Cytoplasm Resistivity of Mammalian Atrial Myocardium Determined by Dielectrophoresis and Impedance Methods

Abstract

Many cardiac arrhythmias are caused by slowed conduction of action potentials, which in turn can be due to an abnormal increase of intracellular myocardial resistance. Intracellular resistivity is a linear sum of that offered by gap junctions between contiguous cells and the cytoplasm of the myocytes themselves. However, the relative contribution of the two components is unclear, especially in atrial myocardium, as there are no precise measurements of cytoplasmic resistivity, R_c. In this study, R_c was measured in atrial tissue using several methods: a dielectrophoresis technique with isolated cells and impedance measurements with both isolated cells and multicellular preparations. All methods yielded similar values for R_c, with a mean of 138 ± 5 Ω·cm at 23°C, and a Q₁₀ value of 1.20. This value is about half that of total intracellular resistivity and thus will be a significant determinant of the actual value of action potential conduction velocity. The dielectrophoresis experiments demonstrated the importance of including divalent cations (Ca²⁺ and Mg²⁺) in the suspension medium, as their omission reduced cell integrity by lowering membrane resistivity and increasing cytoplasm resistivity. Accurate measurement of R_c is essential to develop quantitative computational models that determine the key factors contributing to the development of cardiac arrhythmias.

Introduction

Atrial fibrillation (AF) is an arrhythmia that sustains a high morbidity and mortality, most importantly as a risk factor for stroke and other vascular accidents (1,2). The appearance of abnormal propagation of the cardiac action potential (AP) is a key cause of AF (3,4) therefore it is necessary to quantify those factors that determine this variable. In principle, AP conduction depends on the magnitudes of both the ionic currents that contribute to the AP upstroke as well as the intracellular resistivity of myocardium, R_i; for example, an increase of R_i slows AP conduction velocity. Computational methods offer the best approach to ascertain the relative importance of the different components of this myocardial electrical substrate that lead to arrhythmias (5). Crucial to this is to have accurate values of these electrical components, which in many cases are lacking, in particular a value for R_i.

The value of intracellular resistivity, R_i, depends on two series components: the cytoplasm resistivity, R_c, and that of gap junctions (GJs) between contiguous myocytes, R_j. Gap junction resistivity is a function of the density, distribution, and the properties of individual GJs. The latter is determined by the different isoforms of their component connexin proteins, their phosphorylation state, as well as the concentration of intracellular ions (6,7). Changes to these factors have been described and were evident in several cardiac pathologies such as atrial fibrillation and ventricular hypertrophy (8).

However, there are no quantitative studies available that have measured the absolute value of the cytoplasmic resistivity, R_c so that modeling studies generally assume a standard value. One measurement in frog ventricle at room temperature gave a mean value for R_c of 282 Ω·cm (9), but is likely to be an overestimate as it is comparable to estimates of total R_i by cable analysis (10). The method used the principle that microelectrode resistance varies with solution conductivity. However, it suffered the disadvantage that tissue input resistance had to be measured independently and was therefore prone to significant errors. Therefore, it is crucial to measure accurately R_c to generate an accurate substrate for modeling studies of AP propagation.

To measure R_c we have used two different methodological approaches, dielectrophoresis (DEP) and impedance methods, and a variety of tissue preparations, cultures, and freshly isolated cells, as well as multicellular (tissue) strips, to reduce bias that a single method or preparation might introduce. DEP measures the electrical properties of suspended particles such as a cell with a conducting centre surrounded by an insulating membrane. It involves placing cells in a nonuniform electrical field to generate a dipole within each cell; subsequent movement of the cells in the electrical field depends on their permittivity and conductance properties. Recent developments have increased the scope of the technique to measure cytoplasmic resistivity (11). We have therefore used DEP to measure R_c in atrial cells. We validated the DEP technique by estimating R_c from measurement of the frequency-dependent impedance of atrial cell suspensions and tissue strips by several methods. At high frequencies the impedance offered by capacitative elements in the cell are short-circuited so that net impedance depends only on the conductance of the suspending medium and the cytoplasm within the cell. Although impedance techniques have been used previously (8) they have not been applied to atrial tissue. We show that DEP and impedance approaches provide a similar and accurate value for R_c.

Methods

Preparations

Atrial myocardium was used either as an immortalized cultured cell line (HL-1 (12)) or freshly isolated myocytes and multicellular strips from guinea-pigs. HL-1 is a well-differentiated subclone of an original HL-1 cell line that can be maintained indefinitely. They were grown in gelatin/fibronectin flasks using Claycomb Medium, (see 14) supplemented with fetal bovine serum (10% v/v), penicillin (100 U·ml⁻¹); streptomycin (100 mg·ml⁻¹); L-glutamine (4 mM) and noradrenaline (100 μM) in a 5%CO₂/95% air atmosphere at 37°C in a humidified incubator. Cells were judged to be maximally differentiated when they were spontaneously contracting by microscopic examination. For the experiments cultured cells were dissociated with trypsin-containing Claycomb Medium and stored on ice in fresh Dulbecco’s modified Eagle medium (Life Technologies, UK).

Freshly isolated cells and tissue strips were from guinea-pig atria

Animals were killed by the Schedule-1 method, according to The Animal Act, UK (1986) and UK Home Office guidelines, the heart rapidly removed and atria dissected in Tyrode’s solution. To isolate cells atria were cut into pieces (<1 mm³) and incubated successively in three low-Ca²⁺ solutions containing protease (0.8 mg·ml⁻¹, 1 min), type-I collagenase (0.3 mg·ml⁻¹, 35 min), and type-I collagenase (1.2 mg·ml⁻¹, 10 min). After incubation and gentle trituration the cell suspension was centrifuged (×8 g) and the cells were then stored in low-Ca²⁺ solution. Strips (<0.5 mm diameter, 3–5 mm length) were dissected from separate atria.

Solutions

Tyrode’s solution contained (mM): NaCl, 118, KCl, 4.0, NaHCO₃, 24; NaH₂PO₄, 0.4, MgCl₂, 1.0, CaCl₂, 1.8; glucose 6.0; Na pyruvate, 5.0; pH 7.4 with 5%CO₂/95% O₂. Low-Ca²⁺ solution contained (mM): NaCl, 132; KCl, 5.0; NaH₂PO₄, 3.0; MgSO_4, 1mM; glucose, 10; taurine, 10; HEPES, 4.2 and albumin, 1 mg·ml⁻¹; buffered to pH 7.4 with NaOH. For DEP experiments HL-1 cells were resuspended in an isotonic sucrose/glucose (250/17 mM; 285 mOsm·kg⁻¹) medium with 0.1 mM CaCl₂ and 0.25 mM MgCl₂; phosphate-buffered saline (PBS, Invitrogen, UK) was added to generate a solution conductivity of 2.5 mS·m⁻¹. The CaCl₂ and MgCl₂ were omitted for some experiments, and in others the sucrose and glucose concentrations were also reduced by 40% to generate a hypotonic solution (170 mOsm·kg⁻¹). For impedance experiments cells were resuspended in (mM): sucrose, 257; glucose, 11; Na acetate; 5.0; KCl, 0.5; MgCl₂, 1.0, CaCl₂, 1.0; pH 7.4 with NaOH (conductivity 2.9 mS·m⁻¹; 290 mOsm·kg⁻¹).

DEP

Determination of cell electrical constants by DEP has been described (13). Briefly, cells suspended in the isotonic sucrose/glucose medium were exposed to a nonuniform alternating (0.001–20 MHz, 10 V p-p) electric field generated by gold-plated copper rings, formed by drilling through copper sheets separated by a polyamide insulator (Fig. 1 A) to form a 1 mm diameter well. Cells, within a suspension placed in the well, migrated either toward or from the electrodes upon energizing the field at different frequencies. Cell movement was detected by measuring light intensity of a beam directed through the well. A decrease of intensity represented cell migration from the electrodes and was equivalent to the DEP force, F_DEP. F_DEP, is a force on a particle of radius r in a medium of permittivity ε_s within an electric field, E.

$${\mathbf{F}}_{\text{DEP}}=2\pi {\epsilon }_s{r}^3\mathfrak{R}\left[K\left(\omega \right)\right]\nabla {\mathbf{E}}^2.$$ (1)

Cells were approximated to spheres of cytoplasm (conductivity, σ_c; permittivity, ε_c) surrounded by a membrane (σ_m, ε_m), all suspended in a medium (σ_med, ε_med). This single shell approximation gives a soluble number of parameters for a DEP spectrum. With these assumptions, the Clausius-Mossotti factor, K(ω) is represented as a function of membrane, cytoplasm, and medium conductances and permittivities, thus:

$$K\left(\omega \right)=\left(\frac{{\epsilon }_{\text{cell}}^{\ast }-{\epsilon }_{\text{med}}^{\ast }}{{\epsilon }_{\text{cell}}^{\ast }+2{\epsilon }_{\text{med}}^{\ast }}\right),$$

$${\epsilon }_{\text{cell}}^{\ast }={\epsilon }_{\mathrm{m}}^{\ast }\frac{{\left(\frac{r}{r-d}\right)}^3+2\frac{{\epsilon }_{\mathrm{c}}^{\ast }-{\epsilon }_{\mathrm{m}}^{\ast }}{{\epsilon }_{\mathrm{c}}^{\ast }+2{\epsilon }_{\mathrm{m}}^{\ast }}}{{\left(\frac{r}{r-d}\right)}^3-\frac{{\epsilon }_{\mathrm{c}}^{\ast }-{\epsilon }_m^{\ast }}{{\epsilon }_{\mathrm{c}}^{\ast }+2{\epsilon }_{\mathrm{m}}^{\ast }}},$$ (2)

where ${\epsilon }^{\ast }$ represents complex permittivity, given by ${\epsilon }^{\ast }=\epsilon -\text{j}\sigma /\omega$; a function of permittivity ε, conductivity σ, angular frequency of the energizing field ω, j is √−1. Parameter ${\epsilon }_{\text{cell}}^{\ast }$ represents the net effective permittivity of the cell, whereas the parameters r and d refer to the radius of the cell and the thickness of the cell membrane (10 nm), respectively.

Figure 1.

Experimental chambers used to measure cytoplasmic resistivity. (A) The DEP chamber showing alternating rings of gold-plated copper electrodes (dark rings) and insulators (light rings). The light path to the detectors below the chamber is shown. Below, top view of chamber showing cells migrating to the electrodes (positive DEPs). (B) Chamber for measuring the impedance of cell suspensions. Alternating current is passed between two Pt-black electrodes and connected to a Wien bridge (cell suspension is the darker shaded area at the bottom of the chamber). (C) Measurement of the impedance of atrial strips in an oil-gap chamber between two Pt-black electrodes placed in outer chambers filled with physiological (Tyrode’s) solution.

Light intensity distribution, at each frequency, was measured for 60 s in 3-s bins; prototype 3DEP reader (Deptech, Ringmer, UK), a spectrum generated (Fig. 2) and fitted to Eqs. 1 and 2. These equations may be parameterized by the constants in Eq. 2, namely the conductivity and permittivity values of the cell membrane and cytoplasm, as well as the suspension medium. The spectra were fitted by an iterative least squares method using MATLAB (The MathWorks, Cambridge, UK) to yield individual values of cytoplasm and membrane conductivity, as well as membrane capacitance, c (= ε _m/d) where d is an assumed membrane thickness (13). The correlation coefficients for the curve fits were always greater than 0.95.

Figure 2.

DEP spectra of atrial HL-1-6 cells in (A) isotonic sucrose/glucose medium with divalent cations. (B) Isotonic sucrose/glucose medium with divalent cations omitted. (C) Hypotonic medium with divalent cations omitted. The curves are fits of Eqs. 1 and 2, see Methods. The y-axis relative polarizability is a function of the K(ω) factor with positive values equivalent to a positive DEP. The respective correlation coefficient values for parts A–C are 0.951; 0.981; 0.973.

Impedance methods

An aliquot of isolated cells suspended in the sucrose/glucose medium was placed in a chamber with embedded Pt electrodes (Fig. 1 B), followed by centrifugation (10 × g, 8 min) to increase cell packing. Electrodes were coated with Pt black by electrolysis with Kohlrausch’s solution (3% w/v PtCl₆; 0.025% Pb acetate) at 36 C·cm⁻². Alternating current (0.1–60 kHz, 10 mV p-p) was passed between the electrodes and total capacitance and resistance recorded with a Wien bridge (model 6425, Wayne-Kerr, UK). Electrode capacitance and resistance were measured separately in Tyrode’s solution and subtracted from total values (8). Impedance at low frequencies (Z₁, 0.1–1.0 kHz) was constant and modeled as current flow through medium alone, at the highest frequencies (Z₂) was modeled as parallel current flow through medium and the additional volume of myocyte cytoplasm. R_c was calculated as

$$R_{\mathrm{c}}=p·\frac{Z_1{Z}_2}{Z_1-Z_2},p=\frac{1-\left(Z_{\text{med}}/Z_1\right)}{1+0.5\left(Z_{\text{med}}/Z_1\right)},$$ (3)

where p is the proportional myocyte packing fraction (14), maximum = 1; Z_med is the medium resistivity filling the chamber. System admittance (Y = 1/Z) at each frequency, f (ω = 2 πf) was transformed into conductance, G, and susceptance, B, components (Fig. 3):

$$Y=G+\text{j}B=\frac{R}{R^2+X^2}-\text{j}\frac{X}{R^2+X^2}\left(\mathrm{j}=\sqrt{-1}\right),\text{with}R=\frac{\left|Z\right|}{\sqrt{1+{\tan }^2\omega }};X=\frac{\left|Z\right|}{\sqrt{1+\left(1/{\tan }^2\omega \right)}}.$$ (4)

R_c in atrial strips (<0.5 mm diameter, 3–5 mm length) was measured with a validated method (8) using an oil-gap chamber (Fig. 1 C). Alternating current (0.02–100 kHz, 10 mV p-p) was passed between Pt-black electrodes in outer chambers filled with Tyrode’s solution. Current flowed through the intracellular pathway and a small extracellular shunt around the strip. At high frequencies intracellular current flowed through the cytoplasm resistance of the preparation, R_c, bypassing the capacitative gap junction impedance (8). A small extracellular shunt resistance was estimated separately by measuring the impedance between two Pt-black needle electrodes in the preparation within the oil gap.

Figure 3.

(A) Frequency-dependent impedance of a myocyte suspension (Ω·cm). (B) Values of susceptance, B, and conductance, G, from the data points in part A. f^∗ marks the frequency of maximum susceptance.

Histology and immunofluorescence

Atrial strips as used for impedance measurements were dissected and snap frozen in liquid N₂ either immediately after the heart was removed (t-0) or after 45 min (t-45; duration of an impedance experiment): strips were then stored at –80°C. Cryostat sections (12 μm) were fixed with ice cold 4% paraformaldehyde (for 15 min at room temperature) and then stained with hematoxylin and eosin.

For immunofluorescence staining, paraformaldehyde-fixed cryosections from t-0 and t-45 specimens were permeabilized with 0.1% TritonX-100 in physiological buffer solution (PBS) for 20 min, and then incubated for 1 h with blocking buffer (PBS and 2% bovine serum albumin, 0.1% TritonX-100). Cell membranes were labeled with fluorescein isothiocyanate-conjugated wheat germ agglutinin (1:200; Millipore, UK) and nuclei were labeled with the fluorescent nucleic acid binding dye TO-PRO-3 (Molecular Probes; 1:500), respectively. Sections were treated with both markers for 1 h at room temperature and then mounted with Vectashield fluorescence mounting medium (Vectashield, Vector Laboratories, UK), viewed, and analyzed at ×40 magnification using a Zeiss LSM 510 laser scanning confocal microscope.

Measurement of tissue [ATP]

Tissue strips were dissected 15 or 45 min after removal of the heart (t-15, t-45; corresponding to when impedance experiments started or finished) and after 360 min (t-360). Intracellular [ATP] was measured with a colorimetric assay kit (Abcam, UK) as per the manufacturer’s instructions. Tissue was lysed in ATP Assay Buffer (100 μl/mg wet weight of tissue) and then the tissue homogenate was transferred to 10 kDalton spin column and centrifuged (15,000 × g; 2 min) to pellet insoluble materials. The supernatant was added to a 96-well plate followed by the addition of 50 μl of the Reaction Mix to each well. The absorbance was read at 550 nm using a microplate reader (VersaMax, Molecular Devices). Tissue ATP content was calculated from a standard curve generated at the same time and normalized to the amount (mg) of tissue.

Data analysis

Cytoplasm data values are presented as specific resistivity (R, Ω·cm) rather than conductance (G = 1/R) to conform with most of the physiological literature in this context; all data are mean values ± SD. Differences between data sets were tested by analysis of variance with Bonferonni post hoc tests. The null hypothesis was rejected when p < 0.05.

Results

DEP: measurement of cell electrical constants using cultured atrial cells

Table 1 shows values of specific intracellular and membrane electrical properties and cell radius, i.e., normalized to unit cell cross section and membrane area. Values are for cells suspended in three solutions; standard isotonic sucrose/glucose solution with CaCl₂ and MgCl₂ and divalent cation-free (DCF) isotonic or hypotonic sucrose/glucose solutions. Isotonic DCF solution was used as many DEP experiments are reported using this medium; a hypotonic DCF solution was used to cause cell swelling that may alter intracellular conductivity to test system sensitivity. Specific resistance, R, values are also quoted as reciprocal specific conductances (σ = 1/R). Typical DEP spectra and model fits for cell suspensions in the three media are shown in Fig. 2: all measurements were at 23°C.

Table 1.

Electrical constants of atrial cells measured by DEP

	Isotonic sucrose/glucose medium + CaCl2 and MgCl2	Isotonic sucrose/glucose medium	Hypotonic sucrose/glucose medium
Cytoplasm resistivity, Ω·cm	126 ± 11 (5)	271 ± 49 (5)∗	389 ± 66 (6)∗
Cytoplasm conductance, S·m−1	0.80 ± 0.07(5)	0.38 ± 0.08 (5)∗	0.26 ± 0.04 (6)∗
Membrane resistance, kΩ·7cm2	6.70 ± 0.57 (5)	4.90 ± 1.32 (6)∗	3.10 ± 1.28 (6)∗
Membrane capacitance, μF·cm−2	1.01 ± 0.38 (5)	0.98 ± 0.25 (6)	1.06 ± 0.25 (6)
Cell diameter, μm	12.4 ± 0.6 (5)	12.1 ± 0.7 (6)	14.8 ± 0.4 (6)∗

Values for membrane conductance refer to that offered by the cell suspension. Values for membrane conductance refer to that offered by the cell suspension.

^∗p < 0.05 with respect to values in sucrose/glucose medium.

Intracellular resistivity, R_c, was 126 ± 11 Ω·cm (G_c = 0.80 ± 0.07 S·cm⁻¹) in isotonic sucrose/glucose suspension medium containing CaCl₂/MgCl₂. The value of R_c was significantly greater when cells were suspended in the DCF isotonic sucrose/glucose medium; cell dimensions were not significantly altered. A larger increase of R_c, was measured in the DCF hypotonic solution, in this instance associated with a larger mean cell radius.

Larger values of R_c in both DCF isotonic and hypotonic media may be due to altered membrane solute permeability and/or water fluxes. Omission of extracellular divalent ions from suspension media increases ion channel conductance in different cells (15) and cell swelling with hypotonic solutions may alter the membrane structural integrity. Table 1 shows that specific membrane resistance was significantly reduced in DCF isotonic and hypotonic media. Cell capacitance values, normalized to unit membrane area, C_m, were similar in all three media, with a mean value of 1.01 ± 0.38 μF·cm⁻²; assuming cells were smooth spheres in all the suspension media.

Cytoplasm resistivity by impedance analysis of suspensions of freshly isolated atrial cells

A more direct estimate of R_c is obtained by measuring the impedance of cell suspensions at high alternating current frequencies. Atrial cells were suspended in an isotonic solution with divalent cations, similar to that used previously and suspension impedance was measured between 0.02–30 kHz. Data were obtained at 23°C, with an average suspension packing fraction of 0.110 ± 0.026 (n = 4; see Methods). Low frequency system impedance, when current bypassed the cells, was 622 ± 40 Ω·cm. This was reduced to 420 ± 52 Ω·cm at high frequencies when current flowed through the additional volume offered by the cytoplasm of cells in the suspension, as the membrane offered no impedance: a sample experiment is shown in Fig. 3 A. The value of R_c was estimated to be 146 ± 42 Ω·cm.

These impedance data offer an alternative approach to estimate R_c from determination of the frequency at which maximum susceptance (B = 1/X, complex reactance) is measured when suspension admittance, Y (= 1/Z), was calculated as a function of frequency, f (16), (Fig. 3 B):

$$R_{\mathrm{c}}=\frac{2.48a}{\pi f{C}_{\mathrm{m}}{l}^2},$$ (5)

where l and a are myocyte length and radius, assuming it to be a cylinder, and C_m the specific membrane capacitance (1 μM cm⁻²). The frequency at maximum susceptance was 49.2 ± 18.3 kHz. For a cell length of 100 μm and radius 7.5 μm a value for R_c = 136 ± 59 Ω·cm is obtained.

Measurement of intracellular resistivity of atrial strips by impedance analysis

Problems with using isolated cells are potential damage during their isolation or a change of properties during cell culture. Their use may be validated by estimating R_c in tissue strips, where these problems are minimized. Atrial strips were mounted in an oil-gap chamber and alternating current (0.02–100 kHz) passed between the two ends, forcing a fraction of current flow along the intracellular compartment. Previous studies revealed a component of impedance with a maximum reactance at ∼10 kHz and attributed to GJs (8). At higher frequencies total impedance attains a steady value that may be attributed to the cytoplasmic resistance, similar to that observed in the experiment illustrated in Fig. 3 A. The value of intracellular resistivity was calculated to be 145 ± 37 Ω·cm at 23°C (n = 9). A complementary series of experiments was carried out at 35°C and obtained a value for R_c of 116 ± 10 Ω·cm (n = 6). These experiments also permitted estimation of gap junction resistance (Fig. 3 A) and values of 144 ± 25 and 114 ± 21 Ω·cm were obtained at 23°C and 37°C, respectively. At both temperatures, the contributions from cytoplasmic and gap junction resistances to the total intracellular resistance are therefore similar.

A potential problem using tissue strips is development of a hypoxic core over the time course of the experiment, which might distort measured electrical constants. Tissue structure, by histology and immunohistochemistry, and metabolic integrity, by measuring intracellular [ATP], were examined in similar strips immediately after dissection from the heart (t-0), just prior and just after impedance experiments were conducted (t-15; t-45) and after 360 min (t-360).

Fig. 4 A shows hematoxylin and eosin stained sections from the i), surface of the preparation at t-0 or ii), t-45 or iii), from the center of the strip at t-45. In all sections the gross morphology was similar with distinctive nuclei arranged in a regular array. Fig. 4 B shows confocal images of three further sections obtained under identical conditions; cell boundaries are labeled green and nuclei dark blue. At this greater magnification a similar morphology was also seen in all sections with parallel arrays of cells and no sign of tissue disruption. Fig. 4 C shows ATP levels in atrial strips measured at t-15, t-45, and t-360. The [ATP] was constant between t-15 and t-45, but there was a significant fall at t-360. Therefore, the metabolic state of the tissue did not alter over the time of the experiment. We interpret these data as showing that during the time-course of the impedance experiment preparations remained metabolically and structurally intact.

Figure 4.

Biochemical and histological examination of atrial strips as used for impedance experiments at the following times: immediately after dissection (t-0), and at 15 min (t-15), 45 min (t-45), or 360 min (t-360). (A) Tissue [ATP] in multicellular strips before the experiment (t-15), at the end of experiment (t-45), and at (t-360); preparations in Tyrode’s solution throughout. (B) Hematoxylin and Eosin stained sections of atrial strips t-0 (i, surface section) and t-45 (ii, surface section; iii, from center of preparation). (C) Confocal images labeled for cell membranes and nuclei from similar sections as used for part B.

Summary of R_c values and implications for AP conduction velocity

Fig. 5 A summarizes the four independent values of R_c at room temperature (23°C), using three different experimental methods. Values were not significantly different by these various methods. The average of the four values at 23°C was 138 ± 5 Ω·cm, mean ± SE n = 4. Analysis of the variances of the values by the four methods (Fisher’s F-test) showed it was significantly smaller with the DEP method compared to the three impedance analyses. Also shown is the significantly smaller value obtained at 35°C using tissue strips. Measurement of R_c at 23 and 35°C enables a Q₁₀ for R_c of 1.20 to be obtained from

$${\log }_{10}{Q}_{10}=\frac{10}{35-23}\log \frac{R_{\mathrm{c},23}}{R_{\mathrm{c},35}};$$

similar to a value of 1.23 for 0.15 M isotonic KCl solution (17), but smaller than 1.37 for skeletal muscle (18).

Figure 5.

(A) Values of cytoplasmic resistivity at 23°C as determined by DEP and impedance methods. (B) Dependence of conduction velocity as a function of gap junction resistance, R_j, in atrial (solid line) and ventricular (dashed line) myocardium. Values are quoted as proportional changes from those in control conditions. Conduction velocity was calculated from Eq. 6 using values for a = 6.2 and 12.0 μm; C_m = 1 μF·cm⁻²; R_m = 6700, and 5500 Ω·cm²; τ_m = C_mR_m (values from Table 1 and (8,30)). The thin solid lines show the reduction of conduction velocity for a 2.5-fold increase of R_j in atrial and ventricular myocardium. (C) Dependence of the space constant (λ² = aR_m/2R_i) as a function of gap junction resistance, R_j, in atrial (solid line) and ventricular (dashed line) myocardium: values for the parameters as in B.

Fundamental active and passive electrophysiological properties of myocardium, such as AP propagation velocity (CV) and the space constant, λ, depend on the value of total intracellular resistivity, R_i: itself, the sum of R_c and gap junction resistance, R_j. In turn, R_j is increased under conditions such as ischemia and contributes to the arrhythmogenic substrate in myocardium by slowing CV (19). An accurate value of R_c permits precise evaluation of how changes to R_j influence variables such as CV and λ. For an AP propagating in a one-dimensional substrate, CV is related to R_i by Eq. 6: a = cell radius; τ_m, τ_ap are experimentally derived parameters (9):

$$\mathrm{C}{\mathrm{V}}^2=\frac{a}{2R_{\mathrm{i}}{C}_{\mathrm{m}}{\tau }_{ap}}·{\left(1+\frac{{\tau }_{ap}}{{\tau }_m}\right)}^{-1}.$$ (6)

Fig. 5 B shows the proportional variation of CV for changes to R_j, using the relationship R_i = R_j + R_c, using a value for R_i in atrium of 230 Ω·cm (above) and in ventricle of 415 Ω·cm (8). assuming the same value for R_c. Thus, R_c contributes 50% of total R_i in atrium but only 28% in ventricle. Furthermore, proportional changes to R_j have a much smaller effect on overall R_i and hence CV in atrium compared to ventricle. Fig. 5 B shows that increasing R_j 2.5-fold reduces CV by 69% in ventricle, but by only 77% in atrium. Alternatively, a fall of CV by 40% would require R_j to increase by ≈3.5-fold in ventricle but as much as ≈5.0-fold in atrium. The space constant, λ, is a measure of how far electrotonic currents can spread in tissues (λ² = aR_m/2R_i), a larger value increases intercellular coupling in a functional syncitium such as myocardium. Fig. 5 C shows that, as with CV, variation of R_j has less influence on changes to the value of λ. Overall, because the value of R_c is more similar to that of R_j in atrial myocardium, this tissue is more resistant than ventricle to changes of CV and λ when gap junction properties alter.

Discussion

The value of cytoplasmic resistivity, R_c

R_c has been estimated in large axons and skeletal muscle fibers from their linear cable properties (20,21) or by direct measurement on insertion of large electrodes into the cell (22,23) or extrusion of the cytoplasm for external analysis (24). Values, usually obtained at room temperature, were 1–2 times that of isotonic extracellular solution and were also similar to those from suspensions of single cells by recording frequency-dependent impedance (see 25). The greater value may be due to limited ion mobility in the cytoplasm and also because the cell interior is partly occupied by low conductivity organelles. For simplicity, the resistivity of the cell interior is termed cytoplasmic resistivity, R_c, with the appreciation that this is an imperfect description.

Comparable measurements of R_c in cardiac muscle cells are lacking; there are no recordings with single cell suspensions and estimation of intracellular electrical impedance with microelectrodes is precluded in a functional syncitium of myocardium because it has contributions from both the cytoplasm and gap junction resistances. One direct measurement of R_c in frog myocardium, by using the variation of microelectrode resistance as a function of solution conductivity, yielded a value of 282 Ω·cm at room temperature (9). However, as described in the Introduction, the method was prone to significant errors.

Therefore, these experiments provide the first, to our knowledge, direct reliable measurement of R_c in atrial myocardium and several methods gave similar values with a mean of 138 Ω·cm; furthermore, about twice that of isotonic extracellular solution (17). The consistent values with different methods and preparations (cultured and primary cells or muscle strips) suggests that cell isolation per se does not cause significant changes to R_c and cell culture also yields values similar to freshly isolated cells. The most error-prone method was from the frequency of maximum admittance with isolated cells, as the calculated value is crucially dependent on cell length. HL-1 myocytes were chosen as cultured cells because they display resting and action potentials comparable to intact myocytes (12).

The smaller variance of data using DEP allowed interventions to be used that might alter cellular electrical properties. The standard medium for cell suspension with DEP contains sucrose and glucose and a small amount of PBS to generate finite solution conductance. However, the R_c value was significantly greater than that obtained by impedance methods and only normalized when small concentrations of Ca²⁺ and Mg²⁺ were added to the medium, similar to those used to suspend isolated cells for impedance measurements. A hypotonic suspending medium, also DCF, further increased R_c, possibly due to water influx correcting the imbalance of transmembrane osmolality. The change of R_c by 30% reflects a 35% change of medium osmolality and indicates the cells are effective osmometers.

Values of membrane electrical constants

The DEP method also allowed estimation of specific membrane resistance, R_m, and capacitance, C_m, i.e., normalized to unit membrane area assuming cells were spherical with a smooth surface. Other methods have yielded a value for C_m of ∼1 μF·cm⁻² (25) and our mean value of 1.01 μF·cm⁻² in the divalent ion-containing isotonic solution (Table 1) is in close harmony. The value for R_m of 6.70 kΩ·cm² in Ca and Mg-containing isotonic solution is also similar to those from multicellular atrial preparations (26).

In the DCF isotonic and hypotonic solutions C_m was not significantly altered indicating no gross structural changes to the membrane. However, R_m was significantly reduced in both DCF solutions and the value in the hypotonic solution significantly less than in the isotonic solution. Divalent cations significantly affect ion channel conductance (15,27) and the greater value of R_c and smaller value of R_m in their absence may reflect degradation of membrane function. Thus, it is essential that small concentrations of divalent cations are added to DEP suspension media to ensure cellular integrity.

Implications for cardiac conduction and modeling

The value for R_c in atrial tissue is equivalent to that offered by GJs and therefore offers a significant fraction of the overall intracellular resistance. This is different from ventricular myocardium where R_c and gap junction resistance have also been estimated from impedance measurements of tissue strips and where gap junction resistance represent a greater fraction of the total resistance pathway. Several factors increase the resistance offered by GJs including an increase of intracellular [Ca²⁺] and decrease of pH, as well as dephosphorylation of the component connexin proteins during ischemia for example (28), all conditions associated with decreased myocardial metabolic function. The calculations in Fig. 5, B and C, show that changes to gap junction properties would have a smaller effect in atria than in ventricle on several important electrophysiological properties of myocardium such as action potential conduction velocity and the tissue electrical space constant. Thus, atria should be more resistant to conditions that slow conduction and hence generate reentrant circuits. The lower overall value of intracellular resistance in atrium will also explain the more rapid AP conduction velocity compared to ventricle, despite the smaller size of atrial myocytes. However, slowed conduction is associated with the development of atrial fibrillation (29) and emphasizes the necessity to determine the precise factors that determine this abnormal electrophysiological function.

Limitations

Limitations of the study are that measurements of cytoplasmic resistivity by DEP or impedance techniques were not carried out in tissue from hearts undergoing atrial fibrillation; furthermore, freshly isolated and HL-1 myocytes were not each subjected to DEP and impedance techniques.