Langevin-Debye model for nonlinear electrostatic screening of solvated ions

Minimal Coarse-Grained Models of Polar Solvent for Electrolytes: Stockmayer Versus Dumbbell

This study explores the potential of the dumbbell solvent as a minimal model for understanding electrolyte solutions in polar solvents. Our investigation involves a comparative analysis of the dumbbell model and the Stockmayer model, focusing on ion solvation and ion-ion correlations. We examine electrolytes containing symmetric monovalent salts dissolved in polar solvents while varying the ion density and solvent polarity. Both models predict an augmented solvent coordination number around ions as the solvent polarity increases, with the dumbbell solvent displaying a more pronounced effect. Notably, radial distribution functions (RDFs) between solvent and ions yield differing trends; Stockmayer models exhibit a nonmonotonic relationship due to strong dipole-dipole interactions at higher polarity, while RDFs for ions and dumbbell solvents consistently rise. In response to increased solvent polarity, Stockmayer solvents within the ion's solvation shell undergo continuous dipole orientation shifts, whereas the dumbbell solvent predominantly adopts pointing-away dipole orientations, diminishing pointing-to orientations. This underscores the significance of the interplay between the solvent molecular orientation and dipole rotation. Both models qualitatively predict ion pairing and clustering behaviors across varying solvent dipole strengths and salt concentrations. The Stockmayer solvent generally provides stronger electrostatic screening than the dumbbell solvent due to its neglect of the coupling between molecular orientation and dipole rotation. What's more, at a high dipole moment regime, ion-ion correlations in Stockmayer solvent can become stronger with increasing dipole moment due to stronger solvent-solvent correlations. This study underscores the effectiveness of the dumbbell solvent model in systematically elucidating the fundamental principles governing electrolytes and offers potential applications in the rational design of electrolyte systems.

Effects of temperature on microstructures of MSA-type electroplating solution: a coarse-grained molecular dynamics simulation

In this study, we employ coarse-grained molecular dynamics simulations to explore the microstructure of MSA (methanesulfonic acid)-type electroplating solution, containing Sn(MSA)2 as the primary salt, MSA as the stabilizer, amphiphilic alkylphenol ethoxylate (APEO) as surfactants and cinnamaldehyde (CA) as the brightener agents, as well as water as the solvent. Our simulation indicates that temperature variations can significantly affect the structural properties of the electroplating solution and the adsorption behavior of its key components onto the substrate. Specifically, at low temperatures, the primary salt ions aggregate into ionic clusters, and the amphiphilic APEO surfactants and CA molecules form micelles composed of hydrophobic cores and hydrophilic shells, which reduces the uniformity of the solution and hinders the adsorption of ions, CA and surfactants onto the substrate. Appropriately increasing the temperature can weaken the aggregation of these components in bulk solution due to the accelerated molecular movements and arouse their adsorption. However, on further increasing the temperature, the elevated kinetic energy of the components thoroughly overwhelms the adsorption interactions, and therefore, the ions, surfactants, and CA desorb from the substrate and redissolve into the solution. We systematically analyze the complex interactions between these components at different temperatures and clarify the mechanism of the non-monotonic dependence of adsorption strength on the temperature at the molecular level. Our simulations demonstrate that there is low-temperature scope for reprocessing/recycling and intermediate-temperature scope for substrate-adsorptions of the key components. This study confers insights into a fundamental understanding of the microscopic mechanism for electroplating and can provide guidance for the development of precise electroplatings.

Solvation Thermodynamics of Solutes in Water and Ionic Liquids Using the Multiscale Solvation-Layer Interface Condition Continuum Model

Molecular assembly processes are generally driven by thermodynamic properties in solutions. Atomistic modeling can be very helpful in designing and understanding complex systems, except that bulk solvent is very inefficient to treat explicitly as discrete molecules. In this work, we develop and assess two multiscale solvation models for computing solvation thermodynamic properties. The new SLIC/CDC model combines continuum solvent electrostatics based on the solvent layer interface condition (SLIC) with new statistical thermodynamic models for hydrogen bonding and nonpolar modes: cavity formation, dispersion interactions, combinatorial mixing (CDC). Given the structures of 500 solutes, the SLIC/CDC model predicts Gibbs energies of solvation in water with an average accuracy better than 1 kcal/mol, when compared to experimental measurements, and better than 0.8 kcal/mol, when compared to explicit-solvent molecular dynamics simulations. The individual SLIC/CDC energy mode values agree quantitatively with those computed from explicit-solvent molecular dynamics. The previously published SLIC/SASA multiscale model combines the SLIC continuum electrostatic model with the solvent-accessible surface area (SASA) nonpolar energy mode. With our new, improved parametrization method, the SLIC/SASA model now predicts Gibbs energies of solvation with better than 1.4 kcal/mol average accuracy in aqueous systems, compared to experimental and explicit-solvent molecular dynamics, and better than 1.6 kcal/mol average accuracy in ionic liquids, compared to explicit-solvent molecular dynamics. Both models predict solvation entropies, and are the first implicit-solvation models capable of predicting solvation heat capacities.

Structures of cationic and anionic polyelectrolytes in aqueous solutions: the sign effect

In this study, we use molecular dynamics simulation to explore the structures of anionic and cationic polyelectrolytes in aqueous solutions. We first confirm the significantly stronger solvation effects of single anions compared to cations in water at the fixed ion radii, due to the reversal orientations of asymmetric dipolar H₂O molecules around the ions. Based on this, we demonstrate that the solvation discrepancy of cations/anions and electrostatic correlations of ionic species can synergistically cause the nontrivial structural difference between single anionic and cationic polyelectrolytes. The cationic polyelectrolyte shows an extended structure whereas the anionic polyelectrolyte exhibits a collapsed structure, and their structural differences decline with increasing the counterion size. Furthermore, we corroborate that multiple cationic polyelectrolytes or multiple anionic polyelectrolytes can exhibit largely differential molecular architectures in aqueous solutions. In the solvation dominant regime, the polyelectrolyte solutions exhibit uniform structures; whereas, in the electrostatic correlation dominant regime, the polyelectrolyte solutions exhibit heterogeneous structures, in which the likely charged chains microscopically aggregate through counterion condensations. Increasing the intrinsic chain rigidity causes polyelectrolyte extension and hence moderately weakens the inter-chain clustering. Our work highlights the various, unique structures and molecular architectures of polyelectrolytes in solutions caused by the multi-body correlations between polyelectrolytes, counterions and asymmetric dipolar solvent molecules, which provides insights into the fundamental understanding of ion-containing polymers.

Explicit solvent theory of salt-induced dielectric decrement

Within the framework of an electrolyte model composed of structured solvent molecules and salt ions coupled by electrostatic and hard-core interactions, we characterize the physical mechanism behind salt-induced dielectric decrement.

Structure and dynamics of ions in dipolar solvents: a coarse-grained simulation study

We employ the coarse-grained molecular dynamics simulation to investigate the fundamental structural and dynamic properties of the ionic solution with and without the application of an external electric field. Our simulations, in which the solvent molecules are treated as Stockmayer fluids and the ions are modeled as spheres, can effectively account for the multi-body correlations between ion-ion, ion-dipole, and dipole-dipole interactions, which are often ignored by the mean-field theories or coarse-grained simulations based on a dielectric continuum. By focusing on the coupling between effects of ion solvation, electrostatic correlations and applied electric field, we highlight some nontrivial microscopic molecular features of the systems, such as the reorganization of the dipolar solvent, clustering of the ions, and diffusions of ions and dipolar solvent molecules. Particularly, our simulation indicates the nonmonotonic dependence of the ionic clustering and ion diffusion rates on the dipolar nature of the solvent molecules, as well as the amplification of these tendencies caused by the electric field application. This work provides insights into the fundamental understanding of physicochemical properties for ion-containing liquids and contributes to the design and development of ion-containing materials.

Solvation Energy of Ions in a Stockmayer Fluid

We calculate the solvation energy of monovalent and divalent ions in various liquids with coarse-grained molecular dynamics simulations. Our theory treats the solvent as a Stockmayer fluid, which accounts for the intrinsic dipole moment of molecules and the rotational dynamics of the dipoles. Despite the simplicity of the model, we obtain qualitative agreement between the simulations and experimental data for the free energy and enthalpy of ion solvation, which indicates that the primary contribution to the solvation energy arises mainly from the first and possibly second solvation shells near the ions. Our results suggest that a Stockmayer fluid can serve as a reference model that enables direct comparison between theory and experiment and may be invoked to scale up electrostatic interactions from the atomic to the molecular length scale.

Charge polarization near dielectric interfaces and the multiple-scattering formalism

Interfacial charge polarization is ubiquitous in systems with sharp dielectric contrast. Fully resolving the interfacial charges often relies on demanding numerical algorithms to solve the boundary value problem. The recent development of an analytical multiple-scattering formalism to solve the interfacial charge polarization problem for particles carrying monopolar, dipolar, and multipolar charges is reviewed. Every term produced in this formalism has a simple interpretation, and terms for spherical particles can be rapidly evaluated using an image-line construction. Several practical applications of this formalism are illustrated. A dielectric virial expansion for polarizable particles based on this formalism is also described. The origins of singular polarization charges for particles in close contact are explained and evaluated for both dielectric and conducting spheres.

Discrete and Continuum Models for the Salt in Crowded Environments of Suspended Charged Particles

Electrostatic forces greatly affect the overall dynamics and diffusional activities of suspended charged particles in crowded environments. Accordingly, the concentration of counter- or co-ions in a fluid-''the salt"-determines the range, strength, and order of electrostatic interactions between particles. This environment fosters engineering routes for controlling directed assembly of particles at both the micro- and nanoscale. Here, we analyzed two computational modeling schemes that considered salt within suspensions of charged particles, or polyelectrolytes: discrete and continuum. Electrostatic interactions were included through a Green's function formalism, where the confined fundamental solution for Poisson's equation is resolved by the general geometry Ewald-like method. For the discrete model, the salt was considered as regularized point-charges with a specific valence and size, while concentration fields were defined for each ionic species for the continuum model. These considerations were evolved using Brownian dynamics of the suspended charged particles and the discrete salt ions, while a convection-diffusion transport equation, including the Nernst-Planck diffusion mechanism, accounted for the dynamics of the concentration fields. The salt/particle models were considered as suspensions under slit-confinement conditions for creating crowded "macro-ions", where density distributions and radial distribution functions were used to compare and differentiate computational models. Importantly, our analysis shows that disparate length scales or increased system size presented by the salt and suspended particles are best dealt with using concentration fields to model the ions. These findings were then validated by novel simulations of a semipermeable polyelectrolyte membrane, at the mesoscale, from which ionic channels emerged and enable ion conduction.

Model for screened, charge-regulated electrostatics of an eye lens protein: Bovine gammaB-crystallin

We model screened, site-specific charge regulation of the eye lens protein bovine gammaB-crystallin (γB) and study the probability distributions of its proton occupancy patterns. Using a simplified dielectric model, we solve the linearized Poisson-Boltzmann equation to calculate a 54×54 work-of-charging matrix, each entry being the modeled voltage at a given titratable site, due to an elementary charge at another site. The matrix quantifies interactions within patches of sites, including γB charge pairs. We model intrinsic pK values that would occur hypothetically in the absence of other charges, with use of experimental data on the dependence of pK values on aqueous solution conditions, the dielectric model, and literature values. We use Monte Carlo simulations to calculate a model grand-canonical partition function that incorporates both the work-of-charging and the intrinsic pK values for isolated γB molecules and we calculate the probabilities of leading proton occupancy configurations, for 4<pH<8 and Debye screening lengths from 6 to 20 Å. We select the interior dielectric value to model γB titration data. At pH 7.1 and Debye length 6.0 Å, on a given γB molecule the predicted top occupancy pattern is present nearly 20% of the time, and 90% of the time one or another of the first 100 patterns will be present. Many of these occupancy patterns differ in net charge sign as well as in surface voltage profile. We illustrate how charge pattern probabilities deviate from the multinomial distribution that would result from use of effective pK values alone and estimate the extents to which γB charge pattern distributions broaden at lower pH and narrow as ionic strength is lowered. These results suggest that for accurate modeling of orientation-dependent γB-γB interactions, consideration of numerous pairs of proton occupancy patterns will be needed.

A molecularly based theory for electron transfer reorganization energy

Using field-theoretic techniques, we develop a molecularly based dipolar self-consistent-field theory (DSCFT) for charge solvation in pure solvents under equilibrium and nonequilibrium conditions and apply it to the reorganization energy of electron transfer reactions. The DSCFT uses a set of molecular parameters, such as the solvent molecule's permanent dipole moment and polarizability, thus avoiding approximations that are inherent in treating the solvent as a linear dielectric medium. A simple, analytical expression for the free energy is obtained in terms of the equilibrium and nonequilibrium electrostatic potential profiles and electric susceptibilities, which are obtained by solving a set of self-consistent equations. With no adjustable parameters, the DSCFT predicts activation energies and reorganization energies in good agreement with previous experiments and calculations for the electron transfer between metallic ions. Because the DSCFT is able to describe the properties of the solvent in the immediate vicinity of the charges, it is unnecessary to distinguish between the inner-sphere and outer-sphere solvent molecules in the calculation of the reorganization energy as in previous work. Furthermore, examining the nonequilibrium free energy surfaces of electron transfer, we find that the nonequilibrium free energy is well approximated by a double parabola for self-exchange reactions, but the curvature of the nonequilibrium free energy surface depends on the charges of the electron-transferring species, contrary to the prediction by the linear dielectric theory.

Charge-regulation phase transition on surface lattices of titratable sites adjacent to electrolyte solutions: An analog of the Ising antiferromagnet in a magnetic field

We report a charge-patterning phase transition on two-dimensional square lattices of titratable sites, here regarded as protonation sites, placed in a low-dielectric medium just below the planar interface between this medium and a salt solution. We calculate the work-of-charging matrix of the lattice with use of a linear Debye-Hückel model, as input to a grand-canonical partition function for the distribution of occupancy patterns. For a large range of parameter values, this model exhibits an approximate inverse cubic power-law decrease of the voltage produced by an individual charge, as a function of its in-lattice separation from neighboring titratable sites. Thus, the charge coupling voltage biases the local probabilities of proton binding as a function of the occupancy of sites for many neighbors beyond the nearest ones. We find that even in the presence of these longer-range interactions, the site couplings give rise to a phase transition in which the site occupancies exhibit an alternating, checkerboard pattern that is an analog of antiferromagnetic ordering. The overall strength W of this canonical charge coupling voltage, per unit charge, is a function of the Debye length, the charge depth, the Bjerrum length, and the dielectric coefficients of the medium and the solvent. The alternating occupancy transition occurs above a curve of thermodynamic critical points in the (pH-pK,W) plane, the curve representing a charge-regulation analog of variation of the Néel temperature of an Ising antiferromagnet as a function of an applied, uniform magnetic field. The analog of a uniform magnetic field in the antiferromagnet problem is a combination of pH-pK and W, and 1/W is the analog of the temperature in the antiferromagnet problem. We use Monte Carlo simulations to study the occupancy patterns of the titratable sites, including interactions out to the 37th nearest-neighbor category (a distance of √74 lattice constants), first validating simulations through comparison with exact and approximate results for the nearest-neighbor case. We then use the simulations to map the charge-patterning phase boundary in the (pH-pK,W) plane. The physical parameters that determine W provide a framework for identifying and designing real surfaces that could exhibit charge-patterning phase transitions.

A new lattice Monte Carlo simulation for dielectric saturation in ion-containing liquids

We develop a new, rapid method for the lattice Monte Carlo simulation of ion-containing liquids that accounts for the effects of the reorganization of solvent dipoles under external electrostatic fields. Our results are in reasonable agreement with the analytical solutions to the dielectric continuum theory of Booth for single ions, ion pairs, and ionic cross-links. We also illustrate the substantial disparity between the dielectric functions for like and unlike charges on the nanometer scale. Our simulation rationalizes the experimental data for the dependence of the bulk dielectric value of water on ion concentrations in terms of saturated dipoles near ions.

Synergistic effects of ion pairs on the dielectric properties of diblock copolymer melts

We study the solvation of ion pairs in diblock copolymer melts. Our theory accounts for the size of the ions, the permanent dipole moment and the molecular polarizability of the monomers, the Kuhn length, the compressibility of the liquid mixtures, and the degrees of polymerization. We demonstrate that the electrostatic field near an ion pair causes marked, synergistic effects on the volume fractions of the two blocks and hence the dielectric function. In particular, we illustrate the oscillatory behavior of the dielectric function near an ion pair and the disparity of the dielectric functions between like and unlike charges. These results depend significantly on the chain length and Kuhn length of the diblock copolymers on the nanometer scale.

Communication: modeling charge-sign asymmetric solvation free energies with nonlinear boundary conditions

We show that charge-sign-dependent asymmetric hydration can be modeled accurately using linear Poisson theory after replacing the standard electric-displacement boundary condition with a simple nonlinear boundary condition. Using a single multiplicative scaling factor to determine atomic radii from molecular dynamics Lennard-Jones parameters, the new model accurately reproduces MD free-energy calculations of hydration asymmetries for: (i) monatomic ions, (ii) titratable amino acids in both their protonated and unprotonated states, and (iii) the Mobley "bracelet" and "rod" test problems [D. L. Mobley, A. E. Barber II, C. J. Fennell, and K. A. Dill, "Charge asymmetries in hydration of polar solutes," J. Phys. Chem. B 112, 2405-2414 (2008)]. Remarkably, the model also justifies the use of linear response expressions for charging free energies. Our boundary-element method implementation demonstrates the ease with which other continuum-electrostatic solvers can be extended to include asymmetry.

A novel implicit solvent model for simulating the molecular dynamics of RNA

Although molecular dynamics simulations can be accelerated by more than an order of magnitude by implicitly describing the influence of the solvent with a continuum model, most currently available implicit solvent simulations cannot robustly simulate the structure and dynamics of nucleic acids. The difficulties become exacerbated especially for RNAs, suggesting the presence of serious physical flaws in the prior continuum models for the influence of the solvent and counter ions on the nucleic acids. We present a novel, to our knowledge, implicit solvent model for simulating nucleic acids by combining the Langevin-Debye model and the Poisson-Boltzmann equation to provide a better estimate of the electrostatic screening of both the water and counter ions. Tests of the model involve comparisons of implicit and explicit solvent simulations for three RNA targets with 20, 29, and 75 nucleotides. The model provides reasonable agreement with explicit solvent simulations, and directions for future improvement are noted.

General approach for engineering small-molecule-binding DNA split aptamers

Here we report a general method for engineering three-way junction DNA aptamers into split aptamers. Split aptamers show significant potential for use as recognition elements in biosensing applications, but reliable methods for generating these sequences are currently lacking. We hypothesize that the three-way junction is a "privileged architecture" for the elaboration of aptamers into split aptamers, as it provides two potential splitting sites that are distal from the target binding pocket. We propose a general method for split aptamer engineering that involves removing one loop region, then systematically modifying the number of base pairs in the remaining stem regions in order to achieve selective assembly only in the presence of the target small molecule. We screen putative split aptamer sequence pairs using split aptamer proximity ligation (StAPL) technology developed by our laboratory, but we validate that the results obtained using StAPL translate directly to systems in which the aptamer fragments are assembling noncovalently. We introduce four new split aptamer sequences, which triples the number of small-molecule-binding DNA split aptamers reported to date, and the methods described herein provide a reliable route for the engineering of additional split aptamers, dramatically advancing the potential substrate scope of DNA assembly based biosensors.

Affine-response model of molecular solvation of ions: Accurate predictions of asymmetric charging free energies

Two mechanisms have been proposed to drive asymmetric solvent response to a solute charge: a static potential contribution similar to the liquid-vapor potential, and a steric contribution associated with a water molecule's structure and charge distribution. In this work, we use free-energy perturbation molecular-dynamics calculations in explicit water to show that these mechanisms act in complementary regimes; the large static potential (∼44 kJ/mol/e) dominates asymmetric response for deeply buried charges, and the steric contribution dominates for charges near the solute-solvent interface. Therefore, both mechanisms must be included in order to fully account for asymmetric solvation in general. Our calculations suggest that the steric contribution leads to a remarkable deviation from the popular "linear response" model in which the reaction potential changes linearly as a function of charge. In fact, the potential varies in a piecewise-linear fashion, i.e., with different proportionality constants depending on the sign of the charge. This discrepancy is significant even when the charge is completely buried, and holds for solutes larger than single atoms. Together, these mechanisms suggest that implicit-solvent models can be improved using a combination of affine response (an offset due to the static potential) and piecewise-linear response (due to the steric contribution).

Polarization of ions under geometric confinement in the strong-interacting regime: a two-ring model study

We study the polarization and electrostatic interactions of an ionic system under geometric confinement in the strong-interacting regime. The geometric confinement is introduced via a simple two-ring model, where ions of each species are confined on a ring, respectively. The observed polarization curve exhibits staircase transitions in the low-temperature regime, due to the crossover between energy local minima. We examine the criterion for the validity of the linear response theory and introduce a simple two-state picture that illustrates the signatures of the crossover phenomena.

Influence of nonelectrostatic ion-ion interactions on double-layer capacitance

Recently a Poisson-Helmholtz-Boltzmann (PHB) model [Bohinc et al., Phys. Rev. E 85, 031130 (2012)] was developed by accounting for solvent-mediated nonelectrostatic ion-ion interactions. Nonelectrostatic interactions are described by a Yukawa-like pair potential. In the present work, we modify the PHB model by adding steric effects (finite ion size) into the free energy to derive governing equations. The modified PHB model is capable of capturing both ion specificity and ion crowding. This modified model is then employed to study the capacitance of the double layer. More specifically, we focus on the influence of nonelectrostatic ion-ion interactions on charging a double layer near a flat surface in the presence of steric effects. We numerically compute the differential capacitance as a function of the voltage under various conditions. At small voltages and low salt concentrations (dilute solution), we find out that the predictions from the modified PHB model are the same as those from the classical Poisson-Boltzmann theory, indicating that nonelectrostatic ion-ion interactions and steric effects are negligible. At moderate voltages, nonelectrostatic ion-ion interactions play an important role in determining the differential capacitance. Generally speaking, nonelectrostatic interactions decrease the capacitance because of additional nonelectrostatic repulsion among excess counterions inside the double layer. However, increasing the voltage gradually favors steric effects, which induce a condensed layer with crowding of counterions near the electrode. Accordingly, the predictions from the modified PHB model collapse onto those computed by the modified Poisson-Boltzmann theory considering steric effects alone. Finally, theoretical predictions are compared and favorably agree with experimental data, in particular, in concentrated solutions, leading one to conclude that the modified PHB model adequately predicts the diffuse-charge dynamics of the double layer with ion specificity and steric effects.

Nonlinear Poisson equation for heterogeneous media

The Poisson equation is a widely accepted model for electrostatic analysis. However, the Poisson equation is derived based on electric polarizations in a linear, isotropic, and homogeneous dielectric medium. This article introduces a nonlinear Poisson equation to take into consideration of hyperpolarization effects due to intensive charges and possible nonlinear, anisotropic, and heterogeneous media. Variational principle is utilized to derive the nonlinear Poisson model from an electrostatic energy functional. To apply the proposed nonlinear Poisson equation for the solvation analysis, we also construct a nonpolar solvation energy functional based on the nonlinear Poisson equation by using the geometric measure theory. At a fixed temperature, the proposed nonlinear Poisson theory is extensively validated by the electrostatic analysis of the Kirkwood model and a set of 20 proteins, and the solvation analysis of a set of 17 small molecules whose experimental measurements are also available for a comparison. Moreover, the nonlinear Poisson equation is further applied to the solvation analysis of 21 compounds at different temperatures. Numerical results are compared to theoretical prediction, experimental measurements, and those obtained from other theoretical methods in the literature. A good agreement between our results and experimental data as well as theoretical results suggests that the proposed nonlinear Poisson model is a potentially useful model for electrostatic analysis involving hyperpolarization effects.

Thermodynamics of ion-containing polymer blends and block copolymers

We develop a theory for the thermodynamics of ion-containing polymer blends and diblock copolymers, taking polyethylene oxide (PEO), polystyrene and lithium salts as an example. We account for the tight binding of Li^{+} ions to the PEO, the preferential solvation energy of anions in the PEO domain, the translational entropy of anions, and the ion-pair equilibrium between EO-complexed Li^{+} and anion. Our theory is able to predict many features observed in experiments, particularly the systematic dependence in the effective χ parameter on the size of the anions. Furthermore, comparison with the observed linear dependence in the effective χ on salt concentration yields an upper limit for the binding constant of the ion pair.

Model for evaluating patterned charge-regulation contributions to electrostatic interactions between low-dielectric spheres

We study the electrostatic contribution to the effective potential between two spherical low-dielectric particles that carry proton-titratable sites within a linearized setting. To evaluate the needed work of charging for each possible proton occupancy configuration, together with its crucial dependence on sphere separation, we numerically solve a coarse-grained linear Debye-Hückel model that incorporates nonuniform dielectric and ionic solution properties at a series of intersphere separations and for chosen titratable charge locations on each sphere. We combine the resulting work-of-charging matrix with site-specific chemical potentials of proton binding to construct the Boltzmann-weighted probabilities of each possible occupancy pattern of the titratable sites as functions of intersphere separation. With the use of these probabilities we find that a nonmonotonic average electrostatic potential can result that is repulsive at larger sphere separations but attractive at close separations. The nonmonotonic potential corresponds to particular choices of site-specific unoccupied charge values and their corresponding proton affinities, and its occurrence is dependent on pH in relation to the pKa values of the titratable groups. For the chosen titratable groups, we identify the particular change from repulsive to attractive proton occupancy patterns with decreasing intersphere separation that gives rise to the modeled nonmonotonic dependence and derive more general conditions under which such a nonmonotonic dependence can occur. Within the present model we find that stationary points of the charge-regulated average electrostatic potential, considered as a function of intersphere separation, occur when a normalized Boltzmann-averaged intersphere charge number product equals its covariance with an average free energy of charging divided by k(B)T. We derive more general conditions for the location and nature of critical points in the electrostatic intersphere potential, which are not dependent on the validity of the present linear model. Analysis of the present simple prototype model can be a helpful step toward developing a framework for predicting when (i) patterned charge-regulated occupancy patterns, (ii) orientation-dependent attractions due to relatively fixed heterogeneous charging patterns, and (iii) screened net protein charge could separately dominate the electrostatic portion of the interactions between model biological macromolecules and other nanoparticles.

Electrostatic solvation energy for two oppositely charged ions in a solvated protein system: salt bridges can stabilize proteins

Born-type electrostatic continuum methods have been an indispensable ingredient in a variety of implicit-solvent methods that reduce computational effort by orders of magnitude compared to explicit-solvent MD simulations and thus enable treatment using larger systems and/or longer times. An analysis of the limitations and failures of the Born approaches serves as a guide for fundamental improvements without diminishing the importance of prior works. One of the major limitations of the Born theory is the lack of a liquidlike description of the response of solvent dipoles to the electrostatic field of the solute and the changes therein, a feature contained in the continuum Langevin-Debye (LD) model applied here to investigate how Coulombic interactions depend on the location of charges relative to the protein/water boundary. This physically more realistic LD model is applied to study the stability of salt bridges. When compared head to head using the same (independently measurable) physical parameters (radii, dielectric constants, etc.), the LD model is in good agreement with observations, whereas the Born model is grossly in error. Our calculations also suggest that a salt bridge on the protein's surface can be stabilizing when the charge separation is < or =4 A.

Understanding the protonation behavior of linear polyethylenimine in solutions through Monte Carlo simulations

The success of polyethyleneimine (PEI) as a nonviral-based gene delivery vector has been attributed to its proton buffering capacity. Despite the great interest in PEI for its use in nonviral-based gene delivery, the protonation behavior of PEI in solution is not well understood. Earlier experimental studies have reported inconsistent values of the protonation state of PEI. In this work, we report our investigation of the protonation behavior of a realistic linear PEI (lPEI) with computational approaches. Reported experimental pK(a) values of several diamine compounds are first examined. A screened Coulombic interaction with a distance dependence dielectric is shown to reproduce the shifted pK(a) values of the model diamine compounds. Then atomistic molecular dynamic simulations of lPEI chain with 20 repeating units are performed and the results are used to provide parameters for a coarse-grained polyamine model. The screened Coulombic interaction is then incorporated in the coarse-grained lPEI chain and computational titrations are performed. The obtained computational titration curves of lPEI in solutions were found to be in best agreement with experimental results by Smits et al., but the computational titration curves have too strong of a dependence on salt concentration compared to the experimental results by Smits et al. Disregarding the discrepancy in the salt dependence, our computational titrations reveal that approximately 55% of the lPEI amine groups are protonated under physiological conditions in solution with a nearly alternating arrangement of protonated and nonprotonated amines. Titrations of lPEI in the presence of a polyanion are also performed to determine how the charge state of lPEI could be affected by complexation with DNA in gene therapy preparations. While the presence of the polyanion increases the degree of protonation of the PEI, many of PEI amines remain unprotonated under physiological conditions, providing evidence that PEI complexed with DNA could still have proton buffering capacity. Potential sources of error that have resulted in the inconsistency of previously reported protonation states of PEI were also discussed.

A single molecule as a dielectric medium

For three molecules with weak or negligible charge overlap, we prove that the three-body interaction energy obtained from quantum perturbation theory (to leading order) fits a dielectric model with a nonlocal electronic screening function. The electronic charge cloud of each molecule acts as a dielectric medium for the interaction of the remaining two with the nonlocal dielectric function epsilon(r,r') obtained by O. S. Jenkins and K. L. C. Hunt [J. Chem. Phys. 119, 8250 (2003)], by considering the charge redistribution induced in a single molecule by an external perturbation. The dielectric function depends parametrically on the coordinates of the nuclei, within the Born-Oppenheimer approximation. We also prove that the force on each nucleus in molecule A depends on intramolecular dielectric screening within A. The potential from the charge distribution of B, screened by C acting as a dielectric medium, is further screened linearly within A; and similarly, with the roles of B and C reversed. In addition, the potential due to the unperturbed charge distribution of B and the potential due to the unperturbed charge distribution of C, acting simultaneously, are screened nonlinearly within A. The results show that nonlocal dielectric theory holds on the molecular level, provided that the overlap of the electronic charge distributions is weak.