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Effect of Input Resistance Voltage-Dependency on DC Estimate of Membrane Capacitance in Cardiac Myocytes

M. Zaniboni ^*, F. Cacciani ^* and M. Groppi 

^* Dipartimento di Biologia Evolutiva e Funzionale-Sezione Fisiologia, and Dipartimento di Matematica, Università degli Studi di Parma, Parma, Italy

Correspondence: Address reprint requests to M. Zaniboni, Tel.: 05-21-905623; E-mail: zaniboni{at}biol.unipr.it.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
The measure of membrane capacitance (C_m) in cardiac myocytes is of primary importance as an index of their size in physiological and pathological conditions, and for the understanding of their excitability. Although a plethora of very accurate methods has been developed to access C_m value in single cells, cardiac electrophysiologists still use, in the majority of laboratories, classical direct current techniques as they have been established in the early days of cardiac cellular electrophysiology. These techniques are based on the assumption that cardiac membrane resistance (R_m) is constant, or changes negligibly, in a narrow potential range around resting potential. Using patch-clamp whole-cell recordings, both in current-clamp and voltage-clamp conditions, and numerical simulations, we document here the voltage-dependency of R_m, up to –45% of its resting value for 10-mV hyperpolarization, in resting rat ventricular myocytes. We show how this dependency makes classical protocols to misestimate C_m in a voltage-dependent manner (up to 20% errors), which can dramatically affect C_m-based calculations on cell size and on intracellular ion dynamics. We develop a simple mechanistic model to fit experimental data and obtain voltage-independent estimates of C_m, and we show that accurate estimates can also be extrapolated from the classical approach.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
The value of membrane capacitance (C_m) is a measure of how much charge is needed to displace cell polarization, how fast cell membrane passively responds to current/voltage changes, and how large is the membrane surface area. C_m and input resistance (R_m) of excitable cells are usually referred to as passive electrical properties and derived referring to an equivalent circuit, parallel combination of the two electrical elements. Microelectrode and patch-clamp techniques which have been developed, under current-clamp (CC) and voltage-clamp (VC) conditions, to solve this parallel RC combination to calculate C_m (1,2), can be grouped in two categories: the alternating current (AC) and the direct current (DC) methods. Very accurate estimates of C_m are achieved using AC methods, by means of single or multiple sinusoidal command voltages, used to clamp the cell with two-phase lock-in amplifiers (3,4). Square-wave stimulations are also employed and analyzed in the frequency domain (5), and ramp command voltage protocols have been recently improved (6) and applied to cardiac myocytes (7,8). These techniques have been originally established and largely employed mainly to study changes in membrane area during exocytosis and endocytosis (3,9,10), but they are still too technically demanding for a more general use (6) like the estimate of cell size in cardiac preparations.

The measure of specific C_m (µF/cm²) in cardiac tissue and, since the introduction of enzymatic isolation procedures, of C_m (pF) in single cardiac myocytes, has been pursued from the early days of cardiac electrophysiology (11–13), given the pivotal role of this parameter in cardiac excitability and the physiopathological interest in monitoring cell size in the heart. Systematic errors in the measure of C_m will lead, for example, to erroneous evaluation of cardiomyocytes remodeling in studies where this parameter is measured in hypertrophied hearts (7,14,15), failing hearts (16), dilated cardiomyopathy (17), diabetes (18), and myocardial infarction (19). Such errors will also affect studies on intercellular electrotonic interactions (20–22) where C_m plays a primary role in determining source-sink properties of interacting cells. Finally, they will affect those studies where C_m is measured to normalize transsarcolemmal ion currents and fluxes through carriers to cell surface and, indirectly, to cell volume (23).

Definitely the more common tool adopted by cardiac electrophysiologists to measure C_m remains the use of time-domain DC methods, where constant current (or voltage) steps are imposed to the cell membrane and the study of the resulting voltage (or current) displacements is performed in terms of mono-exponential functions (1), a procedure which, as recently pointed out (6), has not been modified substantially since its introduction 30 years ago (24). This approach is based on the assumption that C_m is constant for a given cell, and that R_m is fairly constant over membrane potential (V_m) changes (1), at least in the voltage range of application of the mentioned protocols. Since R_m changes have been documented around the resting potential (V_r) (cited below), the second assumption is actually that these changes are negligible with respect to C_m estimate.

The fact that R_m changes during the cardiac action potential follows directly from the Hodgkin-Huxley theory of membrane excitability, has been described by Weidmann in early days (25), and recently measured in guinea pig ventricular myocytes (22). In addition, Weidmann, by means of constant current injections into cardiac fibers, noticed that R_m continuously decreased, starting from threshold V_m toward rest and for increasingly hyperpolarizing current injections. The decrease of R_m as membrane potential hyperpolarizes with respect to threshold potential, is accounted for in cardiac myocytes mainly by inward rectification of I_K1 (26,27) and has already been measured in isolated rat ventricular myocytes (13,28).

In the present work, we show that R_m voltage-changes around V_r are indeed not negligible when standard CC/VC protocols are used to calculate C_m, which therefore results to be voltage-dependent as well. To this end, we measure the R_m(V_m) function around V_r and incorporate it in a mechanistic RC equivalent model, which better resembles membrane passive electrical properties. The numerical solution of such circuit allows accurate estimates of C_m which do not depend on voltage. We suggest that I_K1 rectification, as the main responsible for the input resistance voltage dependency, is also the primary source for the error in C_m estimate through the classical approach. This study, performed in turn, on real isolated rat ventricular myocytes, on five different mathematical models of the cardiac action potential, and on a simple RC mathematical model, also suggests a simplified procedure that allows to derive accurate estimates of C_m with the classical constant CC and VC protocols.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Cell isolation
Single cells were obtained by enzymatic dispersion of adult (6 months, 400–500 g) male Wistar rat left ventricles. After thoracotomy, the heart was rapidly removed, mounted on a Langendorff apparatus, and perfused at 37°C with the following sequence of solutions: Ca²⁺-free (control, no added calcium) Tyrode solution for 5 min to remove the blood, low-Ca²⁺ (0.1 mM) solution containing 1 mg/ml type 2 collagenase (Worthington, Lakewood, NJ) and 0.1 mg/ml type XIV protease (Sigma Aldrich, Milan, Italy) for 20 min, and enzyme-free low- Ca²⁺ solution for 5 min. The left ventricle was then minced and shaken for 10 min in the low-Ca²⁺ solution. Myocytes were stored at room temperature in the control solution with 0.5 mM Ca²⁺. All experiments were performed within 2–8 h after isolation. All myocytes used in this study had well-defined striations and did not spontaneously contract.

Solutions
Isolation solution contained 126 mM NaCl, 22 mM dextrose, 5.0 mM MgCl₂, 4.4 mM KCl, 20 mM taurine, 5 mM creatine, 5 mM Na pyruvate, 1 mM NaH₂PO4, and 24 mM HEPES (pH = 7.4 adjusted with NaOH). The solution was gassed with 100% O₂. Control solution for cell bathing during experiments contained 126 mM NaCl, 11 mM dextrose, 5.4 mM KCl, 1.0 mM MgCl₂, 1.08 mM CaCl₂, and 24 mM HEPES (pH-adjusted to 7.4 with NaOH). The pipette filling solution contained 113 mM KCl, 10 mM NaCl, 5.5 mM dextrose, 5 mM K₂ATP, 0.5 mM MgCl₂, and 10 mM HEPES (pH-adjusted to 7.1 with KOH). A drop of cells was placed in the experimental chamber (2.5 ml) and superfused by gravity at a flow rate of 2 ml/min. The temperature of the solutions in the cell bath was 37°C.

Electrical recordings
Suction pipettes were made from borosilicate capillary tubing (Harvard Apparatus, Edenbridge, United Kingdom) and had a resistance, when filled, of 2–4 M. Transmembrane potential and current (i) were recorded by means of an Axoclamp 2B amplifier (Axon Instruments, Union City, CA) adopting the whole-cell configuration of the patch-clamp technique.

Classical DC protocols to measure C_m
Here we summarize time-domain DC methods for the estimate of C_m which have been adopted and criticized in this study. For the sake of clarity, we will consider the resting membrane potential V_r set to zero.

CC approach
Cells were held in whole-cell configuration (amplifier in Bridge mode) and injected at a frequency of 2 Hz, with 200-ms constant current pulses (i_p) usually starting from –0.260 nA and incrementing with 5-pA steps up to 0.235 nA. Subthreshold voltage deflections were sampled at 5 kHz, digitized (Digidata 1200 Series Interface, Axon Instruments, Union City, CA), and best fitted with mono-exponential functions (Kaleidagraph, Synergy Software, Reading, PA) as solutions

(1)

of the equation

(2)

relative to the linear equivalent parallel RC circuit. It is straightforward to derive the plateau value of V_m(t) as V = i_pxR_m. The description does not include any additional series pipette resistance R_s, which was electrically compensated with bridge balance. Passive electrical properties R_m and C_m were derived as best-fitting parameters of experimental voltage traces with Eq. 1.

VC approach, curve fit
To minimize filtering effects due to pipette capacitance, pipettes tips were coated with a hydrophobic compound (Sylgard, Dow Corning, Midland, MI) and the cell bath was maintained at a low level (1 mm). Protocol features were analogous to current clamp; with the amplifier in discontinuous single-electrode voltage-clamp mode, cells were initially clamped at their V_r and then stepped with 40-ms constant voltage-clamps (V_c) usually starting from V_r – 20 mV and incrementing with 1-mV steps up to V_r + 15 mV. Current traces were best fitted with mono-exponentials as linear functions

(3)

of the solutions

(4)

of the equation

(5)

relative to the linear equivalent parallel RC circuit in series with an additional access resistance R_s. R_m, C_m, and R_s were derived as best fitting parameters. Membrane potential V, corresponding to the plateau value of the current (i = V_c / (R_m + R_s)), was derived as V_c ^_ ixR_s, being the additional R_s uncompensated during the protocol.

In the following discussion, we will use the notation V_m = V – V_r = V for the steady-state value reached by V_m during CC and VC protocols.

VC approach, area of the transient
An alternative VC method to measure C_m, to which we refer in this work, is to measure the charge under the current transient and divide it by the V = V_c – V_r. This is often approximated as

(6)

where Q is the area under the current transient and above i. Recently a better approximation for this is adopted (16) as

(7)

where Q is the area under the current transient and above i and ix is an approximation for the area under i and above the exponential resistive component of the current response.

Computer simulations
The electrical properties of single resting cardiomyocytes were simulated, in turn, using membrane equations from the mathematical models listed in Table 1, or solving Eq. 2 and Eq. 5 for a simple RC circuit including a series resistance R_s, which was set to zero in CC simulations. Equations were numerically solved using an adaptive Euler method for the LR91 model, and a fifth-order Runge-Kutta method for PD01 model and for the RC circuit. Simulations on the LR94, PB01, and RM00 were performed on the Cell Electrophysiology Simulation Environment (CESE Version 1.3.4, available on http://cese.sourceforge.net). All simulations were performed on a Pentium IV processor and codes for LR91, PD01, and for the RC circuit implemented in MatLab language (The MathWorks, Natick, MA).

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Estimate of an input resistance voltage-function
The membrane input resistance (R_m) of single rat ventricular myocytes was measured within ±10 mV around resting potential (V_r) through CC and VC protocols (see Materials and Methods). Fig. 1 A shows voltage deflections from a CC protocol and the function R_m(V_m) well fitted by a parabola with c₀ = 58.35, c₁ = 3.64, and c₂ = 0.14. The same analysis performed on 12 myocytes gave an average parabolic fitting of c₀ = 55.99 ± 3.67, c₁ = 2.37 ± 0.25, and c₂ = 0.06 ± 0.01. Current traces from a VC protocol are reported in Fig. 1 B with the function R_m(V_m), which is similar to the one measured in CC, and well fitted by a parabola with c₀ = 54.22, c₁ = 4.51, and c₂ = 0.12. The same analysis performed on 12 myocytes gave an average fitting of c₀ = 58.76 ± 6.95, c₁ = 5.22 ± 0.59, and c₂ = 0.15 ± 0.02. When the two protocols were numerically simulated on a LR91 model, they returned a qualitatively analogous function R_m(V_m), well fitted by a parabola with c₀ = 21.71, c₁ = 1.10, and c₂ = 0.027 . Similar R_m(V_m) behavior was found in all five mathematical models employed (see below).

View larger version (17K): [in this window] [in a new window]   FIGURE 1  Measure of Rm voltage-dependency. Left column: (A) voltage deflections from a CC protocol on a rat myocyte. Only one in every four traces is shown for clarity. (B) Current traces from a VC protocol on a different myocyte. Only one in every second trace is shown. (C) Analogous CC and VC protocols simulated on the LR91 model (Rs = 5 M in VC). Right column: Rm, calculated from plateau values of each voltage and current trace, is reported (solid circles) as a function of Vm, fitted by parabolic functions (dotted lines).

In Fig. 2 we show examples in which C_m was calculated (see Materials and Methods) by mono-exponential fittings of the voltage/current traces reported in Fig. 1. The measured C_m varies with V_m, decreasing with membrane polarization. The same analysis performed on seven myocytes led to C_m changes, as measured from a linear fitting for –10 mV < V_m < 0 mV, of 2.95 ± 0.74 pF/mV in CC and 1.35 ± 0.54 pF/mV in VC. Analogous results are shown, for example, on the more general LR91 model. Analyzing a non-Ohmic RC as if it was Ohmic clearly leads to an error in C_m estimate, which increases with the displacement of membrane potential from V_r, doing more so in CC than in VC. The CC protocol was applied also on the other mathematical models listed in Table 1, and results reported in Fig. 3. In the lower panel of the same figure, the relationship between the relative error in C_m estimate and R_m rectification is also reported for the five mathematical models and for the experimental data. To better isolate this effect, we solved two simple mathematical models of a resting myocyte in whole-cell configuration, which included only R_s, C_m, and R_m (Fig. 4). R_m was kept constant in a first case (Fig. 4 A) and voltage-dependent in a second case (Fig. 4 B). Fig. 4 A shows the symmetrical behavior (with respect to the resting potential/current) of voltage and current traces obtained when the Ohmic circuit (Fig. 4 A) was solved, in turn, in CC and VC conditions. When the constant R_m of the circuit was replaced (as in Fig. 4 B) with an experimental R_m(V_m) function of the type shown in Fig. 1, the voltage and current deflections lost their symmetry and appeared more like those measured in real cells. Traces were all analyzed in terms of mono-exponential fittings (see Materials and Methods), assuming each time, for R_m, the constant value measured at plateau, and C_m values derived and reported as solid circles in Fig. 4 C. The Ohmic assumption, in the case of a physiologically non-Ohmic RC circuit, leads to an error in the estimate of C_m which qualitatively reproduces what we found in real myocytes and in the mathematical model cells (Fig. 2 and Fig. 3), with C_m varying with a sigmoidal-law around the resting potential.

View larger version (19K): [in this window] [in a new window]   FIGURE 2  Estimated Cm as a function of Vm displacement from Vr. (Top) Each solid circle represents a DC estimate of Cm from experimental traces of Fig. 1, A and B. (Bottom) Same for the traces from the LR91 model in Fig. 1 C, where Cm was set to 153.44 pF (horizontal lines) on the basis of the geometrical considerations of Luo and Rudy (43). Dotted lines in each panel represent linear fittings of the experimental data, 1.6, 1.1, 3.1, and 1.7 pF/mV, respectively.

View larger version (13K): [in this window] [in a new window]   FIGURE 3  Voltage-dependency of Rm and of the estimated Cm in five mathematical models of cardiomyocytes. (Top) Rm(Vm) as obtained through the CC protocol applied to the five mathematical models listed in Table 1. (Middle) Cm(Vm) obtained with the same protocol. Results were normalized to the extrapolated resting values of Rm () and Cm (), which were respectively set to 1 for comparison. (Bottom) The above traces were linearly fitted within the –10/0 mV Vm range, to plot the slope of Cm(Vm) versus the slope of Rm(Vm) for the five models. The cross symbol represents the corresponding averaged estimate made on seven current-clamped rat ventricular myocytes (1.89 ± 0.39 mV–1 vs. 3.26 ± 0.25 mV–1), also cited in the Results.

View larger version (26K): [in this window] [in a new window]   FIGURE 4  DC estimate of Cm in Ohmic and non-Ohmic RC circuits. (A) Current-clamped (from –0.25 to 0.25 nA, step 0.025 nA) voltage displacements and voltage-clamped (from –20 to +20 mV, step 2 mV) current traces, mathematically simulated on the circuit reported on the left, with Rm = 56.5 M and Cm = 250 pF. Rs was set to 0 M in CC and 5 M in VC. (B) Identical protocols simulated on the same circuit were (C) Cm versus Vm, as calculated from mono-exponential fittings of the traces in B (solid circles).

View larger version (16K): [in this window] [in a new window]   FIGURE 5  Comparison of Cm estimate with Ohmic and non-Ohmic assumption in real ventricular myocytes. Cm, first calculated through mono-exponential fittings of CC-elicited voltage traces (top), and VC-elicited current traces (bottom), plotted as a function of Vm (solid circles). Cm is also reported as derived through the algorithm which assumes non-Ohmicity of the circuit (open circles). Solid line represents the average value of Cm in the non-Ohmic assumption. This value, together with calculated Rs and Rm(Vm), is incorporated in an equivalent circuit whose solutions, in CC and VC simulations, are fitted with mono-exponentials to derive Cm for each Vm (dotted lines).

View larger version (13K): [in this window] [in a new window]   FIGURE 6  Comparison of Cm estimate with Ohmic and non-Ohmic assumption in the LR91 model cell. Cm, as calculated adopting the Ohmic assumption (solid circles) and the non-Ohmic assumption (open circles) through a CC and a VC protocol on the LR91 model. Horizontal lines represent the actual Cm of the model. A series resistance of 10 M was added in VC simulations.

View larger version (17K): [in this window] [in a new window]   FIGURE 7  Ohmic and non-Ohmic analysis of a current-clamped voltage deflection. (A) Mono-exponential fitting (dotted line) ( = 5.7 ms) of a current-clamped (–0.24 nA) voltage displacement (solid line) from a ventricular myocyte. (Inset) Linear fitting (dotted line) of the logarithm of Vm (solid circles; only one in every four experimental samples is plotted). (B) Dotted lines are solutions of Eq. 2 with experimentally derived from the same cell, and for Cm values from 100 pF to 280 pF (only 1 in every 20 solutions is reported); solid line is the experimental voltage deflection.

View larger version (18K): [in this window] [in a new window]   FIGURE 8  Effect of Cm and Rm(Vm)-slope on the accuracy of Cm estimate in a simple RC circuit. (A and B) CC and VC estimate of Cm through Ohmic assumption on an RC model including an and an Rs = 5 M (in VC), for three different values of Cm (horizontal dotted lines). (C) Error in Cm estimate (percent underestimate from the actual Cm), as measured at Vm = –10 mV, for different values of Cm, in CC (solid line) and VC (dotted lines) simulated tests, and for two values of Rs. (D) Three Rm(Vm) parabolic functions differing for the slope. (E and F) CC and VC estimates of Cm by Ohmic assumption on an RC model including, in turn, the three functions. The double-arrow symbols represent the typical experimental noise level in Cm estimate.

Effect of the slope of R_m(V_m) function on the error in the estimate of C_m using the Ohmic assumption

Good enough estimates of C_m with the Ohmic assumption
From the simulations performed on the non-Ohmic RC model in Fig. 4, it appears that the sigmoidal function of the estimated C_m(V_m), derived assuming a constant R_m, has the property that it crosses the horizontal line corresponding to the actual value of C_m in the resting membrane potential (V_m = 0). Moreover the slope of this function is fairly constant in the V_m range of interest (–10 mV < V_m < 0 mV), where it can be approximated with a straight line. It follows that the proper value of cell capacitance could be, in principle, extrapolated with only two current injections (or VC pulses) experiments, using mono-exponential fittings, and ignoring therefore the voltage-dependency of R_m. Histograms in Fig. 9 show, for both CC and VC, average estimates of C_m, taken ignoring R_m voltage-dependency and adopting mono-exponential fittings (first and second columns in each panel), deriving R_m voltage-dependency and using the least-squares fitting procedure (third column), or assuming the function C_m(V_m) to be linear and extrapolating the zero-potential value from two measures, taken approximately at V_m = –10 mV and V_m = –5 mV, for each cell (fourth column). The average value obtained with the extrapolation procedure does not significantly differ in CC and VC from the one measured with the non-Ohmic assumption.

View larger version (14K): [in this window] [in a new window]   FIGURE 9  Histograms comparing CC and VC estimate of Cm with Ohmic and non-Ohmic assumption in real ventricular myocytes. (First and second bars) Estimate made through the Ohmic assumption at Vm = –10 mV and Vm = –5 mV, respectively. (Third bar) Estimate made through the non-Ohmic assumption (). (Fourth bar) Estimate made through extrapolation from pairs of Ohmic-assumption measurements. All bars are normalized to third bar, which is set to 1 (the asterisk symbol indicates significant differences).

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

It should be noted that a frequently adopted variant of constant step VC protocol to measure C_m is the one (see Materials and Methods) based on the calculation of the charge underlying the capacitive current transient. An earlier version of this approach simply calculated C_m as in Eq. 6. Such calculation is not accurate and has been more recently replaced by Eq. 7, where the mono-exponentially derived time constant appears explicitly, and therefore makes this protocol suffer as well from the voltage-dependency of R_m. Indeed, errors in C_m calculations using Eq. 7 have been derived for the voltage-clamp simulation reported in Fig. 4 B without finding any difference from those derived with the classical protocol (data omitted for the sake of clarity). Therefore we did not discuss in a separate section errors in C_m estimate derived with the numerical integration of the current transient during voltage steps.

Limits of the Ohmic assumption
R_m voltage-dependency for subthreshold potentials is known in cardiac tissue at a cellular level (13,27), and has a part in explaining subthreshold behavior of extracellular membrane polarization (29). We limited the majority of our experimental work and analysis on R_m changes in the 10 mV hyperpolarized V_m-range of well polarized (V_r = –73.45 ± 0.69 mV) rat left-ventricular cells (n = 24). With this, we deliberately wanted to avoid eliciting the time-dependent processes that develop in cardiac membrane for depolarizing potentials (e.g., activation of inward sodium current, inward L-type calcium current, calcium-independent transient outward potassium current, steady-state outward potassium current; Ref. 30) and for more hyperpolarized potentials (e.g., I_K1 time-dependent block and I_f activation; Ref. 31). In fact, voltage traces always reached a plateau within CC steps, as well as current traces never showed time-dependent components other than the capacitive peak during VC steps on tested ventricular myocytes.

The 40–50% decrease of R_m when membrane potential was current- or voltage-clamped from V_r to (see Fig. 1) is reflected into the progressive compression/broadening of the elicited voltage/current traces. In the case of the mathematical LR91 model, the 40% decrease of R_m for V_m = –10 mV, corresponds to a 72% increment of g_K1 whereas other ionic conductances included in the model do not change (carried simulations not shown). Indeed, this is also the case for the other more complex mathematical models, where, in this subthreshold region, the I-V relationship with the highest degree of nonlinearity is that of I_K1 which, moreover, overwhelms all the others in absolute intensity. In fact, the simple analysis of published steady-state I_K1-V curves recorded in rat and other mammals ventricular myocytes (e.g., Refs. 32–35) shows that I_K1 starts rectifying at potentials that are at least 10- or 15-mV hyperpolarized, with respect to the typical V_r for this cell type, which has to be reflected in some R_m voltage-dependency. From our simulation work, we can rule out the direct contribution to R_m rectification of the electrogenic sodium-potassium pump, whose I-V relationship is linear in the subthreshold region of interest of this study (see, for example, I_NaK equations in PD01, LR94, RM00, and Ref. 36). For the same reason, we can exclude calcium, sodium, and potassium background currents which, although flowing in this voltage range, are usually formulated as linear leakage currents therefore not possibly contributing directly to any R_m rectification (see, for example, related equations in PD01 and LR94). It is tempting to hypothesize a significant role of the sodium-calcium exchanger current, whose IV relationship slightly deviates from linearity in the same region (30,37), but this would need further mathematical and experimental work to be confirmed and quantified. Furthermore, it is possible that some other rectifying mechanisms would be present that are still not included in the models and, therefore, in our knowledge of membrane electrical properties.

A major point of the present work is to show that the measure of C_m within the Ohmic assumption is voltage-dependent in a non-negligible manner. Indeed such dependency was always present in tested ventricular myocytes (Figs. 2 and 5), being less dramatic in VC than in CC, and only slightly depending on the actual value of C_m (see experiment in Fig. 8). In this regard, it is interesting to note the 26-pF difference, reported by Tseng and co-authors in canine ventricular myocytes, between average C_m estimates when measurements were performed in CC (voltage displacements up to –10 mV) and VC (–5 or –10 mV steps) by microelectrode impalement (38). According to our findings, both values underestimate the actual C_m, being the CC error at 54 pF worst than the VC error at 80 pF.

A voltage-dependency in the estimate of C_m can actually become hard to detect and therefore negligible, for example, in hyperkalemic conditions where I_K1-V curve shifts horizontally toward depolarized potentials and vertically toward more positive currents, becoming therefore more linear around its reversal potential (see [K⁺]_o-dependency of I_K1 equation in Ref. 33, and in Ref. 30). The same [K⁺]_o-dependency can be viewed in the total steady-state I-V relationship (e.g., Ref. 38). When [K⁺]_o increases, the slope of R_m(V_m) function becomes smaller and errors in the estimate of C_m will be negligible compared with experimental noise. If we consider, for example, the experiment in Fig. 8 D, a cardiac myocyte with an R_m(V_m) function a, characterized by a very weak voltage-dependency, will bring about, when analyzed in VC through the Ohmic assumption (Fig. 8 F), an underestimate of C_m, as measured at V_m = –10 mV, <1%, which will be masked within the typical noise level (5–6%) of our experiments. On the other hand, an increase in series resistance, especially in cells with higher R_m(V_m) slope, can produce appreciable differences also in VC, whereas in CC, differences are always measurable practically in all physiological conditions. This is noteworthy, particularly if we consider that, although the great majority of recent cardiac cellular electrophysiological studies adopts VC constant pulses to measure C_m (e.g., Refs. 39 and 40), many still use the CC approach (21,22), which will then lead to much larger errors.

C_m measurements within homogeneous cell populations suffer frequently for a consistent dispersion (SD often up to 50% of the average) which could be attenuated with the non-Ohmic assumption, allowing better resolution of C_m changes following different pathological conditions or pharmacological treatments. Moreover, errors in C_m evaluation can complicate calculations on intracellular ion dynamics. For example a 10% underestimate of C_m (pF) leads to an 11% overestimate of current density (pA/pF), a 4% underestimate of cell volume in rat ventricular cardiomyocyte (23), and therefore an analogous overestimate of intracellular ion concentration changes. Cell volume and intracellular ion concentration changes will be further underestimated (up to 10%) in species like rabbit, where volume/surface relation is steeper (23).

The correlation, found in the four ventricular models and the one atrial model analyzed, between the slope of the relative error in C_m estimate and the slope of R_m(V_m) (Fig. 3, bottom panel), furthermore emphasizes the generality of the principle: the input resistance of resting cardiomyocytes rectifies in the subthreshold voltage range under study, and such rectification brings about a proportional error in C_m estimate. This proportionality is fully satisfied by our experimental data, as shown by their position on the correlation line of Fig. 3 (bottom panel). Interesting to note is the failure of the PD01 model to exactly reproduce rat data in this correlation plot, most likely due to the particular I_K1 equation chosen in the original work (30), where, on the other hand, the authors themselves recognize the nonuniform properties of I_K1 across the ventricle as one of the potential limitations of their mathematical description (see also Ref. 41).

A mechanistic non-Ohmic assumption
We call the model described in Fig. 4 B "mechanistic" because it is based only on properties of an experimental observable (R_m), independently on its biological determinants (ion channels, pumps, exchangers, etc.). This simple model shows that the property of R_m to vary parabolically with V_m is enough, by itself, to qualitatively explain voltage and current asymmetry in response to symmetric current-/voltage-clamp protocols. It also accounts for the voltage-dependency of C_m estimate in vivo and in the mathematical model cells. Finally it shows that, even though mono-exponential functions fit quite well the solutions of Eq. 2 and Eq. 5 for the non-Ohmic circuit, nevertheless they are solutions of a wrong model of the cell electrical properties and they lead therefore to wrong C_m estimates. If, instead, numerical solutions of Eq. 2 and Eq. 5 including the R_m(V_m) relation are used, the C_m measure becomes independent from V_m and gives estimates that are only a little dispersed around a mean value (Fig. 5), as can also be appreciated in the cell model simulations (e.g., Fig. 6). A further validation of the consistency of the least-square algorithm (see Appendix) to measure C_m comes from the experiment in Fig. 5. A mechanistic mathematical model of a real rat ventricular myocyte, built with experimentally derived R_m(V_m), R_s, and a constant C_m calculated with the least-square protocol on the same cell, was challenged with CC and VC constant steps and analyzed in terms of mono-exponentials. The fact that the resulting C_m values fit well those derived with the Ohmic assumption on the real cell, demonstrates that the voltage-dependency of R_m is enough to explain the error in C_m estimate, and no other effects like leaky seals, additional uncompensated stray capacitances, series resistances, or different equivalent circuits (11), in principle, need to be considered.

It could be argued that the least-square algorithm presented in this study as a tool to calculate C_m in a voltage-independent manner is too complicated and demanding, especially for what concerns off-line numerical processing of the data, making it therefore not suitable for the general purpose of monitoring passive electrical properties, which is often ancillary in most cardiac cellular electrophysiological studies. To answer this concern, we propose a procedure where an algebraic property of the C_m calculation with the Ohmic assumption is used to extrapolate the real value of C_m without need of any further experimental or mathematical work, as summarized in Fig. 9. Some authors use protocols where symmetrical (depolarizing and hyperpolarizing) VC steps are imposed to the membrane and C_m averaged from the analysis of the two opposite current transients (e.g., Ref. 15). It should be noted that this procedure is quite different from the one we are suggesting here and can also lead to mis-estimate of C_m, because C_m error in the Ohmic assumption is not symmetrical with respect to V_r (see curve c in Fig. 8 F).

Why VC gives a better C_m estimate
The typical first-order linear differential equation generating mono-exponential functions as solutions is of the type

(8)

where a and b are constants. This is, for example, the form of Eq. 2 of CC in the Ohmic case when R_m is constant. Conversely, when R_m becomes a function of V_m, Eq. 2 assumes a qualitatively different form, whose solutions will always deviate from mono-exponentials as much as R_m is voltage-dependent.

In the case of Eq. 5 of VC, this can be conveniently rewritten as

(9)

where it becomes evident that the linear form of Eq. 8 tends to be restored when R_m >> R_s.

This explains qualitatively why, assuming the same voltage-dependency of R_m, VC-derived current traces deviate less from mono-exponentials than CC-derived voltage displacements. In other words, provided R_s is kept much lower than R_m, VC gives best estimates of C_m, as we demonstrated in this study with both experimental and numerical approaches.

Limitations of the study
A limitation of this work is that, although the very general mechanism of subthreshold R_m rectification is presented with its implications on C_m estimate, the relative role of the ionic mechanisms underlying this property have not been unambiguously defined in vivo nor in silico. Although, as in previous studies (26,27), a primary contribution of I_K1 is strongly suggested, additional experimental and numerical work will help to better define the possible role of sodium-calcium exchanger or other ionic mechanisms in R_m rectification. Also, we do not report here on any intervention made to modify the different ionic mechanisms of R_m voltage-dependency. Work in this direction will be required, for example, to investigate whether there are conditions where the membrane of resting cardiomyocytes behaves Ohmically, helping to better explain its subthreshold electrical properties in physiological and pathological states.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
This study emphasizes that attention should be paid to the subthreshold voltage-dependency of R_m if C_m is to be measured with classical DC protocols. Large errors always arise from the CC approach, but VC can also be substantially affected. Accurate measures can be achieved by taking R_m(V_m) into account in the analysis of CC and VC experimental results or by ignoring this voltage-dependency, and extrapolating results obtained from at least two different V_m values to V_m = 0.

It is straightforward to hypothesize that the voltage-dependency of R_m would also play a role in other techniques to measure C_m, especially those based on complex impedance analysis, widely adopted in the literature and based on the AC study of the same Eq. 2 and Eq. 5.

	APPENDIX

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
The non-Ohmic form of Eq. 2 and Eq. 5 includes an R_m which is not a constant, but is, instead, a function of the membrane potential. Whatever this function (we used second-order polynomials from experimental fittings in this work), it is not possible to find explicitly the solutions for these new equations, which we should solve numerically.

Fitting procedure with the non-Ohmic solutions
Current-clamp
A set of traces like those in Fig. 1 A was fitted with mono-exponentials to derive the parabolic functions R_m(V_m) and C_m(V_m). R_m(V_m) was replaced into Eq. 2, which was then solved numerically for each current i_p, for all the C_m values of an n-long vector _m = [C_m,1, C_m,2, C_m,3, ..., C_m,n], where C_m,n and C_m,1 were, respectively, the central value of the derived C_m range ± 50 pF. In the case of the experiment shown in Fig. 5, for example, the non-Ohmic form of Eq. 2 was solved for 42 i_p values and, for each i_p, 100 times for C_m values from 65 to 165 pF. For each current injection, the sum of the squared-errors between the numerically integrated solutions and the experimental vector was calculated n times to obtain an n-long vector LS. The actual C_m was then derived as the element of _m that minimizes LS (least-squares method). An example of this procedure is reported in Fig. 7 B.

Voltage-clamp
Same approach as in current-clamp, where Eq. 5 was solved for a given set of voltage steps V_c and, for each voltage-clamped V_c, for all the elements of the vector _m.

	ACKNOWLEDGEMENTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
The authors thank Emilio Macchi and Ken Spitzer for helpful comments and suggestions on the article.

This study was supported by grants from the Italian Ministry of Education, University, and Research (grant No. MIUR-COFIN 2003), and the San Paolo di Torino Foundation.

Submitted on March 15, 2005; accepted for publication June 22, 2005.

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