Membrane admittance

Determination of Potassium- and Sodium-Channel Relaxation Times in Squid Nerve Fibers from Membrane Admittance Analysis... 

An expression for the complex admittance, Y(jf), of an axon membrane is obtained by linearizing the Hodgkin-Huxley (HH) equations (I) and by applying a Laplace transformation (13, 14). The membrane admittance is then given by the general expression... 

Membrane capacitance and conductance were measured using the Wayne-Kerr admittance bridge and measurements were made between 100 Hz and 20 KHz using a PAR lock-in amplifier model 124. One of the results obtained is shown in Figure 1. The dotted curve shown in this figure indicates measured values at high frequencies and the downward slope arises from the presence of electrolyte solutions between the membrane and electrodes. 

Progress was recently made in the construction of an AC admittance modulation system for surface-stabilized lipid membrane biosensors that operated on the basis of the control of the ion permeation by artificial ion channels [40]. A portable admittance modulation measurement device was designed to measure both the in-phase and out-of-phase signal components for determination of the effective ion current and membrane capacitance, respectively [40]. The sensitivity and detection limit of this AC system were tested by studying the interaction of valinomycin with planar BLMs. The electrochemical phenomena were monitored through the in-phase component and measured as conductance changes of the membrane, providing a detection limit of 1 nM for valinomycin. 

Nonequilibrium noise generated by carrier-mediated ion transport was studied in lipid bilayers modified by tetranactin (41). As expected, deviations of measured spectral density from the values calculated from the Nyquist formula 1 were found. The instantaneous membrane current was described as the superposition of a steady-state current and a fluctuating current, and for the complex admittance in the Nyquist formula only a small-signal part of the total admittance was taken. The justification of this procedure is occasionally discussed in the literature (see, for example, Tyagai (42) and references cited therein), but is unclear. 

Description of ion-channel kinetics via admittance analysis provides a framework within which linear kinetic models can be compared to macroscopic data (from a population of channels) in a membrane. Analysis of conduction via driving-point-function determinations also provides proper data (from a true linear analysis) for comparison with the relaxation times obtained from microscopic data from one or a small number of channels in a membrane patch isolated by a micropipette (4). In Markov modeling, the open- and closed-time distributions are fitted to sums of exponential functions (15). 

Data Analysis. Complex admittance determinations were fitted by an admittance function (13, 14, 16) based on the linearized HH equations (I). Admittance measurements were made under steady-state conditions (see Figures 2 and 4). Series resistance (Rs), the access resistance between the two voltage electrodes and up to the inner and outer surfaces of the axon membrane was not removed from measurements. Instead Rs was included and determined in the fit of the steady-state admittance model to the data. The measured complex admittance, therefore, is... 

Figure 3. Admittance data from a K +-conducting membrane and curve fits (solid curves) of eqs 2, 3, and 4 with Y /jf,) = 0 plotted in the complex plane [X(f) vs. R(f)] as impedance [Z(jf) = R(f) + jX(f) = Y 1(jf/)] loci (400 frequency points) over the 12.5 5000-Hz frequency range. These data were acquired rapidly as complex admittance data, as illustrated in Figure 1, at premeasurement intervals of 0.1 and 0.5 s after step voltage clamps to each of the indicated membrane potentials from a holding of —65 mV. The near superposition and similarity in shape of the two loci at 0.1 and 0.5 s, at each voltage, indicates that the admittance data reflect a steady state in this interval after step clamps. Axon 86-41 internally perfused with buffered KF and externally perfused in ASW + TTX at 12 °C. The membrane area is 0.045 cm2.

Note All values are given in milliseconds and are taken from fits of the admittance data plotted as impedance loci in Figure 3 at the two premeasurement intervals (PMI) 0.1 and 0.5 s after step changes to membrane voltage, V. 

Figure 4. Estimates of the potassium-conductance relaxation time, rn, from fits of eqs 2, 3, and 4 with XNa = 0 to admittance determinations at various membrane voltages, similar to those shown in Figure 3. Filled triangles are from fits of the average (AVE) of the real and imaginary parts of eight separate, successive admittance determinations at each voltage. Open circles and squares are from fits of 1 standard deviation added to ( + SD) or subtracted from ( — SD) the real and imaginary parts of the AVE admittance. Intact axon 87-39 in ASW + TTX(1 piM)at 12.5 °C.

Figure 5. Admittance data plotted as magnitude and phase angle vs. frequency as determined at the three premeasurement intervals (20, 100, and 200 ms) shown in Figure 2 and at the indicated membrane voltages. The superposition of the admittance data at each voltage indicates that the admittance is time-invariant in the interval from 20 to 200 ms after step changes in membrane voltage. Axon 87-19 internally perfused with the perfusate described in the text and externally perfused with ASW at 8 °C.

Application of the averaged admittance methodology to a Na+-conducting axon enabled us to obtain three data points at each membrane voltage. Figure 7 shows these data, which were obtained from best fits of eqs 2, 3, and 5 with... 

Figure 6. Admittance data from a Na+-conducting membrane and curve Jits (solid curves) of eqs 2, 3, and 5 with YK()f) = 0 plotted in the complex plane [X(f) vs. R(f)] as impedance loci (400 frequency points) over the frequency range 5 to 2000 Hz. Same axon and conditions as in Figure 5.

The complex admittance method described here allows data to be analyzed without reference to any particular model. This condition is particularly important at this time, when new data and new concepts are challenging previously accepted concepts. The elucidation of the primary structure of channel proteins (2, 3) has stimulated the development of a number of structurally oriented models (e.g., references 21 and 22). In addition, new physical and mathematical concepts have been brought to bear on the problem of channel gating in excitable membranes. These concepts include... 

However, as illustrated in Figure 10.2, living tissue has capacitive properties because of, for example, cell membranes. Therefore, the u and i signals are not in-phase, and impedance and admittance are complex quantities. They are written in bold characters and can be decomposed in their in-phase and quadrature components... 

Pore-size dependent conductances are assigned to individual pores and channels, determined by the overall water content and the law of swelhng. Three possible types of bonds between pores exist. The corresponding bond conductances, viz. abb(w), crbr(w), and crrr(w), can be established straightforwardly. The model was extended towards calculation of the complex admittance of the membrane by assigning capacitances in parallel to conductances to individual pores. 

The plasma membrane presents cationic polymers with a substantial barrier to successfully gaining entry to the cell, as it is a dynamic and a relatively lipophilic structure that restricts the admittance of large, hydrophilic or charged molecules. The contribution of certain pathways in the uptake of cationic polymer-mediated gene delivery is not well understood it is generally believed that the uptake of polyplexes predominantly oecurs through endocytosis, although multiple mechanisms for endocytosis have been described to date and the current question is which pathway of endocytosis is responsible for cationic polymer uptake. 

Carbohydrates and saccharides are the fuels for cell metabolism. They also form important extracellular structural elements, such as cellulose (plant cells), and have other specialized functions. Bacteria cell membranes are protected by cell walls of covalently bonded polysaccharide chains. Human cell membranes are coated with other saccharides. Some saccharides form a jellylike substance filling the space between cells. Some polysaccharides may have a negative charge at a pH of 7, but many carbohydrates are not believed to contribute dominantly to the admittivity of tissue. 

In Section 3.8 about dispersions, the use of permittivity or conductivity parameters was discussed. From Section 2.3.4, we know that conductivity is dependent on the density of charge carriers and their mobility. In the frequency range less than 10 MHz, tissue admittance is usually dominated by the conductivity of the body electroljrtes, but at higher frequencies it is dominated by the dielectric constant. The electroljrtes without cells, in particular urine and cerebrospinal fluid (CSF), have the highest low-frequency conductivity thus, the higher the cell concentration, the lower the low-frequency conductivity. Tooth, cartilage and bone, lipids, fat, membranes such as the skin stratum comeum (SC), and connective tissue may contain many inorganic materials with low conductivity, but they are very dependent on body liquid perfusion. Tissue conductivity data are tabulated in Table 4.2. 

Epithelia are cells organized as layers, and skin is an example. Cells in epitheUa do form gap junctions. In particularly tight membranes, these junctions are special tight junctions. The transmembrane admittance is dependent on the type of cell junctions and to what extent the epithelium is shunted by channels or specialized organs (e.g., sweat ducts in the skin). 

The parallel version is best characterized by admittance because the time constant then is uniquely defined (Section 12.2). It has been used for cells and living tissue, with C for cell membranes, R for intracellular, and G for extracellular liquids. 

Impedance is the preferred parameter characterizing the two resistors, one capacitor series circuit, because it is defined by one unique time constant Xz (Eq. (12.8)). This time constant is independent of R, as if the circuit was current driven. The impedance parameter therefore has the advantage that measured characteristic frequency determining Xz is directly related to the capacitance and parallel conductance (e.g., membrane effects in tissue), undisturbed by an access resistance. The same is not true for the admittance the admittance is dependent both on xz and X2, and therefore on both R and G. 

The admittance time constant is uniquely defined by ty, independent of G, as if the circuit were voltage driven. The admittance parameter therefore has the advantage that the measured characteristic frequency determining xy is directly related to the capacitance (membrane effects) and series resistance in tissue. The same is not true for impedance the impedanee is defined by both ty and X2. 

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