Theory of the electrokinetic behavior of human erythrocytes

We develop a theory of electrophoresis of human erythrocytes that predicts mobilities significantly smaller than those based on the classical Smoluchowski relation. In the classical treatment the charge is assumed to be spread uniformly on the hydrodynamic surface. The present model takes into account that most of the charge, due mainly to sialic acid, is contained in the glycocalyx. The glycocalyx is modeled as a permeable layer of polyelectrolyte molecules anchored to the cell membrane. The charge is assumed to be uniformly distributed throughout this layer. The fluid flow in the layer is treated as being dominated by Stokes friction arising from idealized polymer segments. The Navier-Stokes equations are solved to give the dependence of electroosomotic velocity with distance from the cell surface. An expression for the electrophoretic mobility is obtained which contains two parameters (a) the thickness of the glycocalyx and (b) the mean polymer segment radius. The best fit to experimental data is obtained if these are given the values 75 A and 7 A, respectively. Deviation from experimental data at low ionic strength (less than 0.05 M) occurs. However, this deviation is in the direction one would expect if at low ionic strength the polyelectrolyte layer expands slightly due to decreased charge shielding.     

Steady-state gel electrophoresis of long polymer molecules: a theoretical study

The velocity of long polymer molecules in a gel and the liquid flow profile in the vicinity of a molecule's surface were studied theoretically by combining the Navier-Stokes equation with the Poisson-Boltzmann equation. The electrophoretic mobility has been calculated in dependence of the ionic strength of the electrolyte solution, its viscosity, the gels' volume friction coefficient, the surface charge and the radius of the polymer molecule. The results are presented in a non-dimensional form and depend on two dimensionless parameters only. The first parameter is the radius of the polymer molecule in units of the Debye length. The second is a parameter comprising the electrolyte's viscosity and the gel density. Thus, by similarity theory, the results apply to any given experimental arrangement. Received: 10 May 1999 / Revised version: 17 November 1999 / Accepted: 6 December 1999     

Potentiometric titration studies on globular proteins

A power series method was applied to solve the Poisson-Boltzmann equation for the spherical polyelectrolyte model and numerical calculation with an electronic computer was performed to obtain surface electric potential on rigid globular proteins. Deviation from the ideal linear relationship in Linderstrom-Lang's plot was found to become noticeable as the surface charge density and the radius of protein increases and ionic strength decreases. The calculated surface potential was compared with potentiometric titration data of several proteins whose radii have been analyzed. Assuming the radius of the counterions to be equal to about 1.0 Å, the data for phenolic groups in ribonuclease and for carboxyl groups in conalbumin were interpreted. Reversible intramolecular transformation was found for α-lactalbumin by comparing the present results with the potentiometric titration data for carboxyl groups. The molecular size of each protein was discussed.     

Effect of finite ionic size on the solution of the Poisson-Boltzmann equation: Application to the binding of divalent metal ions to DNA

The nonlinear Poisson-Boltzmann equation is solved for a cylindrical polyelectrolyte solution containing mono- and divalent counterions and monovalent coions. The finite size of the ions is taken into account by the introduction of the distances of closest approach between the ionic charges and the surface of the polyelectrolyte. The choice of these distances is based on the physicochemical properties of the polyelectrolyte and ions in solution. The effects of the finite ionic size on the distribution of the counterions around the polyelectrolyte and on the local ion concentration and the integrated charge fraction of the divalent cations in the vicinity of the polyelectrolyte are discussed. Theoretical predictions regarding the overall extent of binding and the extent of inner-sphere binding of divalent counterions to rodlike polyions are compared with the results of nmr studies of the binding of divalent metal ions to DNA.     

An evaluation of the outer membrane charge and softness of<Emphasis Type="Italic">Thiobacillus ferrooxidans</Emphasis> by the Ohshima's electrophoretic model of a “soft” particle

The surface charge of bacterial cells plays an important role in their interfacial physiology and adhesion to substrata mediated by the electrostatic double-layer interaction. The surface charge or potential of biological cells is generally calculated from the experimentally measurable electrophoretic velocity of these cells migrating in an external electric field, applying the well-known Smoluchowski equation which is valid for “hard” particles with a sharp interface. However, bacterial cells possessing a structured outer membrane of a finite thickness (dependent on the ionic strength and pH of the surrounding liquid medium) are expected to obey Ohshima's electrophoretic mobility equation derived recently for ‘soft” particles. The electrophoretic mobility ofThiobacillus ferrooxidans was measured here by the fully automated technique of electrophoretic light scattering, based on the proportionality between the mobility and the Doppler shift in the frequency of light scattered by electrophoresing cells. Agreement was obtained between the experimentally determined electrophoretic mobility expressed as a function of low ionic strength (60–6000 μmol/L) at different pH values and the best-fit theoretical predictions of the “soft” particle electrophoresis theory, which is better than in the case of applying the Smoluchowski formula. The best-fit surface-charge and softness parameters predict a rather rigid and low-charge outer membrane of the bacterium examined, as compared to the parameters obtained for other bacteria in media of high ionic strength.     

Glycocalyx electrostatic potential profile analysis: ion, pH, steric, and charge effects

The Poisson-Boltzmann equation is modified to consider charge ionogenicity, steric exclusion, and charge distribution in order to describe the perimembranous electrostatic potential profile in a manner consistent with the known morphology and biochemical composition of the cell's glycocalyx. Exact numerical and approximate analytical solutions are given for various charge distributions and for an extended form of the Donnan potential model. The interrelated effects of ionic conditions, bulk pH, ion binding, local dielectric, steric volume exclusion, and charge distribution on the local potential, pH, and charge density within the glycocalyx are examined. Local charge-induced, potential-mediated pH reductions cause glycocalyx charge neutralization. Under certain conditions, local potentials may be insensitive to ionic strength or may decrease in spite of increasing charge density. The volume exclusion of the glycocalyx reduces the local ion concentration, thereby increasing the local potential. With neutral lipid membranes, the Donnan and surface potential agree if the glycocalyx charge distribution is both uniform and several times thicker than the Debye length (approximately 20 A in thickness under physiological conditions). Model limitations in terms of application to microdomains or protein endo- and ectodomains are discussed.     

Impact of chemical and structural anisotropy on the electrophoretic mobility of spherical soft multilayer particles: the case of bacteriophage MS2

We report a theoretical investigation of the electrohydrodynamic properties of spherical soft particles composed of permeable concentric layers that differ in thickness, soft material density, chemical composition, and flow penetration degree. Starting from a recent numerical scheme developed for the computation of the direct-current electrophoretic mobility (μ) of diffuse soft bioparticles, the dependence of μ on the electrolyte concentration and solution pH is evaluated taking the known three-layered structure of bacteriophage MS2 as a supporting model system (bulk RNA, RNA-protein bound layer, and coat protein). The electrokinetic results are discussed for various layer thicknesses, hydrodynamic flow penetration degrees, and chemical compositions, and are discussed on the basis of the equilibrium electrostatic potential and hydrodynamic flow field profiles that develop within and around the structured particle. This study allows for identifying the cases where the electrophoretic mobility is a function of the inner structural and chemical specificity of the particle and not only of its outer surface properties. Along these lines, we demonstrate the general inapplicability of the notions of zeta potential (ζ) and surface charge for quantitatively interpreting electrokinetic data collected for such systems. We further shed some light on the physical meaning of the isoelectric point. In particular, numerical and analytical simulations performed on structured soft layers in indifferent electrolytic solution demonstrate that the isoelectric point is a complex ionic strength-dependent signature of the flow permeation properties and of the chemical and structural details of the particle. Finally, the electrophoretic mobilities of the MS2 virus measured at various ionic strength levels and pH values are interpreted on the basis of the theoretical formalism aforementioned. It is shown that the electrokinetic features of MS2 are to a large extent determined not only by the external proteic capsid but also by the chemical composition and hydrodynamic flow permeation of/within the inner RNA-protein bound layer and bulk RNA part of the bacteriophage. The impact of virus aggregation, as revealed by decreasing diffusion coefficients for decreasing pH values, is also discussed.     

Modeling the electrophoresis of lysozyme. II. Inclusion of ion relaxation.

In this work, boundary element methods are used to model the electrophoretic mobility of lysozyme over the pH range 2-6. The model treats the protein as a rigid body of arbitrary shape and charge distribution derived from the crystal structure. Extending earlier studies, the present work treats the equilibrium electrostatic potential at the level of the full Poisson-Boltzmann (PB) equation and accounts for ion relaxation. This is achieved by solving simultaneously the Poisson, ion transport, and Navier-Stokes equations by an iterative boundary element procedure. Treating the equilibrium electrostatics at the level of the full rather than the linear PB equation, but leaving relaxation out, does improve agreement between experimental and simulated mobilities, including ion relaxation improves it even more. The effects of nonlinear electrostatics and ion relaxation are greatest at low pH, where the net charge on lysozyme is greatest. In the absence of relaxation, a linear dependence of mobility and average polyion surface potential, (lambda zero)s, is observed, and the mobility is well described by the equation [formula: see text] where epsilon 0 is the dielectric constant of the solvent, and eta is the solvent viscosity. This breaks down, however, when ion relaxation is included and the mobility is less than predicted by the above equation. Whether or not ion relaxation is included, the mobility is found to be fairly insensitive to the charge distribution within the lysozyme model or the internal dielectric constant.     

A biophysical model for interaction of cells with a surface coat (Glycocalyx). I. Electrostatic interaction profile

A model of the primary stage of cell-cell interaction is assumed, including not only the classical electrostatic and electrodynamic energies in the sense of DLVO theory but also steric interaction energy, the energy of specific and non-specific bonds and the energy due to changes of surface potential. Furthermore, in this paper, we exploit recent advances in the understanding of the structure of the cell surface (glycocalyx), assuming the fixed electrostatic charges (dissociated groups of the glycocalyx), to be space charge densities and the glycocalyx itself to be an adsorption layer.In this first part the profile of the electrostatic potential between two cells is calculated on the basis of the linear Poisson-Boltzmann equation (analytical integration) and discussed in dependence on charge densities of the glycocalyx, the separation distance between cells and the ionic strength of the suspension medium.     

Density functional theory for the nonspecific binding of salt to polyelectrolytes: thermodynamic properties

The thermodynamics of the nonspecific binding of salt to a polyelectrolyte molecule is studied using a density functional approach. The polyelectrolyte molecule is modeled as an infinite, inflexible, and impenetrable charged cylinder and the counterions and co-ions are modeled as charged hard spheres of equal diameter. The density functional theory is based on a hybrid approach where the hard-sphere contribution to the one-particle correlation function is evaluated nonperturbatively and the ionic contribution to the one-particle correlation function is evaluated perturbatively. The advantage of the approach is that analytical expressions are available for all the correlation functions. The calculated single ion preferential interaction coefficients, excess free energy, and activity coefficients show a nonmonotonic variation as a function of polyion charge in the presence of divalent ions. These properties display considerable departure from the predictions of the nonlinear Poisson-Boltzmann (NLPB) equation, with qualitative differences in some cases, which may be attributed to correlation effects neglected in the NLPB theory.     

Analysis of electrophoretic mobility data for human erythrocytes according to sublayer models

An attempt was made to analyze the electrophoretic mobility data of human erythrocytes in media of different pH values and ionic strengths through cell surface models in which the surface charge layer consists of several ion-penetrable sublayers with a uniform charge distribution in each sublayer. As a result, the three-sublayer model was found to explain the mobility data much better than the two-sublayer model in a wide range of ionic strength at all pH values studied.     

Application of polyelectrolyte theory to the study of the B-Z transition in DNA (1)

We have used the polyelectrolyte theory to study the ionic strength dependence of the B-Z equilibrium in DNA. A DNA molecule is molded as an infinitely long continuously charged cylinder of radius a with reduced linear charge density q. The parameters a and q for the B and Z forms were taken from X-ray data: aB = 1nm, qB = 4.2, aZ = 0.9 nm and qZ = 3.9. A simple theory shows that at low ionic strengths (when Debye screening length rD much greater than a) the electrostatic free energy difference FelBZ = FelZ - FelB increases with increasing ionic strength since qB greater than qZ. At high ionic strengths (when rD much less than a) the FelBZ would go on growing with increasing ionic strength if the inequality qB/aB greater than qZ/aZ were valid. In the converse case when qZ/qB greater than aZ/aB the FelBZ value decreases with increasing salt concentration at high ionic strength. Since X-ray data correspond to the latter case, theory predicts that the FelBZ value reaches a maximum at an intermediate ionic strength of about 0.1 M (where rD approximately a). We also performed rigorous calculations based on the Poisson-Boltzmann equation. These calculations have confirmed the above criterion of nonmonotonous behaviour of the FelBZ value as a function of ionic strength. Different theoretical predictions for the B-Z transition in linear and superhelical molecules are discussed. Theory predicts specifically that at a very low ionic strength the Z form may prove to be more stable than the B form.(ABSTRACT TRUNCATED AT 250 WORDS)     

The δ-function counterion condensate around a line charge derived from the Fuoss,Katchalsky, and Lifson polyelectrolyte theory

It has recently been proven that the counterion condensate around an isolated line charge in an electrolyte, as characterized by nonlinear Poisson-Boltzmann theory, is an encapsulating δ-function. Here the identical result is shown to hold in the framework of the polyelectrolyte theory of Fuoss, Katchalsky, and Lifson. The proof fully exploits analytic solutions to the differential equation which are not available for the nonlinear, cylindrical Poisson-Boltzmann equation.     

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.     

Electrostatic contribution to the bending of DNA

A model is derived that accounts for the short-range electrostatic contribution to the bending of DNA molecule in solution and in complexes with proteins in terms of the non-linear Poisson-Boltzmann equation. We defined that the short-range electrostatic interactions depend on the changes of the polyion surface charge density under deformation, while the long-range interactions depend on the bending-induced changes in distances between each two points along the polyion axis. After an appropriate simplification of the Poisson-Boltzmann equation, the short-range term is calculated separately giving the lower limit for the electrostatic contribution to the DNA persistence length. The result is compared with the theoretical approaches developed earlier [M. Fixman, J. Chem. Phys. 76 (1982) 6346; M. Le Bret, J. Chem. Phys. 76 (1982) 6243] and with the experimental data. The conclusion is made that the results of Fixman-Le Bret, which took into account both types of the electrostatic interactions for a uniformly bent polyion, give the upper limit for the electrostatic persistence length at low ionic strength, and the actual behavior of the DNA persistence length lies between two theoretical limits. Only the short-range term is significant at moderate-to-high ionic strength where our results coincide with the predictions of Fixman-Le Bret. The bending of DNA on the protein surface that is accompanied by an asymmetric neutralization of the DNA charge is also analyzed. In this case, the electrostatic bending energy gives a significant favorite contribution to the total bending energy of DNA. Important implications to the mechanisms of DNA-protein interactions, particularly in the nucleosome particle, are discussed.     

Electrophoretic detection of reversible chlorpromazine · HCl binding at the human erythrocyte surface

The binding of chlorpromazine · HCl at the human erythrocyte surface has been detected through its effect on cellular electrophoretic mobility. Incubation of erythrocytes (approx. 5 · 10⁶/ml) in 23 μM chlorpromazine · HCl resulted in a reduction of negative electrophoretic mobility from the control value of $\text{?1.11 ± 0.01}$ (μm · s^?1)/(V · cm^?1) to $\text{?1.00 ± 0.02}$ (μm · s^?1)/(V · cm^?1) (pH 7.2, ionic strength 0.155). This mobility change was completely reversed when chlorpromazine · HCl was removed by centrifugal washing. Increasing the drug concentration to 70μM did not affect the mobility change, indicating saturation of the electrophoretically detectable drug binding sites over chlorpromazine · HCl concentration range studied here. The effect of the 23 μM chlorpromazine · HCl on electrophoretic mobility was also measured in isotonic media of reduced ionic strength. The drug-induced reduction in negative surface charge density was found to be independent of ionic strength over the range 0.155 (Debye length, 0.8 nm) to 0.00310 (Debye length, 5.7 nm).Fixation of erythrocytes with glutaraldehyde affected neither the normal electrophoretic mobility of discocytes nor the reduced electrophoretic mobility of chlorpromazine · HCl-induced stomatocytes. When these stomatocytes were first fixed with glutaraldehyde, then washed free of chlorpromazine · HCl, they retained the stomatocyte form while regaining a normal control electrophoretic mobility. Conversely, when discocytes fixed in that form were treated with chlorpromazine · HCl, they showed the same mobility change as did fixed or unfixed stomatocytes. The drug-induced mobility change is therefore independent of the shape change, but reflects a contribution to cellular surface charge density from the membrane-bound chlorpromazine · HCl molecules. From the charge reduction, it is estimated that about 10⁶ chlorpromazine · HCl molecules are bound at the electrokinetic cell surface and occupy approximately 0.4% of the total surface area.     

The effect of cationic electrolytes on the electrostatic behavior of cellular surfaces with ionizable groups

Based on the assumption that the electrostatic charges on the surface of sheep leukocytes arise from the dissociation of ionogenic groups, together with the presence of divalent cation (or trivalent cation) in the suspending medium of low ionic strength (or high ionic strength), the non-linear Poisson-Boltzmann equation for cell interaction with a solid surface with constant potential (or constant charge) is numerically solved in this paper. The cellular surface potential and the repulsive (or attractive) force is expressed as the function of separation distance. Because of shrinking the thickness of the electrostatic double layer at high ionic strength, the presence of cationic electrolyte has a less influential role on both the cellular surface potential and interaction force than at low ionic strength. However, due to the continuous equilibration of the ionogenic groups on the cellular surface as separation distance decreases, the presence of cationic electrolyte will not always reduce the interaction force during the whole adhesion period. The distance at which the cationic electrolyte changes its effect from positive to negative is termed the critical separation distance in this paper.     

The osmotic pressure of chondroitin sulphate solutions: experimental measurements and theoretical analysis

We used equilibrium dialysis to measure the osmotic pressure of chondroitin sulphate (CS) solutions as a function of their concentration and fixed charge density (FCD) and the ionic strength and composition of the solution. Osmotic pressure varied nonlinearly with the concentration of chondroitin sulphate and in 0.15 M NaCl at FCDs typical of uncompressed cartilage (approximately 0.4 mmol/g extrafibrillar H2O) was approximately 3 atmospheres. Osmotic pressure fell by 60% as solution ionic strength increased up to about 1 M, but remained relatively constant at higher ionic strengths. The ratio of Ca2+ to Na+ in the medium was a minor determinant of osmotic pressure. The data are compared with a theoretical model of the electrostatic contribution to osmotic pressure calculated from the Poisson-Boltzmann equation using a rod-in-cell model for CS. The effective radius of the polyelectrolyte rod is taken as a free parameter. The model qualitatively reproduces the non-linear concentration dependence, but underestimates the osmotic pressure by an amount that is independent of ionic strength. This difference, presumably arising from oncotic and entropic effects, is approximately 1/3 of the total osmotic pressure at physiological polymer concentrations and ionic strength.     

Size-dependent mobile surface charge model of cell electrophoresis

A model that accurately predicts the effects of cellular size and electric field strength on electrophoretic mobility has been developed. Previous models have predicted that electrophoretic mobility (EPM) is dependent only on cell surface charge, bath viscosity and ionic strength of the electrolyte. However, careful analysis of experimental data from the literature shows that these models do not accurately depict the relationship between chemically determined surface charge and observed mobility. We propose a new model that accounts for electrically driven redistribution of mobile surface charge islands, such as the recently proposed lipid raft structures. This model predicts electrophoretic mobility as a function of a new dimensionless quantity, A, that incorporates the cell radius, the electric field strength, and the average diameter of charged membrane complexes.