Relationship among the surface potential, Donnan potential and charge density of ion-penetrable membranes

Calculation of the potential distribution across a uniformly charged ion-penetrable membrane that we developed previously is extended to derive a relationship among the surface potential, Donnan potential and charge density of an ion-penetrable membrane in a mixed solution of 2-1 and 1-1 electrolytes. We also present an approximate method for calculating the surface potential/Donnan potential/charge density relationship for membranes with an arbitrary distribution of membrane-fixed charges.     

Electrostatic interaction between ion-penetrable multi-layered membranes

A theory of the double layer interaction regulated by the Donnan potential between two ion-penetrable membranes in an electrolyte solution developed previously by Ohshima and Kondo is extended to the case in which the membranes consist of many layers having different thickness and densities of membrane-fixed charges. The interaction force is found to be determined mainly by the contributions from layers located within the depth of 1/kappa (kappa, Debye-Hückel parameter) from the membrane surface. It is also predicted that the interaction force may alter its sign with changing electrolyte concentration.     

Electrostatic repulsion of ion penetrable charged membranes: role of Donnan potential

A model is proposed for the electrostatic repulsion between two ion-penetrable charged membranes in which fixed charges are uniformly distributed. This model assumes that the electric potential far inside the membrane is always equal to the Donnan potential, independent of the membrane separation. In this respect the present model completely differs from usual models for the electrostatic interaction of colloidal particles which assume that the surface potential or the surface charge density remains constant during interaction. It is shown that the rise in potential in the interacting membranes caused by their approach is greatly suppressed so that the potential in the membrane does not exceed the Donnan potential. Numerical results of the calculation of the repulsion by the non-linear Poisson-Boltzmann equation are displayed as a function of the membrane separation and an approximate formula is also derived.     

Membrane potential and Donnan potential

The Nernst-Planck-Poisson equations for the potential profile across a membrane are exactly solved without recourse to the assumption of constant field within the membrane. It is assumed that the membrane core of thickness dc is covered by a surface layer of thickness ds in which the membrane-fixed charges are distributed at a uniform density N. The membrane boundary potentials as well as the diffusion potentials contribute to the membrane potential. It is shown that for ds greater or similar 1/k, k being the Debye-Hückel parameter, the potential in the membrane surface layer except in the region very near the membrane/solution boundary is effectively equal to the Donnan potential and that its contribution to the membrane potential becomes dominant as N increases. For low N, on the other hand, the membrane potential arises mostly from the diffusion potential.     

Effect of membrane potential on the mechanical equilibrium of biological membranes

The mechanical equilibrium of a membranous sac, whose wall is sandwiched by two oppositely charged fluid layers, is investigated as a mathematical model of a living cell. In so doing, it is assumed that the space charge density in the inner and the outer charged fluid layer is constant. It is also assumed that the fluid inside and outside of the charged fluid layer is a perfect conductor. By solving Maxwell's equation, the electric field and the thickness of the inner and the outer charged fluid layer is determined as a function of the geometry of the sac. Then, the fluid pressure in the charged fluid layer is derived by considering the body force created by the electrostatic field. The condition of mechanical equilibrium of the sac membrane yields an equation which reveals the inter-relation between the geometry, the sac fluid pressure and the membrane potential. According to this equation, the change of membrane potential causes a deformation of the sac. If the wall of the membranous sac is permeable, increase (decrease) of the absolute value of the membrane potential results in swelling (shrinking) of the sac. On the other hand, the mechanical change of the sac volume results in the change of the membrane potential. This analysis provides also an explanation of how the red blood cell maintains the biconcave shape, when the red blood cell is assumed to be a fluid filled membranous sac with non-zero membrane potential.     

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.     

Phospholipid Flip-Flop and the Distribution of Surface Charges in Excitable Membranes

There is now good evidence that most of the lipids in a biological membrane are arranged in the form of a bilayer. Charged lipids in the membrane of an excitable cell are subject to a significant driving force, the gradient of the intramembrane potential, which will tend to redistribute the lipids between the two halves of the bilayer by a “phospholipid flip-flop” mechanism. We have calculated, by combining the Boltzmann relation from statistics and the Gouy equation from the theory of the diffuse double layer, the steady-state distribution of charged lipids in the bilayer. This distribution is completely determined, within the framework of the model, by three experimentally accessible variables; the percentage of charged lipid in the bilayer as a whole, the resting potential and the ionic strength. The known values for the percentage of anionic phospholipids in squid axons (10-15%), the membrane potential (50-100 mV) and ionic strength (0.5 M) imply that the charge density and double layer potential at the outer surface of the nerve will be substantially greater than the charge density and double layer potential at the inner surface, in agreement with the best available evidence from physiological measurements.     

Generation of the streaming potential by liposomes in cylindrical capillary. Experimental data

A model of creation a streaming potential U as a result of colloidal particle movement in flow in a capillary has been described previously (Zawada 1996) as well as the systems for measurement (Zawada 1990, 1991). The filling of capillary with a solution of liposomes results in a labile adsorbance of liposomes on a capillary glass and changes the measured streaming potential. In order to minimalize these adverse effects, the capillary was covered with phospholipid layer of different composition. Some concentrations of stearylamine as a component of the phospholipid layer may fully compensate the surface charge of the glass capillary and can reduce the liposomes adsorption. The streaming potential of the liposomes solution depends on the ionic strength of the electrolyte and is smaller than the zeta potential for similar liposomes. This suggests that only a part of ions of the liposome ion atmosphere participate in creating of the streaming potential. These are the ions from the hydrodynamic slipping layer. The regression analysis of the relationships between streaming potential U and concentration of liposomes and next ionic strength of the electrolyte gave the value of the surface potential psi0 and the thickness of the hydrodynamic slipping layer d, that is independent of the ionic strength.     

Effects of unstirred layers or transport number discontinuities on the transient and steady-state current-voltage relationships of membranes

The effects of current-induced electrolyte accumulation and depletion on the electrical properties of a two-layered membrane system have been examined. The membrane consisted of a charged, ion permselective layer and an uncharged, non-selective layer. The model was designed to reveal the properties of membranes possessing long pores with ionic charges at one end or of ion-selective membranes bounded by highly unstirred aqueous layers. Electrolyte concentration profiles in the inert layer and their time-dependent changes were obtained from solutions of the diffusion equation under the condition of constant current. The profiles were then used to calculate the voltage developed across the membrane at various times after the current is switched on. The theoretical results are presented in the form of $\text{i-V}$ curves with reduced coordinates that can be used to obtain time-current-voltage relationships for membranes of the type considered having any thickness of the non-selective layer and bathed in any concentration of any 1:1 electrolyte. Experimental results on a model composite membrane were in good agreement with calculations that assume that ion transport occurs only under the influence of electrical potential and concentration gradients, suggesting that in such systems, the combined effects of convection, osmosis, electro-osmosis, and concentration-dependence of diffusion coefficients, activity coefficients, and transference numbers are small. Voltage fluctuations in the form of periodic spikes were observed experimentally at the limiting current density (the current density at which the electrolyte concentration at one surface of the selective layer goes to 0). These phenomena were not seen when the current was in the direction leading to accumulation of electrolyte in the non-selective (unstirred) layer. Such composite membranes can exhibit S-shaped and N-shaped $\text{i-V}$ curves under ramp-voltage and ramp-current clamps, respectively.     

Theoretical evaluation of a possible nature of the outer membrane potential of mitochondria

A possibility of generation of the outer membrane potential in mitochondria has been suggested earlier in the literature, but the potential has not been directly measured yet. Even its nature, metabolic impact and a possible range of magnitudes are not clear, and require further theoretical and experimental analysis. Here, using simple mathematical model, we evaluated a possible contribution of the Donnan and metabolically derived potentials to the outer membrane potential, concluding that the superposition of both is most probable; exclusively Donnan origin of the potential is doubtful because unrealistically high concentrations of charged macromolecules are needed for maintaining its relatively high levels. Regardless of the mechanism(s) of generation, the maximal possible potential seems to be less than 30 mV because significant osmotic gradients, created at higher values, increase the probability of the outer membrane rupture. New experimental approaches for direct or indirect determination of true value of the outer membrane potential are suggested here to avoid a possible interference of the surface electrical potential of the inner membrane, which may change as a result of the extrusion of matrix protons under energization of mitochondria.     

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.     

Poisson-Boltzmann Theory for Membranes with Mobile Charged Lipids and the pH-Dependent Interaction of a DNA Molecule with a Membrane

We consider a planar stiff model membrane consisting of mobile surface groups whose state of charge depends on the pH and the ionic composition of the adjacent electrolyte solution. To calculate the mean-field interaction potential between a charged object and such a model membrane, one needs to solve a Poisson-Boltzmann boundary value problem. We here derive and discuss the boundary condition at the membrane surface, a condition that is generally appropriate for biological membranes where two charge-regulating mechanisms are present at the same time: the pH-dependent chemical charge regulation and a regulation through the in-plane mobility of the surface groups. As an application of this general formalism, we consider the specific example of a single DNA molecule, approximated by a cylinder with smeared-out surface charges, interacting with such a model membrane. We study the effect that the two competing charge-regulating mechanisms have on the DNA/membrane interaction and the distribution of surface ions in the plane of the membrane. We find that, at short DNA-membrane distances, membrane fluidity can have a considerable impact on the DNA adsorption behavior and can lead to such counterintuitive phenomena as the adsorption of a negatively charged DNA onto a (on average) negatively charged membrane.     

Transport of charged macromolecules across a biological charged membrane

The glomerular capillary wall of the kidney behaves as an electronegatively charged structure consisting of three layers, the lamina densa and the two laminae rarae, which are differently charged. Thus, a three layer model is proposed to analyse the transport of charged macromolecules across this wall. A modified Nernst-Planck equation describes the macromolecule flux across the wall and a Donnan equilibrium is assumed at each interface. For a given value of the fixed charge concentration in each layer, the local sieving coefficient of the macromolecule, i.e. the ratio between the concentrations in the filtrate and in the plasma, is calculated. A sieving curve which relates the sieving coefficient to the Einstein-Stokes radius of the macrosolute is obtained. The fixed charge concentrations in each layer are iteratively modified until simultaneous adjustment is achieved between calculated and experimental curves, for positively and negatively charged tracers and their neutral equivalent.     

Porin channels in intact cells of Escherichia coli are not affected by Donnan potentials across the outer membrane

Donnan potential (interior negative) across the outer membrane of Escherichia coli was measured by the distribution of [14C]choline in a mutant with a deletion through the genes for the active transport of choline. Calculation showed that the presence of membrane-derived oligosaccharides in the periplasm could quantitatively explain the magnitude of the Donnan potential and the periplasmic volume. By measuring the permeability of porin channels in intact cells suspended in solutions of widely different ionic strengths, it was shown that changing Donnan potential from 5 mV to approximately 100 mV had no effect on the permeability of either OmpF or OmpC porin channel toward a zwitterionic compound, cephaloridine. Thus, the "voltage-dependent gating" of porin channel, previously reported from another laboratory, is likely to be an artifact of in vitro reconstitution. The influx of negatively charged compounds, however, was affected by the Donnan potential as expected from the electrolyte diffusion theory.     

Membrane electrostatics

In conclusion, charged membrane together with their adjacent electrolyte solution form a thermodynamic and physico-chemical entity. Their surfaces represent an exceptionally complicated interfacial system owing to intrinsic membrane complexity, as well as to the polarity and often large thickness of the interfacial region. Despite this, charged membranes can be described reasonably accurately within the framework of available theoretical models, provided that the latter are chosen on the basis of suitable criteria, which are briefly discussed in Section A. Interion correlations are likely to be important for the regular and/or rigid, thin membrane-solution interfaces. Lateral distribution of the structural membrane charge is seldom and charge distribution perpendicular to the membranes is nearly always electrostatically important. So is the interfacial hydration, which to a large extent determines the properties of the innermost part of the interfacial region, with a thickness of 2-3 nm. Fine structure of the ion double-layer and the interfacial smearing of the structural membrane charge decrease whilst the surface hydration increases the calculated value of the electrostatic membrane potential relative to the result of common Gouy-Chapman approximation. In some cases these effects partly cancel-out; simple electrostatic models are then fairly accurate. Notwithstanding this, it is at present difficult to draw detailed molecular conclusions from a large part of the published data, mainly owing to the lack of really stringent controls or calibrations. Ion binding to the membrane surface is a complicated process which involves charge-charge as well as charge-solvent interactions. Its efficiency normally increases with the ion valency and with the membrane charge density, but it is also strongly dependent on the physico-chemical and thermodynamic state of the membrane. Except in the case of the stereospecific ion binding to a membrane, the relatively easily accessible phosphate and carboxylic groups on lipids and integral membrane proteins are the main cation binding sites. Anions bind preferentially to the amine groups, even on zwitterionic molecules. Membrane structure is apt to change upon ion binding but not always in the same direction: membranes with bound ions can either expand or become more condensed, depending on the final hydrophilicity (polarity) of the membrane surface. The more polar membranes, as a rule, are less tightly packed and more fluid. Diffusive ion flow across a membrane depends on the transmembrane potential and concentration gradients, but also on the coulombic and hydration potentials at the membrane surface.(ABSTRACT TRUNCATED AT 400 WORDS)     

Ion concentration and charge adjacent to electrically charged cell membrane

Equations describing ion concentration profiles and electric charge in electrolyte solutions adjacent to an electrically charged cell membrane model in the electrochemical equilibrium state are developed and completely solved. The membrane system model consists of an infinitely large planar sheet of finite thickness separating two electrolyte solutions. Electric charges in the membrane model consist of planes of charge parallel to the surfaces of the planar sheet. The charge in solution adjacent to each surface of the membrane is due to differences in the total anion and cation concentrations in each solution.Expressions of concentration and charge are functions of the quantity and location of charge in the membrane, the various permittivities and thickness of the membrane, and the ionic compositions, permittivities, and temperature of the electrolyte solutions.The validity and relation of the model to real membranes are discussed.     

The effect of lipid demixing on the electrostatic interaction of planar membranes across a salt solution

We study the effect of lipid demixing on the electrostatic interaction of two oppositely-charged membranes in solution, modeled here as an incompressible two-dimensional fluid mixture of neutral and charged mobile lipids. We calculate, within linear and nonlinear Poisson-Boltzmann theory, the membrane separation at which the net electrostatic force between the membranes vanishes, for a variety of different system parameters. According to Parsegian and Gingell, contact between oppositely-charged surfaces in an electrolyte is possible only if the two surfaces have exactly the same charge density (sigma(1) = -sigma(2)). If this condition is not fulfilled, the surfaces can repel each other, even though they are oppositely charged. In our model of a membrane, the lipidic charge distribution on the membrane surface is not homogeneous and frozen, but the lipids are allowed to freely move within the plane of the membrane. We show that lipid demixing allows contact between membranes even if there is a certain charge mismatch, /sigma(1)/ not equal /sigma(2)/, and that in certain limiting cases, contact is always possible, regardless of the value of sigma(1)/sigma(2) (if sigma(1)/sigma(2) < 0). We furthermore find that of the two interacting membranes, only one membrane shows a major rearrangement of lipids, whereas the other remains in exactly the same state it has in isolation and that, at zero-disjoining pressure, the electrostatic mean-field potential between the membranes follows a Gouy-Chapman potential from the more strongly charged membrane up to the point of the other, more weakly charged membrane.     

ON THE CAUSE OF THE INFLUENCE OF IONS ON THE RATE OF DIFFUSION OF WATER THROUGH COLLODION MEMBRANES. II

1. It had been shown in previous publications that when pure water is separated from a solution of an electrolyte by a collodion membrane the ion with the same sign of charge as the membrane increases and the ion with the opposite sign of charge as the membrane diminishes the rate of diffusion of water into the solution; but that the relative influence of the oppositely charged ions upon the rate of diffusion of water through the membrane is not the same for different concentrations. Beginning with the lowest concentrations of electrolytes the attractive influence of that ion which has the same sign of charge as the collodion membrane upon the oppositely charged water increases more rapidly with increasing concentration of the electrolyte than the repelling effect of the ion possessing the opposite sign of charge as the membrane. When the concentration exceeds a certain critical value the repelling influence of the latter ion upon the water increases more rapidly with a further increase in the concentration of the electrolyte than the attractive influence of the ion having the same sign of charge as the membrane. 2. It is shown in this paper that the influence of the concentration of electrolytes on the rate of transport of water through collodion membranes in electrical endosmose is similar to that in the case of free osmosis. 3. On the basis of the Helmholtz theory of electrical double layers this seems to indicate that the influence of an electrolyte on the rate of diffusion of water through a collodion membrane in the case of free osmosis is due to the fact that the ion possessing the same sign of charge as the membrane increases the density of charge of the latter while the ion with the opposite sign diminishes the density of charge of the membrane. The relative influence of the oppositely charged ions on the density of charge of the membrane is not the same in all concentrations. The influence of the ion with the same sign of charge increases in the lowest concentrations more rapidly with increasing concentration than the influence of the ion with the opposite sign of charge, while for somewhat higher concentrations the reverse is true.     

The net electric charge of proteins. A comparison of determinations by Donnan potential measurements and by gel electrophoresis

We compare a new method for the determination of the net charge of proteins based on Donnan potential measurements, as described briefly by Ojteg, G., Nygren, K. and Wolgast, M. (1987) Acta Physiol. Scand. 129, 277-286, with a conventional method using polyacrylamide gel electrophoresis. The new technique utilizes the Donnan potential, which develops over a semipermeable membrane that separates the non-permeating protein from the surrounding bath of the same ionic composition as the protein solution, to determine the net valency. The advantages of this method, besides its simplicity, are that it can determine the charge of, e.g., a protein in a free-fluid phase and that the pH and ionic composition of the bathing fluid can be varied over a broad range. The Donnan potential decreased to half its original value when the ionic strength was doubled. Usually a protein concentration of 1-10 mg.ml-1 must be used. The Donnan potential method was applied to determine the net charges of a series of proteins with different isoelectric points. The values showed close agreement with the data obtained by gel electrophoresis.