The Double Fixed Charge Membrane: Solution-Membrane Ion Partition Effects and Membrane Potentials

An analysis is made of the effect of solution-membrane partition of ions on the electrostatic potential and ion concentration profiles in fixed charge membranes. It is shown that the inclusion of partition effects gives rise to large solution-membrane “Donnan” potentials even when the concentration of fixed charges is of the same order as the concentration of the external solution. This effect renders the system and the simplified analysis of the double fixed charge membrane (FCM) previously given more applicable to biological membranes. An analysis is also given of the voltage dependence of the fluxes of individual ion species in the double FCM when it separates different ionic solutions and an expression is deduced for the membrane resting potential. Although the latter is similar in form to the Goldman-Hodgkin-Katz (GHK) equation the corresponding value of the permeability ratio P_C1/P_K is under certain specified conditions both concentration and potential dependent.     

On the relation between displacement currents and activation of the sodium conductance in the squid giant axon.

The early time course of the current passing across the membrane in squid giant axons in which the ionic currents have been blocked reveals substantial asymmetries during and after the application of hyperpolarizing and depolarizing voltage-clamp pulses of identical size. Since the integral of the 'on' and 'off' current transients is zero, these currents must result from charge movements confined to the membrane and, therefore, they are nonlinear displacement currents. The steady state rearrangement of the charges as a consequence of sudden displacements of the membrane potential is consistent with a Boltzmann distribution of charges between two states characterized by different energy levels. Following changes in membrane potential the charges undergo a first order transition between these states. The relaxation time constant for the transition at a given temperature is a function of membrane potential. We propose that these displacement currents arise from a redistribution of the charges involved in the sodium gating system.     

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.     

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.     

Periaxonal surface calcium binding and distribution of charge on the faces of squid axon potassium channel molecules

Summary The calcium binding constant associated with external surface charge in a position to influence the voltage sensing charges for potassium channel gating appears to be 30 molar^–1, a value much larger than previously thought and in approximate agreement with that found for artificial membranes composed of the lipid brain phosphatidylserene. Fixed charge on the periaxonal membrane surface is distributed in such a way that much larger charges occur at a distance of at least 8 angstroms from the channel pore openings. The separation between the ion pathway and the channel gating charge appears to be greater than or equal to 8 angstroms. Periaxonal surface charge which is in a position to determine the surface potential for gating has a magnitude greater than or equal to one (negative) electronic charge per 182 square angstrom before calcium binding, which is reduced to –e/625 Å in a normal divalent ionic environment. With the normal divalent ionic composition of seawater the surface potential at a position to influence the gating voltage sensor is –15 millivolts relative to the bulk external potential. The external surface potential is –3 mV at the pore mouth. There appears to be a negligible amount of fixed charge on the axoplasmic surface in the vicinity of the ion channel opening. Further, our results confirm earlier measurements that have given a negligible amount of axoplasmic surface fixed charge whose field components would be in a position to influence the channel gating charges.     

Modeling the Electric Potential across Neuronal Membranes: The Effect of Fixed Charges on Spinal Ganglion Neurons and Neuroblastoma Cells

We present a model for the electric potential profile across the membranes of neuronal cells. We considered the resting and action potential states, and analyzed the influence of fixed charges of the membrane on its electric potential, based on experimental values of membrane properties of the spinal ganglion neuron and the neuroblastoma cell. The spinal ganglion neuron represents a healthy neuron, and the neuroblastoma cell, which is tumorous, represents a pathological neuron. We numerically solved the non-linear Poisson-Boltzmann equation for the regions of the membrane model we have adopted, by considering the densities of charges dissolved in an electrolytic solution and fixed on both glycocalyx and cytoplasmic proteins. Our model predicts that there is a difference in the behavior of the electric potential profiles of the two types of cells, in response to changes in charge concentrations in the membrane. Our results also describe an insensitivity of the neuroblastoma cell membrane, as observed in some biological experiments. This electrical property may be responsible for the low pharmacological response of the neuroblastoma to certain chemotherapeutic treatments.     

Effect of buried potential barrier in label-less electrochemical immunodetection of glutathione and glutathione-capped gold nanoparticles

The influence of potential barriers, introduced to the immunoglobulin-based sensory films, on voltammetric signals of a redox ion probe has been investigated. Films with positive and negative barriers have been examined by depositing charged self-assembled thiol monolayers as the basal layers of a sensory film. The studies performed with monoclonal anti-glutathione antibody-based sensors using ferricyanide ion probe have shown stronger sensor response to the layer components, as well as to the glutathione-capped gold nanoparticles acting as the antigen, for films with positive potential barrier buried deep in the film than for negative barrier films. The larger changes in differential resistance, peak separation and peak heights observed for films with positive barrier have been attributed to different depth and width of the charge distributions in these films. A buried positive barrier with narrow charge distribution width provides the best conditions for film stability and prevents fouling (less ion-exchanges with the medium). This conclusion has been confirmed by calculations of the electric field distribution and potential profiles in immunosensing films performed by numerical integration of Poisson equation for Gaussian distributions of fixed charges of covalently bound components. The proposed fixed-charge model can aid in rapid evaluation of sensory films in sensor development work. The implications of potential barriers in sensory film design are discussed.     

An Analysis of the Surface Fixed-Charge Theory of the Squid Giant Axon Membrane

The observed shift in threshold potential, after perfusion of the squid giant axon with solutions of low ionic strength, can be predicted by assuming a fixed negative charge on the inside of the membrane. The constant field equation, together with the double-layer potential due to this charge, has been used to determine the change in resting potential during perfusion with solutions of low ionic strength. Neither the modified constant field equation nor Planck's diffusion equation can successfully predict the observed shift in resting potential. It is suggested that a positive charge distribution exists about the sodium channel on the outside of the membrane. The double-layer potential due to this positive charge, together with the independence principle, has been used to predict the relationship between sodium current and membrane potential when the ionic strength and sodium activity of the external solution are decreased. These predictions have been compared with the available experimental observations.     

How does vestibule surface charge affect ion conduction and toxin binding in a sodium channel?

We describe various models for the dielectric geometry and pore mouth charge distribution of a Na channel. The electric potential due to the vestibule charges is then computed on the basis of the nonlinear Possion-Boltzmann equation. The results are used to account for the effect of permeant ion concentration and ionic strength on channel conductance and on toxin association rate constants for Na channels. We find that a single negatively charged group near the entrance to the channel constriction is adequate to account for deviations from Michaelis-Menten conductance kinetics and for the concentration dependence of toxin-binding coefficients. We find further that only a limited range of vestibule geometries and pore mouth charge distributions are consistent with experiment.     

Energization-induced redistribution of charge carriers near membranes

The electric field arising from proton pumping across a topologically closed biological membrane causes accumulation close to the membrane of ionic charges equivalent to the charge of the pumped protons, positive on the side towards which protons are pumped, negative on the other side. We shall call this the 'active surface charge'. We here use the Poisson-Boltzmann equation to evaluate the effects of zwitterionic buffer molecules and uncharged proteins in the aqueous phase bordering the membrane on the magnitude and ionic composition of the active surface charge. For the positive side of the membrane, the main results are: (1) If the membrane is freely accessible to bulk phase ions, pumped protons exchange with these ions, such that the active surface charge consists of salt cations. (2) If a significant fraction of the ions in bulk solution consists of buffer molecules, then some of the pumped protons will remain close to the membrane and constitute a major fraction of the active surface charge. (3) If a protein layer borders the membrane, a significant part of the transmembrane electric potential difference exists within that protein layer and protons inside this layer dominate the active surface charge. (4) On the negative side of the membrane the corresponding phenomena would occur. (5) All these effects are strictly dependent on the transmembrane electric potential difference arising from proton pumping and would come in addition to the well known effects of buffers and electrically charged proteins on the retention of scalar protons. (6) No additional proton diffusion barrier may be required to account for a deficit in number of protons observed in the aqueous bulk phase upon aeration-induced proton pumping.     

A theory for the effects of neutral carriers such as the macrotetralide actin antibiotics on the electric properties of bilayer membranes

Summary To develop a quantitiative theoretical treatment for the effects of neutral macrocyclic antibiotics on the electrical properties of phospholipid bilayer membranes, this paper proceeds from the known ability of such molecules to form stoichiometric, lipid-soluble complexes with cations and deduces the electrical properties that a simple organic solvent phase would have if it were made into a membrane of the thinness of the phospholipid bilayer. In effect, we postulate that the essential barrier to ion movement across a bilayer membrane is its liquid-like hydrocarbon interior and that the neutral macrocyclic antibiotics bind monovalent cations and solubilize them in the membrane as mobile positively charged complexes. Using the Poisson-Boltzmann equation to describe the equilibrium profile of the electrical potential, it is shown that an excess of the positive complexes over all the other ions is expected in the membrane as a net space charge for appropriate conditions of membrane thickness and values of the partition coefficients of the various ionic species and without requiring the presence of fixed charges. Describing the fluxes of these complexes by the Nernst-Planck equation and neglecting the contribution to the electric current of uncomplexed ions, theoretical expressions are derived for the membrane potential in ionic mixtures, as well as for the limiting value of the membrane conductance at zero current when the membrane is interposed between identical solutions. The expressions are given in terms of the ionic activities and antibiotic concentrations in the aqueous solutions so as to be accessible to direct experimental test. Under suitable experimental conditions, the membrane potential is described by an equation recognizible as the Goldman-Hodgkin-Katz equation, in which the permeability ratios are combinations of parameters predicted from the present theory to be independently determinable from the ratio of membrane conductances in single salt solutions. Since this identity between permeability and conductance ratios is expected also for systems obeying the Independence Principle of Hodgkin and Huxley, the applicability of this principle to membranes exposed to antibiotics is discussed, and it is shown that this principle is compatible with the permeation mechanism proposed here.     

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.     

pH dependence of electrostatic interaction between ion-penetrable membranes

A theory on the electrostatic repulsion between ion-penetrable membranes proposed previously by us is extended by taking into account the degree of dissociation of the membrane-fixed ionizable groups. A system of equations which determines the pH dependence of the membrane interaction is presented. The density of membrane-fixed charges is consistently determined as a function of the electric potential so that both the membrane-fixed charge density and the potential are not constant but functions of the membrane separation. The pH at the surface of interacting membranes is also calculated as a function of the membrane separation.     

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.     

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.     

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.     

How pore mouth charge distributions alter the permeability of transmembrane ionic channels.

This paper investigates the effects that surface dipole layers and surface charge layers along the pore mouth-water interface can have on the electrical properties of a transmembrane channel. Three specific molecular sources are considered: dipole layers formed by membrane phospholipids, dipole layers lining the mouth of a channel-forming protein, and charged groups in the mouth of a channel-forming protein. We find, consistent with previous work, that changing the lipid-water potential difference only influences channel conduction if the rate-limiting step takes place well inside the channel constriction. We find that either mouth dipoles or mouth charges can act as powerful ion attractors increasing either cation or anion concentration near the channel entrance to many times its bulk value, especially at low ionic strengths. The effects are sufficient to reconcile the apparently contradictory properties of high selectivity and high conductivity, observed for a number of K+ channel systems. We find that localizing the electrical sources closer to the constriction entrance substantially increases their effectiveness as ion attractors; this phenomenon is especially marked for dipolar distributions. An approximate treatment of electrolyte shielding is used to discriminate between the various mechanisms for increasing ionic concentration near the constriction entrance. Dipolar potentials are far less sensitive to ionic strength variation than potentials due to fixed charges. We suggest that the K+ channel from sarcoplasmic reticulum does not have a fixed negative charge near the constriction entrance; we suggest further that the Ca+2-activated K+ channel from transverse tubule does have such a charge.     

A Few Comments on Electrostatic Interactions in Cell Physiology

The role of fixed charges present at the surface of biological membranes is usually described by the Gouy-Chapman-Grahame theory of the electric double-layer where the Grahame equation is applied independently on each side of the membrane and where the capacitive charges (linked to the transmembrane ionic currents) are disregarded. In this article, we generalize the Gouy-Chapman-Grahame theory by taking into account both intrinsic charges (resulting from the dissociation of membrane constituents) and capacitive charges, in the density value of the membrane surface charges. In the first part, we show that capacitive charges couple electrostatic potentials present on both sides of the membrane. The intensity of this coupling depends both on the value of the membrane specific capacitance and the transmembrane electric potential difference. In the second part, we suggest some physiological implications of membrane electric double-layers.     

Electrostatic Potential of B-DNA: Effect of Interionic Correlations

Modified Poisson-Boltzmann (MPB) equations have been numerically solved to study ionic distributions and mean electrostatic potentials around a macromolecule of arbitrarily complex shape and charge distribution. Results for DNA are compared with those obtained by classical Poisson-Boltzmann (PB) calculations. The comparisons were made for 1:1 and 2:1 electrolytes at ionic strengths up to 1 M. It is found that ion-image charge interactions and interionic correlations, which are neglected by the PB equation, have relatively weak effects on the electrostatic potential at charged groups of the DNA. The PB equation predicts errors in the long-range electrostatic part of the free energy that are only ∼1.5 kJ/mol per nucleotide even in the case of an asymmetrical electrolyte. In contrast, the spatial correlations between ions drastically affect the electrostatic potential at significant separations from the macromolecule leading to a clearly predicted effect of charge overneutralization.     

Lipid demixing and protein-protein interactions in the adsorption of charged proteins on mixed membranes

The adsorption free energy of charged proteins on mixed membranes, containing varying amounts of (oppositely) charged lipids, is calculated based on a mean-field free energy expression that accounts explicitly for the ability of the lipids to demix locally, and for lateral interactions between the adsorbed proteins. Minimization of this free energy functional yields the familiar nonlinear Poisson-Boltzmann equation and the boundary condition at the membrane surface that allows for lipid charge rearrangement. These two self-consistent equations are solved simultaneously. The proteins are modeled as uniformly charged spheres and the (bare) membrane as an ideal two-dimensional binary mixture of charged and neutral lipids. Substantial variations in the lipid charge density profiles are found when highly charged proteins adsorb on weakly charged membranes; the lipids, at a certain demixing entropy penalty, adjust their concentration in the vicinity of the adsorbed protein to achieve optimal charge matching. Lateral repulsive interactions between the adsorbed proteins affect the lipid modulation profile and, at high densities, result in substantial lowering of the binding energy. Adsorption isotherms demonstrating the importance of lipid mobility and protein-protein interactions are calculated using an adsorption equation with a coverage-dependent binding constant. Typically, at bulk-surface equilibrium (i.e., when the membrane surface is "saturated" by adsorbed proteins), the membrane charges are "overcompensated" by the protein charges, because only about half of the protein charges (those on the hemispheres facing the membrane) are involved in charge neutralization. Finally, it is argued that the formation of lipid-protein domains may be enhanced by electrostatic adsorption of proteins, but its origin (e.g., elastic deformations associated with lipid demixing) is not purely electrostatic.