Electrophoretic mobility of human erythrocytes. On the applicability of the charged layer model.

The limitations of previous linear electrokinetic theories are discussed. A special model of the surface charge distribution, based on the minimum condition of the interfacial electrostatic free energy, is introduced. The model describes the electrophoretic mobility, taking into account the electroosmotic flow through the surface macromolecular layer and the surface conductivity. This nonlinear electrophoretic theory describes experimental data obtained with human erythrocytes. Numerical results for an uniformly distributed space charge are also presented.     

A mechanism for the establishment of polar cell morphology based on the cytoskeleton-derived forces exerted on the cell boundary

A mechanism for the establishment of polar cell morphology is presented, based on the internal forces that the cytoskeletal structures exert on the cell boundary. Cell shapes are determined by postulating that they correspond to the minimum of the total energy of the system, which is the sum of the bending energy of the cell boundary and the potential energies of the involved forces. Axisymmetrical cell shapes are considered, and it is assumed that the cytoskeletal structures exert an extensional axial force and are involved in controlling the area of the cell boundary. The dependence of cell shapes on the axial force is presented for different values of this area. The results show that, at increasing axial force, the cell undergoes a discontinuous transition from an oval shape, exhibiting an equatorial mirror symmetry into a polar shape. The proposed mechanism is related to previously documented specific effects of microtubule- and actin-modifying drugs on polar shapes of developing isolated retinal photoreceptor cells. Received: 28 January 1998 / Revised version: 25 July 1998 / Accepted: 29 July 1998     

Electrostatic free energy and spontaneous curvature of spherical charged layered membrane

A biophysical model for the equilibrium curvature of a composite membrane element is derived taking into account the mechanical bilayer properties and the adjacent charged protein layers. The minimum of the total free energy density with respect to the curvature of such a membrane curved was estimated from the sum of the electrostatic free energy density of the charges of the membrane and the elastic surface energy density due to bending the lipid bilayer membrane. It was shown that the equilibrium curvature, i.e. the spontaneous curvature, of such a charged composite sandwich-like membrane depends inversely on the bending stiffness of the lipid membrane itself and directly on the charge amount inside and outside the membrane to the second power. Furthermore the geometric and electrostatic structure of the protein layers and the physico-chemical environment conditions are involved. Corresponding to the model developed a "standard RBC" membrane element has a negative spontaneous curvature, accounting for a discocyte RBC shape. The shape change from a discocyte to a more stomatocytic shape (increase in the negative spontaneous curvature) after reducing the charges in the glycocalyx is also explained within this model.     

Looping charged elastic rods: applications to protein-induced DNA loop formation

We analyze looping of thin charged elastic filaments under applied torques and end forces, using the solution of linear elasticity theory equations. In application to DNA, we account for its polyelectrolyte character and charge renormalization, calculating electrostatic energies stored in the loops. We argue that the standard theory of electrostatic persistence is only valid when the loop’s radius of curvature and close-contact distance are much larger than the Debye screening length. We predict that larger twist rates are required to trigger looping of charged rods as compared with neutral ones. We then analyze loop shapes formed on charged filaments of finite length, mimicking DNA looping by proteins with two DNA-binding domains. We find optimal loop shapes at different salt amounts, minimizing the sum of DNA elastic, DNA electrostatic, and protein elastic energies. We implement a simple model where intercharge repulsions do not affect the loop shape directly but can choose the energy-optimized shape from the allowed loop types. At low salt concentrations more open loops are favored due to enhanced repulsion of DNA charges, consistent with the results of computer simulations on formation of DNA loops by lac repressor. Then, we model the precise geometry of DNA binding by the lac tetramer and explore loop shapes, varying the confined DNA length and protein opening angle. The characteristics of complexes obtained, such as the total loop energy, stretching forces required to maintain its shape, and the reduction of electrostatic energy with increment of salt, are in good agreement with the outcomes of more elaborate numerical calculations for lac-repressor-induced DNA looping.     

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.     

Continuum electrostatic model for the binding of cytochrome c2 to the photosynthetic reaction center from Rhodobacter sphaeroides

Electrostatic interactions are important for protein-protein association. In this study, we examined the electrostatic interactions between two proteins, cytochrome c(2) (cyt c(2)) and the reaction center (RC) from the photosynthetic bacterium Rhodobacter sphaeroides, that function in intermolecular electron transfer in photosynthesis. Electrostatic contributions to the binding energy for the cyt c(2)-RC complex were calculated using continuum electrostatic methods based on the recent cocrystal structure [Axelrod, H. L., et al. (2002) J. Mol. Biol. 319, 501-515]. Calculated changes in binding energy due to mutations of charged interface residues agreed with experimental results for a protein dielectric constant epsilon(in) of 10. However, the electrostatic contribution to the binding energy for the complex was close to zero due to unfavorable desolvation energies that compensate for the favorable Coulomb attraction. The electrostatic energy calculated as a function of displacement of the cyt c(2) from the bound position showed a shallow minimum at a position near but displaced from the cocrystal configuration. These results show that although electrostatic steering is present, other short-range interactions must be present to contribute to the binding energy and to determine the structure of the complex. Calculations made to model the experimental data on association rates indicate a solvent-separated transition state for binding in which the cyt c(2) is displaced approximately 8 A above its position in the bound complex. These results are consistent with a two-step model for protein association: electrostatic docking of the cyt c(2) followed by desolvation to form short-range van der Waals contacts for rapid electron transfer.     

Electrostatic effects in a large enzyme complex: subunit interactions and electrostatic potential field of the icosahedral beta 60 capsid of heavy riboflavin synthase

The role of the electrostatic interactions in the stability of the icosahedral beta 60 capsid of heavy riboflavin synthase from Bacillus subtilis has been investigated using an approach based on the theory of Kirkwood and Tanford. The pH dependence of the electrostatic subunit interactions agrees well with experimental data. The electrostatic subunit interaction energy has a pronounced minimum at pH 8.2 for both the ligated and ligand-free capsid. The latter is characterized by a reduction of the magnitude and the pH range of the electrostatic attraction. It is found that only 8 charged groups, which form one cluster and two ion pairs, provide a significant contribution to the capsid stability. The analysis has shown that the aggregation/disaggregation equilibrium seems to be regulated by electrostatic interactions between beta-subunits forming dimers, which connect the relatively stable pentamers in the beta-60 capsid. The release of the ligand causes a reduction of the electrostatic attraction of the dimers, which may induce disaggregation of the capsid. The electrostatic potential field due to the titratable groups and alpha-helix macrodipoles has been calculated on the basis of the Coulomb relation. Two different values of the dielectric constant have been used for the protein and the surrounding solvent, respectively. The electrostatic potential shows a radially polar distribution with a positive pole at the inner capsid wall and a negative pole outside the capsid. An interesting feature of the electrostatic field is the formation of positive potential "channels" that coincide with the channels constituted by the pentameric and trimeric beta-subunit aggregates. It is supposed that the electrostatic potential field plays a role in enzyme-substrate recognition.     

Shapes of bilayer vesicles with membrane embedded molecules

The interdependence of the lateral distribution of molecules which are embedded in a membrane (such as integral membrane proteins) and the shape of a cell with no internal structure (such as phospholipid vesicles or mammalian erythrocytes) has been studied. The coupling of the lateral distribution of the molecules and the cell shape is introduced by considering that the energy of the membrane embedded molecule at a given site of the membrane depends on the curvature of the membrane at that site. Direct interactions between embedded molecules are not considered. A simple expression for the interaction of the membrane embedded molecule with the local membrane curvature is proposed. Starting from this interaction, the consistently related expressions for the free energy and for the distribution function of the embedded molecules are derived. The equilibrium cell shape and the corresponding lateral distribution of the membrane embedded molecules are determined by minimization of the membrane free energy which includes the free energy of the membrane embedded molecules and the membrane elastic energy. The resulting inhomogeneous distribution of the membrane embedded molecules affects the cell shape in a nontrivial manner. In particular, it is shown that the shape corresponding to the absolute energy minimum at given cell volume and membrane area may be elliptically non-axisymmetric, in contrast to the case of a laterally homogeneous membrane where it is axisymmetric.     

Bacterial adhesion: A physicochemical approach

The adhesion of bacteria to solid surfaces was studied using a physicochemical approach. Adhesion to negatively charged polystyrene was found to be reversible and could be described quantitatively using the DLVO theory for colloidal stability, i.e., in terms of Van der Waals and electrostatic interactions. The influence of the latter was assessed by varying the electrolyte strength. Adhesion increased with increasing electrolyte strength. The adhesion Gibbs energy for a bacterium and a negatively charged polystyrene surface was estimated from adhesion isotherms and was found to be 2–3 kT per cell. This low value corresponds to an adhesion in the secondary minimum of interaction as described by the DLVO theory. The consequences of these findings for adhesion in the natural environment are discussed.     

Electrostatic theory of the assembly of PAMAM dendrimers and DNA

The electrostatic interactions mediated by counterions between a cationic PAMAM dendrimer, modelized as a sphere of radius and cationic surface charge highly increasing with generation, and a DNA, modelized as an anionic elastic line, are analytically calculated in the framework of condensation theory. Under these interactions the DNA is wrapped around the sphere. For excess phosphates relative to dendrimer primary amines, the free energy of the DNA‐dendrimer complex displays an absolute minimum when the complex is weakly negatively overcharged. This overcharging opposes gene delivery. For a highly positive dendrimer and a DNA fixed by experimental conditions to a number of phosphates less than the number of dendrimer primary amines, excess amine charges, the dendrimer may at the same time bind stably DNA and interact with negative cell membranes to activate cell transfection in fair agreement with molecular simulations and experiments. © 2016 Wiley Periodicals, Inc. Biopolymers 105: 276–286, 2016.     

Electrostatic interactions in proteins: Calculations of the electrostatic term of free energy and the electrostatic potential field

The pH-dependence of the electrostatic energy of interactions between titratable groups is calculated for some well studied globular proteins: basic pancreatic trypsin inhibitor, sperm whale myoglobin and tuna cytochrome c. The calculations are carried out using a semi-empirical appraach in terms of the macroscopic model based on the Kirkwood-Tanford theory. The results are discussed in the light of their physicochemical and biological properties. It was found that the pH-dependence of the electrostatic energy correlates with the III–IV transition of cytochrome c. The electrostatic field of the cysteine proteinase inhibitor, cystatin, was calculated in two ways. In the first one, the electrostatic field created by the pH dependent charges of the ionizable groups and peptide dipoles was calculated using the approach proposed. In the second one, the finite-difference method was used. The results obtained by the two methods are in overall agreement. The calculated field was discussed in terms of the binding of cystatin to papain.     

On the multiple-minima problem in the conformational analysis of polypeptides. II. An electrostatically driven Monte Carlo method--tests on poly(L-alanine)

A new approach to the multiple-minima problem in protein folding is presented. It is assumed that the molecule is driven toward the native structure by three types of mechanism. The first one involves an optimization of the electrostatic interactions, whereby the molecule evolves toward conformations in which the charge distribution becomes energetically more favorable. The second mechanism involves a Monte Carlo–energy minimization approach, and the third one is a backtrack mechanism that acts in the opposite direction, increasing the energy—the third type of movement provides a means to perturb the molecule when it is trapped in a stable but energetically unfavorable local energy minimum. This paper describes the implementation of a model based on these mechanisms, and illustrates its effectiveness by computations on different arbitrary starting conformations of a terminally blocked 19-residue chain of poly(L -alanine) for which the global minimum apparently corresponds to the right-handed α-helix. In all cases, the global minimum was attained, even when the starting conformation was a left-handed α-helix. In the latter case, the trajectory of conformations passed through partially melted forms of the left-handed α-helix (because of electrostatic defects at the ends), and then through the formation of structures leading to the more stable right-handed α-helix.     

Two-phase formation in solutions of tobacco mosaic virus and the problem of long-range forces

In a nearly salt-free medium, a dilute tobacco mosaic virus solution of rod-shaped virus particles of uniform length forms two phases; the bottom optically anisotropic phase has a greater virus concentration than has the top optically isotropic phase. For a sample containing particles of various lengths, the bottom phase contains longer particles than does the top and the concentrations top and bottom are nearly equal. The longer the particles the less the minimum concentration necessary for two-phase formation. Increasing the salt concentration increases the minimum concentration. The formation of two phases is explained in terms of geometrical considerations without recourse to the concept of long-range attractive forces. The minimum concentration for two-phase formation is that concentration at which correlation in orientation between the rod-shaped particles begins to take place. This concentration is determined by the thermodynamically effective size and shape of the particles as obtained from the concentration dependence of the osmotic pressure of the solutions measured by light scattering. The effective volume of the particles is introduced into the theory of Onsager for correlation of orientation of uniform size rods and good agreement with experiment is obtained. The theory is extended to a mixture of non-uniform size rods and to the case in which the salt concentration is varied, and agreement with experiment is obtained. The thermodynamically effective volume of the particles and its dependence on salt concentration are explained in terms of the shape of the particles and the electrostatic repulsion between them. Current theories of the hydration of proteins and of long-range forces are critically discussed. The bottom layer of freshly purified tobacco mosaic virus samples shows Bragg diffraction of visible light. The diffraction data indicate that the virus particles in solution form three-dimensional crystals approximately the size of crystalline inclusion bodies found in the cells of plants suffering from the disease.     

pH-dependent stability of sperm whale myoglobin in water-guanidine hydrochloride solutions

An experimental-theoretical approach for the elucidation of protein stability is proposed. The theoretical prediction of pH-dependent protein stability is based on the macroscopic electrostatic model for calculation of the pH-dependent electrostatic free energy of proteins. As a test of the method we have considered the pH-dependent stability of sperm whale metmyoglobin. Two theoretical methods for evaluation of the electrostatic free energy and p K values are applied: the finite-difference Poisson-Boltzmann method and the semiempirical approach based on the modified Tanford-Kirkwood theory. The theoretical results for electrostatic free energy of unfolding are compared with the experimental data for guanidine hydrochloride unfolding under equilibrium conditions over a wide pH range. Using the optical parameters of the Soret absorbance to monitor conformational equilibrium and Tanford's method to estimate the resulting data, it was found that the conformational free energy of unfolding of metmyoglobin is 16.3 kcal mol(-1) at neutral pH values. The total unfolding free energies were calculated on the basis of the theoretically predicted electrostatic unfolding free energies and the experimentally measured midpoints (pH(1/2)) of acidic and alkaline denaturation transitions. Experimental data for alkaline denaturation were used for the first time in theoretical analysis of the pH-dependent unfolding of myoglobin. The present results demonstrate that the simultaneous application of appropriate theoretical and experimental methods permits a more complete analysis of the pH-dependent and pH-independent properties and stability of globular proteins.     

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.     

Calculation of the total electrostatic energy of a macromolecular system: solvation energies, binding energies, and conformational analysis

In this report we describe an accurate numerical method for calculating the total electrostatic energy of molecules of arbitrary shape and charge distribution, accounting for both Coulombic and solvent polarization terms. In addition to the solvation energies of individual molecules, the method can be used to calculate the electrostatic energy associated with conformational changes in proteins as well as changes in solvation energy that accompany the binding of charged substrates. The validity of the method is examined by calculating the hydration energies of acetate, methyl ammonium, ammonium, and methanol. The method is then used to study the relationship between the depth of a charge within a protein and its interaction with the solvent. Calculations of the relative electrostatic energies of crystal and misfolded conformations of Themiste dyscritum hemerythrin and the VL domain of an antibody are also presented. The results indicate that electrostatic charge-solvent interactions strongly favor the crystal structures. More generally, it is found that charge-solvent interactions, which are frequently neglected in protein structure analysis, can make large contributions to the total energy of a macromolecular system.     

Investigating the dielectric effects of channel pore water on the electrostatic barriers of the permeation ion by the finite difference Poisson-Boltzmann method

In this paper, the finite difference Poisson-Boltzmann (FDPB) method with four dielectric constants is developed to study the effect of dielectric saturation on the electrostatic barriers of the permeation ion. In this method, the inner shape of the channel pore is explicitly represented, and the fact that the dielectric constant inside the channel pore is different from that of bulk water is taken into account. A model channel system which is a right-handed twist bundle with four α-helical segments is provided for this study. From the FDPB calculations, it is found that the difference of the ionic electrostatic solvation energy for wider domains depends strongly on the pore radius in the vicinity of the ion when the pore dielectric constant is changed from 78 to 5. However, the electrostatic solvation energy of the permeation ion can not be significantly affected by the dielectric constant in regions with small pore radii. Our results indicate that the local electrostatic interactions inside the ion channel are of major importance for ion electrostatic solvation energies, and the effect of dielectric saturation on the electrostatic barriers is coupled to the interior channel dimensions. Received: 28 January 1997 / Accepted: 24 September 1997     

The mechanical behaviour of cell membranes as a possible physical origin of cell polarity

The mechanical behaviour of a closed layered membrane enclosing a structureless interior is considered. The shape of such an object in flaccid conditions is determined theoretically by assuming that it corresponds to the minimum value of the membrane bending energy. The symmetry breaking properties of this system are revealed. It is suggested that a continuous transition from an axisymmetrical shape involving mirror symmetry with regard to the equatorial plane of the object to the shape with polar asymmetry could be the primary event in establishing cell polarity.