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.     

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 voltage curves of biomolecular lipid membranes.

The first part of this paper describes the current voltage curves of bimolecular membranes of oxidized cholesterol formed between two aqueous solutions of tetrabutylammonium chloride. These membranes are selectively permeable for cations and the membrane interfaces are electrically uncharged. The dependence of the membrane conductivity on the membrane potential can be described as the product of the conductivity at zero current ("zero conductivity") and a function called "overlinearity". The zero conductivity increases linearly with the concentration of tetrabutylammonium chloride. The overlinearity is independent of the concentration of tetrabutylammonium chloride. In the second part the Nernst-Planck and Poisson equations are integrated numerically for a three-phase system consisting of an aqueous electrolyte solution, a membrane and an aqueous electrolyte solution. Each phase is characterized by material constants. Appropriate boundary conditions cause the electric current to build up electrical double layers on both sides of the membrane. The opposing double layers with opposite electrical signs inject the soluble ions into the membrane. This ion injection accounts for the overlinearity of the current voltage curves, thus explaining the measured characteristics.     

Current noise parameters derived from voltage noise and impedance in embryonic heart cell aggregates.

We have recorded membrane impedance and voltage noise in the pacemaker range of potentials (-70 to -59 mV) from spheroidal aggregates of 7-d embryonic chick ventricle cells made quiescent by exposure to tetrodotoxin in medium containing 4.5 mM K+. The input capacitance is proportional to aggregate volume and therefore to total membrane area. The specific membrane capacitance is 1.24 microF/cm2. The input resistance at constant potential is inversely proportional to aggregate volume and therefore to total membrane area. The specific membrane resistance in 18 k omega . cm2 at -70 mV and increases to 81 k omega . cm2 at -59 mV. The RC time constant is 22 ms at -70 mV and increases to 146 ms at -59 mV. The aggregate transmembrane small-signal impedance can be represented by a parallel RC circuit itself in parallel with an inductive branch consisting of a resistor (rL) and an inductor (L) in series. The time constant of the inductive branch (L/rL) is 340 ms, and is only weakly dependent on potential. Correlation functions of aggregate voltage noise and the impedance data were modeled by a population of channels with simple open-close kinetics. The time constant of a channel (tau s) derived from the noise analysis is 300 ms. The low frequency limit of the pacemaker current noise (SI[0]), derived from the voltage noise and impedance, increases from 10(-20) A2/Hz . cm2 at -67 mV to 10(-19) A2/Hz . cm2 at -61 mV.     

Thickness fluctuations in black lipid membranes.

Because a black lipid membrane is compressible, there will be spontaneous fluctuations in its thickness. Qualitative arguments are given that the preferred configuration of the membranes is flat and that thickness fluctuations are smaller in amplitude than the differences in mean thickness observed using different hydrocarbon solvents. Fluctuations with short characteristic lengths will not be large as a result of the large amounts of oil-water contact these would entail. Quantitative analysis based on an extension of the treatment for soap films, predicts that the root mean square (rms) amplitude for fluctuations of wavelength longer than approximately 10 nm is negligible for glyceryl monooleate membranes with squalene (less than 3%) but may be approximately 20% with n-decane. rms fluctuations of 20% would lead to a discrepancy between the rms thickness of the core and the mean reciprocal thickness of only 6%.     

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.     

Monte-Carlo-simulations of voltage fluctuations in biological membranes in the case of small numbers of transport units

Some years ago a theory of non-equilibrium voltage fluctuations in biological membranes was developed (Frehland and Solleder 1985, 1986) under a linearisation condition which is valid for a great number of transport units. In order to get an insight into the stochastic behaviour of such systems, consisting of small numbers of transport units, we carried out Monte-Carlo-simulations and compared the mean voltage course and the spectral density with the results of the previous theory. Under parameter conditions of biological relevance no significant differences from the behaviour of systems with large numbers, as predicted from the earlier theory, could be found in the case of rigid pores and ion carriers. However, in the case of small numbers, channels with open-closed-kinetics showed great deviations. With increasing number of transport units agreement with the previous theory was obtained. Offprint requests to: B. Kleutsch     

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.     

Structural fluctuations and current noise of ionic channels.

In the open state of the acetylcholine-receptor channel an increased current noise is observed, which may result from conformational fluctuations of the channel protein (Sigworth, F.J., 1985, Biophys. J. 47:709-720). In this study the spectrum of the current noise is analyzed assuming that low-frequency motions of structural domains of the protein give rise to conductance fluctuations. The movement of a domain is treated as the motion of an elastically bound Brownian particle, which is described by the Langevin equation. The current-noise spectrum predicted by this model is given by a sum of Lorentzians; it agrees with the observed spectrum when it is assumed that only the slowest process can be resolved in the experiment. The large value of the friction coefficient, which is obtained from the corner frequency, indicates that domain motion is restricted mainly by peptide-peptide interactions.     

Protein attraction in membranes induced by lipid fluctuations.

The nonspecific lipid-mediated attraction between two proteins embedded in a bilayer membrane have been investigated for a model system using Monte Carlo simulations. We found two types of attraction with different regimes. A depletion-induced attraction in the range r < sigmaL, where sigmaL is the diameter of a lipid and r is the distance between the surfaces of the two proteins, and a fluctuation-induced attraction in the range 1 < r/sigmaL < 6, which originates from the gradients of density and orientational fluctuations of the lipids around each protein. The effective potential of the latter type of attraction decays exponentially with U(r) approximately exp(r/vi) where the correlation length is vi/sigmaL approximately 3.2 in the present model system.     

Structural equilibrium fluctuations in mesophilic and thermophilic alpha-amylase

By comparing a mesophilic alpha-amylase with its thermophilic homolog, we investigated the relationship between thermal stability and internal equilibrium fluctuations. Fourier transform infrared spectroscopy monitoring hydrogen/deuterium (H/D) exchange kinetics and incoherent neutron scattering measuring picosecond dynamics were used to study dynamic features of the folded state at room temperature. Fairly similar rates of slowly exchanging amide protons indicate about the same free energy of stabilization DeltaG(stab) for both enzymes at room temperature. With respect to motions on shorter time scales, the thermophilic enzyme is characterized by an unexpected higher structural flexibility as compared to the mesophilic counterpart. In particular, the picosecond dynamics revealed a higher degree of conformational freedom for the thermophilic alpha-amylase. The mechanism proposed for increasing thermal stability in the present case is characterized by entropic stabilization and by flattening of the curvature of DeltaG(stab) as a function of temperature.     

Dielectric noise in membranes

Most theoretical and experimental studies of electrical fluctuations in membranes so far have been devoted to noise associated with conduction processes. In this paper a different type of noise is described which results from dipolar transitions in the membrane. Two mechanisms for the generation of such dielectric noise are analyzed: (a) conformational transitions of membrane proteins involving changes in dipolar moment and/or polarizibility, and (b) rotation of dipolar molecules dissolved in the lipid. The spectral intensity of current noise calculated for the two models exhibits a characteristic dependence on frequency ω with a decrease proportional to $\text{ω}{}^2$ towards low frequencies and an approach to a frequency-independent (white noise) limit at high frequencies. For a given number of dipolar molecules in the membrane, the spectral intensity is inversely proportional to the square of the membrane thickness.     

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.     

Signal propagation in proteins and relation to equilibrium fluctuations

Elastic network (EN) models have been widely used in recent years for describing protein dynamics, based on the premise that the motions naturally accessible to native structures are relevant to biological function. We posit that equilibrium motions also determine communication mechanisms inherent to the network architecture. To this end, we explore the stochastics of a discrete-time, discrete-state Markov process of information transfer across the network of residues. We measure the communication abilities of residue pairs in terms of hit and commute times, i.e., the number of steps it takes on an average to send and receive signals. Functionally active residues are found to possess enhanced communication propensities, evidenced by their short hit times. Furthermore, secondary structural elements emerge as efficient mediators of communication. The present findings provide us with insights on the topological basis of communication in proteins and design principles for efficient signal transduction. While hit/commute times are information-theoretic concepts, a central contribution of this work is to rigorously show that they have physical origins directly relevant to the equilibrium fluctuations of residues predicted by EN models.     

Differentiation between equilibrium and nonequilibrium kinetic systems by noise analysis.

Methods for the differentiation between equilibrium and nonequilibrium steady-state kinetic mechanisms based on fluctuation and noise analysis are discussed. Specifically, the "sharpening" in the auto noise power spectrum is shown to be a useful indicator in identifying a nonequilibrium steady state.     

Transport of local anaesthetics across chromatophore membranes.

1. Both simple amines and tertiary amino local anaesthetics give rise to an accelerated decay of the absorption change of added pH indicator dyes and a decelerated decay of the endogenous carotenoid absorption band shift, following short flash excitation of Rhodopseudomonas sphaeroides chromatophores. 2. With increasing medium pH, lower concentrations of amine or local anaesthetics are effective. 3. The order of potency of the local anaesthetics concurs with their reported membrane/buffer partition coefficients and concentrations required for action potential blockade in nerve fibres. 4. The data are taken as evidence for rapid transport of the free base across the chromatophore membrane and relatively slow penetration of the protonated local anaesthetic. Protolytic reactions complete the effective dissipation of the trans-membrane pH gradient. 5. Benzocaine, with its unusually low pKa and the quaternary derivative, chloropromazine methiodide do not display this type of behaviour. 6. In the presence of membrane potential-collapsing agents, such as valinomycin/K+ or thiocyanate ions, local anaesthetics decelerate the decay of the cresol red change but have no effect on the carotenoid shift decay. It appears that transport of the unprotonated local anaesthetic although electrically neutral, requires the presence of a membrane potential. 7. In contrast, the non-anaesthetic amines act independently of the membrane potential. 8. Ca2+ interferes with the mechanism of local anaesthetic deceleration of the cresol red change decay in the presence of valinomycin/K+ or thiocyanate but not with other anaesthetic or amine reactions.     

Thermal fluctuations in membranes and the problem of mechanoreception

In order to estimate thermal fluctuations in erythrocyte and mechanoreceptor membranes the transverse fluctuations of plane and spherical bilayer membranes and the fluctuations of the surface of a part of such a membrane, possessing disc shape of the radius R are calculated. The obtained values of the transverse fluctuations are two orders smaller than in the paper [5]. Total plane fluctuations of the disc with R=5.10(-7) cm are some orders higher than the threshold values epsilonpi of relative deformation in mechanoreceptor membranes, but their value in the frequency range of 0 dividied by 20 kHz is of the same order as epsilonpi. The estimates of fluctuations are also produced for Pacinian corpuscle membrane and for the globular protein molecule. The conditions necessary for high sensitivity of mechanoreceptor membranes are the large value of Young modulus E and low membrane viscosity eta.     

Models for 1/f noise in nerve membranes.

A recently proposed model for 1/f(w-1) noise in nerve membrane (Clay and Schlesinger, 1976; Lundström and McQueen, 1974) is shown to be mathematically inconsistent in several respects. A self-consistent model based on similar membranes lipid orientation fluctuation effects is proposed.     

Cellular noise regulons underlie fluctuations in Saccharomyces cerevisiae

Stochasticity is a hallmark of cellular processes, and different classes of genes show large differences in their cell-to-cell variability (noise). To decipher the sources and consequences of this noise, we systematically measured pairwise correlations between large numbers of genes, including those with high variability. We find that there is substantial pathway variability shared across similarly regulated genes. This induces quantitative correlations in the expression of functionally related genes such as those involved in the Msn2/4 stress response pathway, amino-acid?biosynthesis, and mitochondrial maintenance. Bioinformatic analyses and genetic perturbations suggest that fluctuations in PKA and Tor signaling contribute to pathway-specific variability. Our results argue that a limited number of well-delineated "noise regulons" operate across a yeast cell and that such coordinated fluctuations enable a stochastic but coherent induction of functionally related genes. Finally, we show that pathway noise is a quantitative tool for exploring pathway features and regulatory relationships in un-stimulated systems.     

Open channel noise. I. Noise in acetylcholine receptor currents suggests conformational fluctuations.

The random passage of ions through an open channel is expected to result in shot noise fluctuations in the channel current. The patch-clamp technique now allows fluctuations of this size to be observed in single-channel currents. In the experiments reported here the acetylcholine-induced currents in cultured rat muscle cells were analyzed; fluctuations were found that were considerably larger than expected for shot noise. A low-frequency component, which was fitted with a Lorentzian, was examined in detail; it appears to arise from fluctuations in channel conductance of approximately 3% on a time scale of 1 ms. The characteristic relaxation time is voltage dependent and temperature dependent (Q10 approximately equal to 3) suggesting that the fluctuations arise from conformational fluctuations in the channel protein.