Divergent estimates of the ratio between Na⁺-Ca²⁺ current densities in t-tubular and surface membranes of rat ventricular cardiomyocytes

Recent advances in experimental techniques for cardiac cell electrophysiology, as well as the elaboration of a method using osmotic shock to physically detach cardiac t-tubules from the surface membrane (a process known as ‘detubulation’), have enabled researchers to assess the density of individual ion currents in the t-tubular and surface membranes (Orchard et al., 2009). Data from adult rat ventricular cardiomyocytes published to date indicate that the mean density of Na⁺-Ca²⁺ exchange current (J_NaCa [pA pF⁻¹]) in the t-tubular membrane (J_NaCa,t) is higher than in the surface membrane (J_NaCa,s) (Despa et al., 2003; Gadeberg et al., 2016; Thomas et al., 2003; Yang et al., 2002). However, the values of the ratio J_NaCa,t/J_NaCa,s (referred to from now on as ‘mean J_NaCa-ratio’) computed from data reported in these studies range between 1.7 and 40 (Pásek et al., 2017).

The reasons for such marked differences are unclear. They might be caused by different experimental conditions, or by differences in the properties of ventricular myocytes from animals of different ages (and therefore different weights) or from different strains of rat. However, inaccuracies in the measurement and evaluation of experimental data and their consequent scatter may also play an important role (Bébarová et al., 2020).

The assessment of the mean J_NaCa-ratio is based on the measurement of Na⁺-Ca²⁺ exchange currents and membrane capacitances in intact and detubulated myocytes, and on the optical measurement of the fraction of t-tubules resisting detubulation. Therefore, the reliability of the assessment is dependent on the accuracy of the measurement of each of these parameters. The aim of this study was to explore: (1) whether the propagation of errors of individual parameters in the computation of the mean J_NaCa-ratio could be responsible for the reported differences; (2) what the possible sources of errors and biases in the experimental data could be; and (3) which range of values of the mean J_NaCa-ratio in adult rat cardiomyocytes might be considered plausible.

The mean values of membrane capacitance (C) and J_NaCa as computed from data measured in rat ventricular myocytes before (C_intact,J_NaCa,intact) and after detubulation (C_detub,J_NaCa,detub) by various groups are summarised in Table 1. Marked differences in the values of J_NaCa,intact and J_NaCa,detub reflect different experimental conditions under which the currents were measured; for example, internal or external solutions, absence or presence of exogenous intracellular Ca²⁺ buffers, different voltage clamp protocols or different temperatures (for comparison see Table 4 in Discussion).

The mean J_NaCa-ratio was calculated from Eqn 4 (see Materials and Methods) and the experimental data specified in Table 1. The fraction of t-tubules resisting detubulation (f_t,res) exhibited different values in various studies; either nearly 0, which represented almost total detubulation (Thomas et al., 2003), 0.08 (Pásek et al., 2008) or even 0.16 (Bryant et al., 2015). Therefore, in each case, the mean J_NaCa-ratio was computed for f_t,res ranging between 0 and 0.2. The traces illustrated in Fig. 1A show that the ratio increases progressively with f_t,res in all cases, but surprisingly exhibits very different values (1.7, 3.8, 5.2 and 18 for f_t,res=0, and 1.8, 5.3, 7.4 and 39.5 for f_t,res=0.16).

Our previous study (Pásek et al., 2017) showed that calculating the mean J_NaCa-ratio from the values of J_NaCa,intact and J_NaCa,detub may result in substantially different values, depending on the way J_NaCa is assessed (using the average of the current densities directly assessed in individual cells versus the ratio of the average current magnitude I to the average C for a given set of cells). To evaluate the influence of these differing methods of J_NaCa density assessment on the value of mean J_NaCa-ratio within the above mentioned span of f_t,res, comparative computations were performed based on data measured by Thomas et al. (2003) and Gadeberg et al. (2016) (Fig. 1B,C, respectively). Although the differences in the mean J_NaCa-ratios computed in both ways from data given by Thomas et al. (2003) were small (Fig. 1B), the data from Gadeberg et al. (2016) produced sizeable differences (25.9 and 57.4% at f_t,res of 0 and 0.16, respectively; Fig. 1C).

To explore whether the different values of mean J_NaCa-ratio calculated from different studies could arise from a large scatter of experimental data, we used the statistical parameters of C, J_NaCa, and f_t,res (Table 1), and reconstructed the distribution of the mean J_NaCa-ratio related to each study using Monte Carlo simulations (for details see Materials and Methods). Two types of distribution of measured data, normal and log-normal, were used. The normal distribution was used because the values from the compared studies were computed within the hypothesis of normality. However, the log-normal distribution appears to be more realistic because C and f_t,res are always positive and J_NaCa must keep a constant sign, conditions that are not strictly observed when using the normal distribution to generate artificial data (for deeper explanation see Limpert and Stahel, 2011 and Kula et al., 2020a).

The simulations showed that, if all the measured quantities (C, J_NaCa, and f_t,res) were normally distributed, distributions of the mean J_NaCa-ratio were asymmetric (Fig. 2A), with medians of 24.5, 4.8, 4.3 and 1.7 (Table 2, upper) when evaluated from data measured by Gadeberg et al. (2016), Yang et al. (2002), Despa et al. (2003) and Thomas et al. (2003), respectively. If these quantities were given log-normal distribution, distributions of the mean J_NaCa-ratio were asymmetric as well (Fig. 2B); the respective medians were 17.0, 4.1, 3.2 and 1.6 (Table 2, lower).

Analysis of 95% confidence intervals of the mean J_NaCa-ratio revealed considerable differences between individual studies whenever normal or log-normal distribution were assumed (Table 2). The interval of values common for all these studies spanned from 1 to 3 in the case of normal distribution, and from 0.3 to 4.3 in the case of log-normal distribution. The latter range covers values of 1.6, 3.2 and 4.1, computed from three studies (Thomas et al., 2003; Despa et al., 2003; Yang et al., 2002). As also follows from Table 2, a large scatter of experimental data could even lead to negative values of the ratio in a considerable number of cases (expressed in percentage, P_n). It is also worth noting that the confidence interval, as well as P_n, gradually increased with an increase of the mean J_NaCa-ratio, suggesting that higher values of the mean J_NaCa-ratio (over 4.3) are less reliable than the lower ones.

This view was further supported by comparing indices P₃₀, which characterise probabilities that the real values of mean J_NaCa-ratio lie within ±30% from the values in Table 2. As expected, when using data from Thomas et al. (2003), Despa et al. (2003), Yang et al. (2002) and Gadeberg et al. (2016), P₃₀ gradually decreased from 74% to 37%, 25%, and 16%, respectively, when normal distribution was assumed, and from 52% to 31%, 20%, and 10%, respectively, when log-normal distribution was assumed. Analysis of these data showed a negative correlation between the mean J_NaCa-ratio and P₃₀ (the Pearson′s coefficients of correlation were −0.68 and −0.78, respectively, for normal and log-normal distribution of the data; not illustrated).

The distribution of the Na⁺-Ca²⁺ exchanger between the t-tubule and surface membranes was shown to have a marked effect on the intracellular Ca²⁺ load and Ca²⁺ transient in a model of rat ventricular cardiomyocyte (Pásek et al., 2017). In particular, these effects were dependent on the value of J_NaCa-ratio. Unfortunately, computation of the mean J_NaCa-ratio from electrophysiological data obtained in intact and detubulated rat ventricular cells (Yang et al., 2002; Despa et al., 2003; Thomas et al., 2003; Gadeberg et al., 2016; Pásek et al., 2017) results in values ranging from 1.7 to nearly 40 (Fig. 1). The discrepancy between the mean J_NaCa-ratios as evaluated from experimental data might be due to methodological differences or due to variations of the quantitative aspects related to the age/weight or the strain of rats. However, we have found that the degree of uncertainty of the mean ratio gradually increases with its value and that the values higher than 4.8 are encumbered by a considerable error resulting from very large confidence intervals, which include a considerable fraction of non-realistic negative values.

The mean J_NaCa-ratio is a decreasing function of c_id and an increasing function of j_id in the whole range of values determined in the compared studies. From the results of studies by Gadeberg et al. (2016), Yang et al. (2002), Despa et al. (2003), and Thomas et al. (2003) summarized in Table 1, the following values could be calculated (for f_t,res=0): J_NaCa-ratio=[18, 5.2, 3.8, 1.7], c_id=[1.1649, 1.3497, 1.4717, 1.3600] and j_id=[3.3908, 2.0833, 1.9000, 1.1905], respectively. The particularly high value of the mean J_NaCa-ratio in the work of Gadeberg et al. (2016) results from a combination of the smallest c_id value and the largest j_id value. The dependence on c_id is less pronounced than the dependence on j_id, and in contrast to c_id, the values of j_id increase in parallel with the values of mean J_NaCa-ratio sorted by size.

One of the factors that could affect the value of j_id is age/weight or strain of the animals. Indeed, the highest j_id and thus J_NaCa-ratio corresponds to the oldest rats (25 weeks, Gadeberg et al., 2016). However, the estimated values of the mean J_NaCa-ratio do not correlate with age/weight at lower values of this parameter; the rats used by Thomas et al. (2003), although older than those used by Yang et al. (2002), exhibited a smaller mean J_NaCa-ratio (1.7 versus 6 according to Eqn 4). This inconsistency in age-dependency of the experimental data suggests that the mean J_NaCa-ratio in adult rat ventricular cardiomyocytes is affected by other factors.

Differences in the experimental conditions, particularly in intracellular and extracellular Na⁺ and Ca²⁺ concentrations, I_NaCa blockers, Ca²⁺ buffers, temperature and resistance of the pipette may also contribute to different values of assessed mean J_NaCa-ratio. The closest conditions (except for different I_NaCa-blocker and temperature, Table 4) were used in experiments performed by Thomas et al. (2003) and Despa et al. (2003), which might explain the closest values of mean J_NaCa-ratio (1.7 and 4.3 according to Eqn 4).

Yang et al. (2002) used higher concentrations of Ca²⁺ in perfusion solution (2.5 versus 1 mM in the other studies) and of Na⁺ in the pipette (20 versus 10 mM), but despite that, J_NaCa was 4-50× smaller than in other studies, even when assessed at the highest value of the voltage ramp (50 mV). Under these conditions, even minimal sensitivity of other currents to Ni²⁺ might affect the resulting mean J_NaCa values, which may be the reason for a higher mean J_NaCa-ratio (6 according to Eqn 4).

As for the measurement performed by Gadeberg et al., 2016, the main difference compared to the other three studies was that I_NaCa was determined at [Ca²⁺]_i of 400 nM during the declining phase of Ca²⁺ transient induced by 10 mM of caffeine. The concentration of Na⁺ in the pipette was not specified in their study, which raises the question of whether the intracellular Na⁺ was well controlled during measurements on intact and detubulated cells. If not, the variation of this ion concentration might have contributed to the high dispersion of measured I_NaCa and thus to such different values of the mean J_NaCa-ratio (for explanation, see section ‘Potential causes of large scatter of experimental data’). More detailed data would be needed to assess the accuracy of this approach.

To answer this question, we performed a random generation of simulated experimental data based on their statistical characteristics (Table 1) and computed the mean values of J_NaCa-ratio that could be obtained if a particular study was repeated. Having generated 50,000 sets of data, we evaluated the 95% confidence interval and the probability of negative values of the mean J_NaCa-ratio for each study (Table 2). The analysis showed that if the data measured in individual studies were distributed normally, the confidence intervals of mean J_NaCa-ratio shared values between 1 and 3. If the distribution was log-normal, the interval of common values of the ratio was between 0.3 and 4.3. Considering that the reduction of J_NaCa after detubulation (Table 1) supports values higher than 1, it is very likely that the real values of the mean J_NaCa-ratio in all four studies were between 1 and 4.3, and that values outside this interval arose from the scatter of experimental data.

The large scatter of experimental data used to assess J_NaCa may have various causes. Firstly, transmural differences in the density and distribution of Na⁺-Ca²⁺ exchanger between the t-tubular and surface membranes might contribute to the large dispersion of the measured I_NaCa and impaired I_NaCa-C proportionality, a prerequisite of a proper assessment of mean J_NaCa (Kula et al., 2020b). A transmural gradient in ventricular J_NaCa was observed in canine hearts by Zygmunt et al. (2000) and Xiong et al. (2005). Similarly, transmural differences have been reported in rat ventricles for cell size (Campbell and Gerdes, 1987) and I_Ca current properties (Volk and Ehmke, 2002). Thus, it is very likely that transmural differences in ventricular J_NaCa exist in rat. Secondly, our reconstruction of experiments performed by Gadeberg et al. (2016) using a model of rat ventricular myocyte incorporating the t-tubular system (Pásek et al., 2012) showed that the intracellular Na⁺ might have varied considerably depending on the inner diameter of the pipette tip and obstruction of its orifice after membrane rupture. The model also indicated that the Ca²⁺ and Na⁺ concentrations under the t-tubular and surface membranes might have differed from their cytosolic levels due to transmembrane Ca²⁺ and Na⁺ fluxes. Both of these factors could contribute to the high variability of J_NaCa among individual cells. As follows from Eqn 4, the high relative s.e. of J_NaCa,intact and J_NaCa,detub (∼ 30%) could then lead to values of mean J_NaCa-ratio that are higher than 1000 (e.g. if J_NaCa,intact=−0.32 pA pF⁻¹ and J_NaCa,detub=−0.06 pA pF⁻¹).

Another source of scatter of experimental data could be the variability in size of the cells that were used in experiments. The larger the differences in cell size, the larger the scatter of membrane capacitances. This may substantially affect the calculated value of mean J_NaCa-ratio. For example, a change of only C_intact within s.e. from the value reported by Yang et al. (2002) (193±41 pF, n=25, relative s.d.=106%) could result in the mean J_NaCa-ratio to span from 4.5 up to 21.5. Furthermore, the fraction of membrane in t-tubules parallels the size of rat ventricular cells (Nakamura et al., 1986; Christé et al., 2020), with small ones having less t-tubules than large ones, which may also contribute to the scatter of J_NaCa values.

Regarding f_t,res, it may vary from cell to cell, owing to a non-homogeneous efficiency of the detubulation procedure, which may have contributed to the scatter of f_t,res. Possible biases in the estimation of f_t,res have been discussed by Pásek et al. (2008). A consequent inaccuracy in the assessed f_t,res may also contribute to the divergent mean J_NaCa-ratio among individual studies, as demonstrated in Fig. 1.

To answer this question, we compared mean J computed from averaging the values obtained in individual cells with the values computed from the ratio of average I to average C. To achieve this, randomly generated data with normal distribution and a Monte Carlo approach were used. If the number of data was ten or more and their scatter was small (relative s.d. below 30%), the assessment of mean J by both ways converged to comparable values, even if both quantities were independent (Table 3, upper). Therefore, the accuracy of both ways of mean J assessment can be considered comparable under these conditions. However, if the scatter of data was larger (relative s.d. over 30%) and correlation of both quantities was low (Pearson coefficient of correlation between I and C below 0.6), the mean J assessed by both ways provided notably divergent values. In this case, the mean J calculated from the ratio of average I to average C was substantially more accurate (Table 3). It is also worth noting that the relative s.d. of I and C higher than 50% strongly indicates that the data are not normally distributed. If this is proved by a statistical test, their means should be calculated using the median or geometric mean (Kula et al., 2020a).

As shown in Table 2, statistical evaluation of the mean J_NaCa-ratio computed according to Eqn 4 may result in very different values and an enormous confidence interval CI₉₅. This might seem to be a consequence of the negative values of the input variables C_intact and C_detub, and non-real values of J_intact and J_detub, resulting from the assumption of their normal distribution. However, using the log-normal distribution that excludes the occurrence of the false values, the confidence interval even increased (Table 2, lower), as well as the probability of unrealistic negative values of the calculated mean J_NaCa-ratio.

We therefore tried to find out whether the mean J_NaCa-ratios from the compared studies could be reconciled and whether a smaller CI₉₅ could be achieved if the mean J_NaCa was computed from the ratio of average I_NaCa to average C. This way of mean J_NaCa assessment would be more accurate if the original I_NaCa and C data were not sufficiently proportional (see section ‘Which way of mean current densities assessment is more accurate?’), which could be caused, for example, by using cells exhibiting transmural differences in J_NaCa, as reported by Zygmunt et al. (2000) and Xiong et al. (2005). The values needed to compute the mean J_NaCa from the ratio of average I to average C were only available in two cases (Thomas et al., 2003; Gadeberg et al., 2016). The Monte Carlo simulations under the assumption of log-normally distributed data then resulted in a mean J_NaCa-ratio of 1.8 and 12.9, for those two studies, respectively, thus approaching each other (see Table 2 for comparison with the former values). Interestingly, the value of the mean J_NaCa-ratio estimated from the results available in the study of Gadeberg et al. (2016) would further decrease to 6.5, and the resulting CI₉₅ would be essentially reduced when using the less dispersed values of C_intact and C_detub measured by Bryant et al. (2015) on the same population of cells (C_intact=260±9 pF, n=37, and C_detub=178±9 pF, n=28).

These considerations lead to the conclusion that the differences between results of the compared studies are not essential and could be reduced by larger sets and more precise evaluation of the experimental data. Taking into account the asymmetric distribution of measured data (Kula et al., 2020a), the common confidence interval of the mean J_NaCa-ratio between 0.3 and 4.3, the reduction of J_NaCa after detubulation common to all compared studies (which suggests a ratio higher than 1) and the substantially smaller CI₉₅ related to the mean ratios assessed from the data measured by Thomas et al. (2003) and Despa et al. (2003), the values from 1.6 to 3.2 appear to represent the most accurate estimates. Assuming that t-tubules are responsible for ∼30% of total membrane capacitance in rat ventricular myocytes (Brette and Orchard, 2007; Pásek et al., 2008), such values implicate that 40 to 60% of Na⁺-Ca²⁺ exchanger proteins are located there. This range agrees well with the results of immunolabelling studies performed on rat myocytes (Kieval et al., 1992; Thomas et al., 2003; Jayasinghe et al., 2009), as well as with the approximately halved rate of decay of caffeine-induced Ca²⁺ transient in detubulated cells (Yang et al., 2002; Gadeberg et al., 2016).

We reproduced what could be potentially obtained as a result of repeating a particular study by sampling a small number of cells out of a large population of either intact or detubulated cells. A random generation of sets of data was performed, using the published statistical parameters (average, s.e. and number of measured data) specified in Table 1. When log-normal distribution was used, the parameters of the data generator were adjusted to produce datasets with an arithmetic mean and s.e. close to those in Table 1. Thus, values were randomly generated for each of the five quantities (C_intact, C_detub,J_NaCa,intact, J_NaCa,detub, and f_t,res), and their averages were computed and stored as a dataset. For each dataset, the value of the mean J_NaCa-ratio was computed using Eqn 4. This was repeated 50,000 times. The distribution of the mean J_NaCa-ratio was reconstructed from the ensemble of these values. The whole procedure was performed using the computational software MATLAB v.7.2 (MathWorks, Natick, MA, USA).

The peer review history is available online at https://journals.biologists.com/jcs/article-lookup/doi/10.1242/jcs.258228