An effective exciton Hamiltonian for all amide bands is used to calculate the linear absorption and photon echo spectra of a 17 residue helical peptide (YKKKH17). The cross peak bandshapes are sensitive to the inter-band couplings. Fluctuations of the local amide frequencies of the all amide fundamental and their overtone and combination states are calculated using the multipole electric field induced by environment employing the electrostatic DFT map of N-methyl acetamide. Couplings between neighboring peptide units are obtained using the anharmonic vibrational Hamiltonian of glycine dipeptide (GLDP) at the BPW91/6-31G(d,p) level. Electrostatic couplings between non-neighboring units are calculated by the fourth rank transition multipole coupling (TMC) expansion including 1/R3 (dipole-dipole), 1/R4 (quadrupole-dipole), and 1/R5 (quadrupole-quadrupole and octapole-dipole) interactions.
2-Dimensional Infrared-Spectroscopy; molecular dynamics; Multidimensional Femtosecond Spectroscopies; Amide-I Modes; Amide-II Modes; Amide-III Modes; Amide-A Modes; Molecular-Dynamics; Alpha-Helix; Proteins; 310-Helix; Spectra
The development and implementation of a tree code (TC) and fast multipole method (FMM) for the efficient, linear-scaling calculation of long-range electrostatic interactions of particle distributions with variable shape and multipole character are described. The target application of these methods are stochastic boundary molecular simulations with polarizable force fields and/or combined quantum mechanical/molecular mechanical potentials. Linear-scaling is accomplished through the adaptive decomposition of the system into a hierarchy of interacting particle sets. Two methods for effecting this decomposition are evaluated: fluc-splitting and box-splitting, for which the latter is demonstrated to be generally more accurate. In addition, a generalized termination criterion is developed that delivers optimal performance at fixed error tolerance that, in the case of quadrupole-represented Drude water, effects a speed-up by a factor of 2–3 relative to a multipole-independent termination criteria. The FMM is shown to be approximately 2–3 times faster than the TC, independent of the system size and multipole order of the particles. The TC and FMM are tested for a variety of static and polarizable water systems, and for the the 70S ribosome functional complex containing an assembly of transfer and messenger RNAs.
In this article, we clearly demonstrate that the electric potential and the magnetic field can contain different information about current sources in three-dimensional conducting media. Expressions for the magnetic fields of electric dipole and quadrupole current sources immersed in an infinite conducting medium are derived, and it is shown that two different point dipole distributions that are electrically equivalent have different magnetic fields. Although measurements of the electric potential are not sufficient to determine uniquely the characteristics of a quadrupolar source, the radial component of the magnetic field can supply the additional information needed to resolve these ambiguities and to determine uniquely the configuration of dipoles required to specify the electric quadrupoles. We demonstrate how the process can be extended to even higher order terms in an electrically silent series of magnetic multipoles. In the context of a spherical brain source model, it has been mathematically demonstrated that the part of the neuronal current generating the electric potential lives in the orthogonal complement of the part of the current generating the magnetic potential. This implies a mathematical relationship of complementarity between electroencephalography (EEG) and magnetoencephalography (MEG), although the theoretical result in question does not apply to the non-spherical case (Dassios (2008), Math Med Biol, vol. 25, p. 133). Our results have important practical applications in cases where electrically silent sources that generate measurable magnetic fields are of interest. Moreover, electrically silent, magnetically active moments of higher order can be useful when cancellation due to superposition of fields can occur, since this situation leads to a substantial reduction in the measurable amplitude of the signal. In this context, information derived from magnetic recordings of electrically silent, magnetically active multipoles can supplement electrical recordings for the purpose of studying the physiology of the brain. Magnetic fields of the electric multipole sources in a conducting medium surrounded by an insulating spherical shell are also presented and the relevance of this calculation to cardiographic and encephalographic experimentation is discussed.
A proposed coarse-grained model of nucleic acids demonstrates that average interactions between base dipoles, together with chain connectivity and excluded-volume interactions, are sufficient to form double-helical structures of DNA and RNA molecules. Additionally, local interactions determine helix handedness and direction of strand packing. This result, and earlier research on reduced protein models, suggest that mean-field multipole-multipole interactions are the principal factors responsible for the formation of regular structure of biomolecules.
The long stator track for high speed maglev trains has a tooth-slot structure. The sensor obtains precise relative position information for the traction system by detecting the long stator tooth-slot structure based on nondestructive detection technology. The magnetic field modeling of the sensor is a typical three-dimensional (3-D) electromagnetic problem with complex boundary conditions, and is studied semi-analytically in this paper. A second-order vector potential (SOVP) is introduced to simplify the vector field problem to a scalar field one, the solution of which can be expressed in terms of series expansions according to Multipole Theory (MT) and the New Equivalent Source (NES) method. The coefficients of the expansions are determined by the least squares method based on the boundary conditions. Then, the solution is compared to the simulation result through Finite Element Analysis (FEA). The comparison results show that the semi-analytical solution agrees approximately with the numerical solution. Finally, based on electromagnetic modeling, a difference coil structure is designed to improve the sensitivity and accuracy of the sensor.
relative position detection sensor; electromagnetic modeling; multipole theory; new equivalent source method; second-order vector potential; difference coil structure
Although magnetic fields interact weakly with biological tissues, at high fields, this interaction is sufficiently strong to cause measurable shifts in the Larmor frequency among various tissue types. While measuring frequency shift and its anisotropy has enabled NMR spectroscopy to determine structures of large molecules, MRI has not been able to fully utilize the vast information existing in the frequency to elucidate tissue microstructure. Using a multipole analysis of the complex MRI signal in the Fourier spectral space, we developed a fast and high-resolution method that enables the quantification of tissue’s magnetic response with a set of magnetic susceptibility tensors of various ranks. The Fourier spectral space, termed p-space, can be generated by applying field gradients or equivalently by shifting the k-space data in various directions. Measuring these tensors allows the visualization and quantification of tissue architecture. We performed 3D whole-brain multipole susceptibility tensor imaging in simulation, on intact mouse brains ex vivo and on human brains in vivo. We showed that these multipole susceptibility tensors can be used to image orientations of ordered white matter fibers. These experiments demonstrate that multipole tensor analysis may enable practical mapping of tissue microstructure in vivo without rotating subject or magnetic field.
MRI; Magnetic susceptibility anisotropy; Susceptibility tensor imaging; Brain connectivity; p-space
Resonant electromagnetic properties of nanoparticles fabricated from high-index semiconductor or dielectric materials are very promising for the realization of novel nanoantennas and metamaterials. In this paper we study optical resonances of Si nanocylinders located on a silica substrate. Multipole analysis of the experimental scattering spectra, based on the decomposed discrete dipole approximation, confirms resonant excitation of electric and magnetic dipole modes in the Si nanocylinders. Influences of light polarization and incident angle on the scattering properties of the nanocylinders are studied. It is shown that the dependence of resonant excitation of the electric and magnetic modes in the nanocylinders on incident angle and polarization of light allows controlling and manipulating the scattered light in this system. The demonstrated properties of Si nanocylinders can be used for the realization of dielectric metasurfaces with different functional optical properties.
The maximum-entropy charge densities of six amino acids and peptides reveal systematic dependencies of the properties at bond critical points on bond lengths. MEM densities demonstrate that low-order multipoles (l
max = 1) and isotropic atomic displacement parameters for H atoms in the multipole model are insufficient for capturing all the features of charge densities in hydrogen bonds.
Charge densities have been determined by the Maximum Entropy Method (MEM) from the high-resolution, low-temperature (T ≃ 20 K) X-ray diffraction data of six different crystals of amino acids and peptides. A comparison of dynamic deformation densities of the MEM with static and dynamic deformation densities of multipole models shows that the MEM may lead to a better description of the electron density in hydrogen bonds in cases where the multipole model has been restricted to isotropic displacement parameters and low-order multipoles (l
max = 1) for the H atoms. Topological properties at bond critical points (BCPs) are found to depend systematically on the bond length, but with different functions for covalent C—C, C—N and C—O bonds, and for hydrogen bonds together with covalent C—H and N—H bonds. Similar dependencies are known for AIM properties derived from static multipole densities. The ratio of potential and kinetic energy densities |V(BCP)|/G(BCP) is successfully used for a classification of hydrogen bonds according to their distance d(H⋯O) between the H atom and the acceptor atom. The classification based on MEM densities coincides with the usual classification of hydrogen bonds as strong, intermediate and weak [Jeffrey (1997) ▶. An Introduction to Hydrogen Bonding. Oxford University Press]. MEM and procrystal densities lead to similar values of the densities at the BCPs of hydrogen bonds, but differences are shown to prevail, such that it is found that only the true charge density, represented by MEM densities, the multipole model or some other method can lead to the correct characterization of chemical bonding. Our results do not confirm suggestions in the literature that the promolecule density might be sufficient for a characterization of hydrogen bonds.
topological properties; hydrogen bonding; maximum entropy method; charge densities; peptides; amino acids
PHEPS (pH-dependent Protein Electrostatics Server) is a web service for fast prediction and experiment planning support, as well as for correlation and analysis of experimentally obtained results, reflecting charge-dependent phenomena in globular proteins. Its implementation is based on long-term experience (PHEI package) and the need to explain measured physicochemical characteristics at the level of protein atomic structure. The approach is semi-empirical and based on a mean field scheme for description and evaluation of global and local pH-dependent electrostatic properties: protein proton binding; ionic sites proton population; free energy electrostatic term; ionic groups proton affinities (pKa,i) and their Coulomb interaction with whole charge multipole; electrostatic potential of whole molecule at fixed pH and pH-dependent local electrostatic potentials at user-defined set of points. The speed of calculation is based on fast determination of distance-dependent pair charge-charge interactions as empirical three exponential function that covers charge–charge, charge–dipole and dipole–dipole contributions. After atomic coordinates input, all standard parameters are used as defaults to facilitate non-experienced users. Special attention was given to interactive addition of non-polypeptide charges, extra ionizable groups with intrinsic pKas or fixed ions. The output information is given as plain-text, readable by ‘RasMol’, ‘Origin’ and the like. The PHEPS server is accessible at .
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.
The concept of fluidic multipoles, in analogy to electrostatics, has long been known as a particular class of solutions of the Navier-Stokes equation in potential flows, however, experimental observations of fluidic multipoles and of their characteristics have not been reported yet. Here we present a two-dimensional microfluidic quadrupole and a theoretical analysis consistent with the experimental observations. The microfluidic quadrupole was formed by simultaneously injecting and aspirating fluids from two pairs of opposing apertures in a narrow gap formed between a microfluidic probe and a substrate. A stagnation point was formed at the center of the microfluidic quadrupole, and its position could be rapidly adjusted hydrodynamically. Following the injection of a solute through one of the poles, a stationary, tunable, and movable – i.e. “floating” – concentration gradient was formed at the stagnation point. Our results lay the foundation for future combined experimental and theoretical exploration of microfluidic planar multipoles including convective-diffusive phenomena.
PMID: 21897375 CAMSID: cams3944
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
Magnetic Split-flow thin (SPLITT) fractionation is a newly developed technique for separating magnetically susceptible particles. Particles with different field-induced velocities can be separated into two fractions by adjusting applied magnetic forces and flow-rates at inlets and outlets.
Magnetic particles, Dynabeads, were used to test this new approach of field-induced velocity for susceptibility determination using magnetic SF at different magnetic field intensities. Reference measurements of magnetic susceptibility were made using a superconducting quantum interference device (SQUID) magnetometer. Various ion-labeled red blood cells (RBC) were used to study susceptibility determination and throughput parameters for analytical and preparative applications of magnetic SPLITT fractionation (SF), respectively. Throughputs were studied at different sample concentrations, magnetic field intensities, and channel flow-rates.
The susceptibilities of Dynabeads determined by SPLITT fractionation (SF) were consistent with those of reference measurement using a superconducting quantum interference device (SQUID) magnetometer. Determined susceptibilities of ion-labeled RBC were consistent within 9.6% variations at two magnetic intensities and different flow-rates. The determined susceptibilities differed by 10% from referenced measurements. The minimum difference in magnetic susceptibility required for complete separation was about 5.0 × 10-6 [cgs]. Sample recoveries were higher than 92%. The throughput of magnetic SF was approximately 1.8 g/h using our experimental setup.
Magnetic SF can provide simple and economical determination of particle susceptibility. This technique also has great potential for cell separation and related analysis. Continuous separations of ion-labeled RBC using magnetic SF were successful over 4 hours. The throughput was increased by 18 folds versus early study. Sample recoveries were 93.1 ± 1.8% in triplicate experiments.
Acute rejection in organ transplant is signaled by the proliferation of T-cells that target and kill the donor cells requiring painful biopsies to detect rejection onset. An alternative non-invasive technique is proposed using a multi-channel superconducting quantum interference device (SQUID) magnetometer to detect T-cell lymphocytes in the transplanted organ labeled with magnetic nanoparticles conjugated to antibodies specifically attached to lymphocytic ligand receptors. After a magnetic field pulse, the T-cells produce a decaying magnetic signal with a characteristic time of the order of a second. The extreme sensitivity of this technique, 105 cells, can provide early warning of impending transplant rejection and monitor immune-suppressive chemotherapy.
Magnetic nanoparticle; SQUID sensor; Remanence field; Transplant rejection; Antibody
The biomagnetic inverse problem has captured the interest of both mathematicians and physicists due to its important applications in the medical field. As a result of our experience in analyzing the electrical activity of the gastric smooth muscle, we present here a theoretical model of the magnetic field in the stomach and a computational implementation whereby we demonstrate its realism and usefulness. The computational algorithm developed for this purpose consists of dividing the magnetic field signal input surface into centroid-based grids that allow recursive least-squares approximations to be applied, followed by comparison tests in which the locations of the best-fitting current dipoles are determined. In the second part of the article, we develop a multiple-regression analysis of experimental gastric magnetic data collected using Superconducting QUantum Interference Device (SQUID) magnetometers and successfully processed using our algorithm. As a result of our analysis, we conclude on statistical grounds that it is sufficient to model the electrical activity of the GI tract using only two electric current dipoles in order to account for the magnetic data recorded non-invasively with SQUID magnetometers above the human abdomen.
gastrointestinal electrical activity; biomagnetic inverse problem; current dipole
Biological polar molecules and polymer structures with energy supply (such as microtubules in the cytoskeleton) can get excited and generate an endogenous electromagnetic field with strong electrical component in their vicinity. The endogenous electrical fields through action on charges, on dipoles and multipoles, and through polarization (causing dielectrophoretic effect) exert forces and can drive charges and particles in the cell. The transport of mass particles and electrons is analyzed as a Wiener-Lévy process with inclusion of deterministic force (validity of the Bloch theorem is assumed for transport of electrons in molecular chains too). We compare transport driven by deterministic forces (together with an inseparable thermal component) with that driven thermally and evaluate the probability to reach the target. Deterministic forces can transport particles and electrons with higher probability than forces of thermal origin only. The effect of deterministic forces on directed transport is dominant.
directed transport; electromagnetic fields in cells; organization in biology
Electro- or magnetoencephalography (EEG/MEG) are of the utmost advantage in studying transient neuronal activity and its timing with respect to behavior in the working human brain. Direct localization of the neural substrates underlying EEG/MEG is commonly achieved by modeling neuronal activity as dipoles. However, the success of neural source localization with the dipole model has only been demonstrated in relatively simple localization tasks owing to the simplified model and its insufficiency in differentiating cortical sources with different extents. It would be of great interest to image complex neural activation with multiple sources of different cortical extensions directly from EEG/MEG. We have investigated this crucial issue by adding additional parameters to the dipole model, leading to the multipole model to better represent the extended sources confined to the convoluted cortical surface. The localization of multiple cortical sources is achieved by the use of the subspace source localization method with the multipole model. Its performance is evaluated with simulated data as compared with the dipole model, and further illustrated with the real data obtained during visual stimulations in human subjects. The interpretation of the localization results is fully supported by our knowledge about their anatomic locations and functional magnetic resonance imaging (fMRI) data in the same experimental setting. Methods for estimating multiple neuronal sources at cortical areas will facilitate our ability to characterize the cortical electrical activity from simple early sensory components to more complex networks, such as in visual, motor and cognitive tasks.
source model; cortical source; extent; electroencephalography; multipole
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
Duty cycle-based trapping and extraction processes have been investigated for linear digitally-driven multipoles by simulating ion trajectories. The duty cycles of the applied waveforms were adjusted so that an effective trapping or ejection electric field was created between the rods and the grounded end cap electrodes. By manipulating the duty cycles of the waveforms, the potentials of the multipole rods can be set equal for part of the waveform cycle. When all rods are negative for this period, the device traps positive ions and when all are positive, it ejects them in focused trajectories. Four Linac II electrodes have been added between the quadrupole rods along the asymptotes to create an electric field along the symmetry axis for collecting the ions near the exit end cap electrode and prompt ejection. This method permits the ions to be collected and then ejected in a concentrated and collimated plug into the acceleration region of a time-of-flight mass spectrometer (TOFMS). Our method has been shown to be independent of mass. Because the resolution of orthogonal acceleration TOFMS depends primarily on the dispersion of the ions injected into the acceleration region and not on the ion mass, this technology will enable high resolution in the ultrahigh mass range (m/z > 20,000).
Parallel imaging techniques have been widely used in high-field magnetic resonance imaging (MRI). Multiple receiver coils have been shown to improve image quality and allow accelerated image acquisition. Magnetic resonance imaging at ultra-low fields (ULF MRI) is a new imaging approach that uses SQUID (superconducting quantum interference device) sensors to measure the spatially encoded precession of pre-polarized nuclear spin populations at microtesla-range measurement fields. In this work, parallel imaging at microtesla fields is systematically studied for the first time. A seven-channel SQUID system, designed for both ULF MRI and magnetoencephalography (MEG), is used to acquire 3D images of a human hand, as well as 2D images of a large water phantom. The imaging is performed at 46 microtesla measurement field with pre-polarization at 40 mT. It is shown how the use of seven channels increases imaging field of view and improves signal-to-noise ratio for the hand images. A simple procedure for approximate correction of concomitant gradient artifacts is described. Noise propagation is analyzed experimentally, and the main source of correlated noise is identified. Accelerated imaging based on one-dimensional undersampling and 1D SENSE (sensitivity encoding) image reconstruction is studied in the case of the 2D phantom. Actual 3-fold imaging acceleration in comparison to single-average fully encoded Fourier imaging is demonstrated. These results show that parallel imaging methods are efficient in ULF MRI, and that imaging performance of SQUID-based instruments improves substantially as the number of channels is increased.
Parallel MRI; ULF MRI; SENSE; MEG; SQUID
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.
One of the challenges in functional brain imaging is integration of complementary imaging modalities, such as magnetoencephalography (MEG) and functional magnetic resonance imaging (fMRI). MEG, which uses highly sensitive superconducting quantum interference devices (SQUIDs) to directly measure magnetic fields of neuronal currents, cannot be combined with conventional high-field MRI in a single instrument. Indirect matching of MEG and MRI data leads to significant co-registration errors. A recently proposed imaging method-SQUID-based microtesla MRI-can be naturally combined with MEG in the same system to directly provide structural maps for MEG-localized sources. It enables easy and accurate integration of MEG and MRI/fMRI, because microtesla MR images can be precisely matched to structural images provided by high-field MRI and other techniques. Here we report the first images of the human brain by microtesla MRI, together with auditory MEG (functional) data, recorded using the same seven-channel SQUID system during the same imaging session. The images were acquired at 46 microtesla measurement field with pre-polarization at 30 mT. We also estimated transverse relaxation times for different tissues at microtesla fields. Our results demonstrate feasibility and potential of human brain imaging by microtesla MRI. They also show that two new types of imaging equipment-low-cost systems for anatomical MRI of the human brain at microtesla fields, and more advanced instruments for combined functional (MEG) and structural (microtesla MRI) brain imaging-are practical.
MEG; MRI; low-field MRI; SQUID; co-registration
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.
Hydration free energy (HFE) is generally used for evaluating molecular solubility, which is an important property for pharmaceutical and chemical engineering processes. Accurately predicting HFE is also recognized as one fundamental capability of molecular mechanics force field. Here we present a systematic investigation on HFE calculations with AMOEBA polarizable force field at various parameterization and simulation conditions. The HFEs of seven small organic molecules have been obtained alchemically using the Bennett Acceptance Ratio (BAR) method. We have compared two approaches to derive the atomic multipoles from quantum mechanical (QM) calculations: one directly from the new distributed multipole analysis (DMA) and the other involving fitting to the electrostatic potential around the molecules. Wave functions solved at the MP2 level with four basis sets (6-311G*, 6-311++G(2d,2p), cc-pVTZ, and aug-cc-pVTZ) are used to derive the atomic multipoles. HFEs from all four basis sets show a reasonable agreement with experimental data (root mean square error 0.63 kcal/mol for aug-ccpVTZ). We conclude that aug-cc-pVTZ gives the best performance when used with AMOEBA, and 6-311++G(2d,2p) is comparable but more efficient for larger systems. The results suggest that the inclusion of diffuse basis functions is important for capturing intermolecular interactions. The effect of long-range correction to van der Waals interaction on the hydration free energies is about 0.1 kcal/mol when the cutoff is 12Å, and increases linearly with the number of atoms in the solute/ligand. In addition, we also discussed the results from a hybrid approach that combines polarizable solute with fixed-charge water in the hydration free energy calculation.
The fast multipole method (FMM) has been shown to have a reduced computational dependence on the size of finest-level groups of elements when the elements are positioned on a regular grid and FFT convolution is used to represent neighboring interactions. However, transformations between plane-wave expansions used for FMM interactions and pressure distributions used for neighboring interactions remain significant contributors to the cost of FMM computations when finest-level groups are large. The transformation operators, which are forward and inverse Fourier transforms with the wave space confined to the unit sphere, are smooth and well approximated using reduced-rank decompositions that further reduce the computational dependence of the FMM on finest-level group size. The adaptive cross approximation (ACA) is selected to represent the forward and adjoint far-field transformation operators required by the FMM. However, the actual error of the ACA is found to be greater than that predicted using traditional estimates, and the ACA generally performs worse than the approximation resulting from a truncated singular-value decomposition (SVD). To overcome these issues while avoiding the cost of a full-scale SVD, the ACA is employed with more stringent accuracy demands and recompressed using a reduced, truncated SVD. The results show a greatly reduced approximation error that performs comparably to the full-scale truncated SVD without degrading the asymptotic computational efficiency associated with ACA matrix assembly.
Fast solvers; iterative methods; acoustic scattering; fast multipole method; adaptive cross approximation; moment methods