Fluctuation and linear analysis of Na-current kinetics in squid axon

The power spectrum of current fluctuations and the complex admittance of squid axon were determined in the frequency range 12.5 to 5,000 Hx during membrane voltage clamps to the same potentials in the same axon during internal perfusion with cesium. The complex admittance was determined rapidly and with high resolution by a fast Fourier transform computation of the current response, acquired after a steady state was attained, to a synthesized signal with predetermined spectral characteristics superposed as a continuous, repetitive, small perturbation on step voltage clamps. Linear conduction parameters were estimated directly from admittance data by fitting an admittance model, derived from the linearized Hodgkin-Huxley equations modified by replacing the membrane capacitance with a "constant-phase-angle" capacitance, to the data. The constant phase angle obtained was approximately 80 degrees. At depolarizations the phase of the admittance was 180 degrees, and the real part of the impedance locus was in the left-half complex plane for frequencies below 1 kHz, which indicates a steady-state negative Na conductance. The fits also yielded estimates of the natural frequencies of Na "activation" and "inactivation" processes. By fitting Na-current noise spectra with a double Lorentzian function, a lower and an upper corner frequency were obtained; these were compared with the two natural frequencies determined from admittance analysis at the corresponding potentials. The frequencies from fluctuation analyses ranged from 1.0 to 10.3 times higher than those from linear (admittance) analysis. This discrepancy is consistent with the concept that the fluctuations reflect a nonlinear rate process that cannot be fully characterized by linear perturbation analysis. Comparison of the real part of the admittance and the current noise spectrum shows that the Nyquist relation, which generally applies to equilibrium conductors, does not hold for the Na process in squid axon. The Na-channel conductance, gamma Na, was found to increase monotonically from 0.1 to 4.8 pS for depolarizations up to 50 mV from a holding potential of -60 mV, with no indication of a maximum value.     

Associations of human erythrocyte band 4.2. Binding to ankyrin and to the cytoplasmic domain of band 3

We have examined the associations of purified red cell band 4.2 with red cell membrane and membrane skeletal proteins using in vitro binding assays. Band 4.2 bound to the purified cytoplasmic domain of band 3 with a Kd between 2 and 8 X 10(-7) M. Binding was saturable and slow, requiring 2-4 h to reach equilibrium. This finding confirms previous work suggesting that the principal membrane-binding site for band 4.2 lies within the 43-kDa cytoplasmic domain of band 3 (Korsgren, C., and Cohen, C. M. (1986) J. Biol. Chem. 261, 5536-5543). Band 4.2 also bound to purified ankyrin in solution with a Kd between 1 and 3.5 X 10(-7) M. As with the cytoplasmic domain of band 3, binding was saturable and required 4-5 h to reach equilibrium. Reconstitution with ankyrin of inside-out vesicles stripped of all peripheral proteins had no effect upon band 4.2 binding to membranes; similarly, reconstitution with band 4.2 had no effect upon ankyrin binding. This shows that ankyrin and band 4.2 bind to distinct loci within the 43-kDa band 3 cytoplasmic domain. Coincubation of ankyrin and band 4.2 in solution partially blocked the binding of both proteins to the membrane. Similarly, coincubation of bands 4.1 and 4.2 in solution partially blocked binding of both to membranes. In all cases, the data suggest the possibility that domains on each of these proteins responsible for low affinity membrane binding are principally affected. The data also provide evidence for an association of band 4.2 with band 4.1. Our results show that band 4.2 can form multiple associations with red cell membrane proteins and may therefore play an as yet unrecognized structural role on the membrane.     

Fusion pore conductance: experimental approaches and theoretical algorithms.

Time-resolved admittance measurements provide the basis for studies showing that membrane fusion occurs through the formation and widening of an initially small pore, linking two previously separated aqueous compartments. Here we introduce modifications to this method that correct the cell-pipette (source) admittance for attenuation and phase shifts produced by electrophysiological equipment. Two new approaches for setting the right phase angle are discussed. The first uses the displacement of a patch-clamp amplifier C-slow potentiometer for the calculation of phase. This calculation is based on amplitudes of observed and expected (theoretical) changes in the source admittance. The second approach automates the original phase adjustment, the validity of which we prove analytically for certain conditions. The multiple sine wave approach is modified to allow the calculation of target cell membrane parameters and the conductance of the fusion pore. We also show how this technique can be extended for measurements of the resting potential of the first (voltage-clamped) membrane. We introduce an algorithm for calculation of fusion pore conductance despite a concurrent change in the resistance of the clamped membrane. The sensitivity of the capacitance restoration algorithm to phase shift errors is analyzed, and experimental data are used to demonstrate the results of this analysis. Finally, we show how the phase offset can be corrected "off-line" by restoring the shape of the capacitance increment.     

Mobility of water bound to biological membranes. A proton NMR relaxation study.

Water proton nuclear magnetic resonance relaxation measurements have been obtained for aqueous suspensions of red cell membranes. These data support a model in which water molecules are exchanging rapidly between a bound phase with restricted motions and a free phase with dynamic properties similar to liquid water. From this model and these data, estimates are obtained for the relaxation time for bound phase water. Possible relaxation mechanisms for bound phase water are discussed and some support is found for an intermolecular interaction modulated by translational motions characterized by a diffusion constant of 10(-9) cm2/s.     

Curve fitting approach for measurement of cellular osmotic properties by the electrical sensing zone method. I. Osmotically inactive volume

We have investigated the confounding effects of dynamic range limitations on measurement of the osmotically inactive volume using electrical sensing zone instruments (e.g., Coulter counters), and propose an improved approach to parameter estimation. The conventional approach for analysis of cell size distributions measured by such particle sizing instruments requires data truncation: the mean cell volume is computed after exclusion of data below a specified lower bound (typically chosen to remove artifacts due to small-volume noise) and above an upper bound (typically governed by instrument limitations). The osmotically inactive volume is then estimated from a Boyle–van’t Hoff plot of the averaged volume data obtained after exposure to various solution osmolalities. We demonstrate that systematic exclusion of data in the conventional approach introduces bias that results in erroneously high estimates of the osmotically inactive volume fraction. To minimize this source of error, we have devised a new algorithm based on fitting a bimodal distribution model to the non-truncated volume data. In experiments with mouse insulinoma (MIN6) cells, the osmotically inactive volume fraction was estimated to be 0.15 ± 0.01 using the new method, which was significantly smaller than the estimate of 0.37 ± 0.02 obtained using the conventional method (p < 0.05). In silico experiments indicated that the parameter estimate obtained by the new method was accurate within 5%, whereas the error associated with the conventional approach was approximately 150%. Parametric analysis was used to elucidate the sensitivity of errors to variations in instrument dynamic range and cell volume distribution width.     

Toward a multi-membrane model for potassium conduction in squid giant axon.

Possible physical mechanisms are considered which come close to a quantitative explanation for features of the potassium admittance magnitude. At 1–30 Hz there is an elevation of [Y] and positive phase above that obtained from the Hodgkin-Huxley model. Moreover there appears to be a slight negative phase for lower frequencies. An additional important feature for model fitting is the movement of the middle zero-phase crossing to the left with depolarization. Two general classes of subsystems are discussed. (1) Extracellular: potassium accumulation, barriers to diffusion near or adjacent to the excitable membrane, diffusion with volume flow, bulklimited diffusion through the Schwann cell layer and adsorption or absorption by the Schwann cells; (2) processes intrinsic to the excitable membrane: cyclic steady state, co-operative, inactivating and second order. A generalized potassium inactivation is treated in detail which provides fairly quantitative fits to transmembrane transfer data with a voltage-dependent inactivation time constant ranging between 40 and 100 ms. However, potassium accumulation coupled with hypothesized sorptive effects of the greater membrane, particularly the Schwann cell layer, also provide reasonable fits. Based on lack of experimental evidence for an inactivation, the choice is made for a multicompartment model. When an HH membrane element is combined with accumulation-depletion in an extracellular space and with a bulk limited or surface limited diffusion through the Schwann cells good agreement is obtained with measured admittance.     

Modeling the Electric Potential across Neuronal Membranes: The Effect of Fixed Charges on Spinal Ganglion Neurons and Neuroblastoma Cells

We present a model for the electric potential profile across the membranes of neuronal cells. We considered the resting and action potential states, and analyzed the influence of fixed charges of the membrane on its electric potential, based on experimental values of membrane properties of the spinal ganglion neuron and the neuroblastoma cell. The spinal ganglion neuron represents a healthy neuron, and the neuroblastoma cell, which is tumorous, represents a pathological neuron. We numerically solved the non-linear Poisson-Boltzmann equation for the regions of the membrane model we have adopted, by considering the densities of charges dissolved in an electrolytic solution and fixed on both glycocalyx and cytoplasmic proteins. Our model predicts that there is a difference in the behavior of the electric potential profiles of the two types of cells, in response to changes in charge concentrations in the membrane. Our results also describe an insensitivity of the neuroblastoma cell membrane, as observed in some biological experiments. This electrical property may be responsible for the low pharmacological response of the neuroblastoma to certain chemotherapeutic treatments.     

Pseudo-streaming potentials in Necturus gallbladder epithelium. II. The mechanism is a junctional diffusion potential

The mechanisms of apparent streaming potentials elicited across Necturus gallbladder epithelium by addition or removal of sucrose from the apical bathing solution were studied by assessing the time courses of: (a) the change in transepithelial voltage (Vms). (b) the change in osmolality at the cell surface (estimated with a tetrabutylammonium [TBA+]-selective microelectrode, using TBA+ as a tracer for sucrose), and (c) the change in cell impermeant solute concentration ([TMA+]i, measured with an intracellular double-barrel TMA(+)-selective microelectrode after loading the cells with TMA+ by transient permeabilization with nystatin). For both sucrose addition and removal, the time courses of Vms were the same as the time courses of the voltage signals produced by [TMA+]i, while the time courses of the voltage signals produced by [TBA+]o were much faster. These results suggest that the apparent streaming potentials are caused by changes of [NaCl] in the lateral intercellular spaces, whose time course reflects the changes in cell water volume (and osmolality) elicited by the alterations in apical solution osmolality. Changes in cell osmolality are slow relative to those of the apical solution osmolality, whereas lateral space osmolality follows cell osmolality rapidly, due to the large surface area of lateral membranes and the small volume of the spaces. Analysis of a simple mathematical model of the epithelium yields an apical membrane Lp in good agreement with previous measurements and suggests that elevations of the apical solution osmolality elicit rapid reductions in junctional ionic selectivity, also in good agreement with experimental determinations. Elevations in apical solution [NaCl] cause biphasic transepithelial voltage changes: a rapid negative Vms change of similar time course to that of a Na+/TBA+ bi-ionic potential and a slow positive Vms change of similar time course to that of the sucrose-induced apparent streaming potential. We conclude that the Vms changes elicited by addition of impermeant solute to the apical bathing solution are pseudo-streaming potentials, i.e., junctional diffusion potentials caused by salt concentration changes in the lateral intercellular spaces secondary to osmotic water flow from the cells to the apical bathing solution and from the lateral intercellular spaces to the cells. Our results do not support the notion of junctional solute-solvent coupling during transepithelial osmotic water flow.     

Inward rectifier K+-channel kinetics from analysis of the complex conductance of Aplysia neuronal membrane.

Conduction in inward rectifier, K+-channels in Aplysia neuron and Ba++ blockade of these channels were studied by rapid measurement of the membrane complex admittance in the frequency range 0.05 to 200 Hz during voltage clamps to membrane potentials in the range -90 to -40 mV. Complex ionic conductances of K+ and Cl- rectifiers were extracted from complex admittances of other membrane conduction processes and capacitance by vector subtraction of the membrane complex admittance during suppressed inward K+ current (near zero-mean current and in zero [K+]0) from complex admittances determined at other [K+]0 and membrane potentials. The contribution of the K+ rectifier to the admittance is distinguishable in the frequency domain above 1 Hz from the contribution of the Cl- rectifier, which is only apparent at frequencies less than 0.1 Hz. The voltage dependence (-90 to -40 mV) of the chord conductance (0.2 to 0.05 microS) and the relaxation time (4-8 ms) of K+ rectifier channels at [K+]0 = 40 mM were determined by curve fits of admittance data by a membrane admittance model based on the linearized Hodgkin-Huxley equations. The conductance of inward rectifier, K+ channels at a membrane potential of -80 mV had a square-root dependence on external K+ concentration, and the relaxation time increased from 2 to 7.5 ms for [K+]0 = 20 and 100 mM, respectively. The complex conductance of the inward K+ rectifier, affected by Ba++, was obtained by complex vector subtraction of the membrane admittance during blockage of inward rectifier, K+ channels (at -35 mV and [Ba++]0 = 5 mM) from admittances determined at -80 mV and at other Ba++ concentrations. The relaxation time of the blockade process decreased with increases in Ba++ concentration. An open-closed channel state model produces the inductive-like kinetic behavior in the complex conductance of inward rectifier, K+ channels and the addition of a blocked channel state accounts for the capacitive-like kinetic behavior of the Ba++ blockade process.     

Mobility of water bound to biological membranes: A proton NMR relaxation study

Water proton nuclear magnetic resonance relaxation measurements have been obtained for aqueous suspensions of red cell membranes. These data support a model in which water molecules are exchanging rapidly between a bound phase with restricted motions and a free phase with dynamic properties similar to liquid water. From this model and these data, estimates are obtained for the relaxation time for bound phase water. Possible relaxation mechanisms for bound phase water are discussed and some support is found for an intermolecular interaction modulated by translational motions characterized by a diffusion constant of 10^?9 cm²/s.     

Origin of the potassium and voltage dependence of the cardiac inwardly rectifying K-current (IK1).

Using various voltage clamp protocols, we have examined the activation and deactivation kinetics of IK1 recorded in dissociated myocytes obtained from canine purkinje fibers. Exponential current relaxations following step changes of the membrane potential were characterized at several different K levels (5, 12, 42, and 82 mM) and several voltages (K reversal potential +/- 40 mV). We have interpreted our data according to a K-activated, K-channel model of IK1 gating. Our data suggests that at least two binding sites for extracellular K must be occupied before the channel opens and occupancy of about three more higher affinity sites for K on the open channel will slow the closing of that channel. In our model, the voltage dependency of gating arises from a combination of three voltage dependent steps: (a) isomerization between open and closed states, (b) binding of K, and (c) occupancy of the channel by internal Mg. Lowering internal K to 40 mM causes major changes in the voltage and K dependence of IK1 gating. However, these changes could be accounted for in our model by relatively small (approximately 20 to 30 mV) shifts in the voltage dependence of several of the steps that govern gating. Our data further suggest that there is an interaction between both extracellular and intracellular K levels and the ability of intracellular Mg to block the IK1 channel.     

On the admittance of membranes associated with channel conduction. Application to channels at non-equilibrium steady state

The small-signal admittance of membranes associated with channel conduction is derived for a general channel model. A general channel model is represented by a set of chemical reactions with each species of the reactions representing a channel state. The membrane admittance is shown to be related to the phenomenological relaxation matrix of the reactions. If the kinetic reactions are at a non-equilibrium steady state, the relaxation matrix may have complex eigenvalues and the equivalent circuit of the membrane admittance may contain RLC or RLC-like branches. For equilibrium kinetic systems, on the other hand, the equivalent circuit contains only RL or RC branches. Thus, the membrane admittance of equilibrium channels is quite different from that of non-equilibrium channels. In particular, we show that the low frequency feature in the admittance of squid axons as observed by Fishman, Poussart, Moore &; Siebenga (1977) can be obtained easily from a non-equilibrium cycling steady-state model.     

Measuring the stiffness of bacterial cells from growth rates in hydrogels of tunable elasticity

Although bacterial cells are known to experience large forces from osmotic pressure differences and their local microenvironment, quantitative measurements of the mechanical properties of growing bacterial cells have been limited. We provide an experimental approach and theoretical framework for measuring the mechanical properties of live bacteria. We encapsulated bacteria in agarose with a user-defined stiffness, measured the growth rate of individual cells and fit data to a thin-shell mechanical model to extract the effective longitudinal Young's modulus of the cell envelope of Escherichia coli (50-150 MPa), Bacillus subtilis (100-200 MPa) and Pseudomonas aeruginosa (100-200 MPa). Our data provide estimates of cell wall stiffness similar to values obtained via the more labour-intensive technique of atomic force microscopy. To address physiological perturbations that produce changes in cellular mechanical properties, we tested the effect of A22-induced MreB depolymerization on the stiffness of E. coli. The effective longitudinal Young's modulus was not significantly affected by A22 treatment at short time scales, supporting a model in which the interactions between MreB and the cell wall persist on the same time scale as growth. Our technique therefore enables the rapid determination of how changes in genotype and biochemistry affect the mechanical properties of the bacterial envelope.     

Distribution and kinetics of membrane dielectric polarization. II. Frequency domain studies of gating currents

We have studied the admittance of the membrane of squid giant axon under voltage clamp in the absence of ionic conductances in the range of 0-12 kHz for membrane potentials (V) between --130 and 70 mV. The admittance was measured at various holding potentials (HP) or 155 ms after pulsing from a given holding potential. Standard P/4 procedure was used to study gating currents in the same axons. We found that the membrane capacity Cm (omega) is voltage as well as frequency dependent. For any given V, the voltage-dependent part of the membrane capacitance has a maximum as the frequency approaches zero and requires at least a two-time constant equivalent circuit to be described. When the holding potential is varied, the voltage-dependent capacitance follows a bell- shaped curve with a maximum change of 0.15 muF/cm2 at about --60 mV. With the pulse method, the maximum is at --40 mV for HP = --70 and it shifts to --70 mV for HP = 0. The shift in the maximum of the voltage- dependent capacitance is consistent with the shift in the charge (Q) vs. V curve observed in our experiments with regular P/4 procedure when the HP is varied. Our data can be explained qualitatively by a four- state model for the sodium channel gating, where a charged particle can move within the field and interact with another particle not affected by the field.     

System for dynamic measurements of membrane capacitance in intact epithelial monolayers.

Dynamic measurements of exocytosis have been difficult to perform in intact epithelial monolayers. We have designed a system that estimates with +/-1% accuracy (99% confidence) the total membrane capacitance of monolayers represented by a lumped model. This impedance measurement and analysis system operates through a conventional transepithelial electrophysiology clamp, performing all signal measurements as frequently as every 5 s. Total membrane capacitance (the series combination of apical and basolateral membranes) is the inverse of one of three unique coefficients that describe the monolayer impedance. These coefficients are estimated using a weighted, nonlinear, least-squares algorithm. Using the estimated coefficients, solution ranges for individual membrane parameters are calculated, frequently providing results within +/-20% of true values without additional electrophysiological measurements. We determined the measurement system specifications and statistical significance of estimated parameters using 1) analytical testing with circuit simulation software and equation-generated data; 2) a system noise analysis combined with Monte Carlo simulations; and 3) analog model circuits for calibration of the electronic system and to check equation-generated results. Finally, the time course of capacitance changes associated with purinergically stimulated mucin exocytosis are quantified in monolayers of the colonic goblet cell-like cell line HT29-CI.16E.     

Binding, internalization, and intracellular processing of proteins interacting with recycling receptors. A kinetic analysis

We measured time-dependent concentration changes of human interferon-alpha 2a (IFN) and human tumor necrosis factor-alpha (TNF) bound at the plasma membrane and internalized by human lung alveolar carcinoma A549 cells in the presence of excess free ligand. Concentration changes for these two ligands were substantially different. We modified our compartmental kinetic model encompassing receptor synthesis and receptor loss (Myers, A. C., Kovach, J. S., and Vuk-Pavlovi?, S. (1987) J. Biol. Chem. 262, 6494-6499) to include receptor recycling. We solved analytically the equations of three variants of the model of receptor recycling. All parameters (rate constants) were identifiable when the data sets consisted of time-resolved concentrations of IFN and TNF at the cell surface and internalized by cells. By least squares fitting we derived the best fit values for the first order rate constants for internalization of the ligand-receptor complex, receptor recycling, turnover of free receptors, elimination of the ligand from cells, and the rate of insertion of free receptors into the membrane. The best fit to data for interactions of cells with IFN was obtained without inclusion of the term for recycling of receptors to the membrane. The simplest model including receptor recycling was necessary and sufficient for the fit to the respective data for TNF. These results demonstrate that the contribution of receptor recycling to the metabolism of the ligand and the receptor can be quantitated by compartmental modeling. Receptor recycling does not contribute to the kinetics of Type I IFN receptor in A549 cells. In contrast, recycling contributes significantly to endocytosis mediated by the TNF receptor.     

Apoptosis induced in neuronal cells by C-terminal amyloid beta-fragments is correlated with their aggregation properties in phospholipid membranes

A number of findings suggest that lipophilic monomeric Abeta peptides can interact with the cellular lipid membranes. These interactions can affect the membrane integrity and result in the initiation of apoptotic cell death. The secondary structure of C-terminal Abeta peptides (29-40) and the longer (29-42) variant have been investigated in solution by circular dichroism measurements. The secondary structure of lipid bound Abeta (29-40) and (29-42) peptides prepared at different lipid/peptide ratio's, was investigated by ATR-FTIR spectroscopy. Finally, the changes in secondary structure (i.e. the transition of alpha-helix to beta-sheet) of the lipid bound peptides were correlated with the induction of neurotoxic and apoptotic effects in neuronal cells. The data suggest that the C-terminal fragments of the Abeta peptide induce a significant apoptotic cell death, as demonstrated by caspase-3 measurements and DNA laddering, with consistently a stronger effect of the longer Abeta (29-42) variant. Moreover, the induction of apoptotic death induced by these peptides can be correlated with the secondary structure of the lipid bound amyloid beta peptides. Based on these observations, it is proposed that membrane bound aggregated Abeta peptides (produced locally as the result of gamma-secretase cleavage) can accumulate and aggregate in the membrane. These membrane bound beta-sheet aggregated amyloid peptides induce neuronal apoptotic cell death.     

Scale-independent roughness value of cell membranes studied by means of AFM technique

The roughness of cell membrane is a very interesting indicator of cell's health state. Atomic Force Microscopy allows us to investigate the roughness of cell membrane in great detail, but the obtained roughness value is scale-dependent, i.e. it strongly depends on measurement parameters, as scanning area and step size. The scale-dependence of the roughness value can be reduced by means of data filtration techniques, that are not standardized at nanometric scale, especially as far as biological data are concerned. In this work, a new method, based on the changes of values of some roughness parameter (root mean square roughness and skewness) as a function of filtration frequencies, has been implemented to optimize data filtering procedure in the calculation of cell membrane roughness. In this way, a root mean square roughness value independent of cell shape, membrane micro-irregularities and measurement parameters can be obtained. Moreover, different filtration frequencies selected with this method allow us to discriminate different surface regimes (nominal form, waviness and roughness) belonging to the raw cell profile, each one related to different features of the cell surface.     

Time Series Analysis of Particle Tracking Data for Molecular Motion on the Cell Membrane

Biophysicists use single particle tracking (SPT) methods to probe the dynamic behavior of individual proteins and lipids in cell membranes. The mean squared displacement (MSD) has proven to be a powerful tool for analyzing the data and drawing conclusions about membrane organization, including features like lipid rafts, protein islands, and confinement zones defined by cytoskeletal barriers. Here, we implement time series analysis as a new analytic tool to analyze further the motion of membrane proteins. The experimental data track the motion of 40 nm gold particles bound to Class I major histocompatibility complex (MHCI) molecules on the membranes of mouse hepatoma cells. Our first novel result is that the tracks are significantly autocorrelated. Because of this, we developed linear autoregressive models to elucidate the autocorrelations. Estimates of the signal to noise ratio for the models show that the autocorrelated part of the motion is significant. Next, we fit the probability distributions of jump sizes with four different models. The first model is a general Weibull distribution that shows that the motion is characterized by an excess of short jumps as compared to a normal random walk. We also fit the data with a chi distribution which provides a natural estimate of the dimension d of the space in which a random walk is occurring. For the biological data, the estimates satisfy 1<d<2, implying that particle motion is not confined to a line, but also does not occur freely in the plane. The dimension gives a quantitative estimate of the amount of nanometer scale obstruction met by a diffusing molecule. We introduce a new distribution and use the generalized extreme value distribution to show that the biological data also have an excess of long jumps as compared to normal diffusion. These fits provide novel estimates of the microscopic diffusion constant.