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.     

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     

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.     

Nonequilibrium voltage fluctuations in biological membranes. II. Voltage and current noise generated by ion carriers, channels and electrogenic pumps

As applications of the general theoretical framework of charge transport in biological membranes and related voltage and current noise, a number of model calculations are presented for ion carriers, rigid channels, channels with conformational substates and electrogenic pumps. The results are discussed with special reference to the problem of threshold values for sensory transduction processes and their limitations by voltage fluctuations. Furthermore, starting from the special results of model calculations, an attempt is made to determine more general aspects of electric fluctuations generated by charge-transport processes in biological membranes: different frequency dependences of voltage and current noise, and dependence of noise intensities with increasing distance from the equilibrium state.     

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).     

Theory of single-file noise.

A general theoretical approach to the analysis of electric fluctuations generated by the so-called single-file diffusion through narrow channels is presented. The formalism is a slight extension of an approach to electric fluctuations in discrete transport systems with negligible interactions between the particles recently developed by one of the authors. In the single-file transport mechanism interactions between the particles must be taken into account. Three main results of principal interest are: (a) the electric fluctuations around stationary states (at equilibrium and non-equilibrium) are determined by the time-dependent solutions of the macroscopic single-file transport equations, (b) as a direct consquence of the interactions between the ions in the single-file transport the macroscopic time-dependent current and the autocorrelation function of the microscopic current fluctuations can exhibit damped oscillatory behavior, and the current noise spectrum can show peaking, (c) the number of binding sites for the ions within the pores seems to have a strong influence on the oscillatory behavior: with increasing number of binding sites the damping of the oscillations decreases and the peaking of the spectrum becomes stronger.     

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.     

Theory of single-file noise

A general theoretical approach to the analysis of electric fluctuations generated by the so-called single-file diffusion through narrow channels is presented. The formalism is a slight extension of an approach to electric fluctuations in discrete transport systems with negligible interactions between the particles recently developed by one of the authors. In the single-file transport mechanism interactions between the particles must be taken into account. Three main results of principal interest are: (a) the electric fluctuations around stationary states (at equilibrium and non-equilibrium) are determined by the time-dependent solutions of the macroscopic single-file transport equations, (b) as a direct consequence of the interactions between the ions in the single-file transport the macroscopic time-dependent current and the autocorrelation function of the microscopic current fluctuations can exhibit damped oscillatory behavior, and the current noise spectrum can show peaking, (c) the number of binding sites for the ions within the pores seems to have a strong influence on the oscillatory behavior: with increasing number of binding sites the damping of the oscillations decreases and the peaking of the spectrum becomes stronger.     

Fluctuations in interbeat interval in rhythmic heart-cell clusters. Role of membrane voltage noise.

Small clusters of ventricular cells prepared from 7-d chick heart maintain spontaneous, stationary, rhythmic beating in culture for many hours. For clusters containing I-125 cells, mean interbeat interval (IBI) is 0.45 +/- 0.08 s and is independent of cell number (N), whereas, the coefficient of variation of IBI (C) is proportional to N-1/2. Because membrane voltage noise in such clusters would also be expected to vary as N-1/2, we propose a model relating fluctuation in IBI (sigma IBI) to voltage noise (sigma v). A simplified model consisting of random voltage fluctuations superimposed upon a linear pacemaker depolarization of slope a is used to analyze the N-dependent shape of the IBI histogram. Values of sigma v derived from the relation sigma IBI = sigma v/a, or calculated from the skewness of the measured IBI histograms, both agree well with those extrapolated from steady-state noise recorded from resting heart-cell aggregates.     

Linear noise approximation method for calculating nonequilibrium fluctuations of the occupation numbers in radiative-collisional average ion models

A novel method is proposed for calculating nonequilibrium fluctuations of the mean occupation numbers of the electron shells in the radiative-collisional average-ion models of multicharged plasma kinetics. For the class of Slater ionic models, equations are derived for the mean occupation numbers of the electron shells and their fluctuations in the Fokker-Planck approximation. To calculate the fluctuations, the Fokker-Planck equation is linearized in the vicinity of the steady-state nonequilibrium solution to the kinetic equations (linear noise approximation). The method proposed allows one to take into account both the nonequilibrium correlations of the occupation-number fluctuations and the thermodynamically equilibrium statistical correlations related to the Coulomb interaction among bound electrons. The relation among the coefficients in the Fokker-Planck equation for the occupation-number fluctuations of the electron shells is discussed based on the fluctuation-dissipative theorem.     

Acetylcholine-induced current fluctuations in tissue-cultured muscle cells under voltage clamp.

Acetylcholine applied ionophoretically to chick skeletal muscle cells grown in tissue culture produces membrane current fluctuations. Cells treated with vinblastine are transformed to a roughly spherical shape. Such transformed cells can be voltage-clamped with microelectrodes. The frequency spectrum of the current fluctuations at fixed voltage obeys a relation of the Lorentz form. From analysis of the current noise, the conductance of a single ionic channel is estimated to be 39 pmho at a temperature of 28 degrees C, and increases with increasing temperature, exhibiting a Q10 of 1.7. The relaxation time for the channel conductance is more sharply temperature dependent, showing a Q10 of approximately 5. These results are in agreement with the picture of acetylcholine-activated ionic channels determined from experiments on frog end plate (Anderson and Stevens, 1973). The relaxation time for carbachol activation is shorter than for acetylcholine, and appears to be more temperature sensitive.     

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.     

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.     

Plasmalemma Voltage Noise in Chara corallina

Voltage noise analysis is applied to plasmalemma ion transport in Chara corallina. There is a component in the noise power spectrum that is probably associated with current fluctuations within passive transport channels, and another component that may be associated either with fluctuations in the number of open channels, or with active transport. The data allow the calculation of time constants that may be attributable to molecular level events in these transport processes.     

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.     

Charge pulse studies of transport phenomena in bilayer membranes. I. Steady-state measurements of actin- and valinomycin-mediated transport in glycerol monooleate bilayers.

A charge pulse technique has been applied to studies of transport phenomena in bilayer membranes. The membrane capacitance can be rapidly charged (in less than a microsecond). The charge then decays through the membrane's conductive mechanism-no current flows through the solution or external circuitry. The resulting voltage decay is thus a manifestation of membrane and boundary layer phenomena only. There are a number of advantages to this approach over conventional voltage or current-clamp techniques: the rise-time of the voltage perturbation is not limited by the time constant deriving from the membrane capacitance and solution resistance, thus permitting study of extremely rapid rate processes; the membrane is exposed to high voltage for relatively short times and thus can be subjected to higher voltages without breakdown; the steady-state current-voltage behavior of the membrane can be deduced from a single charge pulse experiment; the charge (and therefore the integral of the ion flux through the membrane) is monitored allowing detection of rate processes too rapid to follow directly. In this paper we present what is primarily a steady-state analysis of actin (non-, mon-, din-, trin-)-mediated transport of ammonium ion and valinomycin-mediated transport of cesium and potassium ions through glycerol monooleate bilayers. We introduce the concept of the "intercept discrepancy", a method for measuring charge lost through extremely rapid rate processes. Directly observable pre-steady-state phenomena are also discussed but will be the main subject of part II.     

Quantification of fluorophore copy number from intrinsic fluctuations during fluorescence photobleaching

We present a theoretical technique for quantifying the cellular copy-number of fluorophores that relies on the random nature of the photobleaching process. Our approach does not require single-molecule sensitivity, and therefore can be used with commonly used epifluorescence microscopes. Fluctuations arising from photobleaching can be used to estimate the proportionality between fluorescence intensity and copy-number, which can then be used with subsequent intensity measurements to estimate copy-number. We calculate the statistical errors of our approach and verify them with stochastic simulations. By using fluctuations over the entire photobleaching process, we obtain significantly smaller errors than previous approaches that have used fluctuations arising from cytoplasmic proteins partitioning during cellular division. From the time-dependence of the fluctuations as photobleaching proceeds, we can discriminate between desired photobleach fluctuations and background noise or photon shot noise. Our approach does not require cellular division and the photobleaching rate sets a timescale that is adjustable with respect to cellular processes. We hope that our approach will now be applied experimentally.     

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.