Current noise around steady states in discrete transport systems

Subject of this paper is the transport noise in discrete systems. The transport systems are given by a number (n) of binding sites separated by energy barriers. These binding sites may be in contact with constant outer reservoirs. The state of the system is characterized by the occupation numbers of particles (current carriers) at these binding sites. The change in time of the occupation numbers is generated by individual “jumps” of particles over the energy barriers, building up the flux matter (for charged particles: the electric current). In the limit n → ∞ continuum processes as e.g. usual diffusion are included in the transport model. The fluctuations in occupation numbers and other quantities linearly coupled to the occupation numbers may be treated with the usual master equation approach. The treatment of the fluctuations in fluxes (current) makes necessary a different theoretical approach which is presented in this paper under the assumption of vanishing interactions between the particles. This approach may be applied to a number of different transport systems in biology and physics (ion transport through porous channels in membranes, carrier mediated ion transport through membranes, jump diffusion e.g. in superionic conductors). As in the master equation approach the calculation of correlations and noise spectra may be reduced to the solution of the macroscopic equations for the occupation numbers. This result may be regarded as a generalization to non-equilibrium current fluctuations of the usual Nyquist theorem relating the current (voltage) noise spectrum in thermal equilibrium to the macroscopic frequency dependent admittance.The validity of the general approach is demonstrated by the calculation of the autocorrelation function and spectrum of current noise for a number of special examples (e.g, pores in membrances, carrier mediated ion transport).     

Characterization of a Ca-dependent maxi K channel in the apical membrane of a cultured renal epithelium (JTC-12.P3)

Summary A Ca and potential-dependent K channel of large unit conductance was detected in the apical membrane of JTC-12.P3 cells, a continuous epithelial cell line of renal origin. The open probability of the channel is dependent on membrane potential and cytoplasmic-free Ca concentration. At cell-free configuration of the membrane patch, the open probability shows a bell-shaped behavior as function of membrane potential, which decreases at larger depolarization. With increasing Ca concentration, the width of the bell-shaped curve increases and the maximum shifts into the hyperpolarizing direction. For the first time the kinetics of this channel was analyzed under cell-attached conditions. In this case the kinetics could sufficiently be described by a simple open-closed behavior. The channel has an extremely small open probability at resting potential, which increases exponentially with depolarization. The low probability induces an uncertainty about the actual number of channels in the membrane patch. The number of channels is estimated by kinetic analysis. It is discussed that this K channel is essential for the repolarization of the membrane potential during electrogenic sodium-solute cotransport across the apical membrane.     

Relationship between membrane excitability and single channel open-close kinetics.

We have developed a novel technique for simulating the influence of the effects of single channel kinetics on the voltage changes associated with membrane excitability. The technique uses probability distribution functions for the durations of channel open- and closed-state lifetimes, which can be calculated for any model of the ion conductance process. To illustrate the technique, we have used the Hodgkin and Huxley model of nerve membrane ion conductances to simulate channel kinetics during predetermined voltage changes, such as a voltage jump and an action potential. We have also simulated the influence of channels on voltage changes in a free running, non-voltage-clamped patch of membrane 1 micron2 or less in area. The latter results provide a direct illustration of the relationship between fluctuations of membrane excitability and fluctuations in channel open- and closed-state lifetimes.     

Kinetics of the Opening and Closing of Individual Excitability-Inducing Material Channels in a Lipid Bilayer

The kinetics of the opening and closing of individual ion-conducting channels in lipid bilayers doped with small amounts of excitability-inducing material (EIM) are determined from discrete fluctuations in ionic current. The kinetics for the approach to steady-state conductance during voltage clamp are determined for lipid bilayers containing many EIM channels. The two sets of measurements are found to be consistent, verifying that the voltage-dependent conductance of the many-channel EIM system arises from the opening and closing of individual EIM channels. The opening and closing of the channels are Poisson processes. Transition rates for these processes vary exponentially with applied potential, implying that the energy difference between the open and closed states of an EIM channel is linearly proportional to the transmembrane electric field. A model incorporating the above properties of the EIM channels predicts the observed voltage dependence of ionic conductance and conductance relaxation time, which are also characteristic of natural electrically excitable membranes.     

Kinetic characteristics of the excitability-inducing material channel in oxidized cholesterol and brain lipid bilayer membranes

The kinetic characteristics of the opening and closing of the excitability-inducing material (EIM) channel in oxidized cholesterol and in brain lipid bilayers are compared. The kinetics of the opening and closing of individual ion-conducting channels in bilayers doped with small amounts of EIM are determined from discrete fluctuations in ionic current. The kinetics for approach to steady-state conductance are determined for lipid bilayers containing many channels. Steady-state and kinetic characteristics for the EIM channel incorporated in brain lipid bilayers can be accounted for by the model developed for the EIM channel incorporated in oxidized cholesterol membranes. Relaxation time, calculated from rate constants of single-channel membranes or directly measured in many-channel membranes is strongly temperature dependent, and is always shorter in brain lipid membranes. Changes in temperature do not affect the interaction of the electric field and the open channel, but the open configuration of the EIM channel in brain lipid bilayers is stablized with increasing temperature. The configurational energy difference between the open and closed channel, calculated from temperature studies, is larger in brain lipid bilayers. The energy barrier which separates the two configurations of the channel is larger in oxidized cholesterol bilayers.     

Probing conformational changes of gramicidin ion channels by single-molecule patch-clamp fluorescence microscopy

Complex conformational changes influence and regulate the dynamics of ion channels. Such conformational changes are stochastic and often inhomogeneous, which makes it extremely difficult, if not impossible, to characterize them by ensemble-averaged experiments or by single-channel recordings of the electric current that report the open-closed events but do not specifically probe the associated conformational changes. Here, we report our studies on ion channel conformational changes using a new approach, patch-clamp fluorescence microscopy, which simultaneously combines single-molecule fluorescence spectroscopy and single-channel current recordings to probe the open-closed transitions and the conformational dynamics of individual ion channels. We demonstrate patch-clamp fluorescence microscopy by measuring gramicidin ion channel conformational changes in a lipid bilayer formed at a patch-clamp micropipette tip under a buffer solution. By measuring single-pair fluorescence resonance energy transfer and fluorescence self-quenching from dye-labeled gramicidin channels, we observed that the efficiency of single-pair fluorescence resonance energy transfer and self-quenching is widely distributed, which reflects a broad distribution of conformations. Our results strongly suggest a hitherto undetectable correlation between the multiple conformational states of the gramicidin channel and its closed and open states in a lipid bilayer.     

Correlation analysis of electrical noise in lipid bilayer membranes: kinetics of gramicidin A channels.

If a membrane contains ion-conducting channels which form and disappear in a random fashion, an electric current which is passed through the membrane under constant voltage shows statistical fluctuations. Information on the kinetics of channel formation and on the conductance of the single channel may be obtained by analyzing the electrical noise generated in a membrane containing a great number of channels. For this purpose the autocorrelation function of the current noise is measured at different concentrations of the channel-forming substance. As a test system for the application of this technique we have used lipid bilayer membranes doped with gramicidin A. From the correlation time of the current noise generated by the membrane, the rate constants of formation (k-R) and dissociation (k-D) of the channels could be determined. In addition, the mean square of the current fluctuations yielded the single-channel conductance lambda. The values of k-R, k-D, and lambda obtained from the noise analysis agreed closely with the values determined by relaxation measurments and single-channel experiments.     

Calcium currents in the A7r5 smooth muscle-derived cell line. An allosteric model for calcium channel activation and dihydropyridine agonist action

We have investigated the gating kinetics of calcium channels in the A7r5 cell line at the level of single channels and whole cell currents, in the absence and presence of dihydropyridine (DHP) calcium channel agonists. Although latencies to first opening and macroscopic currents are strongly voltage dependent, analysis of amplitude histograms indicates that the primary open-closed transition is voltage independent. This suggests that the molecular mechanisms for voltage sensing and channel opening are distinct, but coupled. We propose a modified Monod-Wyman-Changeux (MWC) model for channel activation, where movement of a voltage sensor is analogous to ligand binding, and the closed and open channels correspond to inactive (T) and active (R) states. This model can account for the activation kinetics of the calcium channel, and is consistent with the existence of four homologous domains in the main subunit of the calcium channel protein. DHP agonists slow deactivation kinetics, shift the activation curve to more negative potentials with an increase in slope, induce intermingled fast and slow channel openings, and reduce the latency to first opening. These effects are predicted by the MWC model if we make the simple assumption that DHP agonists act as allosteric effectors to stabilize the open states of the channel.     

Single K+ channels in endocrine cells dispersed from the cricket (Gryllus bimaculatus) corpora allata

Single channel currents were recorded from cell-attached patches of endocrine cells of the adult male cricket corpora allata. Three distinct types of K+ channels were identified; a weak inward rectifier (Type 1), a strong inward rectifier (Type 2) and a weak outward rectifier (Type 3). The type 1 channel had a slope conductance of 191 +/- 9 pS (n = 4) at negative membrane potentials (Vm) and 101 +/- 6 pS (n = 6) at positive Vm. In addition, the channel showed fast open-closed kinetics at negative Vm and slow open-closed kinetics at positive Vm. The open probability (Po) of this channel was strongly voltage-dependent at positive Vm, but less voltage-dependent at negative Vm. The reversal potential was not modified significantly by the substitution of gluconate for external Cl- but was modified after N-methyl-D-glucamine (NMDG+) was substituted for external K+, according to the Nernst equation for a K+-selective channel. The type 2 channel had a slope conductance of 44 +/- 2 pS (n = 5) at negative Vm, but no detectable outward current was observed at positive Vm. This channel showed very slow open-closed kinetics at negative Vm and its Po was not voltage-dependent. The type 3 channel had a limit conductance of 55 +/- 12 pS (n = 3) at negative Vm and 88 +/- 10 pS (n = 3) at positive Vm. This channel showed slow open-closed kinetics at negative Vm and fast open-closed kinetics at positive Vm. The Po for the channel was voltage-dependent at positive Vm but was voltage-independent at negative Vm. These three types of K+ channels may be important for the control of the resting membrane potential, and may thus participate in the regulation of Ca2+ influx and juvenile hormone secretion in corpora allata cells.     

Simple model of ion transport through alamethicin channels in lipid membranes

The electrical characteristics of wide membrane channels such as those induced in lipid membranes by alamethicin have been analyzed using an electrodiffusion model. The channel is considered to be a water filled cylinder in which the potential energy barrier is a result of the difference in polarization energy of the ion environment when the ion is located inside as compared to outside of the channel. In addition, an electric field related to the channel structure is assumed. It is shown that without postulating any specific chemical ion-channel interaction one can reproduce experimental membrane potentials for NaCl, KCl, and CaCl₂ concentration gradients with a single set of channel parameters. The calculations also yield experimental J-V characteristics of discrete conduction states. In addition, a simple mechanism of interchannel coupling based on the above model is discussed. The model suggests a unifying approach to the problem of the origin of interionic selectivity of membrane channels induced by polyene antibiotics.     

Beating irregularity of single pacemaker cells isolated from the rabbit sinoatrial node.

Single pacemaker heart cells discharge irregularly. Data on fluctuations in interbeat interval of single pacemaker cells isolated from the rabbit sinoatrial node are presented. The coefficient of variation of the interbeat interval is quite small, approximately 2%, even though the coefficient of variation of diastolic depolarization rate is approximately 15%. It has been hypothesized that random fluctuations in interbeat interval arise from the stochastic behavior of the membrane ionic channels. To test this hypothesis, we constructed a single channel model of a single pacemaker cell isolated from the rabbit sinoatrial node, i.e., a model into which the stochastic open-close kinetics of the individual membrane ionic channels are incorporated. Single channel conductances as well as single channel open and closed lifetimes are based on experimental data from whole cell and single channel experiments that have been published in the past decade. Fluctuations in action potential parameters of the model cell are compared with those observed experimentally. It is concluded that fluctuations in interbeat interval of single sinoatrial node pacemaker cells indeed are due to the stochastic open-close kinetics of the membrane ionic channels.     

Periodic forcing of a single ion channel: dynamic aspects of the open-closed switching

An unconventional approach to studying ion channel kinetics is exploited. Here we describe the effects of a periodically varying membrane potential on the open-closed transitions of single K⁺ channels. The use of cycle histograms allows one to measure the transition probabilities as functions of the stimulus phase. The results show that such probabilities vary with the stimulation frequency and with the stimulus history, thus highlighting a dynamic aspect in the switching of this ion channel. Received: 9 May 1997 / Revised version: 5 January 1998 / Accepted: 12 January 1998     

Effects of pipette geometry on the time course of solution change in patch clamp experiments.

The time course of change in current through KATP channels in inside-out membrane patches, after step change of permeant ion (K+) concentration, was measured. A simple model of the patch as a membrane disc at the base of a cone with the apex removed, was able to describe the time course of channel activity after step change of [K+]. By measuring pipette geometry and using jumps of [permeant ion], it was then possible to estimate the time course of concentration at the membrane for jumps of any other ion or gating ligand. A simple channel block mechanism was used to simulate experiments with concentration jumps of a blocking ligand. The rate constants for ligand-channel interaction were extracted by least-squares fitting of computed mass action responses to those observed in simulated experiments. The simulations showed that even with diffusion delays of hundreds of milliseconds (as may occur in inside-out patch experiments), ligand association and dissociation rates of up to 1,000 s-1 could be accurately extracted by this approach. The approach should be generally applicable to the analysis of ligand concentration jump experiments on any ion channel whose activity is modulated by intracellular ligand.     

Conductivity noise in transmembrane ion channels due to ion concentration fluctuations via diffusion.

A Green's function approach is developed from first principles to evaluate the power spectral density of conductance fluctuations caused by ion concentration fluctuations via diffusion in an electrolyte system. This is applied to simple geometric models of transmembrane ion channels to obtain an estimate of the magnitude of ion concentration fluctuation noise in the channel current. Pure polypeptide alamethicin forms stable ion channels with multiple conductance states in artificial phospholipid bilayers isolated onto tips of micropipettes with gigaohm seals. In the single-channel current recorded by voltage-clamp techniques, excess noise was found after the background instrumental noise and the intrinsic Johnson and shot noises were removed. The noise que to ion concentration fluctuations via diffusion was isolated by the dependence of the excess current noise on buffer ion concentration. The magnitude of the concentration fluctuation noise derived from experimental data lies within limits estimated using our simple geometric channel models. Variation of the noise magnitude for alamethicin channels in various conductance states agrees with theoretical prediction.     

Open channel block by gadolinium ion of the stretch-inactivated ion channel in mdx myotubes.

Currents flowing through single stretch-inactivated ion channels were recorded from cell-attached patches on myotubes from mdx mice. Adding micromolar concentrations of gadolinium to patch electrodes containing normal saline produced rapid transitions in the single-channel current between the fully open and closed states. The kinetics of the current fluctuations followed the predictions of a simple model of open channel block in which the transitions in the current arise from the entry and exit of Gd from the channel pore: histograms of the open and closed times were well fit with single exponentials, the blocking rate depended linearly on the concentration of gadolinium in the patch electrode, and the unblocking rate was independent of the concentration of gadolinium. Hyperpolarizing the patch increased the rate of unblocking (approximately e-fold per 85 mV), suggesting the charged blocking particle can exit the channel into the cell under the influence of the applied membrane field. The rate of blocking was rapid and was independent of the patch potential, consistent with the rate of ion entry into the pore being determined by its rate of diffusion in solution. When channel open probability was reduced by applying suction to the electrode, the blocking kinetics were independent of the extent of inactivation, suggesting that mechanosensitive gating does not modify the structure of the channel pore.     

Estimation of Na+ dwell time in the gramicidin A channel. Na+ ions as blockers of H+ currents

The permeation of Na+ through gramicidin A channels shows a simple saturation with increasing Na+ concentration that can be described by two different models. The first model assumes that one Na+ binds to the channel with high affinity (approximately 30 M-1) and that conduction occurs by a 'knock-on' mechanism requiring double occupancy of the channel; the other model assumes that Na+ binding is of low affinity (less than 1 M-1), and that double occupancy of the channel is rare. NMR measurements have shown tight Na+ binding, favoring the first model, but measurements of flux ratios and water transport support the second model. We present here a relatively model-independent measurement of the dwell time of Na+ inside the channel, in which we characterize the fluctuations in H+ current through the channel induced by 'block' from the more slowly permeating Na+ ions. The mean Na+ dwell time inside the channel is estimated to be approximately 10 ns at a membrane potential of 200 mV. This result is inconsistent with tight Na+ binding, thus favoring the second model.     

A different kind of hemocyanin channel in oxidized cholesterol membranes.

A new kind of hemocyanin channel in oxidized cholesterol black lipid membranes has been characterized using the protein extracted from the mollusc Paludina Vivipara. The channel has constant conductance versus applied voltage and is subjected to open-closed fluctuations that follow fairly well a binomial distribution, independent from the applied potential. Single channel electrical properties are in good agreement with those of many channels membranes. This channel is completely different from that already studied formed by Keyhole Limpet Hemocyanin and for some aspects reminds other channels particularly those obtained with Excitability Inducing Material and Alamethicin.     

Block of single L-type Ca2+ channels in skeletal muscle fibers by aminoglycoside antibiotics

The activity of single L-type Ca2+ channels was recorded from cell- attached patches on acutely isolated skeletal muscle fibers from the mouse. The experiments were concerned with the mechanism by which aminoglycoside antibiotics inhibit ion flow through the channel. Aminoglycosides produced discrete fluctuations in the single-channel current when added to the external solution. The blocking kinetics could be described as a simple bimolecular reaction between an aminoglycoside molecule and the open channel. The blocking rate was found to be increased when either the membrane potential was made more negative or the concentration of external permeant ion was reduced. Both of these effects are consistent with a blocking site that is located within the channel pore. Other features of block, however, were incompatible with a simple pore blocking mechanism. Hyperpolarization enhanced the rate of unblocking, even though an aminoglycoside molecule must dissociate from its binding site in the channel toward the external solution against the membrane field. Raising the external permeant ion concentration also enhanced the rate of unblocking. This latter finding suggests that aminglycoside affinity is modified by repulsive interactions that arise when the pore is simultaneously occupied by a permeant ion and an aminoglycoside molecule.     

Gating kinetics of Ca2+-activated K+ channels from rat muscle incorporated into planar lipid bilayers. Evidence for two voltage- dependent Ca2+ binding reactions

The gating kinetics of a Ca2+-activated K+ channel from adult rat muscle plasma membrane are studied in artificial planar bilayers. Analysis of single-channel fluctuations distinguishes two Ca2+- and voltage-dependent processes: (a) short-lived channel closure (less than 1 ms) events appearing in a bursting pattern; (b) opening and closing events ranging from one to several hundred milliseconds in duration. The latter process is studied independently of the first and is denoted as the primary gating mode. At constant voltage, the mean open time of the primary gating mode is a linear function of the [Ca2+], whereas the mean closed time is a linear function of the reciprocal [Ca2+]. In the limits of zero and infinite [Ca2+], the mean open and the mean closed times are, respectively, independent of voltage. These results are predicted by a kinetic scheme consisting of the following reaction steps: (a) binding of Ca2+ to a closed state; (b) channel opening; (c) binding of a second Ca2+ ion. In this scheme, the two Ca2+ binding reactions are voltage dependent, whereas the open-closed transition is voltage independent. The kinetic constant derived for this scheme gives an accurate theoretical fit to the observed equilibrium open-state probability. The results provide evidence for a novel regulatory mechanism for the activity of an ion channel: modulation by voltage of the binding of an agonist molecule, in this case, Ca2+ ion.