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).     

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.     

Potassium-ion conduction noise in squid axon membrane.

Spectral analysis (1-1000 Hz) of spontaneous fluctuations of potential and current in small areas of squid (Loligo pealei) axon shows two forms of noise: f-1 noise occurs in both excitable and inexcitable axons with an intensity which depends upon the driving force for potassium ions. The other noise has a spectral form corresponding to a relaxation process, i.e. its asymptotic behavior at low frequencies is constant, and at high frequencies it declines with a slope of -2. This latter noise occurs only in excitable axons and was identified in spectra by (1) its disappearance after reduction of K+ current by internal perfusion with solutions containing tetraethylammonium (TEA+), Cs+ or reduced [Ki+] and (2) its insensitivity to block of Na+ conduction and active transport. The transition frequency of relaxation spectra are also voltage and temperature dependent and relate to the kinetics of K+-conduction in the Hodgkin-Huxley formulation. These data strongly suggest that the relaxation noise component arises from the kinetic properties of K+ channels. The f-1 noise is attributed to restricted diffusion in conducting K+ channels and/or leakage pathways. In addition, an induced K+ conduction noise associated with the binding of TEA+ and triethyldecylammonium ion to membrane sites is described. Measurement of the induced noise may provide an alternative means of characterizing the kinetics of interaction of these molecules with the membrane and also suggests that these and other pharmacological agents may not be useful in identifying noise components related to the sodium conduction mechanism which, in these experiments, appears to be much lower in intensity than either the normal K conduction or induced noise components.     

Water transport across plant tissue: Role of water channels

The contribution of water-filled, selective membrane pores (water channels) is integrated into a general concept of water transport in plant tissue. The concept is based on the composite anatomical structure of tissues which results in a composite transport pattern. Three main pathways of water flow have been distinguished, ie the apoplastic, symplastic and transcellular (vacuolar) paths. Since the symplastic and transcellular components can not be distinguished experimentally, these components are summarized as a cell-to-cell component. Water channel activity may control the overall water flow across tissues provided that the contribution of the apoplastic component is relatively low. The composite transport model has been applied to roots where most of the data are available. Comparison of the hydraulic conductivity at the root cell and organ levels shows that, depending on the species, there may be a dominating cell-to-cell or apoplastic water flow. Most remarkably, there are differences in the hydraulic conductivity of roots which depend on the nature of the force used to drive water flows (osmotic or hydrostatic pressure gradients). This is predicted by the model. The composite transport model explains low reflection coefficients of roots, the variability in root hydraulic resistance and differences between herbaceous and woody species. It is demonstrated that there is also a composite transport of water at the membrane level (water channel arrays vs bilayer arrays). This results in low reflection coefficients of plasma membranes for certain test solutes as derived for isolated internodes of Chara. The titration of water channel activity in this alga with mercurials and its dependence on changes in temperature or external concentration show that water channels do not exclusively transport water. Rather, they are permeable to relatively big uncharged organic solutes. The result indicates that, at least for Chara, the concept of an exclusive transport of water across water channels has to be questioned.     

Theory of transport noise in membrane channels with open-closed kinetics

A theoretical approach to transport noise in kinetic systems, which has recently been developed, is applied to electric fluctuations around steady-states in membrane channels with different conductance states. The channel kinetics may be simple two state (open-closed) kinetics or more complicated. The membrane channel is considered as a sequence of binding sites separated by energy barriers over which the ions have to jump according to the usual single-file diffusion model. For simplicity the channels are assumed to act independently. In the special case of ionic movement fast compared with the channel open-closed kinetics the results agree with those derived from the usual Master equation approach to electric fluctuations in nerve membrane channels.For the simple model of channels with one binding site and two energy barries the coupling between the fluctuations coming from the open-closed kinetics and from the jump diffusion is investigated.     

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.     

Mechanotransduction by Membrane Proteins: The speed of the hair cell mechanotransducer channel revealed by fluctuation analysis

Although mechanoelectrical transducer (MET) channels have been extensively studied, uncertainty persists about their molecular architecture and single-channel conductance. We made electrical measurements from mouse cochlear outer hair cells (OHCs) to reexamine the MET channel conductance comparing two different methods. Analysis of fluctuations in the macroscopic currents showed that the channel conductance in apical OHCs determined from nonstationary noise analysis was about half that of single-channel events recorded after tip link destruction. We hypothesized that this difference reflects a bandwidth limitation in the noise analysis, which we tested by simulations of stochastic fluctuations in modeled channels. Modeling indicated that the unitary conductance depended on the relative values of the channel activation time constant and the applied low-pass filter frequency. The modeling enabled the activation time constant of the channel to be estimated for the first time, yielding a value of only a few microseconds. We found that the channel conductance, assayed with both noise and recording of single-channel events, was reduced by a third in a new deafness mutant, Tmc1 p.D528N. Our results indicate that noise analysis is likely to underestimate MET channel amplitude, which is better characterized from recordings of single-channel events.     

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.     

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.     

Stochastic environmental fluctuations drive epidemiology in experimental host–parasite metapopulations

Environmental fluctuations are important for parasite spread and persistence. However, the effects of the spatial and temporal structure of environmental fluctuations on host–parasite dynamics are not well understood. Temporal fluctuations can be random but positively autocorrelated, such that the environment is similar to the recent past (red noise), or random and uncorrelated with the past (white noise). We imposed red or white temporal temperature fluctuations on experimental metapopulations of Paramecium caudatum, experiencing an epidemic of the bacterial parasite Holospora undulata. Metapopulations (two subpopulations linked by migration) experienced fluctuations between stressful (5°C) and permissive (23°C) conditions following red or white temporal sequences. Spatial variation in temperature fluctuations was implemented by exposing subpopulations to the same (synchronous temperatures) or different (asynchronous temperatures) temporal sequences. Red noise, compared with white noise, enhanced parasite persistence. Despite this, red noise coupled with asynchronous temperatures allowed infected host populations to maintain sizes equivalent to uninfected populations. It is likely that this occurs because subpopulations in permissive conditions rescue declining subpopulations in stressful conditions. We show how patterns of temporal and spatial environmental fluctuations can impact parasite spread and host population abundance. We conclude that accurate prediction of parasite epidemics may require realistic models of environmental noise.     

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.     

Kinetics of light-sensitive channels in vertebrate photoreceptors

We have studied the ion channels mediating the light response of vertebrate rod photoreceptors by analysing fluctuations in the current across the rod membrane, using the whole cell patch-clamp technique on rods isolated from the axolotl retina. Light decreases the membrane current fluctuations. Noise analysis reveals two components to this decrease: a low frequency component due to biochemical noise in the transduction mechanism, and a high frequency component we attribute to the random opening and closing of the ion channels in the dark. The probability of any one channel being open in the dark is low. The spectrum of the high frequency component of the current fluctuations indicates that the current through an open channel is 4 X 10(-15)A, that the mean channel open time is 2 ms, and that about 10000 channels are open in each rod in the dark. The effect of light is to reduce the opening rate constant of these channels, with no effect on the closing rate constant.     

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.     

The effect of electrical gradients on current fluctuations and impedance recorded fromNecturus gallbladder

Summary Transepithelial current fluctuations were recorded inNecturus gallbladder, clamped at negative as well as positive potentials up to 64 mV. With NaCl-Ringer's (+10mm TAP) on both sides a mucosa-negative potential enhanced the relaxation noise component, present at zero potential, and produced peaking in the power spectrum at potentials above –36mV. Concomitantly at these potentials an inductive as well as a capacitive low-frequency feature appeared in the impedance locus. Clamping at positive potentials of 18 mV suppressed the relaxation noise component. At potentials above 51mV the spectral values increased predominantly at low frequencies. In this case the power spectrum showed only a 1/f noise component. The experiments confirm the previous finding that a K⁺ efflux through fluctuating apical K⁺ channels exists under normal conditions. With serosal KCl-Ringer's the initial Lorentzian component was enhanced at negative but suppressed at positive potentials. The increase at negative potentials was less pronounced than in experiments with NaCl-Ringer's on both sides, indicating saturation of the fluctuating K⁺ current component. With mucosal KCl-Ringer's a negative potential depressed the initial relaxation noise component, whereas it was enhanced at +18 mV clamp potential. In the latter case an additional Lorentzian component became apparent at higher frequencies. At potentials of 36 mV and above the low-frequency Lorentzian disappeared whereas the corner frequency of the high-frequency component increased. The latter experiments demonstrate that the relaxation noise component inNecturus gallbladder consists of two superimposed Lorentzians. As the relaxation times of these two components behave differently under an electrical field, there may exist two different types of K⁺ channels. It is demonstrated that peaking in the plateau of power spectra can be explained by frequency-dependent attenuation effects, caused by a polarization impedance.     

Calcium-activated potassium conductance noise in snail neurons

Current fluctuations were measured in small, 3-6 micrometers-diameter patches of soma membrane in bursting neurons of the snail, Helix pomatia. The fluctuations dramatically increased in magnitude with depolarization of the membrane potential under voltage clamp conditions. Two components of conductance noise were identified in the power spectra calculated from the membrane currents. One component had a corner frequency which increased with depolarization. This component was blocked by intracellular injection of TEA and was relatively insensitive to extracellular calcium levels (as long as the total number of effective divalent cations remained constant). It was identified as fluctuations of the voltage-dependent component of delayed outward current. The second component of conductance noise had a corner frequency which decreased with depolarization. It was relatively unaffected by TEA injection and was reversibly blocked by substitution of extracellular calcium with magnesium, cobalt, or nickel. This second component of noise was identified as fluctuations of the calcium-dependent potassium current. The results suggest that the two components of delayed outward current are conducted through physically distinct channels.     

Backscattering of gyrotron radiation and short-wavelength turbulence during electron cyclotron resonance plasma heating in the L-2M stellarator

Backscattering of gyrotron radiation (θ = π) by short-wavelength density fluctuations (k _⊥ = 30 cm^?1) in the plasma of the L-2M stellarator was studied under conditions of electron cyclotron resonance (ECR) plasma heating at the second harmonic of the electron gyrofrequency (75 GHz). The scattering of the O-wave emerging due to the splitting of the linearly polarized gyrotron radiation into the X- and O-waves was analyzed. The signal obtained after homodyne detection of scattered radiation is a result of interference of the reference signal, the quasi-steady component, and the fast oscillating component. The coefficients of reflection of the quasi-steady component, R ₌ ² (Y), and fast oscillating component, R _～ ² (Y), of scattered radiation are estimated. The growth of the R _～ ² (Y) coefficient from 3.7 × 10^?4 to 5.2 × 10^?4 with increasing ECR heating power from 190 to 430 kW is found to correlate with the decrease in the energy lifetime from 1.9 to 1.46 ms. The relative density of short-wavelength fluctuations is estimated to be 〈n _～ ² 〉/〈n _e ² 〉 = 3 × 10^?7. It is shown that the frequencies of short-wavelength fluctuations are in the range 10–150 kHz. The recorded short-wavelength fluctuations can be interpreted as structural turbulence, the energy of which comprises ～10% of the total fluctuations energy. Simulations of transport processes show that neoclassical heat fluxes are much smaller than anomalous ones. It is suggested that short-wavelength turbulence plays a decisive role in the anomalous heat transport.     

Hofmeister effect in ion transport: reversible binding of halide anions to the roflamycoin channel.

We have studied the anion-dependent gating of roflamycoin ion channels using spectral analysis of noise in currents through multichannel planar lipid bilayers. We have found that in addition to low frequency current fluctuations that may be attributed to channel switching between open and closed conformations, roflamycoin channels exhibit a pronounced higher frequency noise indicating that the open channel conductance has substates with short lifetimes. This noise is well described by a Lorentzian spectrum component with a characteristic cutoff frequency that depends on the type of halide anions according to their position in the Hofmeister series. It is suggested that transitions between the substates correspond to a reversible ionization of the channel by a penetrating anion that binds to the channel structure, more chaotropic anions being bound for longer times. Within a framework of a two-substate model, the duration of the substate with reduced electrostatic barrier for cation current varies exponentially with anion electron polarizability. This explains two features of the roflamycoin channel reported earlier: the increase in apparent single-channel conductance along the series F- < Cl- < Br- < I- and the reverse of channel selectivity from anionic for KF to cationic for KI.     

Theory of Threshold Fluctuations in Nerves: II. Analysis of Various Sources of Membrane Noise

Threshold fluctuations in axon firing can arise as a result of electrical noise in the excitable membrane. A general theoretical expression for the fluctuations is applied to the analysis of three sources of membrane noise: Johnson noise, excess 1/f noise, and sodium conductance fluctuations. Analytical expressions for the width of the firing probability curve are derived for each of these noise sources. Specific calculations are performed for the node of Ranvier of the frog, and attention is given to the manner in which threshold fluctuations are affected by variations of temperature, ion concentrations, and the application of various drugs. Comparison with existing data suggests that threshold fluctuations can best be explained by sodium conductance fluctuations. Additional experiments directed at distinguishing among the various noise sources are proposed.     

Noise analysis reveals K+ channel conductance fluctuations in the apical membrane of rabbit colon

Summary In this paper we describe current fluctuations in the mammalian epithelium, rabbit descending colon. Pieces of isolated colon epithelium bathed in Na⁺ or K⁺ Ringer's solutions were studied under short-circuit conditions with the current noise spectra recorded over the range of 1–200 Hz. When the epithelium was bathed on both sides with Na⁺ Ringer's solution (the mucosal solution contained 50 m amiloride), no Lorentzian components were found in the power spectrum. After imposition of a potassium gradient across the epithelium by replacement of the mucosal solution by K⁺ Ringer's (containing 50 m amiloride), a Lorentzian component appeared with an average corner frequency,f _c=15.6±0.91 Hz and a mean plateau valueS _o=(7.04±2.94)×10^–20 A² sec/cm². The Lorentzian component was enhanced by voltage clamping the colon in a direction favorable for K⁺ entry across the apical membrane. Elimination of the K⁺ gradient by bathing the colon on both sides with K⁺ Ringer's solutions abolished the noise signal. The Lorentzian component was also depressed by mucosal addition of Cs⁺ or tetraethylammonium (TEA) and by serosal addition of Ba²⁺. The one-sided action of these K⁺ channel blockers suggests a cellular location for the fluctuating channels. Addition of nystatin to the mucosal solution abolished the Lorentzian component. Serosal nystatin did not affect the Lorentzian noise. This finding indicates an apical membrane location for the fluctuating channels. The data were similar in some respects to K⁺ channel fluctuations recorded from the apical membranes of amphibian epithelia such as the frog skin and toad gallbladder. The results are relevant to recent reports concerning transcellular potassium secretion in the colon and indicate that the colon possesses spontaneously fluctuating potassium channels in its apical membranes in parallel to the Na⁺ transport pathway.     

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.