Self-consistent treatment of the local dielectric permittivity and electrostatic potential in solution for polarizable macromolecular force fields

A self-consistent method is presented for the calculation of the local dielectric permittivity and electrostatic potential generated by a solute of arbitrary shape and charge distribution in a polar and polarizable liquid. The structure and dynamics behavior of the liquid at the solute∕liquid interface determine the spatial variations of the density and the dielectric response. Emphasis here is on the treatment of the interface. The method is an extension of conventional methods used in continuum protein electrostatics, and can be used to estimate changes in the static dielectric response of the liquid as it adapts to charge redistribution within the solute. This is most relevant in the context of polarizable force fields, during electron structure optimization in quantum chemical calculations, or upon charge transfer. The method is computationally efficient and well suited for code parallelization, and can be used for on-the-fly calculations of the local permittivity in dynamics simulations of systems with large and heterogeneous charge distributions, such as proteins, nucleic acids, and polyelectrolytes. Numerical calculation of the system free energy is discussed for the general case of a liquid with field-dependent dielectric response.     

Continuum Field Charging of Dielectric Spheroids

Classical continuum theory for field charging is applied in an analysis of the ionic charging of spheroidal dielectrics. Assuming that the particle orientation is fixed during the charging process, the saturation charge and charging rate are determined as functions of the orientation and aspect ratio of spheroids. For spheroids of small dielectric constants the saturation charge becomes the largest when the electric field is directed perpendicular to the major axis of the spheroid. For an ensemble of randomly oriented spheroids the average saturation charge can be approximated as the arithmetic average of the saturation charges for the spheroid with the electric field directed along the three principal axes of the spheroid. In addition, the ensemble average of the dimensionless charging rate of randomly oriented spheroids of moderate axial ratio approximates the dimensionless charging rate of a sphere. Copyright 2000 Academic Press.     

Modeling the surface charge evolution of spherical nanoparticles by considering dielectric discontinuity effects at the solid/electrolyte solution interface

It is well known that the electrostatic repulsions between charges on neighboring sites decrease the effective charge at the surface of a charged nanoparticle (NP). However, the situation is more complex close to a dielectric discontinuity, since charged sites are interacting not only with their neighbors but also with their own image charges and the image charges of all neighbors. Titrating site positions, solution ionic concentration, dielectric discontinuity effects, and surface charge variations with pH are investigated here using a grand canonical Monte Carlo method. A Tanford and Kirkwood approach is used to calculate the interaction potentials between the discrete charged sites. Homogeneous, heterogeneous, and patch site distributions are considered to reproduce the various titrating site distributions at the solid/solution interface of spherical NPs. By considering Coulomb, salt, and image charges effects, results show that for different ionic concentrations, modifications of the dielectric constant of NPs having homogeneous and heterogeneous site distributions have little effect on their charging process. Thus, the reaction field, due to the presence of image charges, fully counterbalances the Coulomb interactions. This is not the case for patch distributions, where Coulomb interactions are not completely counterbalanced by the reaction field. Application of the present model to pyrogenic silica is also performed and comparison is made with published experimental data of titration curves at various ionic concentrations.     

Effects of surface site distribution and dielectric discontinuity on the charging behavior of nanoparticles: a grand canonical Monte Carlo study

The surface site distribution and the dielectric discontinuity effects on the charging process of a spherical nanoparticle (NP) have been investigated. It is well known that electrostatic repulsion between charges on neighbouring sites tends to decrease the effective charge of a NP. The situation is more complicated close to a dielectric breakdown, since here a charged site is not only interacting with its neighbours but also with its own image charge and the image charges of all its neighbours. Coexistence of opposite charges, titration sites positions, and pH dependence are systematically studied using a grand canonical Monte Carlo method. A Tanford and Kirkwood approach has been applied to describe the interaction potentials between explicit discrete ampholytic charging sites. Homogeneous, heterogeneous and patch site distributions were considered to reproduce the titration site distribution at the solid/solution interface of natural NPs. Results show that the charging process is controlled by the balance between Coulomb interactions and the reaction field through the solid-liquid interface. They also show that the site distribution plays a crucial role in the charging process. In patch distributions, charges accumulate at the perimeter of each patch due to finite size effects. When homogeneous and heterogeneous distributions are compared, three different charging regimes are obtained. In homogeneous and heterogeneous (with quite low polydispersity indexes) distributions, the effects of the NP dielectric constant on Coulomb interactions are counterbalanced by the reaction field and in this case, the dielectric breakdown has no significant effect on the charging process. This is not the case in patch distributions, where the dielectric breakdown plays a crucial role in the charging process.     

Modifying the charge state distribution of proteins in electrospray ionization mass spectrometry by chemical derivatization

Electrospray ionization (ESI) of denatured proteins produces a broad distribution of multiply-charged ions leading to multiple peaks in the mass spectrum. We investigated changes in the positive-mode ESI charge state distribution produced by several chemical modifications of denatured proteins. Capping carboxylic acid groups with neutral functional groups yields little change in charge state distribution compared with unmodified proteins. The results indicate that carboxyl groups do not play a significant role in the positive charging of denatured proteins in ESI. The modification of proteins with additional basic sites or fixed positive charges generates substantially higher charge states, providing evidence that the number of ionizable sites, rather than molecular size and shape, determines ESI charging for denatured proteins. Fixed charge modification also significantly reduces the number of protons acquired by a protein, in that the charge state envelope is not increased by the full number of fixed charges appended. This result demonstrates that Coulombic repulsion between positive charges plays a significant role in determining charge state distribution by affecting the gas-phase basicity of ionizable sites. Addition of fixed-charge moieties to a protein is a useful approach for shifting protein charge state distributions to higher charge states, and with further work, it may help limit the distribution of protein ions to fewer charge states.     

Atomistic simulation on charge mobility of amorphous tris(8-hydroxyquinoline) aluminum (Alq3): origin of Poole-Frenkel-type behavior

The atomistic simulation of charge transfer process for an amorphous Alq(3) system is reported. By employing electrostatic potential charges, we calculate site energies and find that the standard deviation of site energy distribution is about twice as large as predicted in previous research. The charge mobility is calculated via the Miller-Abrahams formalism and the master equation approach. We find that the wide site energy distribution governs Poole-Frenkel-type behavior of charge mobility against electric field, while the spatially correlated site energy is not a dominant mechanism of Poole-Frenkel behavior in the range from 2x10(5) to 1.4x10(6) V/cm. Also we reveal that randomly meshed connectivities are, in principle, required to account for the Poole-Frenkel mechanism. Charge carriers find a zigzag pathway at low electric field, while they find a straight pathway along electric field when a high electric field is applied. In the space-charge-limited current scheme, the charge-carrier density increases with electric field strength so that the nonlinear behavior of charge mobility is enhanced through the strong charge-carrier density dependence of charge mobility.     

Dielectric relaxation in proteins: a continuum electrostatics model incorporating dielectric heterogeneity of the protein and time‐dependent charges

A boundary element formulation of continuum electrostatics is used to examine time‐independent dielectric relaxation and screening in two proteins, and time‐dependent relaxation in two simpler solutes. Cytochrome c oxidation is modeled by inserting partial charges on the heme, using one to three dielectric regions in the protein. It was suggested recently that for charge insertion on a protein‐bound ligand, all or part of the ligand should be treated as a cavity within the protein medium. Here, the effect of an internal cavity surrounding the central heme atoms is examined, considering separately the static and relaxation (or reorganization) free energies. The former is the free energy to remove the redox electron while maintaining the rest of the structure and charge distribution fixed; the latter is the free energy associated with the relaxation into the product state after the corresponding constraints are released. The effect of the cavity is found to be small for the static free energy, while for the relaxation free energy it is large, as polarization of groups immediately around the heme dominates the relaxation. If the protein surface groups are treated as a distinct medium with a dielectric of 25 (as suggested by recent molecular dynamics simulations), the relaxation free energy decreases significantly (from −37.0 to −43.9 kcal/mol), compared to a model where the whole protein has a dielectric constant of two. Therefore, with this model, although polarization of groups immediately around the heme still dominates the relaxation, polar groups near the protein surface also contribute significantly, and solvent negligibly. The screening of an applied field within myoglobin is calculated, with the protein surrounded by either a low‐dielectric or a high‐dielectric glass. In the vicinity of the CO ligand, the screening is approximately isotropic with a low‐dielectric glass. It is anisotropic with a high‐dielectric glass, but the applied and local fields are still approximately parallel. This has implications for experiments that probe dielectric screening in proteins with the newly developed technique of vibrational Stark spectroscopy: with a high‐dielectric glass, a single, rotationally averaged screening factor can be used, the local field being about 1.65 times the applied field. Finally, we calculate the time‐dependent relaxation in response to instantaneous charge insertion within a spherical cavity in a Debye solvent, and to photoexcitation of a tryptophan solute, illustrating the extension of the boundary element formulation to time‐dependent problems. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 290–305, 2001     

Further studies on the origins of asymmetric charge partitioning in protein homodimers

Dissociation of gas-phase protonated protein dimers into their constituent monomers can result in either symmetric or asymmetric charge partitioning. Dissociation of alpha-lactalbumin homodimers with 15+ charges results in a symmetric, but broad, distribution of protein monomers with charge states centered around 8+/7+. In contrast, dissociation of the 15+ heterodimer consisting of one molecule in the oxidized form and one in the reduced form results in highly asymmetric charge partitioning in which the reduced species carries away predominantly 11+ charges, and the oxidized molecule carries away 4+ charges. This result cannot be adequately explained by differential charging occurring either in solution or in the electrospray process, but appears to be best explained by the reduced species unfolding upon activation in the gas phase with subsequent separation and proton transfer to the unfolding species in the dissociation complex to minimize Coulomb repulsion. For dimers of cytochrome c formed directly from solution, the 17+ charge state undergoes symmetric charge partitioning whereas dissociation of the 13+ is asymmetric. Reduction of the charge state of dimers with 17+ charges to 13+ via gas-phase proton transfer and subsequent dissociation of the mass selected 13+ ions results in a symmetric charge partitioning. This result clearly shows that the structure of the dimer ions with 13+ charges depends on the method of ion formation and that the structural difference is responsible for the symmetric versus asymmetric charge partitioning observed. This indicates that the asymmetry observed when these ions are formed directly from solution must come about due either to differences in the monomer conformations in the dimer that exist in solution or that occur during the electrospray ionization process. These results provide additional evidence for the origin of charge asymmetry that occurs in the dissociation of multiply charged protein complexes and indicate that some solution-phase information can be obtained from these gas-phase dissociation experiments.     

The charging of a liquid dielectric upon its flow past a flat-plate

The charging of a liquid dielectric upon its flow past a flat-plate is considered. Analytical expressions for the density distribution of electric charge and charging current are derived. The dependences of the current and charge density distribution on the system parameters are investigated. The effect of the electric field that emerges owing to charge separation on the charging process is taken into account. Consideration of the electric field is shown to lead to lower values of the electric charge density and charging current. As the Debye number decreases, the charge density also decreases. The charging current increases with an increase in the Debye number.     

Solvation of ions in liquid solvents and proteins

Despite their well-known drawbacks, the approaches of continuum electrostatics are widely used at the analysis of the energies of solvation and reorganization. We propose a method to check the applicability of these approaches in the determination of the solvation energy, which is based on measuring the difference of redox potentials ΔE of two consecutive redox reactions, e.g. for the pairs Co(Cp) ₂ ⁺ /Co(Cp)₂/Co(Cp) ₂ ^? (here, Cp is cyclopentadienyl). In this difference, the solvophobic effects and the liquid junction potential between the working and reference electrodes, which is impossible to measure, cancel out. From the difference of ΔE in two different solvents, the sum of the electrostatic components of the cation-and anion-transfer energies is determined. It is shown that, for large low-charged ions in aprotic media, the continuum electrostatics proves to be true in a wide range of dielectric permittivities including those typical for proteins. The Stokes shift of fluorescence spectra for proflavine (PF) showed that the water reorganization energy and, hence, the energy of the static dielectric response are anomalously high. To study this effect on the solvation energy, we determined the redox potentials of the Co(Cp) ₂ ⁺ /Co(Cp)₂ pair in a number of water-organic media. The organic cosolvent breaks the water structure and reduces the reorganization energy. Accordingly, the redox potential turns more positive. This allowed us to determine the energy of transfer of Co(Cp) ₂ ⁺ ions (and, hence, of other ions) nonviolated by the water structure specifics. The experimental energies of the acetate transfer exceed those calculated by an order of magnitude. This demonstrates the incorrectness of the widely used semicontinuum calculations of the pK of ionogenic groups of proteins. A new algorithm, which permits overcoming this discrepancy, is proposed, namely, the short-range interactions are taken into account based on the experimental energies of the transfer to a model DMF solvent, while the transfer energy from this solvent to the protein is calculated electrostatically. The energy of the ion charging in a protein consists of two physically different components, namely, the charging energy in the pre-existing field of protein dipoles and charges and the energy of the dielectric response of the medium. The former energy is determined by the electronic polarization of the protein (its optical dielectric permittivity), while the latter is determined by all kinds of polarization (static permittivity). Taking into account all the aforementioned peculiarities leads to reasonable agreement with the experiment when estimating the pK of certain groups in α-chymotrypsin. These calculations as well as experimental data (both our and taken form the literature (molecular dynamics)) point to the enhanced dielectric permittivity of the outer layers of proteins.     

Energy contribution of the solvent to the charge migration in DNA

The authors have investigated the interactions of the reaction centers, participating in the charge transfer reaction within the DNA molecule with the phosphate backbones and the solvent molecules, and have estimated the contribution of these interactions into the charge migration in DNA. They have determined the unequal shift of the energy surfaces of the initial and final transition states of the transfer reaction along the energy axis and the dependence of the magnitude of the energy shift on the nature of the reaction centers and the surrounding environment. The nonuniform distribution of the negative charge in the DNA phosphate backbones results in an increase of the positive shift of the energy surface of the DNA base pairs in the center of the structure, where the maximum density of the negative charge is concentrated. Localization of the positive charge on the guanine and the adenine in the DNA base pairs in the oxidized state results in a dependence of the free energy of reaction in the solvent on the pair sequences and their arrangement in the DNA chain. As an example, for the G-C/A-T configuration the positive charges are localized on the same strand that results in a decrease of the free energy of reaction in the solvent for charge migration from G-C to A-T pair by 0.125 eV.     

Generation and transport of charge carriers in polyvinylcarbazole

The processes of generation and transport of excess charge carriers in polyvinylcarbazole at room temperature and at 353 K were experimentally studied. The polymer was charged with pulses of electrons at energies of 7 and 50 keV, which differed in both linear energy transfer and track structure. A universal method of investigation based on the combination of the time-of-flight technique in the both (surface and bulk) modes with the measurement of radiation-induced conductivity was used. The radiation-chemical yield of free charges was measured using two independent procedures. It was shown that the radiation chemical yield of free charges at 293 K was somewhat smaller for 7-keV than for 50-keV electrons; in the latter case, G = 1.1 in an electric field of 2 × 10⁷ V/m. The parameters of the generalized physical model were determined for polyvinylcarbazole.     

Dielectric breakdown and space charge effects in molecular crystal stark spectroscopy

Stark splittings of guest molecules in host crystals are used as a probe of the internal electric field and by Gauss' law the space and polarization charge in molecular crystal dielectric. It was possible with this technique to study both the static problem of charge distribution and to some extent the dynamic problem of charge formation and dissipation. Information also was obtained abouth the mechanism that cause dielectric breakdown in these experiments.     

Dielectric relaxation in an enzyme active site: molecular dynamics simulations interpreted with a macroscopic continuum model.

Dielectric relaxation plays an important role in many chemical processes in proteins, including acid-base titration, ligand binding, and charge transfer reactions. Its complexity makes experimental characterization difficult, and so, theoretical approaches are valuable. The comparison of molecular dynamics free energy simulations with simpler models such as a dielectric continuum model is especially useful for obtaining qualitative insights. We have analyzed a charge insertion process that models deprotonation or mutation of an important side chain in the active site of the enzyme aspartyl-tRNA synthetase. Complexes with the substrate aspartate and the analogue asparagine were studied. The resulting dielectric relaxation was found to involve both ligand and side chain rearrangements in the active site and to account for a large part of the overall charging free energy. With the continuum model, charge insertion is performed along a two-step pathway: insertion into a static environment, followed by relaxation of the environment. These correspond to different physical processes and require different protein dielectric constants. A low value of approximately 1 is needed for the static step, consistent with the parametrization of the molecular mechanics charge set used. A value of 3-6 (depending on the exact insertion site and the nature of the ligand) is needed to describe the dielectric relaxation step. This moderate value indicates that, for this system, the local protein polarizability in the active site is within at most a factor of 2 of that expected at nonspecific positions in a protein interior.     

Charge carrier field emission determines the number of charges on native state proteins in electrospray ionization

Although multiple charging in electrospray ionization (ESI) is essential to protein mass spectrometry, the underlying mechanism of multiple charging has not been explicated. Here, we present a new theory to describe ESI of native-state proteins and predict the number of excess charges on proteins in ESI. The theory proposes that proteins are ionized as charged residues in ESI, as they retain residual excess charges after solvent evaporation and do not desorb from charged ESI droplets. However, their charge state is not determined by the Rayleigh limit of a droplet of similar size to the protein; rather, their final charge state is determined by the electric field-induced emission of small charged solute ions and clusters from protein-containing ESI droplets. This theory predicts that the number of charges on a protein in ESI should be directly proportional to the square of the gas-phase protein diameter and to E*, the critical electric field strength at which ion emission from droplets occurs. This critical field strength is determined by the properties of the excess charge carriers (i.e., the solute) in droplets. Charge-state measurements of native-state proteins with molecular masses in the 5-76 kDa range in ammonium acetate and triethylammonium bicarbonate are in excellent agreement with theoretical predictions and strongly support the mechanism of protein ESI proposed here.     

The mechanism of the appearance of luminescence and electrification in liquid flows through narrow channels

The mechanism of the effects observed in hydrodynamic unit throttles was studied. These effects included luminescence in the visible range localized in a microscopic toroidal volume and electric pulses when a dielectric liquid flew through a narrow passage orifice. Equations for charging and conduction currents were obtained. The stationary electric charge, potential, and field strength on the internal surface of a passage orifice were calculated. It was shown theoretically that the appearance of luminescence most probably occurred in electrical breakdowns in cavitation bubbles in the initial flow section inside the passage orifice. Electric charge formed not only during hydrodynamic cavitation but also in a laminar throttle in the absence of cavitation in the liquid; the electrokinetic mechanism applied to this phenomenon too. It was shown experimentally that electric charges appeared not only in plastic but also in metallic throttles. The suggested mechanism of light emission and electric charge appearance was in agreement with the experimental results.     

Continuous medium theory for nonequilibrium solvation: I. How to correctly evaluate solvation free energy of nonequilibrium

Considering the influences of electrostatic potential Phi upon the change of solute charge distribution deltarho and rho upon the change deltaPhi at the same time, a more reasonable integral formula of dG = (1/2) integral (V) (rhodeltaPhi + Phideltarho)dV is used to calculate the change of the electrostatic free energy in charging the solute-solvent system to a nonequilibrium state, instead of the one of dG = integral (V) PhideltarhodV used before. This modification improves the expressions of electrostatic free energy and solvation free energy, in which no quantity of the intermediate equilibrium state is explicitly involved. Detailed investigation reveals that the solvation free energy of nonequilibrium only contains the interaction energy between the field due to the solute charge in vacuum, and the dielectric polarization at the nonequilibrium state. The solvent reorganization energies of forward and backward electron transfer reactions have been redefined because the derivations lead to a remarkable feature that these quantities are direction-dependent, unlike the theoretical models developed before. The deductions are given in the electric field-displacement form. Relevant discussions on the reliability of theoretical models suggested in this work have also been presented.     

Particle charging and charge screening in nonpolar dispersions with nonionic surfactants

The electrostatic stabilization of colloidal dispersions is usually considered the domain of polar media only because of the high energetic cost associated with introducing electric charge in nonpolar environments. Nevertheless, some surfactants referred to as "charge control agents" are known to raise the conductivity of liquids with low electric permittivity and to mediate charge stabilization of nonpolar dispersions. Here we study an example of the particularly counterintuitive charging and electrostatic interaction of colloidal particles in a nonpolar solvent caused by nonionic surfactants. PMMA particles in hexane solutions of nonionic sorbitan oleate (Span) surfactants are found to exhibit a field-dependent electrophoretic mobility. Extrapolation to zero field strength yields evidence for large electrostatic surface potentials that decay with increasing surfactant concentration in a fashion reminiscent of electrostatic screening caused by salt in aqueous solutions. The amount of surface charge and screening ions in the nonpolar bulk is further characterized via measurements of the particles' pair interaction energy. The latter is obtained by liquid structure analysis of quasi-2-dimensional equilibrium particle configurations studied with digital video microscopy. In contrast to the behavior reported for systems with ionic surfactants, we observe particle charging and a screened Coulomb type interaction both above and below the surfactant's critical micelle concentration.