The ratio between Na+-Ca2+ exchange current densities in t-tubular and surface membranes of rat ventricular cardiomyocytes (JNaCa-ratio) estimated from electrophysiological data published to date yields strikingly different values between 1.7 and nearly 40. Possible reasons for such divergence were analysed by Monte Carlo simulations assuming both normal and log-normal distribution of the measured data. The confidence intervals CI95 of the mean JNaCa-ratios computed from the reported data showed an overlap of values between 1 and 3, and between 0.3 and 4.3 in the case of normal and log-normal distribution, respectively. Further analyses revealed that the published high values likely result from a large scatter of data due to transmural differences in JNaCa, dispersion of cell membrane capacitances and variability in incomplete detubulation. Taking into account the asymmetric distribution of the measured data, the reduction of mean current densities after detubulation and the substantially smaller CI95 of lower values of the mean JNaCa-ratio, the values between 1.6 and 3.2 may be considered as the most accurate estimates. This implies that 40 to 60% of Na+-Ca2+ exchanger is located at the t-tubular membrane of adult 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 (JNaCa [pA pF−1]) in the t-tubular membrane (JNaCa,t) is higher than in the surface membrane (JNaCa,s) (Despa et al., 2003; Gadeberg et al., 2016; Thomas et al., 2003; Yang et al., 2002). However, the values of the ratio JNaCa,t/JNaCa,s (referred to from now on as ‘mean JNaCa-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 JNaCa-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 JNaCa-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 JNaCa-ratio in adult rat cardiomyocytes might be considered plausible.

Basic experimental data

The mean values of membrane capacitance (C) and JNaCa as computed from data measured in rat ventricular myocytes before (Cintact,JNaCa,intact) and after detubulation (Cdetub,JNaCa,detub) by various groups are summarised in Table 1. Marked differences in the values of JNaCa,intact and JNaCa,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).

Table 1.

Statistical parameters of rat ventricular cells related to compared studies

Statistical parameters of rat ventricular cells related to compared studies
Statistical parameters of rat ventricular cells related to compared studies

Calculation of mean JNaCa-ratio in adult rat cardiomyocytes

The mean JNaCa-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 (ft,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 JNaCa-ratio was computed for ft,res ranging between 0 and 0.2. The traces illustrated in Fig. 1A show that the ratio increases progressively with ft,res in all cases, but surprisingly exhibits very different values (1.7, 3.8, 5.2 and 18 for ft,res=0, and 1.8, 5.3, 7.4 and 39.5 for ft,res=0.16).

Fig. 1.

Relation between the mean JNaCa-ratio and the fraction of t-tubules resisting detubulation according to Eqn 4 using the data presented in Table 1. (A) Traces computed from average values of JNaCa and C assessed in intact (JNaCa,intact and Cintact) and detubulated (JNaCa,detub and Cdetub) ventricular cells. (B,C) Comparison of traces related to the data from Thomas et al. (2003) and Gadeberg et al. (2016) (black-dotted and short-dashed lines from A), with corresponding traces computed when mean JNaCa was assessed from ratios of the average INaCa to the average C (grey dotted and short dashed lines). The grey point in C shows the value of 25 reported by Gadeberg et al. (2016) and valid for 16% of t-tubules resisting detubulation (Bryant et al., 2015).

Fig. 1.

Relation between the mean JNaCa-ratio and the fraction of t-tubules resisting detubulation according to Eqn 4 using the data presented in Table 1. (A) Traces computed from average values of JNaCa and C assessed in intact (JNaCa,intact and Cintact) and detubulated (JNaCa,detub and Cdetub) ventricular cells. (B,C) Comparison of traces related to the data from Thomas et al. (2003) and Gadeberg et al. (2016) (black-dotted and short-dashed lines from A), with corresponding traces computed when mean JNaCa was assessed from ratios of the average INaCa to the average C (grey dotted and short dashed lines). The grey point in C shows the value of 25 reported by Gadeberg et al. (2016) and valid for 16% of t-tubules resisting detubulation (Bryant et al., 2015).

Our previous study (Pásek et al., 2017) showed that calculating the mean JNaCa-ratio from the values of JNaCa,intact and JNaCa,detub may result in substantially different values, depending on the way JNaCa 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 JNaCa density assessment on the value of mean JNaCa-ratio within the above mentioned span of ft,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 JNaCa-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 ft,res of 0 and 0.16, respectively; Fig. 1C).

Statistical analysis of mean JNaCa-ratios calculated from different studies

To explore whether the different values of mean JNaCa-ratio calculated from different studies could arise from a large scatter of experimental data, we used the statistical parameters of C, JNaCa, and ft,res (Table 1), and reconstructed the distribution of the mean JNaCa-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 ft,res are always positive and JNaCa 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, JNaCa, and ft,res) were normally distributed, distributions of the mean JNaCa-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 JNaCa-ratio were asymmetric as well (Fig. 2B); the respective medians were 17.0, 4.1, 3.2 and 1.6 (Table 2, lower).

Fig. 2.

Histograms showing frequency distribution of the mean JNaCa-ratio resulting from Monte Carlo simulations of experimental data published by Gadeberg et al. (2016), Yang et al. (2002), Despa et al. (2003) and Thomas et al. (2003). (A,B) The individual datasets containing values of JNaCa,intact, Cintact, JNaCa,detub, Cdetub, and ft,res were generated randomly using the published values of mean±s.e., and assumption of normal (A) and log-normal (B) distribution of the experimental data. The resulting values of the mean JNaCa-ratio were computed using Eqn 4. The total number of datasets was 50,000 in both cases. Statistical characteristics/parameters of individual histograms are specified in Table 2.

Fig. 2.

Histograms showing frequency distribution of the mean JNaCa-ratio resulting from Monte Carlo simulations of experimental data published by Gadeberg et al. (2016), Yang et al. (2002), Despa et al. (2003) and Thomas et al. (2003). (A,B) The individual datasets containing values of JNaCa,intact, Cintact, JNaCa,detub, Cdetub, and ft,res were generated randomly using the published values of mean±s.e., and assumption of normal (A) and log-normal (B) distribution of the experimental data. The resulting values of the mean JNaCa-ratio were computed using Eqn 4. The total number of datasets was 50,000 in both cases. Statistical characteristics/parameters of individual histograms are specified in Table 2.

Table 2.

Characteristics of the distribution of mean JNaCa-ratio resulting from Monte Carlo simulations of experimental data

Characteristics of the distribution of mean JNaCa-ratio resulting from Monte Carlo simulations of experimental data
Characteristics of the distribution of mean JNaCa-ratio resulting from Monte Carlo simulations of experimental data

Analysis of 95% confidence intervals of the mean JNaCa-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, Pn). It is also worth noting that the confidence interval, as well as Pn, gradually increased with an increase of the mean JNaCa-ratio, suggesting that higher values of the mean JNaCa-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 JNaCa-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 JNaCa-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 JNaCa-ratio. Unfortunately, computation of the mean JNaCa-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 JNaCa-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.

Role of differences in experimental conditions in divergent estimates of mean JNaCa-ratio

An overview of experimental conditions described in the compared studies (Table 4) could help to reveal the causes of large differences in the mean JNaCa-ratio estimates (Fig. 1, Table 2). According to Eqn 4, the mean JNaCa-ratio depends on five variables: JNaCa,intact, JNaCa,detub, Cintact, Cdetub and ft,res. The differences in the mean JNaCa-ratio are reflected at each value of ft,res (Fig. 1), so the condition of complete detubulation (ft,res=0) can be introduced in further considerations. In addition, if we introduce new variables jid=JNaCa,intact/JNaCa,detub and cid=Cintact/Cdetub into Eqn 4, the mean JNaCa-ratio will depend on only two variables, jid and cid:
formula
(1)
Table 3.

Impact of mean current density assessment at different relative s.d. of I and C data on the accuracy of mean J

Impact of mean current density assessment at different relative s.d. of I and C data on the accuracy of mean J
Impact of mean current density assessment at different relative s.d. of I and C data on the accuracy of mean J
Table 4.

Measuring conditions and basic characteristics of rats used in the compared studies

Measuring conditions and basic characteristics of rats used in the compared studies
Measuring conditions and basic characteristics of rats used in the compared studies

The mean JNaCa-ratio is a decreasing function of cid and an increasing function of jid 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 ft,res=0): JNaCa-ratio=[18, 5.2, 3.8, 1.7], cid=[1.1649, 1.3497, 1.4717, 1.3600] and jid=[3.3908, 2.0833, 1.9000, 1.1905], respectively. The particularly high value of the mean JNaCa-ratio in the work of Gadeberg et al. (2016) results from a combination of the smallest cid value and the largest jid value. The dependence on cid is less pronounced than the dependence on jid, and in contrast to cid, the values of jid increase in parallel with the values of mean JNaCa-ratio sorted by size.

One of the factors that could affect the value of jid is age/weight or strain of the animals. Indeed, the highest jid and thus JNaCa-ratio corresponds to the oldest rats (25 weeks, Gadeberg et al., 2016). However, the estimated values of the mean JNaCa-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 JNaCa-ratio (1.7 versus 6 according to Eqn 4). This inconsistency in age-dependency of the experimental data suggests that the mean JNaCa-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, INaCa blockers, Ca2+ buffers, temperature and resistance of the pipette may also contribute to different values of assessed mean JNaCa-ratio. The closest conditions (except for different INaCa-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 JNaCa-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, JNaCa 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 JNaCa values, which may be the reason for a higher mean JNaCa-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 INaCa 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 INaCa and thus to such different values of the mean JNaCa-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.

Could scatter of experimental data explain differences in calculated values of mean JNaCa-ratio?

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 JNaCa-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 JNaCa-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 JNaCa-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 JNaCa after detubulation (Table 1) supports values higher than 1, it is very likely that the real values of the mean JNaCa-ratio in all four studies were between 1 and 4.3, and that values outside this interval arose from the scatter of experimental data.

Potential causes of large scatter of experimental data

The large scatter of experimental data used to assess JNaCa 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 INaCa and impaired INaCa-C proportionality, a prerequisite of a proper assessment of mean JNaCa (Kula et al., 2020b). A transmural gradient in ventricular JNaCa 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 ICa current properties (Volk and Ehmke, 2002). Thus, it is very likely that transmural differences in ventricular JNaCa 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 JNaCa among individual cells. As follows from Eqn 4, the high relative s.e. of JNaCa,intact and JNaCa,detub (∼ 30%) could then lead to values of mean JNaCa-ratio that are higher than 1000 (e.g. if JNaCa,intact=−0.32 pA pF−1 and JNaCa,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 JNaCa-ratio. For example, a change of only Cintact 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 JNaCa-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 JNaCa values.

Regarding ft,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 ft,res. Possible biases in the estimation of ft,res have been discussed by Pásek et al. (2008). A consequent inaccuracy in the assessed ft,res may also contribute to the divergent mean JNaCa-ratio among individual studies, as demonstrated in Fig. 1.

Which way of mean current density assessment is more accurate?

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).

Could a more accurate evaluation of measured data reduce the inconsistency of the mean JNaCa-ratios assessed from different studies?

As shown in Table 2, statistical evaluation of the mean JNaCa-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 Cintact and Cdetub, and non-real values of Jintact and Jdetub, 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 JNaCa-ratio.

We therefore tried to find out whether the mean JNaCa-ratios from the compared studies could be reconciled and whether a smaller CI95 could be achieved if the mean JNaCa was computed from the ratio of average INaCa to average C. This way of mean JNaCa assessment would be more accurate if the original INaCa 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 JNaCa, as reported by Zygmunt et al. (2000) and Xiong et al. (2005). The values needed to compute the mean JNaCa 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 JNaCa-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 JNaCa-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 Cintact and Cdetub measured by Bryant et al. (2015) on the same population of cells (Cintact=260±9 pF, n=37, and Cdetub=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 JNaCa-ratio between 0.3 and 4.3, the reduction of JNaCa 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).

Assessment of mean JNaCa-ratio from experimental data

The assessment of mean JNaCa-ratio from experimentally evaluated values of JNaCa and C before and after detubulation was conducted using the following set of relations:
formula
(2)
where JNaCa,intact, JNaCa,detub, Cintact, Cdetub (measured quantities) are the densities of the Na+-Ca2+ exchange current (pA pF−1) and the membrane capacitances (pF) in intact and detubulated cells, and JNaCa,s, JNaCa,t, Cs, Ct (unknown quantities) stand for the densities of the Na+-Ca2+ exchange current and membrane capacitances at the surface and t-tubular membranes. The ft,res representing fraction of t-tubules resisting detubulation can be estimated from analysis of confocal images of intact and detubulated myocytes membrane-stained with di-8-ANEPPS (Pásek et al., 2008; Bryant et al., 2015).
The solution of the set from Eqn 2 for the unknown quantities Ct, Cs, JNaCa,t, and JNaCa,s yields:
formula
(3)
Hence, the mean JNaCa-ratio can be expressed as:
formula
(4)

Monte Carlo simulation of experimental data

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 (Cintact, Cdetub,JNaCa,intact, JNaCa,detub, and ft,res), and their averages were computed and stored as a dataset. For each dataset, the value of the mean JNaCa-ratio was computed using Eqn 4. This was repeated 50,000 times. The distribution of the mean JNaCa-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).

Author contributions

Conceptualization: M.P.; Methodology: M.P., J.S., G.C.; Software: M.P., G.C.; Formal analysis: M.P., J.S., G.C.; Writing - original draft: M.P.; Writing - review & editing: J.S., M.B., G.C.; Project administration: M.B.; Funding acquisition: M.B.

Funding

This study was supported by the Akademie věd České Republiky (RVO: 61388998) and the Ministerstvo Zdravotnictví Ceské Republiky (16-30571A). Georges Christé's work was supported by Université Claude Bernard Lyon 1 statutory research allowances to the EA4612 Neurocardiology Unit.

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

Bébarová
,
M.
,
Pásek
,
M.
and
Zahradník
,
I.
(
2020
).
Toward more accurate data in cardiac cellular electrophysiology
.
Prog. Biophys. Mol. Biol.
157
,
1
-
2
.
Brette
,
F.
and
Orchard
,
C.
(
2007
).
Resurgence of cardiac t-tubule research
.
Physiology
22
,
167
-
173
.
Bryant
,
S. M.
,
Kong
,
C. H. T.
,
Watson
,
J.
,
Cannell
,
M. B.
,
James
,
A. F.
and
Orchard
,
C. H.
(
2015
).
Altered distribution of ICa impairs Ca release at the t-tubules of ventricular myocytes from failing hearts
.
J. Mol. Cell. Cardiol.
86
,
23
-
31
.
Campbell
,
S. E.
and
Gerdes
,
A. M.
(
1987
).
Regional differences in cardiac myocyte dimensions and number in Sprague-Dawley rats from different suppliers
.
Proc. Soc. Exp. Biol. Med.
186
,
211
-
217
.
Christé
,
G.
,
Bonvallet
,
R.
and
Chouabe
,
C.
(
2020
).
Accounting for cardiac t-tubule increase with age and myocyte volume to improve measurements of its membrane area and ionic current densities
.
Prog. Biophys. Mol. Biol.
157
,
40
-
53
.
Despa
,
S.
,
Brette
,
F.
,
Orchard
,
C. H.
and
Bers
,
D. M.
(
2003
).
Na/Ca exchange and Na/K-ATPase function are equally concentrated in transverse tubules of rat ventricular myocytes
.
Biophys. J.
85
,
3388
-
3396
.
Farrance
,
I.
and
Frenkel
,
R.
(
2012
).
Uncertainty of Measurement: A Review of the Rules for Calculating Uncertainty Components through Functional Relationships
.
Clin. Biochem. Rev.
33
,
49
-
75
.
Gadeberg
,
H. C.
,
Bryant
,
S. M.
,
James
,
A. F.
and
Orchard
,
C. H.
(
2016
).
Altered Na/Ca exchange distribution in ventricular myocytes from failing hearts
.
Am. J. Physiol. Heart Circ. Physiol.
310
,
H262
-
H268
.
Jayasinghe
,
I. D.
,
Cannell
,
M. B.
and
Soeller
,
C.
(
2009
).
Organization of ryanodine receptors, transverse tubules, and sodium-calcium exchanger in rat myocytes
.
Biophys. J.
97
,
2664
-
2673
.
Kieval
,
R. S.
,
Bloch
,
R. J.
,
Lindenmayer
,
G. E.
,
Ambesi
,
A.
and
Lederer
,
W. J.
(
1992
).
Immunofluorescence localization of the Na-Ca exchanger in heart cells
.
Am. J. Physiol.
263
,
C545
-
C550
.
Kula
,
R.
,
Bébarová
,
M.
,
Matejovič
,
P.
,
Šimurda
,
J.
and
Pásek
,
M.
(
2020a
).
Distribution of data in cellular electrophysiology: Is it always normal?
Prog. Biophys. Mol. Biol.
157
,
11
-
17
.
Kula
,
R.
,
Bébarová
,
M.
,
Matejovič
,
P.
,
Šimurda
,
J.
and
Pásek
,
M.
(
2020b
).
Current density as routine parameter for description of ionic membrane current: is it always the best option?
Prog. Biophys. Mol. Biol.
157
,
24
-
32
.
Limpert
,
E.
and
Stahel
,
W. A.
(
2011
).
Problems with using the normal distribution – and ways to improve quality and efficiency of data analysis
.
PLoS ONE
6
,
e21403
.
Nakamura
,
S.
,
Asai
,
J.
and
Hama
,
K.
(
1986
).
The transverse tubular system of rat myocardium: its morphology and morphometry in the developing and adult animal
.
Anat. Embryol.
173
,
307
-
315
.
Orchard
,
C. H.
,
Pásek
,
M.
and
Brette
,
F.
(
2009
).
The role of mammalian cardiac t-tubules in excitation-contraction coupling: experimental and computational approaches
.
Exp. Physiol.
94
,
509
-
519
.
Pásek
,
M.
,
Brette
,
F.
,
Nelson
,
A.
,
Pearce
,
C.
,
Qaiser
,
A.
,
Christé
,
G.
and
Orchard
,
C. H.
(
2008
).
Quantification of t-tubule area and protein distribution in rat cardiac ventricular myocytes
.
Prog. Biophys. Mol. Biol.
96
,
244
-
257
.
Pásek
,
M.
,
Šimurda
,
J.
and
Orchard
,
C. H.
(
2012
).
Role of t-tubules in the control of trans-sarcolemmal ion flux and intracellular Ca2+ in a model of the rat cardiac ventricular myocyte
.
Eur. Biophys. J.
41
,
491
-
503
.
Pásek
,
M.
,
Šimurda
,
J.
and
Christé
,
G.
(
2017
).
Different densities of Na-Ca exchange current in T-tubular and surface membranes and their impact on cellular activity in a model of rat ventricular cardiomyocyte
.
Biomed. Res. Int.
2017
,
6343821
.
Thomas
,
M. J.
,
Sjaastad
,
I.
,
Andersen
,
K.
,
Helm
,
P. J.
,
Wasserstrom
,
J. A.
,
Sejersted
,
O. M.
and
Ottersen
,
O. P.
(
2003
).
Localization and function of the Na+/Ca2+ -exchanger in normal and detubulated rat cardiomyocytes
.
J. Mol. Cell. Cardiol.
35
,
1325
-
1337
.
Volk
,
T.
and
Ehmke
,
H.
(
2002
).
Conservation of L-type Ca2+ current characteristics in endo- and epicardial myocytes from rat left ventricle with pressure-induced hypertrophy
.
Pflugers Arch.
443
,
399
-
404
.
Xiong
,
W.
,
Tian
,
Y.
,
Disilvestre
,
D.
and
Tomaselli
,
G. F.
(
2005
).
Transmural heterogeneity of Na+-Ca2+ exchange: evidence for differential expression in normal and failing hearts
.
Circ. Res.
97
,
207
-
209
.
Yang
,
Z.
,
Pascarel
,
C.
,
Steele
,
D. S.
,
Komukai
,
K.
,
Brette
,
F.
and
Orchard
,
C. H.
(
2002
).
Na+-Ca2+ exchange activity is localized in the T-tubules of rat ventricular myocytes
.
Circ. Res.
91
,
315
-
322
.
Zygmunt
,
A. C.
,
Goodrow
,
R. J.
and
Antzelevitch
,
C.
(
2000
).
INaCa contributes to electrical heterogeneity within the canine ventricle
.
Am. J. Physiol. Heart Circ. Physiol.
278
,
H1671
-
H1678
.

Competing interests

The authors declare no competing or financial interests.