Three-dimensional Poisson-Nernst-Planck theory studies: influence of membrane electrostatics on gramicidin A channel conductance

A recently introduced real-space lattice methodology for solving the three-dimensional Poisson-Nernst-Planck equations is used to compute current-voltage relations for ion permeation through the gramicidin A ion channel embedded in membranes characterized by surface dipoles and/or surface charge. Comparisons to a variety of experimental results, presented herein, have proven largely successful. Strengths and weaknesses of the method are discussed.     

(In)validity of the constant field and constant currents assumptions in theories of ion transport.

Constant electric fields and constant ion currents are often considered in theories of ion transport. Therefore, it is important to understand the validity of these helpful concepts. The constant field assumption requires that the charge density of permeant ions and flexible polar groups is virtually voltage independent. We present analytic relations that indicate the conditions under which the constant field approximation applies. Barrier models are frequently fitted to experimental current-voltage curves to describe ion transport. These models are based on three fundamental characteristics: a constant electric field, negligible concerted motions of ions inside the channel (an ion can enter only an empty site), and concentration-independent energy profiles. An analysis of those fundamental assumptions of barrier models shows that those approximations require large barriers because the electrostatic interaction is strong and has a long range. In the constant currents assumption, the current of each permeating ion species is considered to be constant throughout the channel; thus ion pairing is explicitly ignored. In inhomogeneous steady-state systems, the association rate constant determines the strength of ion pairing. Among permeable ions, however, the ion association rate constants are not small, according to modern diffusion-limited reaction rate theories. A mathematical formulation of a constant currents condition indicates that ion pairing very likely has an effect but does not dominate ion transport.     

A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transport through the gramicidin A channel.

A lattice relaxation algorithm is developed to solve the Poisson-Nernst-Planck (PNP) equations for ion transport through arbitrary three-dimensional volumes. Calculations of systems characterized by simple parallel plate and cylindrical pore geometries are presented in order to calibrate the accuracy of the method. A study of ion transport through gramicidin A dimer is carried out within this PNP framework. Good agreement with experimental measurements is obtained. Strengths and weaknesses of the PNP approach are discussed.     

Magnitude and location of surface charges on Myxicola giant axons

The effects of changes in the concentration of calcium in solutions bathing Myxicola giant axons on the voltage dependence of sodium and potassium conductance and on the instantaneous sodium and potassium current-voltage relations have been measured. The sodium conductance-voltage relation is shifted along the voltage axis by 13 mV in the hyperpolarizing direction for a fourfold decrease in calcium concentration. The potassium conductance-voltage relation is shifted only half as much as that for sodium. There is no effect on the shape of the sodium and potassium instantaneous current-voltage curves: the normal constant-field rectification of potassium currents is maintained and the normal linear relationship of sodium currents is maintained. Considering that shifts in conductances would reflect the presence of surface charges near the gating machinery and that shape changes of instantaneous current-voltage curves would reflect the presence of surface charges near the ionic pores, these results indicate a negative surface charge density of about 1 electronic charge per 120 A2 near the sodium gating machinery, about 1 e/300 A2 for the potassium gating machinery, and much less surface charge near the sodium or potassium pores. There may be some specific binding of calcium to these surface charges with an upper limit on the binding constant of about 0.2 M-1. The differences in surface charge density suggest a spatial separation for these four membrane components.     

The inner membrane barrier of lipid membranes experienced by the valinomycin/Rb+ complex: charge pulse experiments at high membrane voltages.

The kinetic analysis of charge pulse experiments at planar lipid membranes in the presence of macrocyclic ion carriers has been limited so far to the low voltage range, where, under certain simplifying conditions, an analytical solution is available. In the present study, initial voltages of up to 300 mV were applied to the membrane, and the voltage decay through the conductive pathways of the membrane was followed as a function of time. The system of differential equations derived from the transport model was solved numerically and was compared with the experimental data. The generalized kinetic analysis of charge pulse experiments and of steady-state current-voltage curves was used to study the voltage dependence of the individual transport steps and to obtain information on the shape of the inner membrane barrier. The data were found to be consistent with a comparatively broad inner barrier such as a trapezoidal barrier or an image force barrier. The inner barrier was found to sense 70-76% of the voltage applied to the membrane. As a consequence, 24-30% of the voltage acts on the two interfacial barriers between membrane and water. The data refer to membranes formed from monoolein, monoeicosenoin, or monoerucin in n-decane.     

Constant fields and constant gradients in open ionic channels.

Ions enter cells through pores in proteins that are holes in dielectrics. The energy of interaction between ion and charge induced on the dielectric is many kT, and so the dielectric properties of channel and pore are important. We describe ionic movement by (three-dimensional) Nemst-Planck equations (including flux and net charge). Potential is described by Poisson's equation in the pore and Laplace's equation in the channel wall, allowing induced but not permanent charge. Asymptotic expansions are constructed exploiting the long narrow shape of the pore and the relatively high dielectric constant of the pore's contents. The resulting one-dimensional equations can be integrated numerically; they can be analyzed when channels are short or long (compared with the Debye length). Traditional constant field equations are derived if the induced charge is small, e.g., if the channel is short or if the total concentration gradient is zero. A constant gradient of concentration is derived if the channel is long. Plots directly comparable to experiments are given of current vs voltage, reversal potential vs. concentration, and slope conductance vs. concentration. This dielectric theory can easily be tested: its parameters can be determined by traditional constant field measurements. The dielectric theory then predicts current-voltage relations quite different from constant field, usually more linear, when gradients of total concentration are imposed. Numerical analysis shows that the interaction of ion and channel can be described by a mean potential if, but only if, the induced charge is negligible, that is to say, the electric field is spatially constant.     

Permeation through an open channel: Poisson-Nernst-Planck theory of a synthetic ionic channel.

The synthetic channel [acetyl-(LeuSerSerLeuLeuSerLeu)3-CONH2]6 (pore diameter approximately 8 A, length approximately 30 A) is a bundle of six alpha-helices with blocked termini. This simple channel has complex properties, which are difficult to explain, even qualitatively, by traditional theories: its single-channel currents rectify in symmetrical solutions and its selectivity (defined by reversal potential) is a sensitive function of bathing solution. These complex properties can be fit quantitatively if the channel has fixed charge at its ends, forming a kind of macrodipole, bracketing a central charged region, and the shielding of the fixed charges is described by the Poisson-Nernst-Planck (PNP) equations. PNP fits current voltage relations measured in 15 solutions with an r.m.s. error of 3.6% using four adjustable parameters: the diffusion coefficients in the channel's pore DK = 2.1 x 10(-6) and DCl = 2.6 x 10(-7) cm2/s; and the fixed charge at the ends of the channel of +/- 0.12e (with unequal densities 0.71 M = 0.021e/A on the N-side and -1.9 M = -0.058e/A on the C-side). The fixed charge in the central region is 0.31e (with density P2 = 0.47 M = 0.014e/A). In contrast to traditional theories, PNP computes the electric field in the open channel from all of the charges in the system, by a rapid and accurate numerical procedure. In essence, PNP is a theory of the shielding of fixed (i.e., permanent) charge of the channel by mobile charge and by the ionic atmosphere in and near the channel's pore. The theory fits a wide range of data because the ionic contents and potential profile in the channel change significantly with experimental conditions, as they must, if the channel simultaneously satisfies the Poisson and Nernst-Planck equations and boundary conditions. Qualitatively speaking, the theory shows that small changes in the ionic atmosphere of the channel (i.e., shielding) make big changes in the potential profile and even bigger changes in flux, because potential is a sensitive function of charge and shielding, and flux is an exponential function of potential.     

Use of current-voltage diagrams in locating peak energy barriers in cell membranes

Summary The current-voltage relations obtained by integrating the Nernst-Planck equations for a variety of energy profiles are obtained. A simple and approximate method for comparing these relations is described. The method is based on using a linearized transform of current-voltage relations for an Eyring single barrier model. A parameter, , related to the location of the single barrier in the Eyring model, and to the shape of the barrier in other models, is readily obtained from the slopes of the linearized relations. It is then a simple matter to determine whether a given current-voltage relation allows discrimination between any particular energy profiles. The results show that the equivalent Eyring model does not always place the peak energy barrier in the same position as other models and that quite large errors in the assignment of position may be made if such a model is used. The results are also used to test the ability of some experimental current-voltage diagrams to discriminate between various energy profiles.     

Steady-state electrodiffusion. Scaling, exact solution for ions of one charge, and the phase plane.

This is the first of two papers dealing with electrodiffusion theory (the Nernst-Planck equation coupled with Gauss's law) and its application to the current-voltage behavior of squid axon. New developments in the exact analysis of the steady-state electrodiffusion problem presented here include (a) a scale transformation that connects a given solution to an infinity of other solutions, suggesting the po-sibility of direct comparison of electrical data for membranes with different thicknesses and other properties; (b) a first-integral relation between the electric field and ion densities more general than analogous relations previously reported, and (c) an exact solution for the homovalent system, i.e., a membrane system permeated by various ion species of the same charge. The latter is a generalization of the known one-ion solution. The properties of the homovalent solution are investigated analytically and graphically. In particular we study the phase-plane curves, which reduce to the parabolas discussed by K. S. Cole in the special case in which the current-density parameter (a linear combination of the ionic current densities) is zero.     

Imitation, internal absorption and the reversal of local drift in stochastic evolutionary games

Evolutionary game dynamics in finite populations is typically subject to noise, inducing effects which are not present in deterministic systems, including fixation and extinction. In the first part of this paper we investigate the phenomenon of drift reversal in finite populations, taking into account that drift is a local quantity in strategy space. Secondly, we study a simple imitation dynamics, and show that it can lead to fixation at internal mixed-strategy fixed points even in finite populations. Imitation in infinite populations is adequately described by conventional replicator dynamics, and these equations are known to have internal fixed points. Internal absorption in finite populations on the other hand is a novel dynamic phenomenon. Due to an outward drift in finite populations this type of dynamic arrest is not found in other commonly studied microscopic dynamics, not even in those with the same deterministic replicator limit as imitation.     

Charges, currents, and potentials in ionic channels of one conformation.

Flux through an open ionic channel is analyzed with Poisson-Nernst-Planck (PNP) theory. The channel protein is described as an unchanging but nonuniform distribution of permanent charge, the charge distribution observed (in principle) in x-ray diffraction. Appropriate boundary conditions are derived and presented in some generality. Three kinds of charge are present: (a) permanent charge on the atoms of the protein, the charge independent of the electric field; (b) free or mobile charge, carried by ions in the pore as they flux through the channel; and (c) induced (sometimes called polarization) charge, in the pore and protein, created by the electric field, zero when the electric field is zero. The permanent charge produces an offset in potential, a built-in Donnan potential at both ends of the channel pore. The system is completely solved for bathing solutions of two ions. Graphs describe the distribution of potential, concentration, free (i.e., mobile) and induced charge, and the potential energy associated with the concentration of charge, as well as the unidirectional flux as a function of concentration of ions in the bath, for a distribution of permanent charge that is uniform. The model shows surprising complexity, exhibiting some (but not all) of the properties usually attributed to single filing and exchange diffusion. The complexity arises because the arrangement of free and induced charge, and thus of potential and potential energy, varies, sometimes substantially, as conditions change, even though the channel structure and conformation (of permanent charge) is strictly constant. Energy barriers and wells, and the concomitant binding sites and binding phenomena, are outputs of the PNP theory: they are computed, not assumed. They vary in size and location as experimental conditions change, while the conformation of permanent charge remains constant, thus giving the model much of its interesting behavior.     

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

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

Calculation of effective diffusivities for biofilms and tissues.

In this study we describe a scheme for numerically calculating the effective diffusivity of cellular systems such as biofilms and tissues. This work extends previous studies in which we developed the macroscale representations of the transport equations for cellular systems based on the subcellular-scale transport and reaction processes. A finite-difference model is used to predict the effective diffusivity of a cellular system on the basis of the subcellular-scale geometry and transport parameters. The effective diffusivity is predicted for a complex three-dimensional structure that is based on laboratory observations of a biofilm, and these numerical predictions are compared with predictions from a simple analytical solution and with experimental data. Our results indicate that, under many practical circumstances, the simple analytical solution can be used to provide reasonable estimates of the effective diffusivity.     

Interpretation of steady-state current-voltage curves: Consequences and implications of current subtraction in transport studies

Summary A problem often confronted in analyses of chargecarrying transport processesin vivo lies in identifying porterspecific component currents and their dependence on membrane potential. Frequently, current-voltage (I–V)—or more precisely, difference-current-voltage (dI-V)—relations, both for primary and for secondary transport processes, have been extracted from the overall membrane current-voltage profiles by subtracting currents measured before and after experimental manipulations expected to alter the porter characteristics only. This paper examines the consequences of current subtraction within the context of a generalized kinetic carrier model for Class I transport mechanisms (U.-P. Hansen, D. Gradmann, D. Sanders and C.L. Slayman, 1981,J. Membrane Biol. 63:165–190). Attention is focused primarily ondI-V profiles associated with ion-driven secondary transport for which external solute concentrations usually serve as the experimental variable, but precisely analogous results and the same conclusions are indicated in relation to studies of primary electrogenesis. The model comprises a single transport loop linkingn (3 or more) discrete states of a carrier molecule. State transitions include one membrane chargetransport step and one solute-binding step. Fundamental properties ofdI-V relations are derived analytically for alln-state formulations by analogy to common experimental designs. Additional features are revealed through analysis of a reduced 2-state empirical form, and numerical examples, computed using this and a minimum 4-state formulation, illustratedI-V curves under principle limiting conditions. Class I models generate a wide range ofdI-V profiles which can accommodate essentially all of the data now extant for primary and secondary transport systems, including difference current relations showing regions of negative slope conductance. The particular features exhibited by the curves depend on the relative magnitudes and orderings of reaction rate constants within the transport loop. Two distinct classes ofdI-V curves result which reflect the relative rates of membrane charge transit and carrier recycling steps. Also evident in difference current relations are contributions from hidden carrier states not directly associated with charge translocation in circumstances which can give rise to observations of counterflow or exchange diffusion. Conductance-voltage relations provide a semi-quantitative means to obtaining pairs of empirical rate parameters. It is demonstrated thatdI-V relationscannot yield directly meaningful transport reversal potentials in most common experimental situations. Well-defined arramgements of reaction constants are shown to givedI-V curves which exhibit little or no voltage sensitivity and finite currents over many tens to hundreds of millivoltsincluding the true reversal potential. Furthermore, difference currents show apparent Michaelian kinetics with solute concentration atall membrane potentials. These findings bring into question several previous reports of reversal potentials, stoichiometries and apparent current-source behavior based primarily on difference current analysis. They also provide a coherent explanation for anomolous and shallow conductances and paradoxical situations in which charge stoichiometry varies with membrane potential.     

Effect of ion concentration,solution and membrane permittivity on electric energy storage and capacitance

Bio-membranes as capacitors store electric energy, but their permittivity is low whereas the permittivity of surrounding solution is high. To evaluate the effective capacitance of the membrane/solution system and determine the electric energy stored within the membrane and in the solution, we estimated their electric variables using Poisson-Nernst-Planck simulations. We calculated membrane and solution capacitances from stored electric energy. The effective capacitance was calculated by fitting a six-capacitance model to charges (fixed and ion) and associated potentials, because it cannot be considered as a result of membrane and solution capacitance in series. The electric energy stored within the membrane (typically much smaller than that in the solution), depends on the membrane permittivity, but also on the external electric field, surface charge density, water permittivity and ion concentration. The effect on capacitances is more specific. Solution capacitance rises with greater solution permittivity or ion concentration, but the membrane capacitance (much smaller than solution capacitance) is only influenced by its permittivity. Interestingly, the effective capacitance is independent of membrane or solution permittivity, but rises as the ion concentration increases and surface charge becomes positive. Experimental estimates of membrane capacitance are thus not necessarily a reliable index of its surface area.     

Intrinsic rectification of ion flux in alamethicin channels: studies with an alamethicin dimer.

Covalent dimers of alamethicin form conducting structures with gating properties that permit measurement of current-voltage (I-V) relationships during the lifetime of a single channel. These I-V curves demonstrate that the alamethicin channel is a rectifier that passes current preferentially, with voltages of the same sign as that of the voltage that induced opening of the channel. The degree of rectification depends on the salt concentration; single-channel I-V relationships become almost linear in 3 M potassium chloride. These properties may be qualitatively understood by using Poisson-Nernst-Planck theory and a modeled structure of the alamethicin pore.     

Transmembrane molecular transport during versus after extremely large, nanosecond electric pulses

Recently there has been intense and growing interest in the non-thermal biological effects of nanosecond electric pulses, particularly apoptosis induction. These effects have been hypothesized to result from the widespread creation of small, lipidic pores in the plasma and organelle membranes of cells (supra-electroporation) and, more specifically, ionic and molecular transport through these pores. Here we show that transport occurs overwhelmingly after pulsing. First, we show that the electrical drift distance for typical charged solutes during nanosecond pulses (up to 100 ns), even those with very large magnitudes (up to 10 MV/m), ranges from only a fraction of the membrane thickness (5 nm) to several membrane thicknesses. This is much smaller than the diameter of a typical cell (∼16 μm), which implies that molecular drift transport during nanosecond pulses is necessarily minimal. This implication is not dependent on assumptions about pore density or the molecular flux through pores. Second, we show that molecular transport resulting from post-pulse diffusion through minimum-size pores is orders of magnitude larger than electrical drift-driven transport during nanosecond pulses. While field-assisted charge entry and the magnitude of flux favor transport during nanosecond pulses, these effects are too small to overcome the orders of magnitude more time available for post-pulse transport. Therefore, the basic conclusion that essentially all transmembrane molecular transport occurs post-pulse holds across the plausible range of relevant parameters. Our analysis shows that a primary direct consequence of nanosecond electric pulses is the creation (or maintenance) of large populations of small pores in cell membranes that govern post-pulse transmembrane transport of small ions and molecules.     

Computing numerically the access resistance of a pore

The access resistance (AR) of a channel is an important component of the conductance of ion channels, particularly in wide and short channels, where it accounts for a substantial fraction of the total resistance to the movement of ions. The AR is usually calculated by using a classical and simple expression derived by Hall from electrostatics (J.E. Hall 1975 J. Gen. Phys. 66:531-532), though other expressions, both analytical and numerical, have been proposed. Here we report some numerical results for the AR of a channel obtained by solving the Poisson-Nernst-Planck equations at the entrance of a circular pore. Agreement is found between numerical calculations and analytical results from Hall's equation for uncharged pores in neutral membranes. However, for channels embedded in charged membranes, Hall's expression overestimates the AR, which is much lower and can even be neglected in some cases. The weak dependence of AR on the pore radius for charged membranes at low salt concentration can be exploited to separate the channel and the access contributions to the measured conductance.     

Doping‐Induced Screening of the Built‐in‐Field in Organic Solar Cells: Effect on Charge Transport and Recombination

We report on the effects of screening of the electric field by doping‐induced mobile charges on photocurrent collection in operational organic solar cells. Charge transport and recombination were studied using double injection (DI) and charge extraction by linearly increasing voltage (CELIV) transient techniques in bulk‐heterojunction solar cells made from acceptor‐donor blends of poly(3‐n‐hexylthiophene):phenyl‐C61‐butyric acid methyl ester (P3HT:PC60BM). It is shown that the screening of the built‐in field in operational solar cells can be controlled by an external voltage while the influence on charge transport and recombination is measured. An analytical theory to extract the bimolecular recombination coefficient as a function of electric field from the injection current is also reported. The results demonstrate that the suppressed (non‐Langevin) bimolecular recombination rate and charge collection are not strongly affected by native doping levels in this materials combination. Hence, it is not necessary to reduce the level of doping further to improve the device performance of P3HT‐based solar cells.     

GAT1 (GABA:Na+:Cl-) cotransport function. Database reconstruction with an alternating access model.

We have developed an alternating access transport model that accounts well for GAT1 (GABA:Na+:Cl-) cotransport function in Xenopus oocyte membranes. To do so, many alternative models were fitted to a database on GAT1 function, and discrepancies were analyzed. The model assumes that GAT1 exists predominantly in two states, Ein and E(out). In the Ein state, one chloride and two sodium ions can bind sequentially from the cytoplasmic side. In the Eout state, one sodium ion is occluded within the transporter, and one chloride, one sodium, and one gamma-aminobutyric acid (GABA) molecule can bind from the extracellular side. When Ein sites are empty, a transition to the Eout state opens binding sites to the outside and occludes one extracellular sodium ion. This conformational change is the major electrogenic GAT1 reaction, and it rate-limits forward transport (i.e., GABA uptake) at 0 mV. From the Eout state, one GABA can be translocated with one sodium ion to the cytoplasmic side, thereby forming the *Ein state. Thereafter, an extracellular chloride ion can be translocated and the occluded sodium ion released to the cytoplasm, which returns the transporter to the Ein state. GABA-GABA exchange can occur in the absence of extracellular chloride, but a chloride ion must be transported to complete a forward transport cycle. In the reverse transport cycle, one cytoplasmic chloride ion binds first to the Ein state, followed by two sodium ions. One chloride ion and one sodium ion are occluded together, and thereafter the second sodium ion and GABA are occluded and translocated. The weak voltage dependence of these reactions determines the slopes of outward current-voltage relations. Experimental results that are simulated accurately include (a) all current-voltage relations, (b) all substrate dependencies described to date, (c) cis-cis and cis-trans substrate interactions, (d) charge movements in the absence of transport current, (e) dependencies of charge movement kinetics on substrate concentrations, (f) pre-steady state current transients in the presence of substrates, (g) substrate-induced capacitance changes, (h) GABA-GABA exchange, and (i) the existence of inward transport current and GABA-GABA exchange in the nominal absence of extracellular chloride.