Distribution Of Point Charges Research Articles

Intermediate electrostatic field for the generalized elongation method.

An intermediate electrostatic field is introduced to improve the accuracy of fragment-based quantum-chemical computational methods by including long-range polarizations of biomolecules. The point charge distribution of the intermediate field is generated by a charge sensitivity analysis that is parameterized for five different population analyses, namely, atoms-in-molecules, Hirshfeld, Mulliken, natural orbital, and Voronoi population analysis. Two model systems are chosen to demonstrate the performance of the generalized elongation method (ELG) combined with the intermediate electrostatic field. The calculations are performed for the STO-3G, 6-31G, and 6-31G(d) basis sets and compared with reference Hartree-Fock calculations. It is shown that the error in the total energy is reduced by one order of magnitude, independently of the population analyses used. This demonstrates the importance of long-range polarization in electronic-structure calculations by fragmentation techniques.

Strand-biased gene distribution, purine assymetry and environmental factors influence protein evolution in Bacillus

A strong purine asymmetry, along with strand-biased gene distribution and the presence of PolC, prevails in Bacillus and some other members of Firmicutes, Fusobacteria and Tenericutes. The analysis of protein features in 21 Bacillus species of diverse metabolic, virulence and ecological traits revealed that purine asymmetry in conjunction with lineage/niche specific constraints significantly influences protein evolution in Bacillus. All Bacillus species, except for Se-respiring Bacillusselenitireducens, display distinct strand-specific biases in amino acid usage, which may affect the isoelectric point or surface charge distribution of proteins with prevalence of acidic and basic residues in the leading and lagging strand proteins, respectively.

Building Water Models: A Different Approach.

Simplified classical water models are currently an indispensable component in practical atomistic simulations. Yet, despite several decades of intense research, these models are still far from perfect. Presented here is an alternative approach to constructing widely used point charge water models. In contrast to the conventional approach, we do not impose any geometry constraints on the model other than the symmetry. Instead, we optimize the distribution of point charges to best describe the “electrostatics” of the water molecule. The resulting “optimal” 3-charge, 4-point rigid water model (OPC) reproduces a comprehensive set of bulk properties significantly more accurately than commonly used rigid models: average error relative to experiment is 0.76%. Close agreement with experiment holds over a wide range of temperatures. The improvements in the proposed model extend beyond bulk properties: compared to common rigid models, predicted hydration free energies of small molecules using OPC are uniformly closer to experiment, with root-mean-square error <1 kcal/mol.

Intermediate electrostatic field for the elongation method.

A simple way to improve the accuracy of the fragmentation methods is proposed. The formalism was applied to the elongation (ELG) method at restricted open-shell Hartree-Fock (ROHF) level of theory. The α-helix conformer of polyglycine was taken as a model system. The modified ELG method includes a simplified electrostatic field resulting from point-charge distribution of the system’s environment. In this way the long-distance polarization is approximately taken into account. The field attenuates during the ELG process to eventually disappear when the final structure is reached. The point-charge distributions for each ELG step are obtained from charge sensitivity analysis (CSA) in force-field atoms resolution. The presence of the intermediate field improves the accuracy of ELG calculations. The errors in total energy and its kinetic and potential contributions are reduced by at least one-order of magnitude. In addition the SCF convergence of ROHF scheme is improved.

Correlated Ions in a Calcium Channel Model: A Poisson–Fermi Theory

We derive a continuum model, called the Poisson-Fermi equation, of biological calcium channels (of cardiac muscle, for example) designed to deal with crowded systems in which ionic species and side chains nearly fill space. The model is evaluated in three dimensions. It includes steric and correlation effects and is derived from classical hard-sphere lattice models of configurational entropy of finite size ions and solvent molecules. The maximum allowable close packing (saturation) condition is satisfied by all ionic species with different sizes and valences in a channel system, as shown theoretically and numerically. Unphysical overcrowding ("divergence") predicted by the Gouy-Chapman diffuse model (produced by a Boltzmann distribution of point charges with large potentials) does not occur with the Fermi-like distribution. Using probability theory, we also provide an analytical description of the implicit dielectric model of ionic solutions that gives rise to a global and a local formula for the chemical potential. In this primitive model, ions are treated as hard spheres and solvent molecules are described as a dielectric medium. The Poisson-Fermi equation is a local formula dealing with different correlations at different places. The correlation effects are apparent in our numerical results. Our results show variations of dielectric permittivity from bath to channel pore described by a new dielectric function derived as an output from the Poisson-Fermi analysis. The results are consistent with existing theoretical and experimental results. The binding curve of Poisson-Fermi is shown to match Monte Carlo data and illustrates the anomalous mole fraction effect of calcium channels, an effective blockage of permeation of sodium ions by a tiny concentration (or number) of calcium ions.

Numerical methods for the Poisson–Fermi equation in electrolytes

The Poisson–Fermi equation proposed by Bazant, Storey, and Kornyshev [Phys. Rev. Lett. 106 (2011) 046102] for ionic liquids is applied to and numerically studied for electrolytes and biological ion channels in three-dimensional space. This is a fourth-order nonlinear PDE that deals with both steric and correlation effects of all ions and solvent molecules involved in a model system. The Fermi distribution follows from classical lattice models of configurational entropy of finite size ions and solvent molecules and hence prevents the long and outstanding problem of unphysical divergence predicted by the Gouy–Chapman model at large potentials due to the Boltzmann distribution of point charges. The equation reduces to Poisson–Boltzmann if the correlation length vanishes. A simplified matched interface and boundary method exhibiting optimal convergence is first developed for this equation by using a gramicidin A channel model that illustrates challenging issues associated with the geometric singularities of molecular surfaces of channel proteins in realistic 3D simulations. Various numerical methods then follow to tackle a range of numerical problems concerning the fourth-order term, nonlinearity, stability, efficiency, and effectiveness. The most significant feature of the Poisson–Fermi equation, namely, its inclusion of steric and correlation effects, is demonstrated by showing good agreement with Monte Carlo simulation data for a charged wall model and an L type calcium channel model.

Difficulties in applying pure Kohn–Sham density functional theory electronic structure methods to protein molecules

Self-consistency-based Kohn–Sham density functional theory (KS-DFT) electronic structure calculations with Gaussian basis sets are reported for a set of 17 protein-like molecules with geometries obtained from the Protein Data Bank. It is found that in many cases such calculations do not converge due to vanishing HOMO–LUMO gaps. A sequence of polyproline I helix molecules is also studied and it is found that self-consistency calculations using pure functionals fail to converge for helices longer than six proline units. Since the computed gap is strongly correlated to the fraction of Hartree–Fock exchange, test calculations using both pure and hybrid density functionals are reported. The tested methods include the pure functionals BLYP, PBE and LDA, as well as Hartree–Fock and the hybrid functionals BHandHLYP, B3LYP and PBE0. The effect of including solvent molecules in the calculations is studied, and it is found that the inclusion of explicit solvent molecules around the protein fragment in many cases gives a larger gap, but that convergence problems due to vanishing gaps still occur in calculations with pure functionals. In order to achieve converged results, some modeling of the charge distribution of solvent water molecules outside the electronic structure calculation is needed. Representing solvent water molecules by a simple point charge distribution is found to give non-vanishing HOMO–LUMO gaps for the tested protein-like systems also for pure functionals.

Implementation of a Protein Reduced Point Charge Model toward Molecular Dynamics Applications

A reduced point charge model was developed in a previous work from the study of extrema in smoothed charge density distribution functions generated from the Amber99 molecular electrostatic potential. In the present work, such a point charge distribution is coupled with the Amber99 force field and implemented in the program TINKER to allow molecular dynamics (MD) simulations of proteins. First applications to two polypeptides that involve α-helix and β-sheet motifs are analyzed and compared to all-atom MD simulations. Two types of coarse-grained (CG)-based trajectories are generated using, on one hand, harmonic bond stretching terms and, on the other hand, distance restraints. Results show that the use of the unrestrained CG conditions are sufficient to preserve most of the secondary structure characteristics but restraints lead to a better agreement between CG and all-atom simulation results such as rmsd, dipole moment, and time-dependent mean square deviation functions.

Distribution of point charges with small discrete energy

We study the asymptotic equidistribution of points near arbitrary compact sets of positive capacity in Rd, d ≥ 2. Our main tools are the energy estimates for Riesz potentials. We also consider the quantitative aspects of this equidistribution in the classical Newtonian case. In particular, we quantify the weak convergence of discrete measures to the equilibrium measure, and give the estimates of convergence rates for discrete potentials to the equilibrium potential. 1. Asymptotic equidistribution of discrete sets Let E be a compact set in R, d ≥ 2. Denote the Euclidean distance between x ∈ R and y ∈ R by |x− y|. We consider potential theory associated with Riesz kernels kα(x) := |x|α−d, x ∈ R, 0 < α < d. For a Borel measure μ with compact support, define its energy by Iα[μ] := ∫∫ kα(x− y) dμ(x)dμ(y). A central theme in potential theory is the study of the minimum energy problem Wα(E) := inf μ∈M(E) Iα[μ], where M(E) is the space of all positive unit Borel measures supported on E. If Robin’s constant Wα(E) is finite, then the above infimum is attained by the equilibrium measure μE ∈ M(E) [14, p. 131–133], which is a unique probability measure expressing the steady state distribution of charge on the conductor E. The capacity of E is defined by Cα(E) := 1 Wα(E) , where we set Cα(E) = 0 when Wα(E) is infinite. For a more detailed exposition of Riesz potential theory, we refer the reader to the book of Landkof [14]. 2010 Mathematics Subject Classification. Primary 31C20; Secondary 31C15.

Numerical computer calculation of distribution of similar point charges at the dielectric surface

Results of numerical calculation of the distribution of similar point charges over a uniformly charged dielectric plane are presented for different surface charge densities and surface temperatures. It is shown that, in the case of the surface charge density ranging from 2 × 10−6 to 7 × 10−5 C/m2 and temperatures ranging from 300 to 800 K, the Lindemann criterion is fulfilled for the average deviation of the point charge from the equilibrium position, and the collection of point charges at the dielectric surface can be regarded as a 2D Coulomb crystal.

Low Order Physical Multipoles

Point multipole expansions are widely used to gain physical insight into complex distributions of charges and to reduce the cost of computing interactions between such distributions. However, practical applications that typically retain only a few leading terms may suffer from unacceptable loss of accuracy in the near-field. We propose an alternative approach for approximating electrostatic charge distributions, Optimal Physical Multipoles (OPMs), which optimally represent the original charge distribution with a set of point charges. By construction, approximation of electrostatic potential based on OPMs retains many of the useful properties of the corresponding point multipole expansion, including the same asymptotic behavior of the approximate potential for a given multipole order. At the same time, OPMs can be significantly more accurate in the near field: up to 5 times more accurate for some of the charge distributions tested here which are relevant to biomolecular modeling. Unlike point multipoles, for point charge distributions the OPM always converges to the original point charge distribution at finite order. Furthermore, OPMs may be more computationally efficient and easier to implement into existing molecular simulations software packages than approximation schemes based on point multipoles. In addition to providing a general framework for computing OPMs to any order, closed-form expressions for the lowest order OPMs (monopole and dipole) are derived. Thus, for some practical applications Optimal Physical Multipoles may represent a preferable alternative to point multipoles.

Using multipole point charge distributions to provide the electrostatic potential in the variational explicit polarization (X-Pol) potential

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.

Potencial elétrico para distribuições de cargas puntiformes: sobre a convergência de séries infinitas

Neste trabalho calculamos o potencial elétrico para algumas distribuições infinitas de cargas puntiformes. Para cada distribuição discutimos os critérios de convergência das séries obtidas. Calculamos também a densidade de carga superficial induzida em um plano condutor infinito "aterrado" e em uma esfera condutora "aterrada" de raio a por uma distribuição infinita de cargas puntiformes com sinais alternados e magnitude que diminui harmonicamente com n (~ q/n)

A non-linear model for elastic dielectric crystals with mobile vacancies

A framework is developed for electromechanical behavior of dielectric crystalline solids subjected to finite deformations. The theory is formulated in the context of electrostatics; however, vacancies in the lattice may carry an electric charge, and their concentrations may be large. The material is treated as a continuous body with a continuous distribution of point vacancies, but volumes and charges of individual defects enter the description. The deformation gradient is decomposed multiplicatively into terms accounting for recoverable thermoelasticity and irreversible volume changes associated with vacancies. Thermodynamic arguments lead to constitutive relations among electromechanical quantities framed in the elastically unloaded intermediate configuration, with the Cauchy stress tensor consistently non-symmetric as a result of electrostatic effects. The requirement of non-negative dissipation imposes constraints on vacancy migration. Following postulation of a quadratic form for the free energy potential, a kinetic equation for vacancy flux is derived in the intermediate configuration, with diffusion driven by gradients of vacancy concentration, electrostatic potential, hydrostatic pressure, and crystal structure. Effects of geometric non-linearity (i.e. finite elastic strains and large vacancy concentrations) are found to affect vacancy diffusion in a body subjected to biaxial lattice strain, for example a film device with lattice mismatch at its interfaces.

Computational design of electrostatic potential to maximise or minimise adhesion to surfaces

The free energy of adsorption based on electrostatic interactions only has been optimised for glycine on a (111) diamond surface in continuum water to show maximum adhesion or maximum repulsion by adjusting the partial point charge distribution in the diamond surface model. It has been found that the free energy of adhesion is readily maximised by means of increasing the polarity of the electrostatic surface potential in a distinctive dipolar pattern. The maximum adhesion obtained in the present calculations was − 21.02 ± 0.49 kJ mol − 1 . The surface with the maximum repulsion, which would be of particular interest for designing novel surfaces with low protein adsorption, had a mostly uniform, slightly positive, surface electrostatic potential with a specific irregular pattern. The maximum free energy of repulsion achieved in the present simulations was 0.08 ± 0.01 kJ mol − 1 . These values suggest that most real surfaces are more likely to be overall attractive due to electrostatic interactions of the partial surface charges, and that the overall energy of repulsion, although theoretically possible, is likely to be small.

Condition of formation of 2D Coulomb crystal on the surface of dielectric

This study deals with the distribution of point charges on the surface of a uniformly charged dielectric plate; the expression for the criterion of formation of 2D Coulomb crystal is obtained. This criterion is independent of the charge surface density and is defined only by the temperature of dielectric surface. At room temperature, the aggregate of point charges on dielectric surface may be considered as 2D Coulomb crystal.

An analytical approach to computing biomolecular electrostatic potential. I. Derivation and analysis

Analytical approximations to fundamental equations of continuum electrostatics on simple shapes can lead to computationally inexpensive prescriptions for calculating electrostatic properties of realistic molecules. Here, we derive a closed-form analytical approximation to the Poisson equation for an arbitrary distribution of point charges and a spherical dielectric boundary. The simple, parameter-free formula defines continuous electrostatic potential everywhere in space and is obtained from the exact infinite-series (Kirkwood) solution by an approximate summation method that avoids truncating the infinite series. We show that keeping all the terms proves critical for the accuracy of this approximation, which is fully controllable for the sphere. The accuracy is assessed by comparisons with the exact solution for two unit charges placed inside a spherical boundary separating the solute of dielectric 1 and the solvent of dielectric 80. The largest errors occur when the source charges are closest to the dielectric boundary and the test charge is closest to either of the sources. For the source charges placed within 2 A from the boundary, and the test surface located on the boundary, the root-mean-square error of the approximate potential is less than 0.1 kcal/mol/mid R:emid R: (per unit test charge). The maximum error is 0.4 kcal/mol/mid R:emid R:. These results correspond to the simplest first-order formula. A strategy for adopting the proposed method for realistic biomolecular shapes is detailed. An extensive testing and performance analysis on real molecular structures are described in Part II that immediately follows this work as a separate publication. Part II also contains an application example.

Adsorption of Pb(II) and Pb(II)-citric acid on sawdust activated carbon: Kinetic and equilibrium isotherm studies

The removal of Pb(II) and Pb(II)-citric acid (Pb(II)-CA) from aqueous solutions by sawdust activated carbon (SDAC) was investigated. The higher adsorptive removal of Pb(II) from aqueous solutions containing Pb(II)-CA than Pb(II) only was observed due to the presence of CA in the former system. The mechanism of adsorption process was studied by conducting pH as well as kinetic studies. Batch adsorption experiments were conducted to study the effect of adsorbent dose, initial concentration and temperature for the removal of Pb(II) from Pb(II) only and Pb(II)-CA aqueous systems. The adsorption was maximum for the initial pH in the range of 6.5–8.0 and 2.0–5.0 for Pb(II) and Pb(II)-CA, respectively. The solution pH, zero point charge (pH zpc) and species distribution of Pb(II) and Pb(II)-CA were found to play an important role in the adsorption of Pb(II) and Pb(II)-CA onto SDAC from water and wastewaters. SDAC exhibited very high adsorption potential for Pb(II) ions in presence of CA than when Pb(II) ions alone were present. The kinetic and equilibrium adsorption data were well modeled using pseudo-first-order kinetics and Langmuir isotherm model, respectively.

Optimization of cutting schemes for the evaluation of molecular electrostatic potentials in proteins via Moving-Domain QM/MM

This work presents new developments of the moving-domain QM/MM (MoD-QM/MM) method for modeling protein electrostatic potentials. The underlying goal of the method is to map the electronic density of a specific protein configuration into a point-charge distribution. Important modifications of the general strategy of the MoD-QM/MM method involve new partitioning and fitting schemes and the incorporation of dynamic effects via a single-step free energy perturbation approach (FEP). Selection of moderately sized QM domains partitioned between C (alpha) and C (from C=O), with incorporation of delocalization of electrons over neighboring domains, results in a marked improvement of the calculated molecular electrostatic potential (MEP). More importantly, we show that the evaluation of the electrostatic potential can be carried out on a dynamic framework by evaluating the free energy difference between a non-polarized MEP and a polarized MEP. A simplified form of the potassium ion channel protein Gramicidin-A from Bacillus brevis is used as the model system for the calculation of MEP.

The Euclidean gravitational action as black hole entropy, singularities, and spacetime voids

We argue why the static spherically symmetric vacuum solutions of Einstein’s equations described by the textbook Hilbert metric gμν(r) is not diffeomorphic to the metric gμν(∣r∣) corresponding to the gravitational field of a point mass delta function source at r=0. By choosing a judicious radial function R(r)=r+2G∣M∣Θ(r) involving the Heaviside step function, one has the correct boundary condition R(r=0)=0, while displacing the horizon from r=2G∣M∣ to a location arbitrarily close to r=0 as one desires, rh→0, where stringy geometry and quantum gravitational effects begin to take place. We solve the field equations due to a delta function point mass source at r=0, and show that the Euclidean gravitational action (in ℏ units) is precisely equal to the black hole entropy (in Planck area units). This result holds in any dimensions D⩾3. In the Reissner–Nordstrom (massive charged) and Kerr–Newman black hole case (massive rotating charged) we show that the Euclidean action in a bulk domain bounded by the inner and outer horizons is the same as the black hole entropy. When one smears out the point-mass and point-charge delta function distributions by a Gaussian distribution, the area-entropy relation is modified. We postulate why these modifications should furnish the logarithmic corrections (and higher inverse powers of the area) to the entropy of these smeared black holes. To finalize, we analyze the Bars–Witten stringy black hole in 1+1 dimension and its relation to the maximal acceleration principle in phase spaces and Finsler geometries.