Patch-clamp experiments

Experimental setup was built around inverted microscope (Diaphot TMD, Nikon, Japan) placed on anti-vibration base (Micro40 M6, Halcyonics, Germany) and shielded by a Faraday cage. Cells were recorded in a 0.5 ml open air chamber with cover glass bottom (Warner Instruments, LLC, USA) and grounded with Ag/AgCl pellet electrode (Warner Instruments). Membrane current was recorded with a patch-clamp amplifier (Axopatch 200B, Axon Instruments Inc., USA) using a glass micropipette fixed to a standard holder of the head-stage preamplifier (CV-203BU, Axon Instruments Inc., USA) mounted on a 4-axis electrical micromanipulator MX 7600R (Siskiyou, USA). Patch pipettes were pulled from borosilicate glass capillaries (Sutter Instruments Co., USA) on Flaming/Brown Micropipette Puller Model P-97 (Sutter Instruments Co., USA). The pipette resistance was typically 2–4 MΩ. Pipette tips were coated with SYLGARD184 (Dow Corning, USA) to reduce stray pipette capacitance and its fluctuations. Formation of the seal between the patch pipette and the cell membrane was monitored by the pipette potential in cell-attached current-clamp mode [12]. A cell was accepted for measurements if the potential stabilized below -40 mV and it withstood rupturing of the membrane patch. The final seal resistance was typically several tens of GΩs.

Ventricular myocytes of rat myocardia were isolated enzymatically [13] from young adult male or neonatal hearts of Han-Wistar rats (Dobra Voda, Slovak Republic). All anaesthetic and surgical procedures were carried out in accordance with the European directive 2010/63/EU, and were approved by the State Veterinary and Food Administration of the Slovak Republic (Ro-2821/09-221, Ro-354/16-221) and by the Ethical Committee of the Institute of Molecular Physiology and Genetics, Slovak Academy of Sciences.

Experiments on isolated cardiac myocytes were performed at room temperature of about 23°C. The standard bath solution contained (in mmol/l): 135 NaCl, 5.4 CsCl, 1 CaCl₂, 5 MgCl₂, 0.33 NaH₂PO₄, 10 HEPES, 0.01 TTX (pH 7.3; 300 mOsm). The composition of the pipette solution was (in mmol/l): 135 CsCH₃SO₃, 10 CsCl, 1 EGTA, 3 MgSO₄, 3 Na₂ATP, 0.05 cAMP, 10 HEPES (pH 7.1; 300 mOsm). Osmolarity of all solutions was verified using Osmomat 010 (Gonotec GmbH, Berlin, Germany).

Impedance measurements and analysis

Whole cell impedance measurements were made using the patch-clamp amplifier in the whole-cell voltage clamp mode with a feedback resistor of 50 MΩ. The gain was set to 0.2 mV/pA to avoid amplifier saturation. The output current was low-pass filtered by the built-in 4-pole Bessel filter set to 10 kHz, digitized at 100 kHz by a 16-bit data acquisition system (Digidata 1320A, Axon Instruments Inc.) and stored in proprietary Axon Binary Files (.abf) for off-line analysis.

The bipolar square-waves voltage stimuli were generated by 16-bit D/A converter (Digidata 1320A). The stimulation period T_S was set in relation to the time constant τ of measurement as T_S = 12τ (6τ for each half-period [5]) and ranged from 20 to 0.6 ms (corresponding to stimulation frequency of 50–1,670 Hz). During impedance measurements, the holding potential was set to 0 mV. Experiments were controlled by software (pClamp 9, Axon Instruments Inc.).

The recorded current responses to square-wave voltage stimuli were processed by the deconvolution procedure specified in the Results. Where indicated, the standard processing method commonly used in patch-clamp laboratories (e.g., pClamp v.10, Molecular Devices, https://mdc.custhelp.com/euf/assets/content/pCLAMP10_User_Guide.pdf) was applied. We followed original procedures [5, 4], in which the first 60 μs of a current record were disregarded and the remaining current was fitted by Eq 1. The peak current amplitude was determined by extrapolation of the best fit to the current response to the new zero time corrected for filter delay. When such estimate of the peak current amplitude was used for calculation of the access resistance by Eq 2A, the estimates of membrane capacitance C_M strongly correlated with R_A. The cross-correlation could be minimized, as suggested by Thompson et al. [5], by tuning the zero time correction to a value that provided acceptable cross-correlation in C_M traces. At 10 μs sampling interval, the exact zero time correction required its tuning down to a fraction of microsecond in about 5–10 steps. The cross-correlation error arose due to the improper best fit; the errorneous estimates of the τ and R_A led to the zero time being wrongly projected.

The parameters of current responses were estimated by fitting Eq 1 to both half-periods of current responses I_M(t) and averaged. (1) where I_P is the peak current, I_S is the steady current, and τ is the time constant. The parameters of the recorded cell circuit, approximated by membrane resistance R_M, membrane capacitance C_M, and access resistance R_A (Fig 1) were calculated from the estimated parameters of current responses (I_P, I_S, τ) using Eqs 2A–2C.

(2A)(2B)(2C)

Notice that Eq 2A contains I_S since in practise the peak current I_P is not calculated from the zero current level, but due to the use of symmetrical bipolar stimulation it is estimated from the steady state current I_S of the preceding half-period. The analysis was performed off-line using routines written in MATLAB (v.2014b, Mathworks Inc., Natick, MA, USA) and implemented in the software MAT-MECAS (http://mat-mecas.sourceforge.net).

Standard deviation σC_M of the analysed time series of capacitance measurements was determined for the bandwidth of interest B, typically 50 Hz. Since the measurements were performed at a stimulation frequency f_S, the number of periods to be averaged m was found using the relationship m = f_S /B. The theoretical limit of C_M resolution for the given bandwidth was calculated using Eq 3 considering white thermal noise [5]: (3) where k is the Boltzmann constant and T is the absolute temperature.

Simulations and evaluation of errors

All simulations were performed in MATLAB (v.2014b, Mathworks).

To test the performance of the deconvolution procedure, we simulated 500 records of membrane current responses using Eqs 4A–4D and a seeding set of random triplets of circuit parameter values generated uniformly within ranges C_M (pF) ∈ [5, 200], R_M (GΩ) ∈ [0.02, 2], R_A (MΩ) ∈ [2, 20]: (4A) where (4B) and the theoretical values or current parameters are: (4C) (4D)

To simulate real recording conditions, the calculated membrane current responses were digitally filtered with the MATLAB function “filter”. The digital filter was an implementation of the analog 10 kHz 4-pole Bessel filter, using the method of invariance of impulse response “impinvar”, while the analog filter was emulated by the MATLAB function “besself”.

To evaluate the absolute error of estimate of circuit parameters, the simulated, digitized and filtered current responses were analysed in the same way as real records (using Eqs 1 and 2A–2C), and the difference between the seeded and the estimated parameter values was calculated.

For evaluation of the crosstalk error due to R_A variation, we simulated two periods of current responses for each random triplet of parameter values, so that the R_A value of the second period was increased by 1 MΩ. The simulated current responses were analysed in the same way as real records using Eqs 1 and 2A–2C. Crosstalk errors were evaluated as the difference between the estimated parameters of the two sets of simulated current responses.

To assess the effect of R_A variation on the resolution of membrane capacitance measurements, the current responses were calculated for a cell circuit with C_M = 50 pF, R_M = 200 MΩ, and R_A values fluctuating randomly around the mean of 5 MΩ. The resolution was evaluated as standard deviations σC_M estimated at the corresponding standard deviations σR_A (22.6, 45.2, 90.4, 135.6, 180.8, 361.7, and 452.1 kΩ), at the bandwidth normalized to 50 Hz.

The dependence of the circuit parameter estimates on the resistance R_Seal was simulated using Eq 5. (5) The Eq 5 differs from Eq 4A by the last term that includes the R_Seal. The current responses for a cell with C_M = 140 pF, R_M = 500 MΩ, and R_A = 4 MΩ were stimulated with V_Stim amplitude of ±10 mV and a period of 6τ. The seal resistance R_Seal was incremented by 100 MΩ, period by period, from 1 to 150 GΩ.

The deconvolution procedure

The recorded current, I_R(t), represents the convolution of the input current I_M(t) with the impulse response, h(t), of the signal path [11]: (6)

The original input signal I_M(t) can be recovered by a deconvolution process, if the impulse response of the recording system is known. For this purpose, we recorded the impulse response of the recording system (Fig 2A). The voltage impulse was represented by a 10-μs 150-mV voltage stimulus (generated by the digitizer set to 100 kHz). When delivered to the amplifier configured with a resistor between the input and the ground, the impulse generated a complex current response (R+C response), which contained the resistive component (R response) combined with the capacitive component (C response) generated by the parasitic capacitance and the inner circuitry of the amplifier. These two capacitive components are always present in the current recording path. Their contribution to the signal should be eliminated to obtain the correct impulse response, h(t). To this end, the voltage impulse was delivered to the amplifier with the same resistor connected to the input but disconnected from the ground. In this arrangement, the current response contained only the capacitive component (C response). To supress the noise, the R+C and C responses, each of 20 ms duration, were recorded 4096 times and averaged. The average C response was subtracted from the average R+C response and the resulting average R-response was transformed by the fast Fourier transform algorithm (FFT) to the frequency domain and normalized to 0–1 range. This operation yielded the system frequency characteristic H_S in 0.05–50 kHz bandwidth (Fig 2B). A special attention was given to proper grounding of the set-up to eliminate high frequency interference signals, because their presence in the H_S function caused oscillations in the reconstructed currents.

In practice, H_S is applicable to reconstruction of current responses measured with the same sampling and filtering frequency. For best results in cell experiments, the charging current of parasitic capacitance of the recording microelectrode should be properly cancelled either by using the circuitry of the amplifier or by subtraction of pre-recorded capacitive current responses, both in the cell-attached configuration before breaking the patch membrane.

Reconstruction of membrane current responses distorted by the recording path was performed in three steps. First, the recorded current response was transformed to the frequency domain by FFT. Then, the deconvolution was performed in the frequency domain by dividing the recorded current response by H_S. Finally, the result was transformed to the time domain by inverse FFT, which yielded the reconstructed current response (Fig 2C).

The reconstructed current responses to square voltage stimuli followed exponential time course and could be approximated by Eq 1. Consequently, the estimated current parameters corresponded to current parameters of the theoretical circuit and Eqs 2A–2C could be used for calculation of the impedance parameters of the recorded circuit.

The whole processing and analysis procedure for deconvolution of membrane current records was written into an interactive software package MAT-MECAS in MATLAB v. 2014b (see: http://mat-mecas.sourceforge.net), which was used also in this study. The software reads records of current responses to square-wave voltages, saved in an appropriate format (the Axon Binary File abf; Axon Instruments Inc., or a text file), and creates records of the time course of C_M, R_M, R_A, and other specified parameters in xlsx format. MAT-MECAS also provides special tools for secondary analyses, including identification of events and artefacts in C_M records. A record of 1000 current responses, each of 100 samples per period, was analysed in about 60 s at full temporal resolution (1 kHz) or in about 3 s if the bandwidth was reduced to 50 Hz. The software application was tested for correct data processing and for correct mathematical processing using simulated data, as these provide exact control of outputs relative to inputs.

Accuracy and resolution

The errors arising from the deconvolution procedure itself were assessed in the simulated set of filtered current responses generated with the set of random triplets of circuit parameters, as described in Materials and Methods. The simulated responses were processed by the deconvolution procedure and fitted with Eq 1. Differences between the estimated parameters of current responses and the ones calculated for the given triplet of circuit parameters are plotted in Fig 3A. The deconvolution procedure provided reconstructed currents almost identical to the original ones, since the differences Δτ, ΔI_S, and ΔI_P were negligible at any combination of the circuit parameter values.

Differences between the circuit parameter values used for simulation and those estimated by the deconvolution procedure are shown in Fig 3B. We plotted the differences as the percentage of the parameter value in the triplet against the corresponding τ values calculated for each triplet using Eq 4B. The time constant of current response τ is a convenient reference parameter since it incorporates all circuit parameters. It also determines the reliability of the fitting procedure since, due to the constant sampling interval, the number of sampled data points is proportional to τ. The relative errors in C_M were under 0.02% and virtually independent of τ at any combination of circuit parameter values. The relative error in R_M estimates was less than 0.1% for τ > 500 μs and increased to about 1% for τ = 50 μs. The relative error in R_A estimates was very low, under 0.03% for τ > 100 μs at any combination of parameter values.

The resolution limit of the deconvolution procedure for membrane capacitance measurements was estimated on hardware models made of different capacitors combined with membrane and access resistors (Fig 3C). The deconvolution procedure performed near the physical limit of C_M resolution given by thermal noise (Eq 3), even for very small cells (C_M < 20 pF).

Cross-talk error

In measurements on real cells, all circuit parameters (Fig 1A) may vary in a rather wide range. Therefore, the parameter cross-talk is a crucial issue of high-resolution measurements. Of special consideration is the access resistance, which solely defines I_P, contributes substantially to the time constant of measurement, and may vary by up to several MΩ on short as well as long-time scales. At the same time, the cell membrane resistance R_M may change from a few GΩ to a few tens of MΩ due to experimental interventions or membrane impedance nonlinearities. On the other hand, R_M contributes substantially to the time constant only when comparable to R_A, which is not a standard situation.

Propensity of the deconvolution method for cross-talk error was tested by impedance measurements using the same set of parameter triplets as that used for testing the accuracy, to which siblings differing in R_A by 1 MΩ were generated. The Fig 4A shows results of analysis of the simulated current responses related to the time constant of the membrane charging τ, through which the interdependence of parameters arises. Except for the very short τ’s, the cross-talk of a 1 MΩ change in R_A to measured C_M and R_M was negligible. The differences in C_M estimates were typically much less than 1 fF, while the estimates of R_A changes were almost exactly 1 MΩ (0.9995 MΩ). Analysis of the same data set by the standard procedure (see Methods) returned substantially larger errors in estimates of any impedance parameter (Fig 4A).

The role of cross-talk of R_M variation to other parameters was studied in additional simulations. These revealed that even very large variation of membrane resistance does not present a problem, since the method managed to keep errors of C_M estimation below 0.2 fF when R_M was changed by 100 MΩ in the 0.02–2 GΩ range (not shown).

Additional assessment of the crosstalk error was performed using the built-in compensation circuitry of the recording amplifier [5]. Recordings shown in Fig 4B were made with the input of the amplifier open and with the membrane capacitance compensation set to about 25 or 130 pF to simulate a small or a large cell, respectively. The series resistance compensation was set to about 5.5 MΩ in both cases. During recording, the series resistance compensation was changed to about ~6.5 MΩ and back to emulate fast and large R_A variation. Absence of visible correlated changes in the C_M and R_A traces obtained by the deconvolution procedure, but not in traces obtained by the standard procedure, confirms results of numerical simulations.

In real cell experiments, R_A can vary broadly at various time scales. In the presence of cross-talk, variation of R_A would transpire to C_M recordings, increase the variability of C_M estimates and decrease the resolution of C_M measurements. To assess the effect of cross-talk on C_M resolution, we simulated traces of current responses of a typical cell with R_A fluctuating randomly and compared the standard deviation of R_A with the standard deviation of C_M (Fig 4C). With the use of the deconvolution procedure, variance of C_M due to cross-talk from R_A was three orders below the variance that would result from thermal noise. If the current responses were analysed by the standard procedure, the resulting crosstalk error increased fluctuations of membrane capacitance estimates above the level expected for thermal noise substantially for R_A changes exceeding 20 kΩ, as present in majority of experiments.

Effect of parasitic capacitance

The parasitic capacitance, or stray capacitance, of a recording microelectrode, C_P, connects to ground in parallel to cell membrane (Fig 1). It is charged very fast since it is not attenuated by the access resistance (Fig 5A, upper panel). The parasitic charging current is also reconstructed by the deconvolution procedure. Due to Gibbs phenomenon [3], the presence of I_Cp appears as dumped oscillation at the onset of the reconstructed current response.

To assess the problem of fast parasitic capacitance charging current (I_Cp) for the deconvolution procedure, we emulated behaviour of the circuit with parasitic capacitance using compensation circuitry of the recording amplifier (Fig 5B). The relatively large increases in C_P setting by 1.8 pF (Fig 5B, lower panel) caused a small difference in both the C_M and R_A estimates from current records processed by the deconvolution procedure (Fig 5B, the upper and middle panel, respectively, black traces). The C_P dependent error could be excluded when the deconvolution procedure was applied to the current responses with the first 60 μs blanked to omit the oscillations in the reconstructed currents (Fig 5B, upper and middle panel, grey traces). Comparison of black and grey traces in Fig 5B indicates that partially uncompensated C_P affects the estimates of cell impedance parameters minimally. However, if C_P varied in time it would cause small artefacts in C_M and R_A traces, which look similar as if there was crosstalk of R_A to C_M. Since we showed that crosstalk is not present in the case of the deconvolution procedure (cf. Fig 4), appearance of small correlated events in C_M and R_A traces indicates in real experiments instabilities in parasitic charging current. In these tests, the access resistance setting was shortly changed at the two C_P levels (Fig 5B, middle panel). As in previous tests, the crosstalk of R_A to C_M records was below the resolution of the C_M measurement, indicating that the oscillations in the reconstructed current due to uncompensated parasitic capacitance have negligible effect on R_A crosstalk to C_M.

To overcome C_P interference, the current response to step voltage should be recorded in cell-attached configuration and subtracted from the analysed whole-cell current records. This removes oscillations from the deconvolved current responses and leaves unperturbed membrane current responses even at highest frequencies of the available bandwidth. When the input capacitance have changed substantially during experiment, blanking of the first few samples of recorded current responses before the deconvolution analysis could be applied if necessary (Fig 5B, grey traces).

Tests on real cells

The quality of the deconvolution procedure was tested on isolated cardiac myocytes routinely used in our lab. The high-resolution C_M measurements in these cells were difficult by previous methods [14] for their large size (typically above 100 pF), variable membrane resistance (ranging from 50 to 1000 MΩ), and unstable access resistance. Plasmalemma in these cells displays complex spatial network of radial invaginations (t-tubules), caveolae, and endo/exocytotic vesicles. Since secretion of natriuretic peptides was also reported in these cells [14], it could be expected that active membrane processes would be present as in other cell types.

In Fig 6A, a small isolated cardiomyocyte of neonatal rat heart displays spontaneous stepwise increases in membrane capacitance of about 10 to 50 fF in amplitude without larger changes in membrane conductance and with a stable access resistance during the recording period. The theoretically expected noise in the C_M trace for 50 Hz bandwidth (Eq 3) was 1.0 fF, while the standard deviation of signal noise in the C_M trace was 2.6 fF, due to suboptimal period of voltage stimuli set to 1.25 ms, while for the charging time constant of 0.062 ms the optimal stimulation period would be 0.744 ms.

The recording on isolated cardiomyocyte of adult rat heart (Fig 6B) shows capacitive changes under less stable recording conditions when the access resistance increased gradually by 550 kΩ and the membrane conductance contained single channel-like activity of up to 80 pS conductance. Nevertheless, the deconvolution method neatly resolved stepwise capacitive events of 40 to 80 fF in the C_M trace without any cross-talk from R_M and R_A variation. At 50 Hz bandwidth, the standard deviation of signal noise in the C_M trace was 11.5 fF, while expected noise was 6.8 fF. The difference was partially due to suboptimal stimulation period (10 ms vs. 8.12 ms) and increased contribution of 1/f noise. The slow overall decrease in the C_M trace by 0.5 pF was due to relaxation of the intramembrane charge related to gating of voltage sensitive ion channels after changing the holding potential from -50 to 0 mV [7].

In Fig 7 we document the ability of the deconvolution method to resolve membrane capacitance events connected with prolonged opening of a fusion pore [5]. The capacitance event started with a gradual, not stepwise, increase (1.7 s of the trace time). At the same time, R_M trace showed a small negatively correlated change, despite the fact that the deconvolution procedure is free of cross-talk errors. Together with the accompanying increased error of the fit (SSE) of current responses (Fig 7, bottom panel) this indicated that the three-element impedance model (Eq 1) was inappropriate for description of the current. We employed Eq 7 as a five-element model for vesicle fusion [5].

(7) where τ2 = C_v/G_p, C_v is the capacitance of the fusing vesicle equal to the amplitude of the capacitance increase, G_p is the conductance of the fusion pore, and I_P2 is the increase of the peak current amplitude due to fusion event (see [4] for details of the fitting procedure). The estimated fusion pore conductance increased from 0.1 to 0.5 nS within 250 ms and then jumped to an incalculable value, indicating that the pore opened wide and the fusion process had been completed.

Seal resistance variation

There is a well-known but also well-hidden problem of the seal stability common to all patch-clamp measurements, since a stable and well isolating seal resistance is needed to confine the circuit current to the cell membrane (Fig 1A). The seal resistance is difficult to measure [15] but its quality can be assessed indirectly [12, 16]. If the cell membrane resistance reaches the gigaohm range, a small variation of R_Seal may cause the recorded current to be at variance with the membrane current. Thus, changes in R_Seal may affect estimates of current parameters in impedance measurements and generate artefactual events in current records. We tested the effect of R_Seal by simulation of the whole circuit (Fig 1A) using Eq 5. As evidenced in Fig 8A (left panel), reduction of R_Seal below 10 GΩ strongly affected estimates of C_M and R_M but had negligible effect on R_A estimates. Importantly, however, it had no effect at all on the charge under the capacitive membrane current response Q_C, calculated as a simple integral of the current response above the steady current level according to Eq 8.

(8)

The illustrative experiment on a cardiomyocyte (Fig 8B) shows an upsurge of C_M that correlated with the drop in R_M but not in R_A and Q_C records. Since the deconvolution-based procedure is effectively free of the cross-talk error, the observed combination of events indicates that this change did not happen at the cell membrane and should be considered as an artefact caused by variation of R_Seal. Indeed, in the case of true C_M change, as shown in Fig 8B, the large event in the C_M record was not correlated with R_M or R_A but was reflected in Q_C.

The charge Q_C was unaffected by R_Seal as expected, because only I_M but not V_M is affected by R_Seal. This trait is a precious signature of an unstable seal. Independence of Q_C on step change of R_Seal was confirmed in a simulated experiment (Fig 8A, right panel). A step change of R_Seal caused artefactual increase of C_M, decrease of R_M, and no change of R_A, in their respective traces. For comparison, a true increase of C_M caused no change of R_M and R_A but caused a correlated increase in Q_C.