The interaction between two charged rods in an electrolyte solution

A problem on the electrostatic interaction of two homogeneously charged macromolecular rods of a finite length, submerged into an electrolyte solution was considered. An explicit expression for the energy of interaction and rotational moment as a function of the angle of rotation between the long axes of the molecules was obtained. At small angles of rotation, the expression for energy turns into the corresponding formula for parallel rods, and the rotational moment tends to zero, as it follows from geometrical considerations. The possibility is discussed whether the study is applicable to real biological systems, including liquid-crystalline dispersions of DNA.     

Inhomogeneous interfacial regions of biological systems

On the premise that the contributions from ion-ion interaction energy terms to chemical potential of an ion in inhomogeneous regions cannot be neglected, the classical expression for electrochemical potential is modified. Resulting basic differential equations and their implications regarding ion distributions in inhomogeneous regions are presented. It is shown that a necessary consequence is the existence of a dielectric profile in inhomogeneous regions due to nonvanishing interaction energy terms of the chemical potential.     

Equilibrium surface charge distribution in phospholipid vesicles. I. Method of calculation

This paper presents a method of calculation of the surface charge equilibrium distribution between the two surfaces of a spherically closed phospholipid bilayer suspended in aqueous electrolyte solution. The net surface charge is supposed to be provided by the ionized polar groups of the phospholipid molecules. Its equilibrium distribution is found by minimization of the free electrostatic energy. The procedure of minimization utilizes the solution of the Poisson-Boltzmann equation which describes the double electric layers of the membrane and an expression for the membrane potential derived under the assumption of absence of charges in the membrane phase. An analytical solution of the problem in the range of validity of the linearized Poisson-Boltzman equation is obtained. It is shown that in this case an equilibrium transmembrane potential exists, and the surface charge density is greater at the outer surface of the vesicle.     

Ionic permeation free energy in gramicidin: a semimicroscopic perspective

Ion permeation through the gramicidin channel is studied using a model that circumvents two major difficulties inherent to standard simulational methods. It exploits the timescale separation between electronic and structural contributions to dielectric stabilization, accounting for the influence of electronic polarization by embedding the channel in a dielectric milieu that describes this polarization in a mean sense. The explicit mobile moieties are the ion, multipolar waters, and the carbonyls and amides of the peptide backbone. The model treats the influence of aromatic residues and the membrane dipole potential. A new electrical geometry is introduced that treats long-range electrostatics exactly and avoids problems related to periodic boundary conditions. It permits the translocating ion to make a seamless transition from nearby electrolyte to the channel interior. Other degrees of freedom (more distant bulk electrolyte and nonpolar lipid) are treated as dielectric continua. Reasonable permeation free energy profiles are obtained for potassium, rubidium, and cesium; binding wells are shallow and the central barrier is small. Estimated cationic single-channel conductances are smaller than experiment, but only by factors between 2 (rubidium) and 50 (potassium). When applied to chloride the internal barrier is large, with a corresponding miniscule single-channel conductance. The estimated relative single-channel conductances of gramicidin A, B, and C agree well with experiment.     

On the calculation of electrostatic interactions in proteins

In this paper we present a classical treatment of electrostatic interactions in proteins. The protein is treated as a region of low dielectric constant with spherical charges embedded within it, surrounded by an aqueous solvent of high dielectric constant, which may contain a simple electrolyte. The complete analysis includes the effects of solvent screening, polarization forces, and self energies, which are related to solvation energies. Formulae, and sample calculations of forces and energies, are given for the special case of a spherical protein. Our analysis and model calculations point out that any consistent treatment of electrostatic interactions in proteins should account for the following. Solvent polarization is an important factor in the calculation of pairwise electrostatic interactions. Solvent polarization substantially affects both electrostatic energies and forces acting upon charges. No simple expression for the effective dielectric constant, Deff, can generally be valid, since Deff is a sensitive function of position. Solvent screening of pairwise interactions involving dipolar groups is less effective than the screening of charges. In fact for many interactions involving dipoles, solvent screening can be essentially ignored. The self energy of charges makes a large contribution to the total electrostatic energy of a protein. This must be compensated by specific interactions with other groups in the protein. Strategies for applying our analysis to proteins whose structures are known are discussed.     

Strong electrolyte continuum theory solution for equilibrium profiles, diffusion limitation, and conductance in charged ion channels.

The solution for the ion flux through a membrane channel that incorporates the electrolyte nature of the aqueous solution is a difficult theoretical problem that, until now, has not been properly formulated. The difficulty arises from the complicated electrostatic problem presented by a high dielectric aqueous channel piercing a low dielectric lipid membrane. The problem is greatly simplified by assuming that the ratio of the dielectric constant of the water to that of the lipid is infinite. It is shown that this is a good approximation for most channels of biological interest. This assumption allows one to derive simple analytical expressions for the Born image potential and the potential from a fixed charge in the channel, and it leads to a differential equation for the potential from the background electrolyte. This leads to a rigorous solution for the ion flux or the equilibrium potential based on a combination of the Nernst-Planck equation and strong electrolyte theory (i.e., Gouy-Chapman or Debye-Huckel). This approach is illustrated by solving the system of equations for the specific case of a large channel containing fixed negative charges. The following characteristics of this channels are discussed: anion and mono- and divalent cation conductance, saturation of current with increasing concentration, current-voltage relationship, influence of location and valence of fixed charge, and interaction between ions. The qualitative behavior of this channel is similar to that of the acetylcholine receptor channel.     

Fatty acid contamination and dielectric relaxation in phospholipid vesicle suspensions

Aqueous vesicle or micelle suspensions from various synthetic lecithins or surfactants - most of them purified by a simple ion-exchange procedure in methanol - were investigated, some with ionic admixtures. The dielectric permittivity '(nu) between 5 kHz and 100 MHz was determined by different time-and frequency-domain methods, with attention given to electrode polarization below 1 MHz. Pure ether lecithins (used to reduce hydrolysis during preparation) as well as ester lecithins showed no dielectric dispersion below 10 MHz (Delta' 3). In contrast, even dilute colloidal solutions containing about 1 mol% (with respect to solute) ionic amphiphiles normally exhibited large dielectric dispersion (10 < Delta' < 700), especially with electrolyte present. This low-frequency dispersion is sensitive to vesicle coagulation or fusion. Underlying relaxation mechanisms are discussed, and the main relaxation is shown to be the same as for other charged colloids. This conclusion suggest a new interpretation of measurements, previously reported by other authors, who gave an interpretation in terms of correlated zwitterionic head group orientation in multilamellar lecithin liposomes. Possible effects from traces of impurities in lipids are discussed.     

Interaction energy decomposition in protein-protein association: a quantum mechanical study of barnase-barstar complex

Protein-protein interactions are very important in the function of a cell. Computational studies of these interactions have been of interest, but often they have utilized classical modelling techniques. In recent years, quantum mechanical (QM) treatment of entire proteins has emerged as a powerful approach to study biomolecular systems. Herein, we apply a semi-empirical divide and conquer (DC) methodology coupled with a dielectric continuum model for the solvent, to explore the contribution of electrostatics, polarization and charge transfer to the interaction energy between barnase and barstar in their complex form. Molecular dynamic (MD) simulation was performed to account for the dynamic behavior of the complex. The results show that electrostatics, charge transfer and polarization favor the formation of the complex. Our study shows that electrostatics dominates the interaction between barnase and barstar ( approximately 73%), while charge transfer and polarization are approximately 21% and approximately 6%, respectively. Close inspection of the polarization and charge-transfer effects on the charge distribution of the complex reveals the existence of two, well localized, regions in barstar. The first region includes the residues between P27 and Y47 and the second region is between N65 and D83. Since no such regions could be detected in barnase clearly suggests that barstar is well optimized for efficiently binding barnase. Furthermore, using our interaction energy decomposition scheme, we were able to identify all residues that have been experimentally determined to be important for the complex formation and to suggest other residues never have been investigated. This suggests that our approach will be useful as an aid in further understanding protein-protein contacts for the ultimate goal to produce successful inhibitors for protein complexes.     

Pair Potentials from ab initio Calculations for use in MD Simulations of Molten Alkali Carbonates

Ab initio calculations at the Hartree-Fock SCF level have been carried out to determine the pair interaction between the alkali ions and the carbonate ion. A distinction has been made between terms in the metal ion - carbonate ion interaction which have different physical origins, such as static coulomb interaction, short-range repulsion and electronic polarization. The additivity of the pair interaction is investigated in 3-body calculations. It is shown that for these 3-body systems pairwise addition of 2-body interactions from which polarization effects have been omitted is superior to pairwise addition of the full Hartree-Fock interactions. A model potential based on these modified interactions has been constructed. Results of MD simulations show that both structural and dynamical properties are well described by these pair potentials.     

A new method for computing the macromolecular electric potential

A general methodology is developed for the rigorous computation of the electrostatic potential for a protein of arbitrary shape, assuming the presence of linear dielectric media. The theory proceeds by considering the distribution of induced polarization charge at the dielectric interface, rather than by attempting a direct solution of Poisson's equation (as in the finite-difference approach of Warwicker & Watson). The method is applied to a study of two-dimensional model proteins, where it is shown that the presence of a cleft is associated with a region of relatively high potential in the solvent medium. The results of a preliminary calculation in three dimensions for the protein lysozyme are also discussed; again, a region of enhanced potential is observed near the cleft at the active site. Our computational evidence supports the suggestion of Warwicker & Watson that clefts are associated with important electrostatic effects.     

Influence of the solvent structure on the electrostatic interactions in proteins

The proper estimation of the influence of the many-body dynamic solvent microstructure on a pairwise electrostatic interaction (PEI) at the protein-solvent interface is very important for solving many biophysical problems. In this work, the PEI energy was calculated for a system that models the interface between a protein and an aqueous solvent. The concept of nonlocal electrostatics for interfacial electrochemical systems was used to evaluate the contribution of a solvent orientational polarization, correlated by the network of hydrogen bonds, into the PEI energy in proteins. The analytical expression for this energy was obtained in the form of Coulomb's law with an effective distance-dependent dielectric function. The asymptotic and numerical analysis carried out for this function revealed several features of dielectric heterogeneity at the protein-solvent interface. For charges located in close proximity to this interface, the values of the dielectric function for the short-distance electrostatic interactions were found to be remarkably smaller than those determined by the classical model, in which the solvent was considered as the uniform dielectric medium of high dielectric constant. Our results have shown that taking into consideration the dynamic solvent microstructure remarkably increases the value of the PEI energy at the protein-solvent interface.     

Study of thermodynamic properties of symmetric and asymmetric electrolyte systems in mixture with neutral components: Monte Carlo simulation results and integral equation predictions

Monte Carlo simulation is conducted to obtain the structure, excess internal energy and Helmholtz energy for systems containing charged and neutral hard spheres of comparable concentrations. The results are compared with the thermodynamic properties predicted by solving Ornstein–Zernike equation with hypernetted chain (HNC) and mean spherical approximation (MSA) closures. The HNC approximation is found to well represent the simulation results for both structure and excess energy, while the excess energy of MSA deviates from the simulation results in the intermediate- and high-density range. A simple modification of MSA, referred to as KMSA, is proposed to accurately predict the excess internal and Helmholtz energy in the studied density range. KMSA is proved to capture the effects of neutral component, size and charge asymmetry, system temperature and dielectric constant of the background solvent, on the excess energy of electrolyte systems.     

Electrostatic potential and Born energy of charged molecules interacting with phospholipid membranes: calculation via 3-D numerical solution of the full Poisson equation.

Understanding the physicochemical basis of the interaction of molecules with lipid bilayers is fundamental to membrane biology. In this study, a new, three-dimensional numerical solution of the full Poisson equation including local dielectric variation is developed using finite difference techniques in order to model electrostatic interactions of charged molecules with a non-uniform dielectric. This solution is used to describe the electric field and electrostatic potential profile of a charged molecule interacting with a phospholipid bilayer in a manner consistent with the known composition and structure of the membrane. Furthermore, the Born interaction energy is then calculated by appropriate integration of the electric field over whole space. Numerical computations indicate that the electrostatic potential profile surrounding a charge molecule and its resultant Born interaction energy are a function of molecular position within the membrane and change most significantly within the polar region of the bilayer. The maximum interaction energy is observed when the charge is placed at the center of the hydrophobic core of the membrane and is strongly dependent on the size of the charge and on the thickness of the hydrocarbon core of the bilayer. The numerical results of this continuum model are compared with various analytical approximations for the Born energy including models established for discontinuous slab dielectrics. The calculated energies agree with the well-known Born analytical expression only when the charge is located near the center of a hydrocarbon core of greater than 60 A in thickness. The Born-image model shows excellent agreement with the numerical results only when modified to include an appropriate effective thickness of the low dielectric region. In addition, a newly derived approximation which considers the local mean dielectric provides a simple and continuous solution that also agrees well with the numerical results.     

A precise analytical method for calculating the electrostatic energy of macromolecules in aqueous solution

A new method for calculating the total electrostatic free energy of a macromolecule in solution is presented. It is applicable to molecules of arbitrary shape and size, including membranes or macromolecular assemblies with substrate molecules and ions. The method is derived from integrating the energy density of the electrostatic field and is termed the field energy method. It is based on the dielectric model, in which the solute and the surrounding water are regarded as different continuous dielectrics. The field energy method yields both the interaction energy between all charge pairs and the self energy of single charges, effectively accounting for the interaction with water. First, the dielectric boundary and mirror charges are determined for all charges of the solute. The energy is then given as a simple function of the interatomic distances, and the standard atomic partial charges and volumes. The interaction and self energy are shown to result from three-body and pairwise interactions. Both energy terms explicitly involve apolar atoms, revealing that apolar groups are also subject to electrostatic forces. We applied the field energy method to a spherical model protein. Comparison with the Kirkwood solution shows that errors are within a small percentage. As a further test, the field energy method was used to calculate the electrostatic potential of the protein superoxide dismutase. We obtained good agreement with the result from a program that implements the numerical finite difference algorithm. The field energy method provides a basis for energy minimization and dynamics programs that account for the solvent and screening effect of water at little computational expense.     

An optimized potential function for the calculation of nucleic acid interaction energies I. Base stacking

An optimized potential function for base-stacking interaction is constructed. Stacking energies between the complementary pairs of a dimer are calculated as a function of the rotational angle and separation distance. Using several different sets of atomic charges, the electrostatic component in the monopole-monopole approximation (MMA) is compared to the more refined segmented multipole–multipole representation (SMMA); the general features of the stacking minima are found to be correctly reproduced with IEHT or CNDO atomic charges. The electrostatic component is observed to control the location of stacking minima. The MMA, in general, is not a reliable approximation of the SMMA in regions away from minima; however, the MMA is reliable in predicting the location and nature of stacking minima. The attractive part of the Lennard-Jones 6–12 potential is compared to and parameterized against the expression for the second-order interaction terms composed of multipole-bond polarizability for the polarization energy and transition-dipole bond polarizabilities for approximation of the dispersion energy. The repulsive part of the Lennard-Jones potential is compared to a Kitaygorodski-type repulsive function; changing the exponent from its usual value of 12 to 11.7 gives significantly better agreement with the more refined repulsive function. Stacking minima calculated with the optimized potential method are compared with various perturbation-type treatments. The optimized potential method yields results that compare as well with melting data as do any of the more recent and expensive perturbation methods.     

The osmotic flow of an electrolyte through a charged porous membrane

A pore model in which the pore wall has a continuous distribution of electrical charge is used to investigate the osmotic flow through a charged permeable membrane separating electrolyte solutions of unequal concentrations. The pore is treated as a long, circular, cylindrical duct. The analysis is based on a continuum formulation in which a dilute electrolyte solution is described by the coupled Nernst-Planck/Poisson creeping flow equations. Account is taken of the significant size of the electrolyte ions (assumed to be rigid spheres) when compared with the diameter of the membrane pores. Analytical solutions for the ion concentrations, hydrostatic pressure and electrostatic potential in the electrolyte solutions are given and an intra-pore flow solution is derived. A mathematical expression for the osmotic reflection coefficient as a function of the solute ion: pore diameter ratio λ and the solute fluxes is obtained. Approximate solutions are quoted which relate the solute fluxes and the solution electrostatic potentials at the membrane surfaces to the bulk solution concentrations, the membrane pore charge and pore geometry. The osmotic reflection coefficient is thus determined as a function of these parameters.     

Modeling loop reorganization free energies of acetylcholinesterase: a comparison of explicit and implicit solvent models

The treatment of hydration effects in protein dynamics simulations varies in model complexity and spans the range from the computationally intensive microscopic evaluation to simple dielectric screening of charge-charge interactions. This paper compares different solvent models applied to the problem of estimating the free-energy difference between two loop conformations in acetylcholinesterase. Molecular dynamics (MD) simulations were used to sample potential energy surfaces of the two basins with solvent treated by means of explicit and implicit methods. Implicit solvent methods studied include the generalized Born (GB) model, atomic solvation potential (ASP), and the distance-dependent dieletric constant. By using the linear response approximation (LRA), the explicit solvent calculations determined a free-energy difference that is in excellent agreement with the experimental estimate, while rescoring the protein conformations with GB or the Poisson equation showed inconsistent and inferior results. While the approach of rescoring conformations from explicit water simulations with implicit solvent models is popular among many applications, it perturbs the energy landscape by changing the solvent contribution to microstates without conformational relaxation, thus leading to non-optimal solvation free energies. Calculations applying MD with a GB solvent model produced results of comparable accuracy as observed with LRA, yet the electrostatic free-energy terms were significantly different due to optimization on a potential energy surface favored by an implicit solvent reaction field. The simpler methods of ASP and the distance-dependent scaling of the dielectric constant both produced considerable distortions in the protein internal free-energy terms and are consequently unreliable.     

Electrostatic interactions of charged groups in an aqueous medium and their effect on the formation of the secondary structures of polypeptide chains

Analysis of experimental data shows that interaction of charges in an aqueous medium is satisfactorily evaluated by macroscopic dielectric water permeability even at small (approximately 3--4 A) distances. It has been shown that a non-polar molecule located between the charges weakens their interaction. At the same time interaction between charges situated on the non-polar molecules somewhat increases. An estimate is made of the free energy loss of alpha- and beta-structures of polypeptides, resulting from the insertion of charged groups as well as of the free energy of interactions involving charged groups in different secondary structures.