, a structural refinement using aspherical scattering factors from theoretically predicted multipole population parameters, yields accurate geometry and anisotropic displacement parameters, including hydrogen-bonding parameters. All potential hydrogen-bond donors and acceptors are involved in hydrogen bonding, forming an intricate three-dimensional network of N—H⋯O and O—H⋯O bonds.
A new induced dipole polarization model based on interacting Gaussian charge densities is presented. In contrast to the original induced point dipole model, the Gaussian polarization model is capable of finite interactions at short distances. Aspects of convergence related to the Gaussian model will be explored. The Gaussian polarization model is compared with the damped Thole-induced dipole model and the point dipole model. It will be shown that the Gaussian polarization model performs slightly better than the Thole model in terms of fitting to molecular polarizability tensors. An advantage of the model based on Gaussian charge distribution is that it can be easily generalized to other multipole moments and provide effective damping for both permanent electrostatic and polarization models. Finally, a method of parameterizing polarizabilities is presented. This method is based on probing a molecule with point charges and fitting polarizabilities to electrostatic potential. In contrast to the generic atom type polarizabilities fit to molecular polarizability tensors, probed polarizabilities are significantly more accurate in terms of reproducing molecular polarizability tensors and electrostatic potential, while retaining conformational transferability.
induced dipole; polarization; Gaussian; Thole
Electrostatic micro-electro-mechanical system (MEMS) is a special branch with a wide range of applications in sensing and actuating devices in MEMS. This paper provides a survey and analysis of the electrostatic force of importance in MEMS, its physical model, scaling effect, stability, nonlinearity and reliability in detail. It is necessary to understand the effects of electrostatic forces in MEMS and then many phenomena of practical importance, such as pull-in instability and the effects of effective stiffness, dielectric charging, stress gradient, temperature on the pull-in voltage, nonlinear dynamic effects and reliability due to electrostatic forces occurred in MEMS can be explained scientifically, and consequently the great potential of MEMS technology could be explored effectively and utilized optimally. A simplified parallel-plate capacitor model is proposed to investigate the resonance response, inherent nonlinearity, stiffness softened effect and coupled nonlinear effect of the typical electrostatically actuated MEMS devices. Many failure modes and mechanisms and various methods and techniques, including materials selection, reasonable design and extending the controllable travel range used to analyze and reduce the failures are discussed in the electrostatically actuated MEMS devices. Numerical simulations and discussions indicate that the effects of instability, nonlinear characteristics and reliability subjected to electrostatic forces cannot be ignored and are in need of further investigation.
MEMS; Electrostatic force; Scaling effect; Stability; Nonlinearity; Reliability
At room temperature, the m-Nitrophenol (m-NPH) appears in two polymorphic structures: orthorhombic and monoclinic forms. In the present work, we shall focus on the monoclinic form of this compound which has a centrosymmetric structure with the space group P21/n. The molecular dipole moment has been estimated experimentally. High resolution single crystal diffraction experiment was performed at low temperature with MoKα radiation. The crystal structure was refined using the multipolar model of Hansen and Coppens (1978). The molecular electron charge density distribution is described accurately. The study reveals the nature of inter-molecular interactions including charge transfer and hydrogen bonds. In this crystal, hydrogen bonds of moderate strength occur between the hydroxyl group and the O atom in the nitro one.
Electron charge density; M-Nitrophenol; XD program; nonlinear optical compound (NLO)
Ionocovalency (IC), a quantitative dual nature of the atom, is defined and correlated with quantum-mechanical potential to describe quantitatively the dual properties of the bond. Orbiotal hybrid IC model scale, IC, and IC electronegativity scale, XIC, are proposed, wherein the ionicity and the covalent radius are determined by spectroscopy. Being composed of the ionic function I and the covalent function C, the model describes quantitatively the dual properties of bond strengths, charge density and ionic potential. Based on the atomic electron configuration and the various quantum-mechanical built-up dual parameters, the model formed a Dual Method of the multiple-functional prediction, which has much more versatile and exceptional applications than traditional electronegativity scales and molecular properties. Hydrogen has unconventional values of IC and XIC, lower than that of boron. The IC model can agree fairly well with the data of bond properties and satisfactorily explain chemical observations of elements throughout the Periodic Table.
ionocovalency; molecular properties; electronegativity; theoretical chemistry
The maximum entropy method (MEM) has been used in many studies to reliably recover effective lifetimes from kinetics, whether measured experimentally or simulated computationally. Here, recent claims made by Mulligan et al. regarding MEM analyses of kinetics (Anal. Biochem. 421 (2012) 181–190) are shown to be unfounded. Their assertion that their software allows “analysis of datasets too noisy to process by existing iterative search algorithms” is refuted with a MEM analysis of their triexponential test case with increased noise. In addition, it is shown that lifetime distributions recovered from noisy kinetics data with the MEM can be improved by using a simple filter when bootstrapping the prior model. When deriving the bootstrapped model from the lifetime distribution obtained using a uniform model, only the slower processes are represented as Gaussians in the bootstrapped model. Using this new approach, results are clearly superior to those of Mulligan et al. despite the presence of increased noise. In a second example, ambiguity in the interpretation of Poisson kinetics in the presence of scattered excitation light is resolved by filtering the prior model.
Kinetics; Inverse Laplace transform; Rate constants; Lifetime distributions; Maximum entropy method
The mechanism of serine proteases prominently illustrates how charged amino acid residues and proton transfer events facilitate enzyme catalysis. Here we present an ultrahigh resolution (0.93 Å) x-ray structure of a complex formed between trypsin and a canonical inhibitor acting through a substrate-like mechanism. The electron density indicates the protonation state of all catalytic residues where the catalytic histidine is, as expected, in its neutral state prior to the acylation step by the catalytic serine. The carboxyl group of the catalytic aspartate displays an asymmetric electron density so that the Oδ2–Cγ bond appears to be a double bond, with Oδ2 involved in a hydrogen bond to His-57 and Ser-214. Only when Asp-102 is protonated on Oδ1 atom could a density functional theory simulation reproduce the observed electron density. The presence of a putative hydrogen atom is also confirmed by a residual mFobs − DFcalc density above 2.5 σ next to Oδ1. As a possible functional role for the neutral aspartate in the active site, we propose that in the substrate-bound form, the neutral aspartate residue helps to keep the pKa of the histidine sufficiently low, in the active neutral form. When the histidine receives a proton during the catalytic cycle, the aspartate becomes simultaneously negatively charged, providing additional stabilization for the protonated histidine and indirectly to the tetrahedral intermediate. This novel proposal unifies the seemingly conflicting experimental observations, which were previously seen as either supporting the charge relay mechanism or the neutral pKa histidine theory.
Enzyme Catalysis; Enzyme Kinetics; Enzyme Mechanisms; Enzyme Structure; Enzymes; Protease; Protease Inhibitor; Proteolytic Enzymes; Serine Protease; X-ray Crystallography
Determining the total number of charged residues corresponding to a given value of net charge for peptides and proteins in gas phase is crucial for the interpretation of mass-spectrometry data, yet it is far from being understood. Here we show that a novel computational protocol based on force field and massive density functional calculations is able to reproduce the experimental facets of well investigated systems, such as angiotensin II, bradykinin, and tryptophan-cage. The protocol takes into account all of the possible protomers compatible with a given charge state. Our calculations predict that the low charge states are zwitterions, because the stabilization due to intramolecular hydrogen bonding and salt-bridges can compensate for the thermodynamic penalty deriving from deprotonation of acid residues. In contrast, high charge states may or may not be zwitterions because internal solvation might not compensate for the energy cost of charge separation.
In the last two decades mass spectrometry has given an impressive contribution to biochemistry, protein science, proteomics and structural biology. This technique offers powerful insights into protein structure and dynamics along with useful information on the role of solvent in protein stability as it is able to preserve non-covalent interactions and globular structures during the proteins' flight inside the mass spectrometer. Unfortunately, the key issue of the charge state of ionizable groups, presumably different from that in solution, has not been elucidated yet. So far conflicting assumptions and conclusions have been drawn by several groups. In the present work a very accurate structural and energetic analysis of the protonation state of two peptides and a small protein in the gas phase was performed. Results suggest that internal solvation can stabilize charge separation with the formation of zwitterionic states.
We performed density functional calculations to examine the effects of solvation, hydrogen bonding, backbone conformation, and the side chain on 15N chemical shielding in proteins. We used N-methylacetamide (NMA) and N-formyl-alanyl-X (with X being one of the 19 naturally occurring amino acids excluding proline) as model systems. In addition, calculations were performed for selected fragments from protein GB3. The conducting polarizable continuum model was employed to include the effect of solvent in the density functional calculations. Our calculations for NMA show that the augmentation of the polarizable continuum model with the explicit water molecules in the first solvation shell has a significant influence on isotropic 15N chemical shift but not as much on the chemical shift anisotropy. The difference in the isotropic chemical shift between the standard β-sheet and α-helical conformations ranges from 0.8 ppm to 6.2 ppm depending on the residue type, with the mean of 2.7 ppm. This is in good agreement with the experimental chemical shifts averaged over a database of 36 proteins containing >6100 amino acid residues. The orientation of the 15N chemical shielding tensor as well as its anisotropy and asymmetry are also in the range of values experimentally observed for peptides and proteins.
chemical shielding tensor; chemical shift calculation; dipeptides; solvent effect; nitrogen-15; density-functional calculation
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.
In order to analyze where epileptic spikes are generated, we assessed the level of concordance between EEG source localization using distributed source models and simultaneous EEG-fMRI which measures the hemodynamic correlates of EEG activity. Data to be compared were first estimated on the same cortical surface and two comparison strategies were used: (1) MEM-concordance: a comparison between EEG sources localized with the Maximum Entropy on the Mean (MEM) method and fMRI clusters showing a significant hemodynamic response. Minimal geodesic distances between local extrema and overlap measurements between spatial extents of EEG sources and fMRI clusters were used to quantify MEM-concordance. (2) fMRI-relevance: estimation of the fMRI-relevance index α quantifying if sources located in an fMRI cluster could explain some scalp EEG data, when this fMRI cluster was used to constrain the EEG inverse problem. Combining MEM-concordance and fMRI-relevance (α) indexes, each fMRI cluster showing a significant hemodynamic response (p <0.05 corrected) was classified according to its concordance with EEG data. Nine patients with focal epilepsy who underwent EEG-fMRI examination followed by EEG recording outside the scanner were selected for this study. Among the 62 fMRI clusters analyzed (7 patients), 15 (24%) found in 6 patients were highly concordant with EEG according to both MEM-concordance and fMRI-relevance. EEG concordance was found for 5 clusters (8%) according to α only, suggesting sources missed by the MEM. No concordance with EEG was found for 30 clusters (48%) and for 10 clusters (16%) α was significantly negative, suggesting EEG-fMRI discordance. We proposed two complementary strategies to assess and classify EEG-fMRI concordance. We showed that for most patients, part of the hemodynamic response to spikes was highly concordant with EEG sources, whereas other fMRI clusters in response to the same spikes were found distant or discordant with EEG sources.
PMID: 17945511 CAMSID: cams3473
EEG source localization; Simultaneous EEG-fMRI; Concordance; Interictal epileptic spikes