Relaxations, fluctuations and ion transfer across membranes

Megahertz Sampling of Prestin (SLC26a5) Voltage-Sensor Charge Movements in Outer Hair Cell Membranes Reveals Ultrasonic Activity that May Support Electromotility and Cochlear Amplification

Charged moieties in the outer hair cell (OHC) membrane motor protein, prestin, are driven by transmembrane voltage to power OHC electromotility (eM) and cochlear amplification (CA), an enhancement of mammalian hearing. Consequently, the speed of prestin's conformational switching constrains its dynamic influence on micromechanics of the cell and the organ of Corti. Corresponding voltage-sensor charge movements in prestin, classically assessed as a voltage-dependent, nonlinear membrane capacitance (NLC), have been used to gauge its frequency response, but have been validly measured only out to 30 kHz. Thus, controversy exists concerning the effectiveness of eM in supporting CA at ultrasonic frequencies where some mammals can hear. Using megahertz sampling of guinea pig (either sex) prestin charge movements, we extend interrogations of NLC into the ultrasonic range (up to 120 kHz) and find an order of magnitude larger response at 80 kHz than previously predicted, indicating that an influence of eM at ultrasonic frequencies is likely, in line with recent in vivo results (Levic et al., 2022). Given wider bandwidth interrogations, we also validate kinetic model predictions of prestin by directly observing its characteristic cut-off frequency under voltage-clamp as the intersection frequency (Fis), near 19 kHz, of the real and imaginary components of complex NLC (cNLC). The frequency response of prestin displacement current noise determined from either the Nyquist relation or stationary measures aligns with this cut-off. We conclude that voltage stimulation accurately assesses the spectral limits of prestin activity, and that voltage-dependent conformational switching is physiologically significant in the ultrasonic range.SIGNIFICANCE STATEMENT The motor protein prestin powers outer hair cell (OHC) electromotility (eM) and cochlear amplification (CA), an enhancement of high-frequency mammalian hearing. The ability of prestin to work at very high frequencies depends on its membrane voltage-driven conformation switching. Using megahertz sampling, we extend measures of prestin charge movement into the ultrasonic range and find response magnitude at 80 kHz an order of magnitude larger than previously estimated, despite confirmation of previous low pass characteristic frequency cut-offs. The frequency response of prestin noise garnered by the admittance-based Nyquist relation or stationary noise measures confirms this characteristic cut-off frequency. Our data indicate that voltage perturbation provides accurate assessment of prestin performance indicating that it can support cochlear amplification into a higher frequency range than previously thought.

Fractal methods to analyze ion channel kinetics

We describe the traditional nonfractal and the new fractal methods used to analyze the currents through ion channels in the cell membrane. We discuss the hidden assumptions used in these methods and how those assumptions lead to different interpretations of the same experimental data. The nonfractal methods assumed that channel proteins have a small number of discrete states separated by fixed energy barriers. The goal was to determine the parameters of the kinetic diagram, which are the number of states, the pathways between them, and the kinetic rate constants of those pathways. The discovery that these data have fractal characteristics suggested that fractal approaches might provide more appropriate tools to analyze and interpret these data. The fractal methods determine the characteristics of the data over a broad range of time scales and how those characteristics depend on the time scale at which they are measured. This is done by using a multiscale method to accurately determine the probability density function over many time scales and by determining how the effective kinetic rate constant, the probability of switching states, depends on the effective time scale at which it is measured. These fractal methods have led to new information about the physical properties of channel proteins in terms of the number of conformational substates, the distribution of energy barriers between those states, and how those energy barriers change with time. The new methods developed from the fractal paradigm shifted the analysis of channel data from determining the parameters of a kinetic diagram to determining the physical properties of channel proteins in terms of the distribution of energy barriers and/or their time dependence.

Robust, high-resolution, whole cell patch-clamp capacitance measurements using square wave stimulation

High-resolution, whole cell capacitance measurements are usually performed using sine wave stimulation using a single frequency or a sum of two frequencies. We present here a high-resolution technique for whole-cell capacitance measurements based on square-wave stimulation. The square wave represents a sum of sinusoidal frequencies at odd harmonics of the base frequency, the amplitude of which is highest for the base frequency and decreases as the frequency increases. The resulting currents can be analyzed by fitting the current relaxations with exponentials, or by a phase-sensitive detector technique. This method provides a resolution undistinguishable from that of single-frequency sine wave stimulation, and allows for clear separation of changes in capacitance, membrane conductance, and access resistance. In addition, it allows for the analysis of more complex equivalent circuits as associated with the presence of narrow fusion pores during degranulation, tracking many equivalent circuit parameters simultaneously. The method is insensitive to changes in the reversal potential, pipette capacitance, or widely varying cell circuit parameters. It thus provides important advantages in terms of robustness for measuring cell capacitances, and allows analysis of complicated changes of the equivalent circuits.

A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transport through the gramicidin A channel

A lattice relaxation algorithm is developed to solve the Poisson-Nernst-Planck (PNP) equations for ion transport through arbitrary three-dimensional volumes. Calculations of systems characterized by simple parallel plate and cylindrical pore geometries are presented in order to calibrate the accuracy of the method. A study of ion transport through gramicidin A dimer is carried out within this PNP framework. Good agreement with experimental measurements is obtained. Strengths and weaknesses of the PNP approach are discussed.

Cooperative charge fluctuations by migrating protons in globular proteins

A review of the hydrogen bonded network on the protein surface shows the presence of a charged complex system with parallel and competitive interactions, including ionizable side-chains, migrating protons, bound water and nearby backbone peptides. This system displays cooperative effects of dynamical nature, reviewed for lysozyme as a case. By increasing the water coverage of the protein powder, the bound water cluster exhibits a percolative transition, detectable by the onset of large water-assisted displacements of migrating protons, with a parallel emergence of protein mobility and biological function. By lowering the temperature, migrating protons exhibit a glassy dielectric relaxation in the low frequency range, pointing to a frustration by competing interactions similar to that observed in spin glasses and fragile glass forming liquids. The observation of these dissipative processes implies the occurrence of spontaneous charge fluctuations. A simplified model of the protein surface, where conformational and ionizable side-chain fluctuations are averaged out, is used to discuss the statistical physics of these cooperative effects. Some biological implications of this dynamical cooperativity for enzymatic activity are briefly suggested at the end.

A parametric modeling of ionic channel current fluctuations using third-order statistics and its application to estimation of the kinetic parameters of single ionic channels

A parametric modeling of stationary ionic-channel current fluctuations (SICF's) using third-order cumulants is presented and its application to estimation of the kinetic parameters of single ionic channels is discussed. We consider the case where third-order cumulants of SICF's are nonzero, and where SICF's are corrupted by an unobservable additive colored Gaussian noise that is independent of SICF's. First, we construct a virtual synthesizer that yields an output whose third-order cumulants are equivalent to those of SICF's on a specific slice. The synthesizer output is expressed by the sum of N5 - 1 first-order differential equation systems, where N8 denotes the number of states of single ionic channels. Next, discretizing the synthesizer output, we derive a discrete autoregressive (AR(N8 - 1)) process driven by the sum of N8 - 1 moving average (MA(N9 - 2)) processes. Then the AR coefficients are explicitly related to the kinetic parameters of single ionic channels, implying that the kinetic parameters can be estimated by identifying the ARMA coefficients using the third-order cumulants. In order to assess the validity of the proposed modeling and the accuracy of parameter estimates, Monte Carlo simulation is carried out in which the closed-open and closed-open-blocked schemes are treated as specific examples.

Cooperative gating of chloride channel subunits in endothelial cells

New methods are described to detect subconductance levels and to analyse ion channel gating. These methods are applied to simulated and experimental data. Single chloride channel records from inside-out membrane patches excised from human umbilical venous endothelial cells (HUVEC) exhibit, in addition to the full closed and full open configurations, intermediate subconductance levels which are multiple of an elementary conductance of 112.5 pS. Analysis of transitions from one state to another and the comparison of real data with simulated data leads to the proposal of a cooperative model of gating for the observed subunits of a chloride channel.

A source of bias in the analysis of single channel data: assessing the apparent interaction between channel proteins

A recent study of single sodium channel currents in neuroblastoma cells suggested interaction between ion channels in close proximity to one another (T. Kiss and K. Nagy, Eur. Biophys. J. 12, 13, 1985). The opening of one channel appeared to affect the likelihood that neighboring channels might open. Some of the conclusions were based on the analysis of observed channel openings that were segregated depending on whether one channel or more than one channel was open at the same time. We hypothesized that the longer one channel remained open, the more likely another channel operating independently, would open, thereby creating the impression of an apparent coupling of channel behavior. We performed simulations and measurements of single sodium channel currents to determine whether the technique of event segregation could account for apparent channel interactions. The simulations showed that the segregation of overlapping (more than one channel open at the same time) and nonoverlapping events led to a bias in the estimated open time and the derived closing rate. To avoid the bias, we found that random pairing of opening and closing events provided an unbiased estimate of the mean closing rate. Using this random assignment approach, we showed that the mean closing rate of single sodium channels in neonatal rat myocytes decreased with depolarization over a limited range of membrane potential. This suggested that the underlying closure mechanism(s) was voltage dependent. From the analysis of open times, we found no evidence for channel interaction in the time scale of tens of milliseconds. Depolarizing steps without events occurred in runs suggesting the existence of long-lived shut state(s). Double pulse experiments with the prepulse and test pulse above threshold showed significant inactivation of channels that did not open. The rate of inactivation of shut channels was substantially slower than the closure rate of open channels. The rate of inactivation of cardiac sodium channels appeared to be strongly dependent on the initial channel state.

Determination of K(+)-channel relaxation times in squid axon membrane by Hodgkin-Huxley and by direct linear analysis

An assumption in the use of the Hodgkin-Huxley (HH) formulation (A.L. Hodgkin and A.F. Huxley, J. Physiol. 117 (1952) 500) to extract kinetic parameters from ion conductance responses to step voltage changes across biological membranes is that estimates obtained from such an analysis are equivalent to those obtained by direct, small-perturbation analysis. Comparison of the estimates of the K(+)-conductance relaxation time, tau n, derived from HH vs rapid, complex admittance determinations in the same squid giant axons shows significant differences for the same step changes over a 60 mV range from holding (-65 mV). The admittance determinations (2.5-5000 Hz) are shown to satisfy criteria of linear analysis (i.e., estimates are equivalent to a small-step analysis and are time invariant). The discrepancies between the two methods arise from the fact that the HH power-law description for a constant power does not yield a best fit of data over the voltage range examined and thus best estimates of tau n are power dependent. Furthermore, the large step changes in membrane voltage may excite nonlinear modes unrelated to conductance gating that contaminate the data to which the nonlinear formulation is applied to estimate linear kinetic parameters. Thus, the long-standing assumption that application of the HH methodology and empiricism is equivalent to a direct linear analysis is not substantiated. This result suggests that in comparisons between microscopic and macroscopic conduction data, microkinetic parameters derived from analysis of single ion-channel data should not be compared to macrokinetic parameters from a large population of the channel derived by HH analysis.

Using fractals to understand the opening and closing of ion channels

Looking at an old problem from a new perspective can sometimes lead to new ways of analyzing experimental data which may help in understanding the mechanisms that underlie the phenomena. We show how the application of fractals to analyze the patch clamp recordings of the sequence of open and closed times of cell membrane ion channels has led to a new description of ion channel kinetics. This new information has led to new models that imply: (a) ion channel proteins have many conformational states of nearly equal energy minima and many pathways connecting one conformational state to another, and (b) that these many states are not independent but are linked by physical mechanisms that result in the observed fractal scaling. The first result is consistent with many experiments, simulations, and theories of globular proteins developed over the last decade. The second result has stimulated the suggestion of several different physical mechanisms that could cause the fractal scalings observed.

The inactivation of sodium channels in the node of Ranvier and its chemical modification

The many experimental studies reported demonstrate the complexity of what is termed inactivation, the decrease of current flow through sodium channels at maintained depolarization. Even at the normal resting potential of, say, -70 mV for a frog node of Ranvier, ca. 20% of the channels are closed and inactivated, i.e., incapable of passing current on a sudden depolarization, in contrast to the remaining 80% of closed but resting channels. The term inactivation has thus evolved from bulk current ("macroscopic") phenomena and is applied to channels although its single-channel ("microscopic") basis is not entirely clear and may even vary among preparations. It is conceivable that the macroscopic phenomenon may have more than a single microscopic cause; this point will probably not be settled until a physical description of the conformational states of the channel macromolecule becomes available. At any rate, channel transition into an inactivated closed state can be easily affected by numerous reagents of highly diverse chemical nature and, most likely, different primary sites of action as already suggested by the sidedness of effective application, e.g., iodate and endopeptidases to the inside, polypeptide toxins to the outside. But also the search for a common denominator, a secondary target of all these treatments, has not been very successful as demonstrated by the experiments with group-specific reagents. Since modification of inactivation is often accompanied by shifts in the voltage dependence of gating parameters, a target could be the "voltage sensor" of the channel, charged and/or dipolar components of the channel macromolecule that, by being moved in the electric field, somehow induce gating and whose movement is measured as gating current (e.g, Hille, 1984). The fraction of open channels as a function of membrane potential, F(E), may serve as an indicator. It may be simply shifted (to more negative potentials) as by veratridine (Leibowitz et al., 1987) or flattened (reduction of gating charge?) and shifted (in the positive direction) as by Anemonia sulcata toxin II (Ulbricht and Schmidtmayer, 1981) or chloramine-T (Drews, 1987). On the other hand, the steady-state inactivation curve is shifted to more negative potentials by the toxin (Ulbricht and Schmidtmayer, 1981), but to more positive potentials by chloramine-T (Wang, 1984a; Schmidtmayer, 1985). Obviously, modifiers may affect activation and inactivation quite differently, a result that touches on the question as to what extent inactivation derives its potential dependence from activation.(ABSTRACT TRUNCATED AT 400 WORDS)

A parametric modeling of membrane current fluctuations with its application to the estimation of the kinetic properties of single ionic channels

A parametric autoregressive moving average (ARMA) signal modeling of membrane current fluctuations observed in biomembranes is described. Kinetic properties of single ionic channels contributing to membrane current fluctuations and the parameters of the corresponding ARMA process are explicitly related. The model was shown to be effectively applied to the estimation of the kinetic parameters of single ionic channels. Estimation of the parameters via ARMA signal identification was examined in detail for the basic closed-open scheme and the three-state sequential blocking scheme. The estimation accuracy of this method was theoretically evaluated. Computer simulation revealed the validity of the proposed modeling and the effectiveness of the parametric method for the estimation of kinetic parameters when the model was applied to the estimation. The proposed modeling may form a theoretical base for the parametric analysis of membrane current fluctuations in a variety of kinetic schemes.

Statistical discrimination of fractal and Markov models of single-channel gating

A statistical comparison is presented of Markov and fractal models of ion channel gating. The analysis is based on single-channel data from two types of ion channels: open times from a 90 pS Ca-activated K channel from GH3 pituitary cells, and closed times from a nonselective channel from rabbit corneal endothelium (Liebovitch et al., 1987a). Maximum likelihood methods were used to fit the data. For both data sets the best Markov model had three exponential components. The best Markov model had a higher likelihood than the fractal model, and the Asymptotic Information Criterion favored the Markov model for each data set. A more detailed analysis, using the Monte Carlo methods described in Horn (1987), showed that the Markov model was not significantly better than the fractal model for the corneal endothelium channels. The inability to discriminate the models definitively in this case was shown to be due in part to the small size of the data set.

Patch-clamp techniques for time-resolved capacitance measurements in single cells

Two methods are described for estimation of passive cell parameters such as membrane capacitance, membrane conductance and access resistance in tight-seal whole cell recording. Both methods are restricted in their application to cases where the cell under study can be approximated by a simple three-component network with linear properties over some voltage range. One method, referred to as the time domain technique, requires only standard electrophysiological equipment and a computer. Parameters are derived from an analysis of capacitive transients during square wave stimulation. It is readily adaptable to wide variations in experimental parameters. Particularly, it is equally applicable to the "slow whole-cell" configuration (access resistance in the range 100 M omega to 1 G omega) and to normal whole-cell measurements (access resistance typically 10 M omega). The other method applies a sine wave command signal to the cell and employs a lock-in amplifier to analyse the resulting current signal. Two modes of operating the lock-in amplifier are described. One mode provides an output signal directly proportional to small changes in capacitance at maximum resolution (1-10 fF). The other mode, in conjunction with a digital computer, supplies estimates of all passive cell parameters, as does the time domain technique, but with a large amount of data reduction performed by the lock-in amplifier itself. Due to the special hardware, however, this method is not as flexible as the time domain technique.

Indications of the existence of ferroelectric units in excitable-membrane channels

Previous work in excitability has focused primarily on the mathematical description of the phenomena, while mechanisms postulated to explain these were simple mechanical interpretations of the terms of this description. The problem considered here is that of the physical mechanism underlying excitation. The experimental facts to be explained must be not only the electrical behavior of the membrane, but also its electromechanical, electro-optic and thermoelectric behavior. Previous work on the physically grounded electrodiffusion theory foundered not because of the incorrectness of the electrodiffusion approach, but because the assumed description of the dielectric properties of the membrane was too simple. Extension of the dielectric equation of state to a nonlinear polynomial form converts the classical electrodiffusion system of equations into a nonlinear polynomial form converts the classical electrodiffusion system of equations into a ferroelectric electrodiffusion system. The consideration of ferroelectric behavior in excitable channels makes possible straight-forward physical explanation of the phenomena of membrane swelling during action potential, currents induced by temperature changes, transition temperatures, current-voltage hysteresis, nonlinear electrical behavior, voltage-dependent birefringence and rectangular pulses from single channels. The hypothesis is therefore proposed that excitable channels contain ferroelectric transmembrane units. These may be crystals or liquid crystals.

Closed channel-open channel equilibrium of the sodium channel of nerve. Simple models of macromolecular equilibria

The consistency of an electrodiffusion kinetics to describe the time-dependent opening of sodium channels of nerve suggests that motions over relatively long distances (on the atomic scale) are involved in the equilibrium as well. As a result, it is expected that a relatively large fraction of possible macromolecular conformations are unreactive. An equilibrium constant between locally reactive forms and the unreactive conformations is introduced. The consequences of this formalism is investigated in a square well potential, a harmonic potential, and a system consisting of two harmonic potentials with different spatial extents. The limits of knowledge from Nernstian behavior are shown. As an alternative to the Nernstian analysis, the experimental data of the sodium channel's quasi-equilibrium - the probability of the channel's being open as a function of voltage - can be described as resulting from motion caused by an electric field on a charge which is confined by a harmonic potential. A force constant is found from this analysis. (Such Hookian force constants cannot be found from spectroscopic experiments in condensed systems where the large-displacement vibrations are overdamped and, hence, spectroscopically unobservable). From the force constant, an approximate value of the Young's modulus can be calculated. The modulus' value falls in the range for rubber. As for rubbers, the restoring force is, then, expected to be mostly entropic rather than enthalpic in origin. Using the appropriate theory for linear chains of rubber and the Young's modulus, the approximate length of the chain causing the rubber-like force is calculated. The result is found to be near the length suggested for the hydrophilic chains that connect transmembrane sections of the sodium channel.

The effects of n-pentane on voltage-clamped squid nerve sodium currents. A reinterpretation using kinetics of ordered systems

The sodium-current voltage-clamp data of Haydon and Kimura obtained on squid nerves treated with n-pentane (J. Physiol. 312 (1981) 57) are fitted with a previously described model (K.A. Rubinson, J. Physiol. 281 (1978) 14P; Biophys. Chem. 15 (1982) 245). The apparently complex action of the perturbant can be interpreted as due to a shift in shielding of the applied potential jumps, a change in channel conductivity, and an increase in the rate constant of channel shutoff. The shift in shielding due to n-pentane is found to be quantitatively the same for variables describing both kinetic and equilibrium quantities, which are independent. The transmembrane sodium potential remains unchanged, however.