Ion flow in the bath and flux interactions between channels.

We present an exact solution to the linearized Nernst-Planck-Poisson equation for spherically symmetric current flow. This solution differs from Levitt's solution (Levitt, D. G. 1992. Biophys. J., Eq. A5) by its dependence on an additional parameter, which is equal to the net ion flux for monovalent ion-selective channels. For ion-selective channels, this solution may provide better boundary conditions to modelling the flow in the channel pore itself, although only at low salt concentrations. We use the solution to estimate the effects of flux interaction between closely packed channels.     

Observations on Levitt's "new theory of transport".

Levitt (1974) (Biochim. Biophys. Acta 373, 115--131) has recently developed a "New Theory of Transport for Cell Membrane Pores" based on the supposition that equivalent pores in the red cell membrane are so small that water and small solute molecules such as urea can not pass each other. Levitt's concept is based on the implicit assumption that urea and water are spherical molecules. We have shown, using a scale model, that Levitt's supposition is not in agreement with the actual molecular shapes. Levitt has further asserted that there is a serious methodological error in measurements reported fifteen years ago by Goldstein and Solomon (1960) (J. Gen. Physiol. 44, 1--17). We have shown that the supposed "methodological error" lies in the fact that Levitt made his mathematical analysis of the appropriate equations under conditions significantly different from those employed by Goldstein and Solomon. A computer solution of the equations under the actual conditions used shows that Levitt's assertion is not justified.     

Stochastic theory of ion movement in channels with single-ion occupancy. Application to sodium permeation of gramicidin channels.

The electrodiffusion equations were solved for the one-ion channel both by the analytical method due to Levitt and also by Brownian dynamic simulations. For both types of calculations equilibration of ion distribution between the bath and the ends of the channel was assumed. Potential profiles were found that give good fits to published data on Na+ permeation of gramicidin channels. The data were best fit by profiles that have no relative energy maximum at the mouth of the channel. This finding suggests that alignment of waters or channel charged groups inside the channel in response to an ion's approach may provide an energetically favorable situation for entry sufficient to overcome the energy required for removing bulk waters of hydration. An alternative possibility is that the barrier to ion entry is situated outside the region restricted to single-ion occupancy. Replacement of valine with more polar amino acids at the No. 1 location was found to correspond to a deepening of the potential minima near the channel mouths, an increase in height of the central barrier to ion translocation across the channel, and possibly a reduction in the mobility of the ion-water complex in the channel. The Levitt theory was extended to calculate passage times for ions to cross the channel and the blocking effects of ions that entered the channel but didn't cross. These quantities were also calculated by the Brownian dynamics method.     

Electrostatic modeling of ion pores. Energy barriers and electric field profiles

This paper presents calculations of the image potential for an ion in an aqueous pore through lipid membrane and the electric field produced in such a pore when a transmembrane potential is applied. The method used is one introduced by Levitt (1978, Biophys. J. 22:209), who solved an equivalent problem, in which a surface charge density is placed at the dielectric boundary. It is shown that there are singularities in this surface charge density if the model system has sharp corners. Numerically accurate calculations require exact treatment of these singularities. The major result of this paper is the development of a projection method that explicitly accounts for this behavior. It is shown how this technique can be used to compute, both reliably and efficiently, the electrical potential within a model pore in response to any electrical source. As the length of a channel with fixed radius is increased, the peak in the image potential approaches that of an infinitely long channel more rapidly than previously believed. When a transmembrane potential is applied the electric field within a pore is constant over most of its length. Unless the channel is much longer than its radius, the field extends well into the aqueous domain. For sufficiently dissimilar dielectrics the calculated values for the peak in the image potential and for the field well within the pore can be summarized by simple empirical expressions that are accurate to within 5%.     

Experimental and theoretical studies on TI+ interactions with the cation-selective channel of the sarcoplasmic reticulum

Summary This paper presents an experimental study and a theoretical interpretation of the effects of thallous ion on the electrical properties of the cation-selective channel of the sarcoplasmic reticulum (SR channel). The properties of this channel in solutions which do not contain thallous ion are consistent with the predictions of Läuger's theory for singly occupied pores (P. Läuger, 1973,Biochim. Biophys. Acta 311:423–441). However, this theory does not account for SR channel properties in mixtures containing thallous ion. SR channel conductance is less than predicted in mixed salt solutions of thallium with either potassium or ammonium (J. Fox, 1983,Biochim. Biophys. Acta 736:241–245), yet is greater than expected in mixtures of lithium and thallium. In a simple single-ion pore, the ratio of the products of the single-salt binding constants and maximum conductances is equal to the permeability ratio calculated from zero-current potential experiments under near equilibrium conditions. This is not found for the SR channel when thallous ion is present. SR channel properties in the presence of thallous ion can, however, be explained by a model which postulates the existence of two external modulatory sites on the channel, without implying double-occupancy in the permeation pathway. When thallous ion is bound to a modulatory site the maximum conductance of the channel to all permeating ions is altered (thallous included). Two other models (a three-barrier, two-internal-site pore which allows multiple occupancy, and a pore with fluctuating barriers) are discussed, but are found to be unable to fit our conductance data at different concentrations.     

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.     

Confirmation of the predicted source of a slow folding reaction: proline 8 of bovine pancreatic trypsin inhibitor

A major question in protein structural analysis concerns the applicability of results from model systems to other proteins. Theoretical approaches seem the best manner of transferring information from one system to another, but their accuracy in the model systems must first be tested with results from experiment. Since bovine pancreatic trypsin inhibitor (BPTI) is a model system for the evaluation of energy minimization and molecular dynamics routines, we can use folding and stability measurements to examine the reliability of these methods. All two-disulfide mutants of BPTI investigated thus far have two very slow folding reactions which have characteristics of proline isomerization. These reactions may occur because the non-native cis form of two of the four prolines in BPTI significantly destabilizes the folded state of the protein. Previous energy minimization studies of wild-type BPTI suggested that the cis form of Pro8 was the most destabilizing of the four prolines [Levitt,M. (1981) J. Mol. Biol., 145, 251-263]. In this paper, we show that mutation of Pro8----Gln in the two-disulfide bond mutant Val30/Ala51 results in a loss of the slowest folding reaction, consistent with Levitt's prediction.     

Bipartite expressions for diffusional mass transport in biomembranes

Analytical expressions for solute diffusion through a membrane barrier for different initial and boundary conditions are available in the literature. The three commonest initial and boundary conditions are for a membrane without solute respectively immersed in a solution of constant concentration, immersed in such a solution for one side but with the other side isolated, and immersed in such a solution for one side and with the other side kept at zero concentration. The physical quantities for the first two initial and boundary conditions are concentration and average concentration (the total solute entering the membrane) with amperometric current (flux) and solute that permeates through the membrane (charge passed) for the third initial and boundary condition. Expressions for these methods in the literature are inconvenient for practical applications because of the infinite mathematical series required. An investigation of convergence of these expressions was therefore carried out. Simple but accurate bipartite expressions for these methods were constructed and provided theoretical support for studies on mass transport characterization of biomembranes. As a specific application, these expressions enabled a direct fit of the simulated observables to experimental values to obtain diffusion coefficients. For these initial and boundary conditions and corresponding physical quantities, simple one point methods for diffusion coefficient estimation are also suggested. These latter diffusion coefficients can be initial values for numerical fit methods.     

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.     

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.     

Probing the structure of the conduction pathway of the sheep cardiac sarcoplasmic reticulum calcium-release channel with permeant and impermeant organic cations

The sarcoplasmic reticulum Ca(2+)-release channel plays a central role in cardiac muscle function by providing a ligand-regulated pathway for the release of sequestered Ca2+ to initiate contraction following cell excitation. The efficiency of the channel as a Ca(2+)-release pathway will be influenced by both gating and conductance properties of the system. In the past we have investigated conduction and discrimination of inorganic mono- and divalent cations with the aim of describing the mechanisms governing ion handling in the channel (Tinker, A., A.R. G. Lindsay, and A.J. Williams. 1992. Journal of General Physiology. 100:495-517.). In the present study, we have used permeant and impermeant organic cations to provide additional information on structural features of the conduction pathway. The use of permeant organic cations in biological channels to explore structural motifs underlying selectivity has been an important tool for the electrophysiologist. We have examined the conduction properties of a series of monovalent organic cations of varying size in the purified sheep cardiac sarcoplasmic reticulum Ca(2+)-release channel. Relative permeability, determined from the reversal potential measured under bi- ionic conditions with 210-mM test cation at the cytoplasmic face of the channel and 210 mM K+ at the luminal, was related inversely to the minimum circular cation radius. The reversal potential was concentration-independent. The excluded area hypothesis, with and without a term for solute-wall friction, described the data well and gave a lower estimate for minimum pore radius of 3.3-3.5 A. Blocking studies with the impermeant charged derivative of triethylamine reveal that this narrowing occurs over the first 10-20% of the voltage drop when crossing from the lumen of the SR to the cytoplasm. Single-channel conductances were measured in symmetrical 210 mM salt. Factors other than relative permeability determine conductance as ions with similar relative permeability can have widely varying single-channel conductance. Permeant ions, such as the charged derivatives of trimethylamine and diethylmethylamine, can also inhibit K+ current. The reduction in relative conductance with increasing concentrations of these two ions at a holding potential of 60 mV was described by a rectangular hyperbola and revealed higher affinity binding for diethylmethylamine as compared to trimethylamine. It was possible to describe the complex permeation properties of these two ions using a single-ion four barrier, three binding site Eyring rate theory model. In conclusion, these studies reveal that the cardiac Ca(2+)-release channel has a selectivity filter of approximately 3.5-A radius located at the luminal face of the protein.(ABSTRACT TRUNCATED AT 400 WORDS)     

Electrostatic radius of the gramicidin channel determined from voltage dependence of H+ ion conductance.

The results of Decker and Levitt (1987) suggest that the conductance of H+ ion through the gramicidin channel is limited primarily by diffusion in the bulk solution at the channel mouth. It is assumed in this paper that the H+ conductance is 100% diffusion limited. This means that all the factors that influence the H+ flux are external to the channel and are presumed to be known. In particular, the diffusion coefficient of H+ in this region is assumed to be equal to the bulk solution value and the only force acting on the ion is that due to the applied voltage. A model of the H+ flux is derived, based on the Nernst-Planck equation. It has three adjustable parameters: the electrostatic radius, the capture distance, and the radius of the H+ ion. The acceptable range of the parameters was determined by comparing the predictions of the model with the experimental measurements of the H+ conductance at pH 3.75. The best fit was obtained for an electrostatic radius in the range 2.3-2.7 A. This is in good agreement with earlier predictions (2.5 A) based on the assumption that the dielectric constant of the channel water is equal to that of bulk water. The addition of 1 M choline Cl- (an impermeant) increases the H+ current at low voltage and decreases it at high voltage. The increase can be explained by the small surface charge that results from the separation of charge produced by exclusion of the large choline cation (relative to Cl-) from the membrane surface. The decrease at high voltages can be accounted for by the change in the profile of the applied potential produced by the increase in ionic strength.     

The diffusion model for migration and selection in a plant population

 The diffusion approximation is derived for migration and selection at a multiallelic locus in a partially selfing plant population subdivided into a lattice of colonies. Generations are discrete and nonoverlapping; both pollen and seeds disperse. In the diffusion limit, the genotypic frequencies at each point are those determined at equilibrium by the local rate of selfing and allelic frequencies. If the drift and diffusion coefficients are taken as the appropriate linear combination of the corresponding coefficients for pollen and seeds, then the migration terms in the partial differential equation for the allelic frequencies have the standard form for a monoecious animal population. The selection term describes selection on the local genotypic frequencies. The boundary conditions and the unidimensional transition conditions for a geographical barrier and for coincident discontinuities in the carrying capacity and migration rate have the standard form. In the diallelic case, reparametrization renders the entire theory of clines and of the wave of advance of favorable alleles directly applicable to plant populations. Received 30 August 1995; received in revised form 23 February 1996     

Molecular dynamics study of the KcsA channel at 2.0-A resolution: stability and concerted motions within the pore

The stability of the KcsA channel accommodating more than one ion in the pore has been studied with molecular dynamics. We have used the very last X-ray structure of the KcsA channel at 2.0-A resolution determined by Zhou et al. [Nature 414 (2001) 43]. In this channel, six of the seven experimentally evidenced sites have been considered. We show that the protein remains very stable in the presence of four K+ ions (three in the selectivity filter and one in the cavity). The locations and the respective distances of the different K+ ions and water molecules (W), calculated within our KWKWKK sequence, also fits well with the experimental observations. The analysis of the K+ ions and water molecules displacements shows concerted file motions on the simulated time scale (approximately 1 ns), which could act as precursor to the diffusion of K+ ions inside the channel. A simple one-dimensional dynamical model is used to interpret the concerted motions of the ions and water molecules in the pore leading ultimately to ion transfer.     

Molecular sieving characteristics of the cultured endothelial monolayer

We examined the selectivity of the bovine pulmonary artery endothelial monolayer in vitro to molecules of different sizes. The cultured bovine pulmonary endothelial monolayer was grown on a gelatinized filter and the transendothelial transport was studied by determining the permeability of molecules ranging from 182 to 340,000 daltons under diffusion conditions. The permeabilities across the cultured bovine endothelium were modeled according to cylindrical pore theory. The data were best fit by a two-pore model with radii 65 A and 304 A and a ratio of small to large pores of 160:1. The results indicate that the cultured endothelial monolayer is a selective barrier to molecules of different sizes and that the molecular selectivity is consistent with a diffusional pathway through endothelial pore equivalents. The cultured endothelial monolayer is a useful system for studying the permeability characteristics of the endothelial barrier.     

Conduction of Monovalent and Divalent Cations in the Slow Vacuolar Channel

The conduction properties of individual physiologically important cations Na+, K+, Mg2+, and Ca2+ were determined in the slowly activating (SV) channel of sugar beet vacuoles. Current-voltage relationships of the open channel were measured on excised tonoplast patches in a continuous manner by applying a +/-140 mV ramp-wave protocol. Applying KCl gradients of either direction across the patch we have determined that the relative Cl- to K+ permeability was < or =1%. Symmetrical increase of the concentration of tested cation caused an increase of the single channel conductance followed by saturation. Fitting of binding isotherms at zero voltage to the Michaelis-Menten equation resulted in values of maximal conductance of 300, 385, 18, and 13 pS, and of apparent dissociation constants of 64, 103, 0.04, and 0.08 mm for Na+, K+, Mg2+, and Ca2+, respectively. Deviations from the single-ion occupancy mechanism are documented, and alternative models of permeation are discussed. The magnitude of currents carried by divalent cations at low concentrations can be explained by an unrealistically wide (approximately 140 A) radius of the pore entrance. We propose instead a fixed negative charge in the pore vestibules, which concentrates the cations in their proximity. The conduction properties of the SV channel are compared with reported characteristics of voltage-dependent Ca2+-permeable channels, and consequences for a possible reduction of postulated multiplicity of Ca2+ pathways across the tonoplast are drawn.     

Probes of the conduction process of a voltage-gated Cl- channel from Torpedo electroplax

The open-channel conductance properties of a voltage-gated Cl- channel derived from Torpedo californica electroplax and incorporated into planar bilayers were studied by several approaches. In neutral bilayers the channel conductance saturates with Cl- activity according to a rectangular hyperbolic relation with a half-saturation activity of 75 mM and a maximum conductance of 32 pmho. The observation of identical behavior in charged membranes implies that ions permeating the channel do not sense the surface potential of the bulk membrane. The Cl-:Br- permeability ratio, measured under biionic conditions, is independent of salt concentration. SCN- ion reversibly blocks the channel. The voltage dependence of the block implies the existence of two separate blocking sites within the channel: one accessible from the cis side only (the side to which vesicles are added) and the other accessible from the trans side only. The block at each site is competitive with Cl-. The results are consistent with a single-ion Eyring model of the conduction process in which the ion must traverse three kinetic barriers as it permeates the channel and in which the channel can accommodate at most one ion at a time.     

Ion permeation in a K+ channel inChara australis: Direct evidence for diffusion limitation of ion flow in a maxi-K channel

Summary We report a study of a potassium-selective channel in the membrane delineating cytoplasmic drops fromChara australis. The relatively large conductance (170 pS in 150 mol/m³ (mm) KCl), high ion selectivity (P _Cl/P _K=0.015±0.01) and voltagedependent kinetics of this channel indicate that it is a type of maxi-K channel commonly found in animal cells but not previously detected in any plant cell.The current-voltage (I/V) characteristic of these channels was examined in drop-attached and in excised outside-out patches using the patch-clamp technique, over the unusually large voltage range of –250 to 200 mV. TheI/V characteristic is nonlinear and shows saturation at extreme voltages; the current also saturates at high [K⁺]. In solutions with symmetrical KCl concentrations the saturation behavior of the current is asymmetrical. The permeability of the channel depends on whether it is observed in excised or in drop-attached membrane patches.Here we investigate the main factors affecting the permeation of K⁺ ions through this maxi-K channel. We present the first direct evidence for the importance of diffusion external to the pore in limiting ion flow through maxi-K channels. The data are consistent with an ion translocation mechanism whose current is limited (i) at high voltages by ion diffusion external to the pore and (ii) at high [K⁺] by the maximum transport rate of the channel. We fit the data to a diffusion-limited pore model in which the pore exhibits saturation described by Michaelis-Menten kinetics with aK _m=50±25 mol/m³ andG _max=300±20 pS.