Nonequilibrium voltage fluctuations in biological membranes. I. General framework of charge transport in discrete systems and related voltage noise

This paper continues our work on the theory of nonequilibrium voltage noise generated by electric transport processes in membranes. Introducing the membrane voltage as a further variable, a system of kinetic equations linearized in voltage is derived by which generally the time-dependent behaviour of charge-transport processes under varying voltage can be discussed. Using these equations, the treatment of voltage noise can be based on the usual master equation approach to steady-state fluctuations of scalar quantities. Thus, a general theoretical approach to nonequilibrium voltage noise is presented, completing our approach to current fluctuations which had been developed some years ago. It is explicitly shown that at equilibrium the approach yields agreement with the Nyquist relation, while at nonequilibrium this relation is not valid. A further general property of voltage noise is the reduction of low-frequency noise with increasing number of transport units as a consequence of the interactions via the electric field. In a second paper, the approach will be applied for a number of special transport mechanisms, such as ionic channels, carriers or electrogenic pumps.     

Current fluctuations in discrete transport systems far from equilibrium. Breakdown of the fluctuation dissipation theorem

A recently developed theoretical approach to transport fluctuations around stable steady states in discrete biological transport systems is used in order to investigate general fluctuation properties at nonequilibrium. An expression for the complex frequency dependent admittance at nonequilibrium is derived by calculation of the linear current response of the transport systems to small disturbances in the applied external voltage. It is shown that the Nyquist or fluctuation dissipation theorem, by which at equilibrium the macroscopic admittance or linear response can be expressed in terms of fluctuation properties of the system, breaks down at nonequilibrium. The spectral density of current fluctuations is decomposed into one term containing the macroscopic admittance and a second term which is bilinear in current. This second term is generated by microscopic disturbances, which cannot be excited by external macroscopic perturbations. At special examples it is demonstrated that this second term is decisive for the occurrence of excess noise e.g. the 1/f(2)-Iorentzian noise generated by the opening and closing of nerve channels in biological membranes.     

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).     

Nonequilibrium ion transport through pores

In this paper is presented an investigation of the influence of the internal structure of pores in membranes on a) the time dependent macroscopic relaxation current after a voltage jump, b) the macroscopic frequency dependent admittance and c) the microscopic current fluctuations around stationary (nonequilibrium) states. All these quantities are determined by the time dependent transport equations, which are calculated with the use of the eigenvectors and eigenvalues of the matrix of coefficients, occurring in the transport equations. Numerical calculations for channels with up to 31 barriers are presented. The treatment of the fluctuations is done with the use of a general approach to nonequilibrium transport noise recently developed by one of the authors. It is shown that the influence of the internal barrier structure as, e.g., the height of central or decentral barriers in the pores is of great complexity. Nevertheless we hope that the calculations lead to a better understanding especially of the microscopic nonequilibrium transport fluctuations in complex systems.This work has been supported by the Deutsche Forschungsgemeinschaft     

On the Na+,K+ pump in fluctuating membrane potentials

The present work investigates the usefulness of noise in the activity of the Na+,K+ pump. Random gating activity of the neighboring ion channels causes local fluctuations of the electric potential. They are modeled by a Markovian symmetric dichotomic noise, added to the membrane potential. The noise-averaged pump current is calculated for a general rectangular voltage signal and the model parameters of the effective two-state enzyme cycle are tuned to fit experimental results. Then, using these parameters, the amount of transported charge is calculated, and studied as a function of noise intensity. Signal and noise characteristics are identified at which fluctuations enhance pump activity. The biological impact of this phenomenon seems to be absent in physiological conditions for it occurs at noise amplitudes over 50 mV, which are unlikely to appear due to ion channels. However, under some conditions, externally applied dichotomic noise of intensity about 150 mV may sensibly increase the quantity of transported charge.     

1/f noise in membranes.

The present situation of 1/f noise in the passage of ions across membranes is examined. A survey of biological and synthetic membranes is given at which a 1/f frequency dependence has been observed in the spectrum of voltage or current fluctuations. Empirical relations and theories of 1/f noise in membranes are critically discussed.     

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.     

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.     

Analysis of the effects of cesium ions on potassium channel currents in biological membranes

Cesium ions block potassium channels in biological membranes in a voltage dependent manner. For example, external cesium blocks inward current with little or no effect on outward current. Consequently, it produces a characteristic N-shaped current-voltage relationship. We have modeled this result by single file diffusion of ions in a narrow channel spanning the membrane with a special blocking site in the channel for cesium ions. The model enables us to make detailed comparisons of the effects of cesium on potassium channels in different types of biological membranes.     

Attenuation of current and voltage noise signals recorded from epithelia

Transepithelially recorded current and voltage fluctuations are filtered by the impedance of the electrical equivalent parameters of the preparation, in series or in parallel, with the noise source. Fluctuations in voltage and current are assumed to be caused by fluctuations in conductance of the apical membrane. In order to obtain an estimation of the intrinsic noise amplitudes, calculations are presented to correct the transepithelial fluctuations. The influence of different model parameters on the recorded noise spectra is investigated. It is shown that the shape of the transepithelially recorded noise spectra may differ from the intrinsic ones, e.g. “peaking” in the power spectra may be explained by the assumption of a positive (referred to cell inside) e.m.f. at the basolateral membrane or a polarization impedance in series with the epithelium. Furthermore it is demonstrated that the ratio of voltage to current noise power may differ from the squared magnitude of the impedance.     

Transport noise in membranes. Current and voltage fluctuations at equilibrium

A formulism is described for the treatment of noise resulting from the transport of ions in channels containing an arbitrary number of activation energy barriers. The analysis is based on Nyquist's theorem and is therefore restricted to fluctuations around the equilibrium state. Within this limit the spectral intensities of current and voltage noise are given by the frequency-dependent admittance, which in turn is closely linked to the relaxation-time spectrum of the transport system. Explicit expressions for the spectral intensity of current noise are derived for channels with two and three energy barriers. The analysis may be used to predict the spectral intensity of noise from the gating system in nerve.     

Subthreshold Voltage Noise Due to Channel Fluctuations in Active Neuronal Membranes

Voltage-gated ion channels in neuronal membranes fluctuate randomly between different conformational states due to thermal agitation. Fluctuations between conducting and nonconducting states give rise to noisy membrane currents and subthreshold voltage fluctuations and may contribute to variability in spike timing. Here we study subthreshold voltage fluctuations due to active voltage-gated Na⁺ and K⁺ channels as predicted by two commonly used kinetic schemes: the Mainen et al. (1995) (MJHS) kinetic scheme, which has been used to model dendritic channels in cortical neurons, and the classical Hodgkin-Huxley (1952) (HH) kinetic scheme for the squid giant axon. We compute the magnitudes, amplitude distributions, and power spectral densities of the voltage noise in isopotential membrane patches predicted by these kinetic schemes. For both schemes, noise magnitudes increase rapidly with depolarization from rest. Noise is larger for smaller patch areas but is smaller for increased model temperatures. We contrast the results from Monte Carlo simulations of the stochastic nonlinear kinetic schemes with analytical, closed-form expressions derived using passive and quasi-active linear approximations to the kinetic schemes. For all subthreshold voltage ranges, the quasi-active linearized approximation is accurate within 8% and may thus be used in large-scale simulations of realistic neuronal geometries.     

1/f Membrane noise generated by diffusion processes in unstirred solution layers.

A mathematical treatment is given for 1/f noise observed in the ion transport through membranes. It is shown that this noise can be generated by current or voltage fluctuations which occur after step changes of the membrane permeability. Due to diffusion polarization in the unstirred solution layers near the membrane these fluctuations exhibit a 1 square root of t time course which produces noise with a 1/f frequency dependence. The spectral density of 1/f noise is calculated for porous membranes with random switches between a finite and zero pore permeability. A wide frequency range and a magnitude of 1/f noise are obtained which are compatible with experimental data of 1/f noise reported for nerve membranes.     

Current noise of a protein-selective biological nanopore

1/f current noise is ubiquitous in protein pores, porins, and channels. We have previously shown that a protein-selective biological nanopore with an external protein receptor can function as a 1/f noise generator when a high-affinity protein ligand is reversibly captured by the receptor. Here, we demonstrate that the binding affinity and concentration of the ligand are key determinants for the nature of current noise. For example, 1/f was absent when a protein ligand was reversibly captured at a much lower concentration than its equilibrium dissociation constant against the receptor. Furthermore, we also analyzed the composite current noise that resulted from mixtures of low-affinity and high-affinity ligands against the same receptor. This study highlights the significance of protein recognition events in the current noise fluctuations across biological membranes.     

Non-equilibrium voltage noise generated by ion transport through pores

In this paper, we describe a systematic approach to the theoretical analysis of non-equilibrium voltage noise that arises from ions moving through pores in membranes. We assume that an ion must cross one or two barriers in the pore in order to move from one side of the membrane to the other. In our analysis, we consider the following factors: a) surface charge as a variable in the kinetic equations, b) linearization of the kinetic equations, c) master equation approach to fluctuations. To analyze the voltage noise arising from ion movement through a two barrier (i.e., one binding site) pore, we included the effects of ions in the channel's interior on the voltage noise. The current clamp is considered as a white noise generating additional noise in the system. In contrast to what is found for current noise, at low frequencies the voltage noise intensity is reduced by increasing voltage across the membrane. With this approach, we demonstrate explicity for the examples treated that, apart from additional noise generated by the current clamp, the non-equilibrium voltage fluctuations can be related to the current fluctuations by the complex admittance.     

1/f noise in membranes

The present situation of 1/f noise in the passage of ions across membranes is examined. A survey of biological and synthetic membranes is given at which a l/f frequency dependence has been observed in the spectrum of voltage or current fluctuations. Empirical relations and theories of 1/f noise in membranes are critically discussed. Supported by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 38 “Membranforschung”     

Open channel noise. V. Fluctuating barriers to ion entry in gramicidin A channels.

We have measured the fluctuations in the current through gramicidin A (GA) channels in symmetrical solutions of monovalent cations of various concentrations, and compared the spectral density values with those computed using E. Frehland's theory for noise in discrete transport systems (Frehland, E. 1978. Biophys. Chem. 8:255-265). The noise for the transport of NH4+ and Na+ ions in glycerol-monooleate/squalene membranes could be accounted for entirely by "shot noise" in the process of transport through a single-filing pore with two ion binding sites. However, in confirmation of results in a previous paper (Sigworth, F. J., D. W. Urry, and K. U. Prasad. 1987. Biophys. J. 52:1055-1064) currents of Cs+ showed a substantial excess noise at low ion concentrations, as did currents of K+ and Rb+. The excess noise was increased in thicker membranes. The observations are accounted for by a theory that postulates fluctuations of the entry rates of ions into the channel on a time scale of approximately 1 microsecond. These fluctuations occur preferentially when the channel is empty; the presence of bound ions stabilizes the "high conductance" conformation of the channel. The fluctuations are sensed to different degrees by the various ion species, and their kinetics depend on membrane thickness.     

Membrane potential and time requirements for detection of weak signals by voltage‐gated ion channels

The question of minimum detection limits for biological processes sensitive to membrane potential perturbations has arisen in various contexts. Of special interest are the prediction of theoretical limits for sensory perception processes and for possible biological effects of environmental or therapeutic electric and magnetic fields. A new method is presented here, addressing the particular case in which perturbations of membrane potential affect the gating rate probability of voltage‐sensitive ion channels. Using a two‐state model for channel gating, the influence of the perturbing potential on the mean fraction of open channels is approximated by a Boltzmann distribution, and integrated over time to obtain a quantity proportional to the net change in expected charge transfer through the membrane. This change in net charge transfer (the signal, S) is compared to the expected root mean variance in charge transfer (the noise, N) due to random channel gating. Using a nominal criterion of S/N = 1, a model is developed for predicting the minimum time and number of ion channels necessary to detect a given membrane potential. Example calculations, carried out for a gating charge of 6, indicate that a 1 μV induced membrane potential can be detected after 10 ms by an ensemble of less than 10⁸ ion channels. Bioelectromagnetics 20:102–109, 1999. Published 1999 Wiley‐Liss, Inc.     

Nonequilibrium response spectroscopy of voltage-sensitive ion channel gating.

We describe a new electrophysiological technique called nonequilibrium response spectroscopy, which involves application of rapidly fluctuating (as high as 14 kHz) large-amplitude voltage clamp waveforms to ion channels. As a consequence of the irreversible (in the sense of Carnot) exchange of energy between the fluctuating field and the channel protein, the gating response is exquisitely sensitive to features of the kinetics that are difficult or impossible to adequately resolve by means of traditional stepped potential protocols. Here we focus on the application of dichotomous (telegraph) noise voltage fluctuations, a broadband Markovian colored noise that fluctuates between two values. Because Markov kinetic models of channel gating can be embedded within higher-dimensional Markov models that take into account the effects of the voltage fluctuations, many features of the response of the channels can be calculated algebraically. This makes dichotomous noise and its generalizations uniquely suitable for model selection and kinetic analysis. Although we describe its application to macroscopic ionic current measurements, the nonequilibrium response method can also be applied to gating and single channel current recording techniques. We show how data from the human cardiac isoform (hH1a) of the Na+ channel expressed in mammalian cells can be acquired and analyzed, and how these data reveal hidden aspects of the molecular kinetics that are not revealed by conventional methods.     

Noise analysis of ion current through the open and the sugar-induced closed state of the LamB channel of Escherichia coli outer membrane: evaluation of the sugar binding kinetics to the channel interior.

LamB, a sugar-specific channel of Escherichia coli outer membrane was reconstituted into lipid bilayer membranes and the current noise was investigated using fast Fourier transformation. The current noise through the open channels had a rather small spectral density, which was a function of the inverse frequency up to about 100 Hz. The spectral density of the noise of the open LamB channels was a quadratic function of the applied voltage. Its magnitude was not correlated to the number of channels in the lipid bilayer membrane. Upon addition of sugars to the aqueous phase the current decreased in a dose-dependent manner. Simultaneously, the spectral density of the current noise increased drastically, which indicated interaction of the sugars with the binding site inside the channel. The frequency dependence of the spectral density was of Lorentzian type, although the power of its frequency dependence was not identical to -2. Analysis of the power density spectra using a previously proposed simple model (Benz, R., A. Schmid, and G. H. Vos-Scheperkeuter. 1987. J. Membr. Biol. 100: 12-29), allowed the evaluation of the on- and the off-rate constants for the maltopentaose binding to the binding site inside the LamB channels. This means also that the maltopentaose flux through the LamB channel could be estimated by assuming a simple one-site, two-barrier model for the sugar transport from the results of the noise analysis.