INTRODUCTION

Top Abstract Introduction Background and theoretical... Materials and methods Results Discussion Appendix References

The measurement of membrane capacitance has been used widely in biological experiments as a means of assessing changes in membrane area. Such measurements have been of particular interest in studies of epithelial transport, as it has been surmised that regulation of salt and water transport at apical and basolateral membranes of polarized epithelial cells involves targeting and trafficking of channels and transporters between the cytosol and the plasma membranes of the cells.

It is usually assumed that the dielectric properties of the plasma membranes are constant, so that changes in capacitance can be attributed to changes in membrane area. There is, however, an extensive literature dating from the classical papers of Debye (1929), Cole and Cole (1941), and others (see references in Schwan, 1957; Daniel, 1967; Cole, 1968; Gabler, 1978; Pethig, 1979; Jonscher, 1983; Kell and Harris, 1985; Takashima, 1989) that describes the behavior of dipoles in viscous media and documents the existence of audio frequency dielectric dispersions or relaxation processes. Schwan referred to these relaxation processes with the terminology "- and -dispersions," where -dispersions occurred at frequencies below ~100 kHz (Schwan, 1957). To our knowledge, there have been few attempts to determine whether -dispersions exist in epithelial plasma membranes in general (Watanabe et al., 1991), and no attempts for tight epithelia like those of renal distal tubules.

Our laboratories have been interested in determining the way in which tight epithelia regulate the density of apical membrane epithelial Na⁺ channels (ENaCs) and knowing when and under what conditions vesicle trafficking plays a role in shuttling channels between the cytosol and the apical membrane of the cells. In this regard it would be crucial to know whether this membrane exhibits -dispersions, because their existence would seriously complicate the design of experiments and interpretation of data where changes of capacitance may not reflect alone changes of membrane area.

We have examined with dielectric spectroscopy the native apical membrane of the well-studied frog skin (northern R. pipiens), as this membrane contains highly selective and amiloride-sensitive ENaCs. We describe the methods and approaches that we used to determine the complex capacitance of this membrane and report that this plasma membrane exhibits several dielectric relaxation processes at low audio frequencies (<10 kHz) that can be characterized by their capacitive increments, relaxation frequencies, and Cole-Cole power law dependence. An examination of the relationship between the spontaneous rates of Na⁺ transport among tissues as measured by short-circuit currents and the dc capacitance indicated that increases in Na⁺ transport were correlated with increases in dc capacitance. Further examination of the data revealed that increases in dc capacitance were due to selective increases in the lowest audio frequency capacitive increments of the complex capacitance spectrum and their associated static capacitances. However, it remains unknown from capacitance measurements alone whether changes in capacitance are attributable to changes in membrane area associated with changes in dielectric increments.

Preliminary results have been presented at meetings of the Federation of American Societies of Experimental Biology (FASEB) and the Biophysical Society (Awayda et al., 1989, 1991; Awayda and Helman, 1990, 1992).

	BACKGROUND AND THEORETICAL CONSIDERATIONS

Top Abstract Introduction Background and theoretical... Materials and methods Results Discussion Appendix References

Dielectric dispersions

Dielectric dispersions have been observed between subaudio and gigahertz frequencies (Schwan, 1957; Coster and Smith, 1974). By the early part of this century it was widely recognized that dispersions could arise from series structural arrangements of leaky dielectrics that are referred to as Maxwell-Wagner dispersions. In 1929, Debye published his theory of behavior of polar molecules or dipoles in a viscous medium whereby dispersions could also arise from dipolar relaxation processes (Debye, 1929). Cole and Cole (1941) and others examined this theory in a wide range of materials, including biological membranes, and observed dielectric dispersions at audio frequencies. Cole and Cole noted quite generally that dielectric dispersions deviate from ideal behavior, exhibiting a power-law dependence resulting in observation of "depressed" semicircles in Nyquist plots of the complex dielectric constant (*) and hence the complex capacitance (C*). Accordingly, C* = *A/d, where A and d are membrane area and thickness, respectively. Dispersions take the form described by Eq. 1, where for a membrane containing a single dispersion with time constant  and Cole-Cole power-law factor   (1  ),

(1)

which can be rewritten as

(2)

The angular relaxation frequency is 2f_r = 1/_r, and f_r is the relaxation frequency in Hz. _r = ₀   is the dielectric increment between infinite and zero frequencies ( = 2f), where the terminology "infinite frequency" takes on the meaning f f_r.

Dispersions at relaxation frequencies less than ~100 kHz have been referred to as -dispersions. Dispersions at higher radio frequencies have been referred to by Schwan and his colleagues as -dispersions. -Dispersions extend into the gigahertz range (Schwan, 1957; Schwan and Foster, 1980; Foster and Schwan, 1989). Recognizing that multiple dispersions may exist within , ,  and higher ranges of frequency, Eq. 2 can be written as Eq. 3 to indicate the possible existence of several capacitive increments C_i and C_j associated with the ranges of frequency of the - and -dispersions, respectively:

(3)

At zero frequency, C^dc = C_i + C_j + C. In addition to the static dc capacitance (C^dc), we can define static capacitances C and C (see Fig. 1), where C = C_j + C if multiple dispersions exist in the -range of relaxation frequencies. Accordingly, C^dc = C_i + C if multiple dispersions exist in the -range of relaxation frequencies, which is the focus of attention in the present series of experiments.

View larger version (22K): [in this window] [in a new window]   FIGURE 1   Frequency-dependent complex capacitance (C*) due to - and -dielectric dispersions (relaxation processes). Absolute magnitude and phase angle of C* are plotted against frequency according to Eq. 3 (see text). Curves were calculated assuming relaxation frequencies of 50 Hz and 500 kHz with capacitive increments C and C in the audio and radio frequency ranges, respectively. Cdc is the static dc capacitance. C is the static capacitance at frequencies considerably higher than -relaxation processes but at frequencies considerably less than -relaxation processes. C is the static capacitance at frequencies considerably greater than -relaxation processes but less than those at very high frequencies ranging into gigahertz frequencies. The solid lines were calculated assuming ideal Debye dispersions, and the dashed lines were calculated assuming Cole-Cole power law behavior.

Illustrated in Fig. 1 are plots of the magnitude and phase angle (Bode plots) of C*, where it is assumed for simplicity that - and -dispersion ranges of frequency each contain a single relaxation process with ideal Debye ( =  = 1.0) or Cole-Cole ( =  = 0.6) power-law behavior. The static capacitance C was assumed to be unity with capacitive increments of 1 and 8 units for - and -relaxation processes, respectively, so that C^dc is 10 times greater than the static capacitance C, and C is two times greater than C. Because C* is complex it can be represented by its real (Real) and imaginary (Imag) components or by its absolute magnitude |C*| and phase angle (). |C*| and  (degrees) are plotted in Fig. 1.

The theoretical curves in Fig. 1 are also replotted in Fig. 2 A in the form of Nyquist plots (Real versus Imag), where each dispersion gives rise to either an ideal ( = 1) or depressed ( < 1) semicircle. If C* is measured only at audio frequencies (-dispersions), then as indicated by the solid lines, capacitance would decrease with increasing audio frequency, so that extrapolation of C* to the real axis would give the static capacitance C. The semicircles appear depressed when  < 1.0, and this behavior is due presumably to a distribution of time constants or relaxation times of the dipoles associated with the relaxation process (Cole and Cole, 1941; Cole, 1968; Gabler, 1978; Pethig, 1979; Jonscher, 1983). When capacitance is measured at audio frequencies and the membrane contains several -relaxation processes between C^dc and C, C* can be decomposed into a sum of processes as shown in Fig. 2 B with capacitive increments C₁, C₂, ... , C_n at relaxation frequencies f₁, f₂, ... , f_n with static capacitances at the intercepts on the real axis, C₁, C₂, ... , C_n. Accordingly, C_n  C, and it is implied by omission of the  subscript that our measurements will pertain only to relaxation processes in the range of -dispersions. Hence the complex capacitance measured in the audio frequency range can vary not only because of changes in membrane area and thickness, but also because of changes in dielectric increments, relaxation frequencies, and the distribution of relaxation times of each of the relaxation processes. We shall in this paper use the terminology "capacitive increments" and "dielectric increments," recognizing that changes in capacitance can occur because of changes in dielectric increments with or without changes in membrane area.

View larger version (27K): [in this window] [in a new window]   FIGURE 2   Frequency-dependent complex capacitance plotted as Nyquist plots with real (Real) against imaginary (Imag) components of C*. Theoretical plots of Fig. 1 are shown in A. Static capacitances Cdc, C, and C are indicated on the real axis. Note depression of the semicircles when the Cole-Cole power law factor is less than unity. Thick solid lines represent the complex capacitance spectrum observable at audio frequencies. (B) Two -relaxation processes with relaxation frequencies of 20 Hz and 2.0 kHz and with capacitive increments C1 and C2 of 3 and 5 units, respectively. The Cdc as given by Eq. 5 and graphically represented here reflects the sum of the capacitive increments and the static capacitance C2. At frequencies considerably higher than 2 kHz, the static capacitance C2 approaches a value of 2 units and would remain unchanged if the area is unchanged and, in particular, if changes occur in either the relaxation frequencies and/or the capacitive increments. Also indicated is the static capacitance C1 that intercepts the real axis at 7 units. If, for example, the relaxation frequency at 2 kHz is increased to frequencies in the range of MHZ or higher, this very high frequency relaxation process would not be observed at audio frequencies. Instead, the 20-Hz relaxation process would appear as indicated (thin solid line) with a capacitive increment C1 that extrapolates at much higher frequencies to the static capacitance C1. For points of reference, the solid circles mark frequencies at 20 Hz, and the open squares mark frequencies at 2.0 kHz on the thick solid lines of the capacitance spectrum and on the individual relaxation processes illustrated by the thin solid lines that give rise to the spectrum.

	MATERIALS AND METHODS

Top Abstract Introduction Background and theoretical... Materials and methods Results Discussion Appendix References

Experiments were carried out with isolated epithelial preparations of abdominal skins of northern frogs (Rana pipiens; Kons Scientific Co., Germantown, WI) devoid of connective tissue and glands of the corium (Fisher et al., 1980). Tissues were mounted in edge-damage-free chambers (Abramcheck et al., 1985), which were continuously perfused with Ringer's solution at a rate of ~5 ml/min. Tissues were short-circuited, except during measurements of the transepithelial impedance, with a four-electrode (Ag/AgCl, 4.5M NaCl, 3% agar) very low-noise voltage clamp.

Impedance analysis

Transepithelial impedance was measured under voltage-clamp conditions at frequencies between 0.1 Hz and 5.5 kHz. The voltage command signals consisted of two bands of 53 discrete frequencies as described by Mrgineanu and Van Driessche (1990). Command signals applied to the tissues ranged between ~2 and ~20 mV peak to peak (p-p). Because the measured impedance was independent of the magnitude of the command voltage, it could be inferred that the impedance was measured in linear regions of current-voltage relationships. The low-frequency band contained frequencies between 0.1 and 43.1 Hz, whereas the high-frequency band overlapped the low-frequency band and contained frequencies between 12.8 and 5516 Hz. The command signals were applied to the voltage clamp sequentially. Transepithelial voltage and current signals were acquired with a 12-bit analog-to-digital converter after the signals were filtered at their Nyquist frequencies and amplified. Voltage command signals were also filtered before being applied to the voltage clamp. The digitized current and voltage signals were Fourier transformed to yield current and voltage vectors from which the measured impedance (Z_meas) was calculated at each of the 106 discrete frequencies. With a fundamental frequency of 0.1 Hz for the lower frequency band and a fundamental frequency of 12.8 Hz for the higher frequency band, the time for data acquisition was slightly greater than 10 s. In some experiments, the fundamental frequency of the lower frequency band was increased to 0.2 or 0.5 Hz, thereby shifting the entire lower frequency band to higher frequencies and reducing the time for data acquisition. The results were the same.

The solution resistance (R_sol) between the voltage electrodes was measured sometimes before and always at the end of the experiments. Impedance was measured with the electrodes in place, but in the absence of tissue separating apical and basolateral chamber solutions. R_sol was independent of frequency (<100 kHz), as expected for simple electrolyte solutions, and averaged 38.9 ± 1.0  · cm² for our chambers with 0.484 cm² cross-sectional area and the positioning of the voltage electrodes within the chambers. In addition to R_sol, cytoplasmic resistance (R_cyt) exists in series with apical and basolateral plasma membranes for a combined resistance R_ser = R_sol + R_cyt. Assuming a cell layer thickness of 30-60 µm for the electrically coupled basolateral membranes of the multicell layered epithelium of frog skin and a volume resistivity of the Ringer solution of ~100  · cm of the cytoplasmic fluid, R_cyt would be in the range of 0.3-0.6 · cm². If cytoplasmic volume resistivity is about twice that of the extracellular solution volume resistivity and in the range reported by Fricke and Morse (1925) and Bao et al. (1992), R_cyt is near 1  · cm² and is the value we used in our calculations. Accordingly, the transepithelial impedance Z_T = Z_meas  R_ser.

We also examined under current-clamp conditions the Z_meas at frequencies between 10 and 100 kHz, using 18-µA/cm² p-p sinusoids, resulting in <2-mV p-p changes in transepithelial voltage. Amplified current and voltage signals were displayed as Lissajous figures on a Nicolet model 2090 digital oscilloscope (Nicolet Instruments Corp., Madison, WI), and the impedance was determined from measurements of photographic images. These data confirmed that the Z_meas at much higher frequencies than 5.5 kHz approached those of R_ser as indicated above and as expected when the capacitive reactances of apical and basolateral membranes approach zero.

In the absence of tissue, the frequency response (<100 kHz) of the chambers and bridges was purely resistive, so that no correction was required for stray capacitance. The chambers were characterized with Ringer's solution alone and with Lucite gaskets (to replace the tissue) predrilled with small apertures to give values of R_sol between 2 and 25 k · cm². The phase difference between voltage and current signals was <±0.1° under voltage-clamp conditions and <±1.5° under current-clamp conditions.

Experimental design

Transporting conditions

View larger version (24K): [in this window] [in a new window]   FIGURE 3   Transepithelial electrical equivalent circuits. (A) Apical and basolateral membranes are shunted by the paracellular resistance Rp. Ra and Rb are the slope resistances, and Ca and Cb are the capacitances of apical and basolateral membranes, respectively. Not shown is the solution resistance Rsol in series with the tissues. (B) Inhibition of apical membrane Na+ entry (100 µM amiloride and substitution of all apical solution Na+ with NMDG (see text)) cause Ra Rb, and thus Ra is negligible. If, in addition, the impedance of basolateral membranes is considerably less than the reactance of the apical membrane capacitance (Cb Ca and/or the decrease in Rb by K+-depolarization of the basolateral membrane), the transepithelial electrical equivalent circuit reduces to Ca paralleled by Rp.

Transport-inhibited conditions

(4)

Calculation of complex capacitance, C^*_a

With the measured impedance and series resistance and with a preliminary estimate of R_p obtained by extrapolation of (Z_meas  R_ser) to zero frequency, C^*_a was calculated (Eq. 4) at each of the 106 discrete frequencies. This extrapolation to values of R_p could be done by eye or by using TableCurve (Jandel Scientific, San Rafael, CA) to fit the lowest frequency values of Real (Z_meas  R_ser) as a function of frequency to smooth curves that intercepted the impedance ordinate at zero frequency. From a direct graphical examination of the Nyquist capacitance plots, we determined not only the number of relaxation processes, but also the approximate magnitudes of the capacitive increments (C_i) and relaxation frequencies (f_r) that were used as the starting values for nonlinear curve fitting of the impedance data. It may be emphasized that the data in all cases conformed to Cole-Cole relaxation processes, and more complicated phenomena could be excluded.

Final determination of the magnitudes of the capacitive increments, relaxation frequencies, and power-law dependencies was done using a least-squares nonlinear minimization program (MINSQ, now called Scientist; Micromath Scientific, Salt Lake City, UT) to minimize the real and imaginary components of Z_meas over the parameter space of the relaxation processes and the R_p, where for the -dispersions,

(5)

It should be emphasized that all data are normalized to the planar area of the tissues. Actual membrane area, depending on the degree of in- and out-foldings, will accordingly be greater than planar area. Accordingly, the ratio of actual to planar area is variable, and this will be reflected in the values of capacitance reported (µF/cm² of planar area) when changes in actual area occur.

Data are summarized as means ± SE unless noted otherwise. All experiments were carried out at room temperature.

	RESULTS

Top Abstract Introduction Background and theoretical... Materials and methods Results Discussion Appendix References

Transepithelial impedance of transport-inhibited tissues

Impedance was measured before (see Appendix) and after complete inhibition of Na⁺ transport. Illustrated for a typical transport-inhibited tissue in Fig. 4 is the Z_meas plotted as a Nyquist plot at frequencies between 0.1 Hz and 5.5 kHz (Fig. 4 A) and at frequencies greater than or equal to 43 Hz in expanded form in Fig. 4 B. The data are also plotted in Fig. 4, C and D, in the form of Bode plots. All attempts to fit the data to single ideal semicircles over the entire range of frequency failed. With bandwidth limited to low frequencies (<50 Hz), smooth curves could be fit to the impedance vectors, requiring, however, a power-law dependence to account for flattening or depression of the semicircles. The solid lines shown in Fig. 4 were determined by nonlinear curve fitting of the data between 0.5 Hz and 43 Hz to an equation of a depressed impedance semicircle used previously (Van Driessche, 1986) and modified here (Eq. 6) for transport-inhibited tissues:

(6)

where it is explicitly assumed that C_a is constant at all frequencies. In every case, 1   was less than unity (ranging between ~0.80 and ~0.98), indicating depression or power-law dependence of the impedance of transport-inhibited tissues. Similar values of power-law dependence were observed for impedance of tissues studied in their transporting state (see Appendix). Because 1) we could not explain power-law dependence of impedance at very low frequencies less than 50 Hz while assuming constancy of C_a; 2) we could not explain having to exclude data for fitting at frequencies greater than 50 Hz to any model where capacitance is constant; 3) we could not fit data to distributed parameter models consistent with the morphology of this epithelium; and 4) because the theory of dipolar relaxations outlined above could explain power-law dependence of the impedance as well as the more complex behavior of impedance at all frequencies, we rejected the thesis that C_a was constant at audio frequencies.

View larger version (19K): [in this window] [in a new window]   FIGURE 4   Measured impedance (Zmeas) of isolated epithelium of frog skin after inhibition of apical membrane Na+ entry by amiloride and Na+-free apical solution. (A) Nyquist plot of Zmeas at frequencies between 0.1 Hz and 5.5 kHz. Shunt resistance, Rp, extrapolated to the real axis is 37.8 k · cm2. A single depressed semicircle (Eq. 4, solid line) was fit to the data between 0.5 Hz and 43 Hz. The apex of the depressed semicircle is at 1.9 Hz. (B) Expanded view of Zmeas at frequencies  43 Hz. The solid line is the extension of the depressed semicircle shown in A. At 5.5 kHz, Zmeas approaches the value of Rsol. The real axis intercept of the fitted line exceeds the value of Rsol. (C and D) Bode plots of the absolute value of Zmeas and phase angle (). Solid lines correspond to those in A and B for a depressed semicircle fitted to data at frequencies between 0.5 and 43 Hz.

Apical membrane capacitance is frequency dependent (dielectric spectroscopy)

Apical membrane capacitance, C_a*, calculated as described in Materials and Methods, invariably showed a strong dependence on frequency, as illustrated in Fig. 5. Between 0.1 Hz and 5.5 kHz, capacitance fell progressively with increasing frequency. Inspection of the capacitance spectra indicated clearly that frequency-dependent changes in capacitance were associated with at least two or three relaxation processes, as indicated in the spectra shown in Fig. 5, A and B. For the spectra shown in this figure, relaxation frequencies were 9.9 Hz, 152 Hz, and 5.8 kHz (Fig. 5 A) and 67 Hz and 3.2 kHz (Fig. 5 B) with corresponding capacitive increments and static capacitances indicated on the real axis of the Nyquist plots.

View larger version (25K): [in this window] [in a new window]   FIGURE 5   Complex capacitance of apical membrane of frog skin (Ca*). Representative examples are shown of apical membranes exhibiting two (B) or three (A) relaxation processes. (A) Cdc was near 2.5 µF/cm2. The solid line represents the nonlinear least-squares best fit of the impedance vectors. Dashed lines represent the individual relaxation processes at frequencies of 9.9 Hz, 152 Hz, and 5.8 kHz. Capacitive increments (Ci) and static capacitances (Ci) are indicated at the intercepts of the depressed semicircles on the real axis. (B) Cdc was near 1.8 µF/cm2. Relaxation frequencies of the two processes were 67 Hz and 3.2 kHz, with corresponding capacitive increments and static capacitances indicated on the real axis at the intercepts of the individual relaxation processes (dashed lines).

A histogram of relaxation frequencies was generated by log binning the relaxation frequencies from all tissues, as indicated in Fig. 6. The histogram was fit by nonlinear curve fitting to the sum of four Gaussian functions characterized in the usual way by their means ± SD. Relaxation frequencies fell into four populations centered at means of 30.4, 103, 2364, and 6604 Hz (Fig. 6 and Table 1), which we labeled f₁... f₄ with corresponding capacitive increments C₁... C₄ (Table 2).

View larger version (45K): [in this window] [in a new window]   FIGURE 6   Histogram of relaxation frequencies. Observations were log-binned and fit to four populations of relaxation frequencies (fi), assuming normal Gaussian distributions. Mean relaxation frequencies and standard deviations are summarized in Table 1.

Further inspection of the data revealed that the tissues could be grouped as summarized in Table 1 as groups I and II. Tissues in group I characteristically exhibited relaxation frequencies in the range of f₃, averaging 2085 ± 131 Hz (mean ± SE). Tissues in group II exhibited relaxation frequencies in the range of f₄, averaging 6806 ± 393 Hz. In no tissue did we observe relaxation frequencies in the frequency ranges of both f₃ and f₄. Relaxation frequencies were in the range of either f₃ or f₄.

Each group could be subdivided further, depending on the existence of f₁ and/or f₂, as indicated also in Table 1. Relaxation processes in the ranges of f₁ or f₂ could exist alone or in combination. f₁ averaged 24.8 ± 3.1 Hz, and f₂ averaged 142 ± 16.8 Hz.

Retaining the same groupings, we have summarized in Table 2 C_a^dc, C, and the capacitive increments C₁, C₂, C₃, and C₄. C_a^dc and C averaged 1.95 ± 0.06 and 0.14 ± 0.01 µF/cm², respectively, indicating that -dispersions accounted for ~93% of the static dc capacitance of the tissues. Although there is considerable uncertainty in the absolute values of C, owing to the uncertainty of the precise value of the series resistance, and although the absolute area and thickness of the native apical membrane dielectric are unknown, it is of interest to note that with dielectric thicknesses, d, in the range of 40-60 Å, the capacitance of a vacuum (C_vac) would be in the range of 0.22-0.15 µF/cm² and in the range of the calculated values of C (C_vac = 8.85 · 10¹⁴/d, Farads/cm²).

Between groups, the capacitive increment C₃ was greater in value than C₄, averaging 0.88 ± 0.06 and 0.29 ± 0.03 µF/cm², respectively. When the capacitive increments C₁ or C₂ were present alone (groups IA and IB; groups IIA and IIB), their values were similar within groups. When C₁ and C₂ were present together in the same spectrum, there appeared to be an inverse relationship between the values of C₁ and C₂ (Fig. 7).

View larger version (12K): [in this window] [in a new window]   FIGURE 7   Inverse relationship between capacitive increments C1 and C2 () in tissue groups IC (A) and IIC (B) summarized in Table 2. Shown also are the means ± SE () of C1 (group IA) and C2 (group IB) in A, and C1 (group IIA) and C2 (group IIB) in B in those tissues exhibiting either C1 or C2 capacitive increments but not both in the same spectrum.

To ensure that the higher frequency relaxation processes did not arise from critical errors in estimation of the series solution resistance, |Z_meas| was determined in the range of 10-100 kHz. The values of |Z_meas| extrapolated to infinite frequency were found to approach closely those values of R_sol measured in the absence of tissue. To ensure viability of the assumption for frog skins that the impedance of the basolateral membranes was negligible under the conditions of our transport-inhibited studies, basolateral membranes were depolarized within seconds by substitution of basolateral solution Na⁺ with K⁺, which results in marked decreases in R_b and hence Z_b (Tang et al., 1985). The apical membrane dc capacitance remained unchanged from control for 1 h after basolateral membrane depolarization (Fig. 8). Relatively small and slow time-dependent changes in the capacitive increments and relaxation frequencies were observed but were not correlated in time with a decrease in the magnitude of Z_b. No correlation existed between the capacitance spectra and the spontaneous values of the dc shunt resistance (R_p), which ranged between 5.0 and 62.4 k · cm², and the capacitance spectra remained unchanged after needle puncture of the tissues to artificially decrease the dc shunt resistance to low values less than 1 k · cm². Consequently, it was concluded that the observed relaxation processes were attributable to processes associated with the apical membranes of the epithelial cells.

View larger version (23K): [in this window] [in a new window]   FIGURE 8   Changes in complex capacitance C*a after K+-depolarization of basolateral membranes. The control spectrum () consisted of two relaxation processes with relaxation frequencies of 19 Hz and 1.7 kHz. Spectra were measured at 5-min intervals after K+-depolarization () and at 5, 15, 25, 40, and 60 min (shown in this figure). Note absence of change of the dc capacitance and the relatively slow time-dependent changes in capacitance and phase angle at the higher audio frequencies. Relatively small time-dependent increases in the absolute value of capacitance, |C*a|, at 166.4 Hz () and marked time-dependent decreases at 1062 Hz ().

Cole-Cole power-law dependence

Most likely because of distribution of time constants associated with a relaxation process, dielectric dispersions exhibit a power-law dependence that was first recognized by Cole and Cole (1941). _i ranged between 0.5 and 1.0 among all relaxation processes and averaged 0.70 ± 0.03, 0.72 ± 0.02, 0.76 ± 0.02, and 0.95 ± 0.02 for the f₁... f₄ relaxation processes, respectively.

Static dc capacitance varies with short-circuit currents

The static dc capacitance was correlated with the short-circuit currents, which are a measure of the rate of Na⁺ entry into the cells through their apical membranes (Fig. 9). Linear regression analysis of the C_a^dc plotted as a function of the spontaneous I_sc indicated that dc capacitance increased with a slope of 0.028 ± 0.006 (SE) µF/µA and a zero current transport rate intercept of 1.48 ± 0.12 (SE) µF/cm².

View larger version (19K): [in this window] [in a new window]   FIGURE 9   Relationship between short-circuit current (Isc) and dc capacitance among all tissues (n = 49). The linear regression () and the 99% confidence interval () are shown. The slope is 0.028 ± 0.006 (SE) µF/µA with zero current intercept 1.48 ± 0.12 (SE) µF/cm2. The 99% confidence limits are 0.011 and 0.044 µF/µA for the slope and 1.17 and 1.79 µF/cm2 for the intercept.

Contribution of capacitive increments to the static dc capacitance

Because the dc capacitance was correlated with the rate of Na⁺ transport, it was of interest to know which of the dielectric increments contributed to increases in the dc capacitance. To address this question, we plotted the capacitive increments as a function of the static dc capacitance that ranged between 1.25 and 2.72 µF/cm². As indicated in Fig. 10 D, the C₄ capacitive increments (group II tissues) were generally quite small and did not increase significantly with increases in C_a^dc. Although the C₃ capacitive increments (group I) varied considerably among tissues, C₃ did not change significantly with increases in the dc capacitance. In contrast, and as indicated in Fig. 10, A and B, increases in C_a^dc could be attributed to increases in C₁ and/or C₂. Accordingly, the transport-related increases in static dc capacitance were due principally to selective increases in the very low frequency C₁ and/or C₂ capacitive increments.

View larger version (26K): [in this window] [in a new window]   FIGURE 10   Relationships between capacitive increments and the dc capacitance among tissues. Increases in dc capacitance are correlated with increases in either C1 (groups IA and IIA), C2 (groups IB and IIB), or C1 + C2 (groups IC and IIC), as indicated in A and B. Values of C1 and C2 are indicated by solid and open circles, respectively, in A. Solid thick lines are the slopes of the respective linear regressions, and thin lines are the 99% confidence interval, where indicated. Confidence intervals are not shown in A, to preserve clarity. Neither C3 (C) nor C4 (D) changed significantly with increases in dc capacitance.

	DISCUSSION

Top Abstract Introduction Background and theoretical... Materials and methods Results Discussion Appendix References

Apical membrane electrical equivalent circuit and -dispersions

In view of the extensive literature documenting the existence of audio frequency -dispersions in dielectrics, it should not be surprising that a biological plasma membrane like the native apical membrane of frog skin exhibits dielectric relaxation phenomena. We found that ~93% of the static dc capacitance of this membrane was frequency dependent, exhibiting multiple relaxation processes at low and very low audio frequencies. Accordingly, the capacitance of this membrane should be modeled as indicated in Fig. 11 as the parallel sum of capacitive increments (Eq. 5) with time constants R_iC_i = (2f_i)¹ associated with each of the relaxation processes. In contrast to the apical membrane resistance R_a that represents the dc or ionic conductance of the epithelial Na⁺ channels, the R_i of the dielectric relaxation processes are ac resistances that contribute to the membrane resistance (or conductance) only at frequencies greater than zero. These resistances are referred to as ac resistances because the charges giving rise to the relaxations are constrained to motions within the dielectric and thus do not contribute to the dc conductance of the membrane. With mean f_i and C_i taken from Tables 1 and 2, R_i of the four relaxation processes were calculated, which in sequence R₁... R₄ were 8272, 1189, 84, and 76  · cm².

View larger version (13K): [in this window] [in a new window]   FIGURE 11   Electrical equivalent circuit of apical membrane. Ra is the dc ionic resistance to Na+ current through epithelial Na+ channels that are principally amiloride sensitive and some (a few percent) that are amiloride-insensitive. R1... Ri are the ac resistances of the relaxation processes with capacitive increments C1... Ci. Time constants of the relaxation processes are i = RiCi = (2fi)1. Ci is the static capacitance associated with the -relaxation processes at f fi.

Origin of relaxation processes

In principle, the R_i will depend upon the charge density and mobility of the charges and/or dipoles within the dielectric, and so an equivalent volume resistivity (_i) can be calculated. Assuming a maximum dielectric thickness (d) of 5 nm and a uniform distribution of charges within the dielectric, _i = R_i/d. In fact, we do not know how the charges are distributed, and hence _i may be larger than the values summarized in Table 3 if membrane thickness is less than 5 nm. _i ranged between 0.16 and 17.8 M · cm among relaxation processes approaching, at the lower frequency relaxation frequencies, the volume resistivities of 16-18 M · cm distilled water, where at neutral pH charge densities would be in the vicinity of 10⁷ M at aqueous ionic mobilities. Realistically, the mobilities of the dielectric charges are expected to be considerably less than those of an aqueous environment, and charge densities would be scaled upward by one or more orders of magnitude, but not to the extent of reaching the molar range of concentration of the lipids. Because the concentration of lipids within the bilayer and the charge densities associated with the relaxation processes are most likely different by several orders of magnitude, it may be inferred that either an extremely small quantity of charged lipids gives rise to -dispersions, and/or that dispersions may arise from charges associated with the integral transmembrane proteins.

View this table: [in this window] [in a new window]   TABLE 3   Riac and i of relaxation processes

There are no definitive studies that permit unequivocal speculation on the origin of -dispersions in native biological membranes. In this regard, -dispersions have not been observed in studies of planar neutral lipid bilayer membranes with or without adsorbed layers of proteins (Hanai et al., 1964, 1965; White and Thompson, 1973). Proteins studied in aqueous solutions give rise to -dispersions at radio frequencies (Gabler, 1978, and references therein), so it is unlikely that loose protein loops or strands extending from the surfaces of the lipid bilayers can account for the -dispersions of native plasma membranes. Because a large variety of channels, transporters, and other proteins span the bilipid layers of plasma membranes, it is possible and seems likely that low-frequency -dispersions may arise from dipoles associated with integral membrane-spanning proteins that are sensed by the electrical field within the membrane. It has also been pointed out, however, that -dispersions can arise from translational and rotational movements of charged proteins and lipids in vesicles and cells, where unrestricted translation of the lipids and proteins within the plane of the membrane can give rise to low and very low audio frequency dielectric relaxations (Kell and Harris, 1985). It is also well appreciated that dielectric dispersions can arise from charge movements within the membrane that are associated with the gating mechanism of excitable channels in nerve membranes (Armstrong and Bezanilla, 1975).

There has, in fact, been relatively little study of low and very low audio frequency dispersions in biological membranes containing mixtures of proteins and lipids (see the review by Kell and Harris, 1985) and none in epithelial plasma membranes. Our experiments in frog skin are the first of their kind to evaluate the -dispersions at the apical membranes of these cells. Since completion of these experiments, -dispersions have been observed at apical membranes of cell cultured A6 epithelia (Helman et al., 1995; Liu et al., 1995), cell cultured pancreatic ducts (Mangino et al., 1992), and other native tight epithelia (S. I. Helman, unreported observations), so that -dispersions at the apical membrane of frog skin are not exclusive to this tissue. An ultimate understanding of the origin of -dispersions is of particular interest in knowing the interactions and arrangements between the lipids and proteins and their interactions with electrical fields, and the effect of these fields on membrane transport and behavior.

The existence of -dispersions imposes limitations and complications in the design and interpretation of experiments that use measurements of capacitance as a means of assessing changes in membrane area. We refer in part to our own experiments, which were done to determine whether inhibition of apical membrane Na⁺ entry by amiloride caused a change in apical membrane capacitance. It was suggested that amiloride increased C_a (Awayda et al., 1989). We now believe that this suggestion is inconclusive, and we address this issue in the Appendix.

To underscore the issues involved, we illustrate as shown in Fig. 12 for measurements made at a single frequency that increases or decreases in relaxation frequency alone, while all other factors remain the same, give rise to changes in capacitance at the frequency of measurement despite constancy of the dc capacitance, capacitive increments, and membrane area.

View larger version (21K): [in this window] [in a new window]   FIGURE 12   When capacitance is measured at a single frequency in the frequency-sensitive range of a relaxation process, changes of capacitance will occur due to changes of relaxation frequency in the absence of change of dc capacitance, capacitive increments and membrane area. For purpose of illustration, the solid line indicates a relaxation process with absolute capacitance that varies between one and five units (dielectric increment of four units) and a relaxation frequency of 500 Hz. If relaxation frequency of of the process decreases to 100 Hz or increases to 2.5 kHz, as indicated by the dashed lines, and capacitance is measured at a constant frequency, then capacitance will decrease or increase as illustrated at a single frequency of 500 Hz despite constancy of Cdc, C and the dielectric increment.

We illustrate also in Fig. 13, A and C, that capacitance per unit planar area can change because of changes in dielectric increments in the absence of change in actual membrane area. In Fig. 13 A the changes in C^dc are due to changes in the dielectric increment. C_i, which is proportional to area, is unchanged. Similarly in Fig. 13 C, C^dc is increased because of a selective increase in the dielectric increment associated with the C₁ relaxation process. The dielectric increment of the C₂ relaxation process is unchanged, as are the static capacitances C₁ and C₂, which are proportional to area. When changes in area accompany changes in capacitive increments, as illustrated in Fig. 13, B and D, the C_i change together with the C^dc. Thus, despite a more extensive description of the relaxation processes at audio and higher frequencies as revealed by dielectric spectroscopy, the general problem remains, namely, understanding the origin of changes in capacitance, which are due to changes in either area and/or dielectric increments, where the latter can be altered, for example, by phosphorylation of membrane proteins or lipids or other chemical reactions that do not involve changes in area. We know of no absolute or unequivocal procedure to make this assessment based on capacitance measurements in the audio frequency spectrum, because changes in dielectric increments at higher than audio frequency relaxation processes would result in changes in capacitance at audio frequencies indistinguishable from those due to changes in membrane area. Consequently, it is not possible to know unequivocally whether vesicle trafficking involving membrane insertion and retrieval at the apical membranes of the cells is operative based solely on changes of capacitance. The behavior of the dc capacitance and indeed the capacitance at any frequency in the ranges of - and -dispersions may in fact be uncorrelated with changes in membrane area.

View larger version (15K): [in this window] [in a new window]   FIGURE 13   Changes in complex capacitance can occur because of changes in membrane area and/or changes in dielectric increments. (A) A single relaxation process (thick solid line) where, in the absence of change in membrane area, the dc capacitance can either increase or decrease because of change in the dielectric increment without a change in the static capacitance, Ci. If changes in the dielectric increment are accompanied by a change in membrane area, Ci must also change as indicated in B. (C) Two relaxation processes (thick solid line), where the dc capacitance increases because of a selective increase in the C1 dielectric increment, with no change in the C2 dielectric increment or the static capacitances C1 and C2. In this case the membrane area is unchanged. If the change in the C1 dielectric increment is associated with an increase in membrane area, then as indicated in D, the static capacitances C1 and C2 increase together with the dc capacitance.

It remains of particular interest to know the origin of the audio frequency relaxation processes because, as in frog skin, they dominate in determining the membrane capacitance. In the absence of more detailed information of the content and organization of specific membrane lipids, glycolipids, and integral and surface proteins, and recognizing that biological membranes exhibit a great deal of membrane heterogeneity (Jacobson, 1983; Curtain et al., 1988; Sweet and Schroeder, 1988; van Meer and Simons, 1988; Tocanne et al., 1989; Almeida et al., 1992), it will be of interest to examine other epithelial plasma membranes to characterize their -dispersions and to determine how best to differentiate between changes of capacitance due to changes in membrane area and changes in dielectric increments.