Ion-conformational interaction and charge transport through channels of biological membranes

This work proposes a theory of charge transport through channels in biological membranes, based on ion flow interaction with charged groups of protein macromolecules that form the channel. Displacements of the groups are due to conformational changes of the protein molecule, the relaxation times of which are much larger than the average time of ion ocurrence in the channel. Ion flow is assumed to depend on the conformational changes and vice-versa. The resulting self-organizing ion-conformational system is described by a set of nonlinear differential equations for conformational variables and average occupancy of the channel by ions. The system exhibits multistable behaviour in a certain range of control parameters (potential difference, input ion flow). The stationary states of the system may be identified with the states of discrete conductivity of the ionic channels.     

Dependence of ion flow through channels on the density of fixed charges at the channel opening. Voltage control of inverse titration curves

Model calculations were done to investigate the effect of titratable fixed charges at channel openings on ion flow through open channels. The current titration curves (channel current vs. bulk pH) can assume the shape expected from the change of the ionic surface concentration with pH (c-control), or be inverted, i.e., follow the change of the electrical field within the membrane (V-control). The relationships were explored pars pro toto for Goldman-Hodgkin-Katz channels, two-barrier one-site channels and six-barrier five-site channels. With net current flowing in the direction of the concentration gradient and from the titrated fixed charge layer into the channel, c-control is the sign of low channel occupancy (entrance-step limitation) and V-control the sign of high channel occupancy (exit-step limitation). At intermediate occupancy, the current titration curve can be nearly invariant to pH.     

Application of the Poisson-Nernst-Planck theory with space-dependent diffusion coefficients to KcsA

The Poisson-Nernst-Planck electrodiffusion theory serves to compute charge fluxes and is here applied to the ion current through a protein channel. KcsA was selected as an example because of the abundance of experimental and theoretical data. The potassium channels MthK and KvAP were used as templates to define two open channel models for KcsA. Channel boundary surfaces and protein charge distributions were defined according to atomic radii and partial atomic charges. To establish the sensitivity of the results to these parameters, two different sets were used. Assigning the potassium diffusion coefficients equal to the value for free-diffusion in water (1.96 x 10(-9) m(2)/s), the computed currents overestimated the experimental data. Ion distributions inside the channel suggest that the overestimate is not due to an excess of charge shielding. A good agreement with the experimental data was achieved by reducing the potassium diffusion coefficient inside the channel to 1.96 x 10(-10) m(2)/s, a value of substantial motility but nonetheless in accord with the intuitive notion that the channel has a high affinity for the ions and therefore slows them down. These results are independent of the open channel model and the parameterization adopted for atomic radii and partial atomic charges. The method offers a reliable estimate of the channel current with low computational effort.     

Flux, coupling, and selectivity in ionic channels of one conformation.

Ions crossing biological membranes are described as a concentration of charge flowing through a selective open channel of one conformation and analyzed by a combination of Poisson and Nernst-Planck equations and boundary conditions, called the PNP theory for short. The ion fluxes in this theory interact much as ion fluxes interact in biological channels and mediated transporters, provided the theoretical channel contains permanent charge and has selectivity created by (electro-chemical) resistance at its ends. Interaction occurs because the flux of different ionic species depends on the same electric field. That electric field is a variable, changing with experimental conditions because the screening (i.e., shielding) of the permanent charge within the channel changes with experimental conditions. For example, the screening of charge and the shape of the electric field depend on the concentration of all ionic species on both sides of the channel. As experimental interventions vary the screening, the electric field varies, and thus the flux of each ionic species varies conjointly, and is, in that sense, coupled. Interdependence and interaction are the rule, independence is the exception, in this channel.     

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.     

A hydrodynamic theory of ion conductance through Ohmic pores

A method of calculating the size of membrane pores lacking strong ionic selectivity is presented. By treating the flow of ions through a small channel as a hydrodynamic phenomenon, the electrical conductance becomes a function of the ratio of ion radius to channel radius. Thus when both the channel conductance and the ion size are known, the radius of the channel may be estimated. The method gives good agreement among radii predicted from conductances of four different alkali cations in alamethicin pores.     

Tests of continuum theories as models of ion channels. II. Poisson-Nernst-Planck theory versus brownian dynamics

We test the validity of the mean-field approximation in Poisson-Nernst-Planck theory by contrasting its predictions with those of Brownian dynamics simulations in schematic cylindrical channels and in a realistic potassium channel. Equivalence of the two theories in bulk situations is demonstrated in a control study. In simple cylindrical channels, considerable differences are found between the two theories with regard to the concentration profiles in the channel and its conductance properties. These differences are at a maximum in narrow channels with a radius smaller than the Debye length and diminish with increasing radius. Convergence occurs when the channel radius is over 2 Debye lengths. These tests unequivocally demonstrate that the mean-field approximation in the Poisson-Nernst-Planck theory breaks down in narrow ion channels that have radii smaller than the Debye length.     

Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type calcium channels.

L-type Ca channels contain a cluster of four charged glutamate residues (EEEE locus), which seem essential for high Ca specificity. To understand how this highly charged structure might produce the currents and selectivity observed in this channel, a theory is needed that relates charge to current. We use an extended Poisson-Nernst-Planck (PNP2) theory to compute (mean) Coulombic interactions and thus to examine the role of the mean field electrostatic interactions in producing current and selectivity. The pore was modeled as a central cylinder with tapered atria; the cylinder (i.e., "pore proper") contained a uniform volume density of fixed charge equivalent to that of one to four carboxyl groups. The pore proper was assigned ion-specific, but spatially uniform, diffusion coefficients and excess chemical potentials. Thus electrostatic selection by valency was computed self-consistently, and selection by other features was also allowed. The five external parameters needed for a system of four ionic species (Na, Ca, Cl, and H) were determined analytically from published measurements of thre limiting conductances and two critical ion concentrations, while treating the pore as a macroscopic ion-exchange system in equilibrium with a uniform bath solution. The extended PNP equations were solved with these parameters, and the predictions were compared to currents measured in a variety of solutions over a range of transmembrane voltages. The extended PNP theory accurately predicted current-voltage relations, anomalous mole fraction effects in the observed current, saturation effects of varied Ca and Na concentrations, and block by protons. Pore geometry, dielectric permittivity, and the number of carboxyl groups had only weak effects. The successful prediction of Ca fluxes in this paper demonstrates that ad hoc electrostatic parameters, multiple discrete binding sites, and logistic assumptions of single-file movement are all unnecessary for the prediction of permeation in Ca channels over a wide range of conditions. Further work is needed, however, to understand the atomic origin of the fixed charge, excess chemical potentials, and diffusion coefficients of the channel. The Appendix uses PNP2 theory to predict ionic currents for published "barrier-and-well" energy profiles of this channel.     

Surface potentials measure ion concentrations near lipid bilayers during rapid solution changes.

We describe a puffing method for changing solutions near one surface of lipid bilayers that allows simultaneous measurement of channel activity and extent of solution change at the bilayer surface. Ion adsorption to the lipid headgroups and screening of the bilayer surface charge by mobile ions provided a convenient probe for the ionic composition of the solution at the bilayer surface. Rapid ionic changes induced a shift in bilayer surface potential that generated a capacitive transient current under voltage-clamp conditions. This depended on the ion species and bilayer composition and was accurately described by the Stern-Gouy-Chapman theory. The time course of solute concentrations during solution changes could also be modeled by an exponential exchange of bath and puffing solutions with time constants ranging from 20 to 110 ms depending on the flow pressure. During changes in [Cs+] and [Ca2+] (applied separately or together) both the mixing model and capacitive currents predicted [Cs+] and [Ca2+] transients consistent with those determined experimentally from: 1) the known Cs(+)-dependent conductance of open ryanodine receptor channels and 2) the Ca(2+)-dependent gating of ryanodine receptor Ca2+ channels from cardiac and skeletal muscle.     

Anomalous mole fraction effect, electrostatics, and binding in ionic channels.

Ionic channels bathed in mixed solutions of two permeant electrolytes often conduct less current than channels bathed in pure solutions of either. For many years, this anomalous mole fraction effect (AMFE) has been thought to occur only in single-file pores containing two or more ions at a time. Most thinking about channels incorporates this view. We show here that the AMFE arises naturally, as an electrostatic consequence of localized ion specific binding, if the average current through a channel is described by a theory (Poisson-Nernst-Planck, PNP) that computes the average electric field from the average concentration of charges in and near the channel. The theory contains only those ion-ion interactions mediated by the mean field, and it does not enforce single filing. The AMFE is predicted by PNP over a wide range of mean concentrations of ions in the channel; for example, it is predicted when (on the average) less, or much less, than one ion is found in the channel's pore. In this treatment, the AMFE arises, in large measure, from a depletion layer produced near a region of ion-specific binding. The small excess concentration of ions in the binding region repels all nearby ions of like charge, thereby creating a depletion layer. The overall conductance of the channel arises in effect from resistors in series, one from the binding region, one from the depletion zone, and one from the unbinding region. The highest value resistor (which occurs in the depletion zone) limits the overall series conductance. Here the AMFE is not the result of single filing or multiple occupancy, and so previous views of permeation need to be revised: the presence of an AMFE does not imply that ions permeate single file through a multiply occupied pore.     

Linoleic acid inhibits TRP channels with intrinsic voltage sensitivity: Implications on the mechanism of linoleic acid action

Open channel block (OCB) is a process by which ions bind to the inside of a channel pore and block the flow of ions through that channel. Repulsion of the blocking ions by membrane depolarization is a known mechanism for open channel block removal. For the N-methyl-D-aspartate (NMDA) channel, this mechanism is necessary for channel activation and is involved in neuronal plasticity. Several types of Transient Receptor Potential (TRP) channels, including the Drosophila TRP and TRP-Like (TRPL) channels, also exhibit open channel block. For the Drosophila TRP and TRPL channels, removal of open channel block is necessary for the production of the physiological response to light. Recently, we have shown that lipids such as polyunsaturated fatty acids (PUFAs), represented by linoleic acid (LA), alleviate OCB under physiological conditions, from the Drosophila TRP and TRPL channels and from the mammalian NMDA channel. Here we show that OCB removal by LA is not confined to the Drosophila TRPs but also applies to mammalian TRPs such as the heat activated TRPV3 channel. TRPV3 shows OCB alleviation by LA, although it shares little amino acid sequence homology with the Drosophila TRPs. Strikingly, LA inhibits the heat-activated TRPV1 and the cold temperature-activated TRPM8 channels, which are intrinsic voltage sensitive channels and do not show OCB. Together, our findings further support the notion that lipids do not act as second messengers by direct binding to a specific site of the channels but rather act indirectly by affecting the channel-plasma membrane interface.     

Block of tetrodotoxin-resistant Na+ channel pore by multivalent cations: gating modification and Na+ flow dependence

Tetrodotoxin-resistant (TTX-R) Na(+) channels are much less susceptible to external TTX but more susceptible to external Cd(2+) block than tetrodotoxin-sensitive (TTX-S) Na(+) channels. Both TTX and Cd(2+) seem to block the channel near the "DEKA" ring, which is probably part of a multi-ion single-file region adjacent to the external pore mouth and is involved in the selectivity filter of the channel. In this study we demonstrate that other multivalent transitional metal ions such as La(3+), Zn(2+), Ni(2+), Co(2+), and Mn(2+) also block the TTX-R channels in dorsal root ganglion neurons. Just like Cd(2+), the blocking effect has little intrinsic voltage dependence, but is profoundly influenced by Na(+) flow. The apparent dissociation constants of the blocking ions are always significantly smaller in inward Na(+) currents than those in outward Na(+) current, signaling exit of the blocker along with the Na(+) flow and a high internal energy barrier for "permeation" of these multivalent blocking ions through the pore. Most interestingly, the activation and especially the inactivation kinetics are slowed by the blocking ions. Moreover, the gating changes induced by the same concentration of a blocking ion are evidently different in different directions of Na(+) current flow, but can always be correlated with the extent of pore block. Further quantitative analyses indicate that the apparent slowing of channel activation is chiefly ascribable to Na(+) flow-dependent unblocking of the bound La(3+) from the open Na(+) channel, whereas channel inactivation cannot happen with any discernible speed in the La(3+)-blocked channel. Thus, the selectivity filter of Na(+) channel is probably contiguous to a single-file multi-ion region at the external pore mouth, a region itself being nonselective in terms of significant binding of different multivalent cations. This region is "open" to the external solution even if the channel is "closed" ("deactivated"), but undergoes imperative conformational changes during the gating (especially the inactivation) process of the channel.     

Conductance channels in neutral lipid bilayers

Summary A model channel for the conduction of ions (positive or negative but not both) through a lipid bilayer is presented. The transition-state theory is used to relate the current with voltage and ionic concentrations. Sites within the channel are considered to act cooperatively so that the ion is subjected to a ligand field in which it has complete freedom along the channel axis. The ions in the channel are treated as an ionic gas. Effects due to space-charges within the channel arising from the conducting ions are considered whereas surface-charge effects are neglected.The ionic specificity of the channel is indicated and the theory compared to that in which equilibrium free energy changes are the dominant influence.     

Modifications of single acetylcholine-activated channels in BC3H-1 cells. Effects of trimethyloxonium and pH

We have examined the effects of chemical modification with trimethyloxonium (TMO) and changes in external pH on the properties of acetylcholine (ACh)-activated channels in BC3H-1 cells, a clonal muscle cell line. TMO reacts covalently and specifically with carboxylic acid moieties in proteins to convert them to neutral methyl esters. In BC3H-1 cells TMO modification reduces the whole-cell response to ACh measured at negative membrane potentials by approximately 60%. G omega seal patch-clamp recordings of single ACh channel currents showed that the reduction in ACh sensitivity is due to alterations in both the current-carrying and the kinetic properties of the channels. Under all our experimental conditions, i.e., in external solutions of normal or low ionic strength, with or without external divalent cations, and at external pHs between 5.5 and 8.1, TMO treatment reduced ACh single-channel conductance to 70-90% of normal. The effects of TMO on channel kinetics were dependent on the ionic conditions. In normal ionic strength solutions containing both calcium and magnesium ions TMO modification reduced the channel average open time by approximately 25%. A similar reduction in open time was seen in calcium-free solution, but was not present when both calcium and magnesium ions were absent from the external solution. Lowering the ionic strength of the solution increased the mean open time in normal channels by about threefold, but did not affect the kinetics of modified channels. In low ionic strength solutions normal ACh channel open times were maximal at approximately pH 6.7 and decreased by three- to fourfold at both acid and alkaline pH. TMO modification removed the pH dependence of channel kinetics, and average open times were short at all pHs between 5.5 and 8.1. We suggest that TMO modifies normally titratable groups on the external surface of ACh channels that help to determine both the gating and permeability properties of ACh channels.     

Thermodynamics of heat activation of single capsaicin ion channels VR1

Temperature affects functions of all ion channels, but few of them can be gated directly. The vanilloid receptor VR1 provides one exception. As a pain receptor, it is activated by heat >42 degrees C in addition to other noxious stimuli, e.g. acids and vanilloids. Although it is understood how ligand- and voltage-gated channels might detect their stimuli, little is known on how heat could be sensed and activate a channel. In this study, we characterized the heat-induced single-channel activity of VR1, in an attempt to localize the temperature-dependent components involved in the activation of the channel. At <42 degrees C, openings were few and brief. Raising the ambient temperature rapidly increased the frequency of openings. Despite the large temperature coefficient of the apparent activity (Q(10) approximately 27), the unitary current, the open dwell-times, and the intraburst closures were all only weakly temperature dependent (Q(10) < 2). Instead, heat had a localized effect on the reduction of long closures between bursts (Q(10) approximately 7) and the elongation of burst durations (Q(10) approximately 32). Both membrane lipids and solution ionic strength affected the temperature threshold of the activation, but neither diminished the response. The thermodynamic basis of heat activation is discussed, to elucidate what makes a thermal-sensitive channel unique.     

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.     

Single file ion diffusion and the Schmoluchowski equation

The Schmoluchowski equation is introduced into the problem of single file ion diffusion in a channel. The ions mutually interact due to coulomb repulsion and are also subject to a single ion potential due to the channel. The positions of the ions are represented by a continuous co-ordinate. The problem is reduced to the solution of a pair of transfer integral equations. The resistivity of finite and infinite channels is calculated for various dielectric constants and mean ionic separations. The ionic density for finite channels is also calculated. The results clearly demonstrate that strong coulomb interaction leads to a co-operative motion of the ions across channels.