## Abstract

Measurement of gap junction conductance (*g*
_{j}) with patch-clamp amplifiers can, due to series resistance problems, be subject to considerable errors when large currents are measured. Formulas developed to correct for these errors unfortunately depend on exact estimates of series resistance, which are not always easy to obtain. Discontinuous single-electrode voltage-clamp amplifiers (DSEVCs) were shown to overcome series resistance problems in single whole cell recording. With the use of two synchronized DSEVCs, the simulated*g*
_{j} in a model circuit can be measured with a maximum error of <5% in all recording situations investigated (series resistance, 5–47 MΩ; membrane resistance, 20–1,000 MΩ;*g*
_{j}, 1–100 nS). At a very low*g*
_{j} of 100 pS, the error sometimes exceeded 5% (maximum of 15%), but the error was always <5% when membrane resistance was >100 MΩ. The precision of the measurements is independent of series resistance, membrane resistance, and*g*
_{j}. Consequently, it is possible to calculate*g*
_{j} directly from Ohm’s law, i.e., without using correction formulas. Our results suggest that DSEVCs should be used to measure*g*
_{j} if large currents must be recorded, i.e., if cells are well coupled or if membrane resistance is low.

- voltage-clamp technique
- connexins
- ion channels

dual-cell voltage clamp is now widely used to study electrical gap junctional communication between cells. Although at first two-electrode voltage-clamp amplifiers were used, i.e., each cell had to be impaled by two microelectrodes (6, 17, 18), single-electrode voltage-clamp amplifiers have been used ever since the tight seal whole cell recording technique became available (5). Two different amplifier designs are currently available for this kind of experiment. Patch-clamp amplifiers preset the potential at the electrode to the command potential and continuously inject current to correct for voltage deviations from the command potential (5, 11, 16). “Switched” single-electrode voltage-clamp amplifiers change at high frequency between potential measurement and current injection (2, 31). The intracellular potential is measured at the end of the current-free period when the voltage transient occurring at the electrode tip during injection of current has completely relaxed. Because of the discontinuous mode of action, these amplifiers are often referred to as discontinuous single-electrode voltage-clamp amplifiers (DSEVCs).

Up to now, patch-clamp amplifiers have been used almost exclusively in dual-cell voltage-clamp experiments. However, when patch-clamp amplifiers are used for the measurement of gap junction conductance (*g*
_{j}), series resistance (formed by the pipette and the broken membrane patch) can cause serious errors in the measurement (22, 24, 28, 30). Series resistance is a problem mainly in situations in which gap junction resistance (*R*
_{j}) and/or membrane resistance is low, because large currents will flow. According to Ohm’s law, this will result in large voltage drops across the series resistance. In those situations, errors in the measurement of*g*
_{j} can become as large as 70% (22). To correct for the errors introduced by series resistance, several correction formulas were proposed that take into account either series resistance alone (24, 28) or a combination of series resistance and cell input resistance (22). All of these formulas depend critically on an exact estimate of the series resistance, which is not always easy to obtain (24).

As mentioned above, switched single-electrode voltage-clamp amplifiers inject current discontinuously and measure the intracellular potential at a time when no current flows across the pipette. Thus these amplifiers should avoid the problem of errors caused by series resistance. It was shown recently that DSEVCs avoid series resistance problems in voltage-clamp measurements in single cells (4, 7, 14). However, there are no studies on the use of DSEVCs in the measurement of *g*
_{j}.

It was, therefore, the aim of this study to investigate the usefulness of DSEVCs for *g*
_{j}measurement. In most studies, electrodes with direct current resistances of 2–5 MΩ are used. This will usually result in series resistance of >5 MΩ (5, 11). However, series resistance can increase considerably during the course of an experiment, and higher values (10–20 MΩ) can easily result. In perforated patch experiments, series resistance often is as high as 50 MΩ. Cell input resistance (membrane resistance) depends mainly on the cell type and electrode solution used and can vary between ∼15 MΩ in adult ventricular cardiomyocytes (9, 29) and >1 GΩ in neonatal rat heart cells (15). *R*
_{j}can vary between 2–3 MΩ (26, 28) and >10 GΩ when single gap junction channels are investigated (21). Using these data, we designed model circuits with different series resistances, membrane resistances, membrane capacitances, and*R*
_{j} to investigate the accuracy of*g*
_{j} measurements with two synchronized DSEVCs. Our results show that in all recording situations (series resistance 5–47 MΩ; membrane resistance 20–1,000 MΩ)*g*
_{j} (1–100 nS) was precisely measured by the DSEVCs with a maximum error of 5%. At a *g*
_{j} of 100 pS, which is well within the ranges of gap junction single-channel conductance, the error was only <5% when membrane resistance was >100 MΩ. The results strongly suggest that DSEVCs should be used for the measurement of*g*
_{j} in those situations when either membrane resistance is low or*g*
_{j} is high.

## METHODS

#### Equivalent circuit.

The equivalent circuit we used in this study is depicted in Fig.1. Each cell is represented by a parallel combination of a resistor (*R*
_{m1} or*R*
_{m2}) and a membrane capacitance (*C*
_{m1} or*C*
_{m2}). The pipette resistance and the conductance connecting the two cells are represented by resistors [series resistance (*R*
_{s1} and*R*
_{s2}) and junctional resistance (*R*
_{j}), respectively].

To study the influence of each of the components of the model circuit on the outcome of the measurements, several combinations of resistances and capacities were investigated. Specifically, the following values were chosen for the respective components: series resistance, 5, 10, 20, and 47 MΩ; membrane resistance, 20, 100, 500, and 1,000 MΩ;*R*
_{j}, 10, 20, 100, 200, 1,000, and 10,000 MΩ; and*C*
_{m}, 56, 150, and 220 pF. These values cover almost the entire physiological range for these parameters. All possible combinations of resistors were investigated with the different cell capacitances. Resistors of <1 GΩ had tolerances of 5%. The other resistors had tolerances of 10%. Because it was the goal of this study to determine the accuracy of the measurement of *R*
_{j}and *g*
_{j}, we measured the exact values of all resistors used to represent*R*
_{j}. For this purpose, the resistors were put between the amplifier headstage and ground. We then measured the voltage drop across the resistors in response to an applied current of known magnitude. The resistance was then calculated according to Ohm’s law. The exact values were 9.88, 19.7, 99.8, 197, 990, and 9,578 MΩ, corresponding to conductances of 101.21, 50.81, 10.07, 5.09, and 1.01 nS and 100 pS, respectively.

#### Experimental protocol and data evaluation.

As can be seen from Fig. 1, the*R*
_{j} is electrically isolated from ground. Therefore, any current flowing across *R*
_{j} will not lead to a change of the sum of the currents measured by the two resistors (*I*
_{1}+ *I*
_{2}). This means that application of a voltage difference across*R*
_{j} will result in junctional currents (*I*
_{j}) of equal magnitude but opposite polarity in both cells. This principle is exploited for the measurement of*g*
_{j}. When a voltage jump is applied to one cell, this will result in a change in the current recorded from this cell. This current is the sum of the*I*
_{j} and the elicited membrane current (*I*
_{m}). Because the voltage in the other cell was kept constant, the change in the current recorded from this cell (Δ*I*
_{2}) is equal to −*I*
_{j}. Consequently, *g*
_{j}can be calculated according to
Equation 1The membrane potentials of *cells 1* and*2*(*V*
_{1} and*V*
_{2}, respectively) and Δ*I*
_{2} were determined from the recordings, and*g*
_{j} was calculated from *Eq*.*
1
*.

Throughout this paper, we will refer to the pulsed cell as*cell 1* (with*R*
_{m1},*R*
_{s1}, and*I*
_{1}) and to the nonpulsed cell as *cell 2* (with*R*
_{m2},*R*
_{s2}, and*I*
_{2}), regardless of the actual physical situation.

The following experimental protocol was used to measure*R*
_{j} and*g*
_{j}. A voltage command of −100 mV (500 ms in duration) was applied to*cell 1* while *cell 2* was held at 0 mV. Then the “cells” were switched, i.e., the same stimulus was applied to the other cell. This protocol was repeated five times, and the resulting voltage and current traces were averaged.

Data, the voltage and current of each amplifier, were sampled at a rate of 5 kHz/channel after low-pass filtering at 1 kHz. The WinTida (HEKA elektronik, Lambrecht, Germany) data acquisition system, which uses the ITC-16 interface (Instrutech, New York, NY), was used for data acquisition and analysis.

#### Amplifiers and amplifier adjustment.

Two DSEVCs were used for this study (SEC 05 and SEC 10, npi-electronic, Tamm, Germany). DSEVCs use a time-sharing principle, i.e., they switch between current injection and potential measurement with a high frequency. Because the membrane potential is measured at a time when no current flows across the electrode, series resistance problems are totally avoided (Refs. 2, 31; for a review see Ref. 12). Both amplifiers were used with the same (synchronized) switching frequency (*f*
_{sw}) by connecting the SEC 05 system to the internal clock of the SEC 10 system. In this way, current injection and membrane voltage sampling occurred at the same time in both systems, thus avoiding interference. However, to achieve complete suppression of the series resistance, the current injection artifact must decay almost completely before the membrane potential is sampled. To obtain a linear response from the DSEVC, the*f*
_{sw} must be in the range of tens of kilohertz (2), i.e., the current injection artifacts must decay very rapidly (within a few microseconds). In the amplifiers used here, this is achieved by the use of supercharging (1) in addition to the regular feedback-based capacity compensation method (12). Supercharging here is used not for speeding the response of the membrane potential to a current step [as described initially by Armstrong and Chow (1)] but to eliminate stray capacitances around the microelectrode and to reduce the response time of the microelectrode to brief current steps of a few microseconds. With this approach, it is possible to compensate stray capacitances around a microelectrode by injection of a certain amount of charge induced by a spike superimposed on the command signal applied to the electrode. Electronic circuits and calculations are given in detail in the papers by Strickholm (19, 20). The increase in speed is considerable (up to 100-fold; see Refs. 19 and 20). With this approach, it was possible to reduce the settling time of the microelectrode after a current pulse to 2–3 μs (tested and checked in many labs, e.g., see Refs. 8 and14). From these findings, a frequency formula was derived that describes the relationship between the electrode time constant, the*f*
_{sw} of the DSEVC, the sampling frequency of the data acquisition system (*f*
_{s}), the upper cutoff frequency of the low-pass filter (*f*
_{f}), and the time constant of the cell membrane (27)
Equation 2where*f*
_{e} is the upper cutoff frequency (−3 dB frequency) of the microelectrode and*f*
_{m} is the upper cutoff frequency of the membrane.

With the time constant in the range of 1–3 μs recorded for the electrode resistances used in this study,*f*
_{e} is 80–160 kHz; *f*
_{sw} of the DSEVC was set to 30–50 kHz (calculated range 25–53 kHz),*f*
_{s} was 5 kHz, and*f*
_{f} was 1 kHz (see below). These settings are currently used for recordings in our lab. The *f*
_{m} was calculated to be 140 Hz (membrane resistance, 20 MΩ;*C*
_{m}, 56 pF).

The headstage output signal (i.e., the response to the discontinuous current injection at the electrode level) from both amplifiers was monitored during all experiments with a separate oscilloscope. The*f*
_{sw} was adjusted to the optimal value between 30 and 50 kHz, and the electrode artifacts decayed in 1–3 μs after optimal tuning of capacity compensation. Thus the recording conditions satisfied the criteria for reliable single-electrode voltage clamp.

The fast settling of the microelectrodes allows the use of high switching frequencies (20–50 kHz), which in turn allows high voltage-clamp gains of up to 10–20 μA/V. By use of a proportional-integral controller, it is possible to further increase the gain to 100 μA/V without affecting the noise level or the stability of the clamp. The proportional-integral controller yields an instantaneous fast response to changes (proportional gain) while the integral part increases the gain for frequencies below the (adjustable) upper cutoff frequency of the integrator (i.e., for slow signals; Ref.14). Thus the so-called “integrator” improves voltage control considerably, allowing voltage-clamp measurements with an error <1%, which is very important for the accuracy of the calculation of*g*
_{j}. A more detailed description of the amplifiers can be found in the publications by Richter et al. (14) and Draguhn and co-workers (4).

The resistors *R*
_{s1}and *R*
_{s2} were connected to the inputs of the amplifier headstages via shielded cables. Each time the series resistance was changed (i.e., when the*R*
_{s1} and*R*
_{s2} resistors were changed) the electrode capacitance was carefully canceled with the capacity compensation circuit of the amplifiers and the voltage-clamp gain was adjusted. To do so, both cells were simultaneously subjected to voltage jumps of 40 mV. Voltage-clamp gain was adjusted to give the fastest possible voltage responses with an overshoot of <5% (see also *Determinants of voltage control*). The*f*
_{sw} was set to 32 kHz in all experiments (25% duty cycle).

#### Cell culture.

Human cervix carcinoma HeLa cells transfected with connexin 43 (Cx43; kindly provided by K. Willecke, Bonn, Germany) were cultured in DMEM supplemented with 100 μg/ml streptomycin, 100 U/ml penicillin, and 0.5 μg/ml puromycin.

#### Electrophysiology.

For electrophysiological experiments, the cells were grown in plastic petri dishes on coverslips. Immediately before the experiments, the coverslips were removed from the dishes, washed with modified Tyrode solution, and transferred to an experimental chamber containing modified Tyrode solution (in mM: 135 NaCl, 4 KCl, 2 CaCl_{2}, 1 MgCl_{2}, 0.33 NaH_{2}PO_{4}, 10 HEPES, and 10 glucose, pH 7.4). Patch electrodes were pulled from borosilicate glass (GC 150FT, Clark Electromedical, Pangbourne, UK) with a microprocessor-controlled puller (P97, Sutter Instruments, Novato, CA) and had direct current resistances of 3–4 MΩ when filled with intracellular solution (in mM: 125 CsCl, 8 NaCl, 1 CaCl_{2}, 10 EGTA, 2 Na_{2}ATP, 3 MgATP, 0.1 Na_{2}GTP, and 10 HEPES, pH adjusted to 7.2 with CsOH). For the measurement of single-channel gap junction currents, heptanol (2 mM) was added to the modified Tyrode solution to uncouple the cells and reveal single gap junction channel events.

## RESULTS

#### Determinants of voltage control.

For accurate measurement of*g*
_{j}, it is necessary for the voltage of the nonpulsed cell to be kept constant. To find out which parameters and adjustments are important for exact voltage control, we investigated different settings with respect to voltage-clamp gain. Figure 2 illustrates that accurate voltage control is possible only when both the proportional and the integral parts of the voltage-clamp amplifiers were used. With the exception of small voltage transients at the beginning and at the end of the voltage pulse, there was no change in the voltage of the nonpulsed cell. Employment of only the proportional part of the amplifier did not allow clamping of the voltage in the nonpulsed cell (Fig. 2
*B*). Adjustment of the proportional part of the amplifier was, however, important for the speed of the clamp and for adjustment of the overshoot.

#### Effect of series resistance.

The effect of series resistance on the accuracy of the measurement of*g*
_{j} was assessed by using different resistors to represent*R*
_{s1} and*R*
_{s2}. In Fig.3
*A*, the measured fraction of*g*
_{j} is plotted vs.*g*
_{j} for a combination of resistors in which the membrane resistance amounts to 1 GΩ in both cells.*R*
_{s1} was kept constant at 5 MΩ and combined with different values for*R*
_{s2}. No influence of series resistance on the accuracy of the measurements of*g*
_{j} could be observed. The error was always well below 5%. Almost identical results were obtained when*R*
_{s1} was set to 10, 20, and 47 MΩ, respectively (data not shown). For technical reasons, the error can become >5% at a*g*
_{j} of 100 pS if the membrane resistance of both cells is set to 100 or 20 MΩ (Fig.3
*B*; see *Measurement of very low g _{j}
*and discussion for an explanation).

#### Effect of membrane resistance.

To investigate the effect of membrane resistance on the accuracy of the measurement of*g*
_{j}, we investigated all possible combinations of*R*
_{m1} and*R*
_{m2}. Figure4 shows representative results obtained with *R*
_{m1} set to 20 MΩ and variable values of*R*
_{m2}. The*g*
_{j} is accurately measured (error <4%), except when*g*
_{j} is set to 100 pS (see *Measurement of very low g _{j}
* anddiscussion for an explanation). Voltage control and correct assessment of

*g*

_{j}are possible even if membrane resistance changes suddenly in either the stepped or the nonstepped cell, as illustrated in Fig.5. Changing

*R*

_{m1}from 1 GΩ to 500, 100, and 20 MΩ and back again did not lead to loss of voltage control in either cell, nor did it affect the current recorded in the nonpulsed cell.

Measurements in Cx43-transfected HeLa cells also demonstrate that membrane resistance does not affect the measurement of*g*
_{j} with two DSEVCs. The results shown in Fig. 6 were taken from an experiment in which one of the cells developed a large leak current. The membrane resistances were calculated to be 320 and 46 MΩ for the “nonleaky” cell and the “leaky” cell, respectively. Changing the membrane potential of the leaky cell from a holding potential of 0 mV to −40 mV resulted in a large current composed of the*I*
_{m} and the gap junction current in this cell (*I*
_{1} in Fig.6
*A*) and a gap junction current of 1.83 nA in the other cell (*I*
_{2} in Fig.6
*A*). Application of the same voltage jump to the nonleaky cell resulted in a much smaller current in this cell (*I*
_{1} in Fig.6
*B*) and a gap junction current of 1.80 nA in the leaky cell (*I*
_{2} in Fig.6
*B*), demonstrating that the gap junction current was not influenced by the (different) membrane resistances of the cells.

#### Effect of R_{j}.

As can be seen from Figs. 3 and 4, there was no effect of*R*
_{j} on the accuracy of its measurement (see below for an explanation for the variability seen with*g*
_{j} set to 100 pS).

It is well known that, with patch-clamp amplifiers, the error in the measurement of *g*
_{j}depends on the ratio of*R*
_{j} to series resistance (*R*
_{j}/*R*
_{s}) (30) and the ratio of series resistance to membrane resistance (*R*
_{s}/*R*
_{m}) (3, 24). We therefore analyzed the maximum error in the determination of *g*
_{j} as a function of the quotients*R*
_{j}/*R*
_{s}and*R*
_{s}/*R*
_{m}, respectively. It is evident from Fig. 7that with DSEVCs the precision of*g*
_{j} measurement does not depend on either*R*
_{j}/*R*
_{s}or*R*
_{s}/*R*
_{m}.

#### Measurement of very low g_{j}.

During measurement of very small currents (<20 pA), noise introduced by the amplifiers and the headstages became a limiting factor for current resolution. To overcome this obstacle, separate headstages with a 10-fold increased current output gain and an improved signal-to-noise ratio were tested for the measurement of small*I*
_{j}. Measurement of very small *I*
_{j}is only possible if the membrane resistances of both cells are high, because otherwise the membrane currents would introduce too much noise. For this reason, we tested these headstages only with membrane resistances of 500 MΩ and 1 GΩ;*R*
_{j} was set to values between 200 MΩ and 10 GΩ (5 nS to 100 pS).

The results of these measurements are summarized in Fig.8. Independent of series resistance,*g*
_{j} is measured with extreme accuracy (maximum error <4%).

To find out whether the use of these headstages allows measurement of single *I*
_{j}, we performed the corresponding experiments in HeLa cells expressing Cx43 (Fig. 9). The results demonstrate that high-resolution measurements of single gap junction channel events are indeed possible with DSEVCs equipped with these headstages. Event amplitudes calculated from an all-point histogram of the data shown in Fig. 9 reveal transitional amplitudes of ∼60 pS, which correspond very well with the results of other investigators (10,23).

## DISCUSSION

Measurement of *g*
_{j}with the help of “dual whole cell voltage clamp” critically depends on accurate voltage control in both cells of a pair. It is therefore important to record the intracellular voltage. Continuous feedback amplifiers (patch-clamp amplifiers) cannot measure the intracellular voltage, and large currents and/or high series resistances lead to large errors in the estimate of the intracellular voltage. This was shown to cause serious errors in the measurement of*g*
_{j} (22, 30).

In this study, we demonstrate that these errors can be avoided by the use of DSEVCs. Neither series resistance nor membrane resistance had a significant influence on the precision of*g*
_{j} measurement. The maximum error was <5% in all recording situations, except at very high *R*
_{j}values (10 GΩ = 100 pS) when the error was larger (maximum 12%) for technical reasons. The errors seen at an*R*
_{j} of 10 GΩ can be explained by the fact that currents in excess of ±5 nA must be recorded if membrane resistance is low, i.e., 20 MΩ. This will lead to a very low resolution of the digitized current signal. Under the assumption of a current recording range of ±10.24 nA and an effective resolution of 14 bits (Instrutech ITC-16 user manual), 1 significant bit will correspond to ∼1.25 pA. This amounts to ∼12.5% of the expected signal of 10 pA. Together with some noise introduced by the model circuit and the headstages, this error can easily explain the variability in the measurement of the*g*
_{j} of 100 pS seen with low membrane resistances (cf. Fig. 4).

However, this problem was overcome by the use of headstages that, due to higher amplification of the signal, provided an improved signal-to-noise ratio (cf. Fig. 8). Because the simulated*g*
_{j} of 100 pS is well within the range of single-channel conductances of connexins, ranging from 270 pS for Cx37 (13) to ∼26 pS for Cx45 (25), we tried to record gap junction single-channel currents with these headstages. We found that we could reliably record gap junction single-channel events with two synchronized DSEVCs (Fig. 9). Amplitude resolution and background noise were comparable to those of a combination of two patch-clamp amplifiers.

Our data show that DSEVCs allow measurement of the transcellular potential difference and exact, direct measurement of*I*
_{j}. These measurements allow a very accurate calculation of*g*
_{j}. With patch-clamp amplifiers, in many situations, correction formulas must be used to calculate*g*
_{j} (22, 24). However, a very important point should be considered regarding use of these formulas: they critically depend on exact estimates of series resistance. With respect to this prerequisite, Veenstra and Brink (24) demonstrated that estimation of series resistance from either the amplitude of the whole cell capacity transient or from the product of cell capacitance and the time constant of decay of the capacitive transient gave rise to errors of maximally 79 and 27%, respectively. On the other hand, Van Rijen and co-workers (22) calculated that an error of 20% in the estimate of series resistance can give rise to errors of up to 37% in the calculation of*g*
_{j}. Another important point implicit in the use of such correction formulas is the fact that they correct for voltage changes in both the pulsed and the nonpulsed cell. However, because the voltage change in the nonpulsed cell can elicit membrane currents, the current recorded from the nonpulsed cell can contain “contaminating” currents that cannot be distinguished from the*I*
_{j} and thus corrupt the calculation of*g*
_{j}.

We demonstrated that DSEVCs can accurately control the cellular potential and thus the transcellular potential. This feature might be especially useful in studies of the voltage-dependent gating behavior of gap junction channels because DSEVCs can avoid errors in voltage control and cross talk between the amplifiers. Furthermore, DSEVCs could be advantageous in studies investigating signal transduction and the pharmacology of connexins. Normally, membrane ion channels are blocked by heavy metal ions and/or other substances to increase cell input resistance (26, 28). Those substances, however, may interfere with signal transduction and thus spoil such experiments. Because DSEVCs can accurately measure*g*
_{j} even when membrane resistance is low, those interventions may not be necessary, and interactions between channel blockers and signal transduction processes or connexins may be avoided.

In summary, our study shows that DSEVCs with proportional-integral voltage-clamp circuits can accurately control the voltage of two electrically connected cells and, consequently, allow accurate determination of*g*
_{j}. The accuracy of the measurement of*g*
_{j} is independent of series resistance, membrane resistance, and*g*
_{j}. Our results strongly suggest that DSEVCs should be used in situations in which*1*) membrane resistance is low,*2*)*g*
_{j} is large, and*3*) the voltage dependence of*g*
_{j} is important.

## Footnotes

Address for reprint requests and all other correspondence: M. Lauven, Institute of Pharmacology, University of Cologne, Gleueler Str. 24, D-50931 Köln, Germany (E-mail:melani.lauven{at}uni-koeln.de).

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- Copyright © 1999 the American Physiological Society