Intracellular Ca²⁺ Transient Phase II Can be Assessed by Half-Logistic Function Model in Isolated Aequorin-Injected Mouse Left Ventricular Papillary Muscle

Abstract

Background

Myocardial contraction and relaxation are regulated by increases and decreases in intracellular cytoplasmic calcium (Ca²⁺) concentration ([Ca²⁺]_i). In previous studies, we found that a half-logistic (h-L) function, which represents a half-curve of a symmetrical sigmoid logistic function with a boundary at the inflection point, curve-fits the first half of the ascending phase (CaTI) and the second half of the descending phase of the [Ca²⁺]_i transient curve (CaTIV) better than a mono-exponential (m-E) function. In the present study, we investigated the potential application of an h-L function to the analysis of the second half of the ascending phase of the [Ca²⁺]_i transient curve (CaTII).

Methods

The [Ca²⁺]_i transient was measured using the Ca²⁺-sensitive photoprotein aequorin, which was microinjected into 15 isolated left ventricular (LV) papillary muscles of mice. The observed CaTII data during the time duration from the point corresponding to the maximum of the first-order time derivative of Ca²⁺ concentration (dCa/dt_max) to the point corresponding to the peak Ca²⁺ concentration was curve-fitted by the least-squares method using the h-L and m-E function equations.

Results

The mean correlation coefficient (r) values of the h-L and m-E curve-fits for CaTII were 0.9996 and 0.9984, respectively. The Z transformation of h-L r was larger than that of m-E r (p < 0.0001). H-L residual mean square (RMS) was smaller than m-E RMS (p < 0.001).

Conclusions

The h-L function tracks the magnitudes and time courses of CaTII more accurately than the m-E function in isolated aequorin-injected mouse LV papillary muscle. Compared with the m-E time constant, the h-L time constant of CaTII is a more reliable index for evaluating the time duration of the change in the increase in [Ca²⁺]_i during the combination of the middle part of the contraction process and the early part of the relaxation process. CaTII can be assessed by the h-L function model in cardiac muscles. The h-L approach may provide a more useful model for studying each process in myocardial Ca²⁺ handling.

INTRODUCTION

Myocardial contraction and relaxation are regulated by increases and decreases in intracellular cytoplasmic calcium (Ca²⁺) concentration ([Ca²⁺]_i) in myocardial Ca²⁺ handling and excitation-contraction (E-C) coupling in cardiac muscle cells.1 The waveforms of the [Ca²⁺]_i transient provide valuable information for evaluating the change in [Ca²⁺]_i.

Curve-fitting and non-linear regression are valuable tools for elucidating the mechanism, summarizing information, eliminating noise, allowing speculation regarding unmeasured data, and separating the effects of multiple factors. To maximize the amount of useful information extracted from the [Ca²⁺]_i transient curve, we previously curve-fitted the first half of the ascending phase of the [Ca²⁺]_i transient curve (CaTI)2 and the second half of the descending phase of the [Ca²⁺]_i transient curve (CaTIV)3 with a half-logistic (h-L) function, which is a half-curve of a symmetrical sigmoid logistic function with a boundary at the inflection point, better than a mono-exponential (m-E) function. We found that the h-L time constant of the CaTI (Caτ_1L) represents the time duration of the increase in Ca²⁺ concentration from sarcoplasmic reticulum (SR) by Ca²⁺-induced Ca²⁺ release (CICR) during the early part of the contraction process.2 Furthermore, we found that the h-L time constant of CaTIV (Caτ_4L) represents the time duration of the decrease in Ca²⁺ concentration resulting from Ca²⁺ sequestration into SR and Ca²⁺ removal from the cytoplasm to the extracellular space through the Na⁺/Ca²⁺ exchanger during the late part of the relaxation process.3 In addition, the second half of the ascending phase of the [Ca²⁺]_i transient curve (CaTII) probably affects both the middle part of the contraction process and the early part of the relaxation process. We speculated that CaTII can also be represented by the h-L function.

In the present study, we investigated the potential application of the h-L function to the analysis of CaTII in cardiac muscle and sought to determine whether the h-L function curve-fits it more accurately than the m-E function. Improved curve-fit by using the h-L function would provide a reliable time constant and increase our understanding of each process in myocardial Ca²⁺ handling and E-C coupling.

MATERIALS AND METHODS

This study protocol was approved by the Animal Investigation Committee of the Jikei University School of Medicine. All procedures during the experiments were conducted in the Jikei University School of Medicine in conformance with the “Guiding Principles for the Care and Use of Animals in the Field of Physiological Sciences” endorsed by the American Physiological Society and the Physiological Society of Japan.

Surgical preparation

The experimental procedures of our study have been previously described in detail.2-5 In short, 15 mice (C57BL/6; body weight 25-30 g) were anesthetized with intravenous (1.5-2.0 mg/kg) and intraperitoneal (15-20 mg/kg) pentobarbital sodium, respectively. Following removal of the heart, the aorta was cannulated with a blunted 18 G needle, and the heart was mounted on a Langendorff apparatus. The coronary blood was washed out with Tyrode’s solution containing 2 mM Ca²⁺ buffered by N-2-hydroxyethyl-piperazine-N-2-ethanesulfonic acid (HEPES) at a constant pressure for 5 min. The solution was changed to HEPES-buffered Tyrode’s solution containing 2 mM Ca²⁺ and 20 mM 2, 3-butanedione monoxime after the heart beat was stabilized. Once the contraction stopped completely, the heart was removed from the Langendorff apparatus. A thin papillary muscle was dissected out from the mouse left ventricular (LV) walls.

After both ends of the isolated muscle had been tied with silk threads, the muscle was mounted horizontally in an experimental chamber, immersed in a bath and continuously perfused with Tyrode’s solution. One end of the muscle was attached to a fixed hook and the other was attached to the arm of a tension transducer (BG-10; Kulite Semiconductor Products, Leonia, NJ, USA; compliance 2.5 μm/g, unloaded resonant frequency 0.6 kHz). A pair of platinum black electrodes was placed parallel to the muscle, which was regularly stimulated by a single square pulse of 5 ms duration and 0.2 Hz; the strength of the stimulation was 1.5-fold the threshold. The muscle was slowly stretched and adjusted to the length at which the developed tension reached maximum (L_max).

Aequorin injection

Aequorin was dissolved in 150 mM KCl and 5 mM HEPES, pH 7.0, at a final concentration of 50-100 μM. Using glass micropipettes with a resistance of 30-50 MΩ, we injected aequorin into 150-200 superficial cells of the preparation using nitrogen gas. Aequorin light signals were detected using a photomultiplier (EMI 9789A; Thorn EMI, Ruislip, UK) placed just above the muscle.6 In twitch response, the aequorin light signal was recorded through a 500-Hz low-pass filter. Sixty-four aequorin light signals were averaged to improve the signal-to-noise ratio.

The measurement of aequorin’s light signal was essentially similar to those noted in previous reports.7,8 Aequorin light signals were converted to [Ca²⁺]_i using an in vitro calibration curve.9 The constants used in the present experiment were as follows: n, 3.14; K_R, 4,025,000; K_TR, 114.6.10 Ca²⁺ signals were sampled at 1-ms intervals and digitized with an A/D converter. All Ca²⁺ data were stored on tape (NFR-3515W; Sony Magnescale, Tokyo, Japan) and a computer (PC-9801; NEC, Tokyo, Japan) for later analysis.

Tyrode’s solution

Tyrode’s solution buffered with HEPES was used during all experiments, including muscle dissection and aequorin injection. The composition of the solution (in mM) was as follows: NaCl, 136.9; KCl, 5.4; MgCl₂, 0.5; NaH₂PO₄, 0.33; HEPES, 5; glucose, 5. The pH was adjusted to 7.40 ± 0.05 with NaOH at 24 °C, and the solution was equilibrated with 100% O₂. The temperature of the solution was continuously monitored with a thermocouple and maintained at 30 ± 0.5 °C.

Ca²⁺ transients

Ca²⁺ signals were measured from the point before the twitch stimulation procedure was started. The Ca²⁺ signal gradually increased, reached a peak, then decreased and returned to the resting Ca²⁺ concentration before the following twitch stimulation - all within the 1,000-ms sampling window. Signals for CaTII, which extends from the point corresponding to the maximum of the first-order time derivative of Ca²⁺ concentration (dCa/dt_max) to the point corresponding to the peak Ca²⁺ concentration, as shown in Figure 1, were used for subsequent analyses. The dCa/dt was obtained by differentiating the sampled Ca²⁺ data after digital smoothening using an 11-point, non-weighted moving average of digitized Ca²⁺ data signals.

Figure 1.

[Ca²⁺]_i transient curve schema. The [Ca²⁺]_i transient curve is divided into four sequential phases. The first half of the ascending phase of the [Ca²⁺]_i transient curve (CaTI) is the curve from the point corresponding to the beginning of twitch stimulation to the point corresponding to the maximum of the first-order time derivative of Ca²⁺ concentration (dCa/dt_max). The second half of the ascending phase of the [Ca²⁺]_i transient curve (CaTII) is the curve from the point corresponding to dCa/dt_max to the point corresponding to the peak Ca²⁺concentration. The first half of the descending phase of the [Ca²⁺]_i transient curve (CaTIII) is the curve from the point corresponding to the peak Ca²⁺ concentration to the point corresponding to the minimum of the first-order time derivative of Ca²⁺ concentration (dCa/dt_min). The second half of the descending phase of the [Ca²⁺]_i transient curve (CaTIV) is the curve from the point corresponding to dCa/dt_min to the point corresponding to the resting Ca²⁺ concentration.

h-L function equation

The following h-L function was used to curve-fit CaTII data by the least-squares method as shown in Figure 2A.

Figure 2.

Half-logistic (h-L) and mono-exponential (m-E) distribution function curves for the second half of the ascending phase of the [Ca²⁺]_i transient curve (CaTII). (A) The h-L function curve described in Equation 1 (see text). Ca_2A, h-L amplitude constant of CaTII; Caτ_2L, h-L time constant of CaTII; Ca_2B, h-L non-zero asymptote of CaTII; t_aC, time at the point corresponding to the maximum of the first-order time derivative of Ca²⁺ concentration (dCa/dt_max). (B) The m-E function curve described in Equation 2. Ca₂₀, m-E amplitude constant of CaTII; Caτ_2E, m-E time constant of CaTII; Ca_2∞, m-E non-zero asymptote of CaTII.

Ca(t) = 2Ca_2A/{1 + exp[(t - t_aC)/Caτ_2L]} + Ca_2B (Eq. 1)

where t is the time from the beginning of twitch stimulation to the point corresponding to the Ca²⁺ concentration, Ca_2A is the h-L amplitude constant of CaTII, Caτ_2L is the h-L time constant of CaTII, Ca_2B is the h-L non-zero asymptote of CaTII, and t_aC is a constant which represents the time at the point corresponding to dCa/dt_max. It is noted that t_aC is determined before the h-L curve fitting. The h-L function curve given by Eq. 1 increases monotonically from Ca(t_aC) [= (Ca_2A + Ca_2B)] to Ca(∞) [= Ca_2B]. Caτ_2L value corresponds to the time duration for the curves to increase from Ca(t_aC) [= (Ca_2A + Ca_2B)] to Ca(t_aC + Caτ_2L) {= [2Ca_2A/(1 + e) + Ca_2B]~(0.54 Ca_2A + Ca_2B)}.

M-E function equation

An m-E function was also used to curve-fit CaTII data by the least-squares method as shown in Figure 2B.

Ca(t) = Ca₂₀exp[-(t - t_aC)/Caτ_2E] + Ca_2∞ (Eq. 2)

where t is the time from the beginning of twitch stimulation to the point corresponding to the Ca²⁺ concentration, Ca₂₀ is the m-E amplitude constant of CaTII, Caτ_2E is the m-E time constant of CaTII, Ca_2∞ is the m-E non-zero asymptote of CaTII, and t_aC is a constant which represents the time at the point corresponding to dCa/dt_max. It is noted that t_aC is determined before the m-E curve fitting. The m-E function curve given by Eq. 2 increases monotonically from Ca(t_aC) [= (Ca₂₀ + Ca_2∞)] to Ca(∞) [= Ca_2∞]. The Caτ_2E value corresponds to the time duration for the curve to increase from Ca(t_aC) [= (Ca₂₀ + Ca_2∞)] to Ca(t_aC + Caτ_2E) [= (Ca₂₀/e + Ca_2∞)~(0.37 Ca₂₀ + Ca_2∞)].

It should be noted that the h-L function (Eq. 1) has the same number (three) of variable parameters as the m-E function (Eq. 2) and that Ca_2A, Caτ_2L, and Ca_2B in Eq. 1 are conceptually similar to Ca₂₀, Caτ_2E, and Ca_2∞ in Eq. 2, respectively.

Statistical analysis

Goodness of fit was evaluated with the correlation coefficient (r) and residual mean square (RMS) for the h-L and m-E curve-fitting models. Fisher’s Z transformation (Z) of r 11 was calculated with the following equation: Z = 1/2[ln(1 + r) - ln(1 - r)]. Residual values were calculated as the observed CaTII data minus the best-fit h-L or m-E value at each sampling data point. RMS was calculated as the residual sum of squares (RSS) divided by the residual degrees of freedom, which indicates the number of data points analyzed minus the number of parameters in the function.12

Analyses were performed using Statcel (OMS, Saitama, Japan), StatView ver 5.0 (SAS Institute, Cary, NC, USA), and Deltagraph 5.4.5v J (Deltapoint, Monterey, CA, USA) software. Values in the text are expressed as the mean ± standard deviation (SD) unless otherwise noted. The Z of r and RMS values were compared between the goodness of h-L and m-E fits. The Student’s paired t test was used for comparison between h-L and m-E Z and Wilcoxon signed-rank test was used for comparison between h-L and m-E RMS. Simple linear regression analyses were performed between the amplitude constants and between the time constants for the different [Ca²⁺]_i transient phases. A p value of < 0.05 was considered to indicate statistical significance.

RESULTS

Measurements of Ca²⁺ transient

The muscle length and diameter of the 15 isolated aequorin-injected LV papillary muscle specimens of mice were 2.01 ± 0.43 and 0.63 ± 0.11 mm, respectively.

Table 1 summarizes the representative values from the entire magnitude and time course of [Ca²⁺]_i transient data. The time duration from the point corresponding to dCa/dt_max up to the point corresponding to the peak Ca²⁺ concentration, which corresponds to CaTII, was 16.5 ± 1.6 ms.

Table 1. Observed values of the [Ca²⁺]_i transient curve.

Observed value
Ca2+ concentration at dCa/dtmax (nmol/l)	819 ± 189
Peak Ca2+ concentration (nmol/l)	1,694 ± 384
Ca2+ concentration at dCa/dtmin (nmol/l)	1,259 ± 440
Time to dCa/dtmax (ms)	60.1 ± 1.4
Time to peak Ca2+ concentration (ms)	76.1 ± 2.1
Time to dCa/dtmin (ms)	106.4 ± 14.9
dCa/dtmax (nmol/l/ms)	119.8 ± 28.7
dCa/dtmin (nmol/l/ms)	-20.9. ± 4.6

Data are presented as the mean ± standard deviation of the observed values of the [Ca²⁺]_i transient curves in the 15 isolated aequorin-injected left ventricular papillary muscles of mice. dCa/dt _max , maximum of the first-order time derivative of Ca²⁺ concentration; dCa/dt _min , minimum of the first-order time derivative of Ca²⁺ concentration.

H-L and m-E curve-fits

The representative best-fitted h-L and m-E function curves for the observed CaTII data in the muscle of a mouse are shown in Figure 3A and B, respectively. The residual Ca²⁺ concentrations calculated from the observed CaTII data minus the best-fitted h-L and m-E curves are shown in Figure 3C and D, respectively.

Figure 3.

Representative best-fitted half-logistic (h-L) and mono-exponential (m-E) function curves for the observed second half of the ascending phase of the [Ca²⁺]_i transient curve (CaTII) data during the time duration from the point corresponding to the maximum of the first-order time derivative of Ca²⁺ concentration (dCa/dt_max) to the point corresponding to the peak Ca²⁺ concentration in the isolated aequorin-injected left ventricular papillary muscle of a mouse. (A) Representative best-fitted h-L function curve (thin dotted line) for the observed CaTII data (black dots). Ca_2A, h-L amplitude constant of CaTII; Caτ_2L, h-L time constant of CaTII; Ca_2B, h-L non-zero asymptote of CaTII; r, correlation coefficient; Z, Z transformation of r. (B) Representative best-fitted m-E function curve (thin dotted line) for the observed CaTII data (black dots). Ca₂₀, m-E amplitude constant of CaTII; Caτ_2E, m-E time constant of CaTII; Ca_2∞, m-E non-zero asymptote of CaTII. (C) Residual Ca²⁺ concentrations calculated from the observed CaTII data minus the best-fitted h-L curve. RMS, residual mean square. (D) Residual Ca²⁺ concentrations calculated from the observed CaTII data minus the best-fitted m-E curve.

The h-L and m-E function parameters for CaTII in the 15 specimens are shown in Table 2.

Table 2. Calculated half-logistic (h-L) and mono-exponential (m-E) function parameters for the second half of the ascending phase of the [Ca²⁺]i transient curve (CaTII).

h-L		m-E
Ca2A	0-914 ± 249 nmol/l	Ca20	0-997 ± 274 nmol/l
Caτ 2L	03.75 ± 0.46 ms	Caτ 2E	06.06 ± 0.85 ms
Ca2B	1,730 ± 395 nmol/l	Ca2∞	1,790 ± 410 nmol/l

Data are presented as the mean ± standard deviation of the calculated values of the h-L and m-E function parameters for CaTII in the 15 isolated aequorin-injected left ventricular papillary muscles of mice. Ca_2A, h-L amplitude constant of CaTII; Caτ _2L, h-L time constant of CaTII; Ca_2B, h-L non-zero asymptote of CaTII; Ca₂₀, m-E amplitude constant of CaTII; Caτ _2E, m-E time constant of CaTII; Ca_2∞, m-E non-zero asymptote of CaTII.

Goodness of h-L and m-E fits

The mean r value of the h-L and m-E curve-fitted for CaTII in the 15 specimens was 0.9996 and 0.9984, respectively. The Z transformation value of the h-L and m-E r was 4.31 ± 0.93 and 3.57 ± 0.35, respectively. H-L Z was significantly larger than m-E Z (p < 0.0001).

The h-L and m-E RMS values were 106 ± 172 and 416 ± 433 (nmol/l)², respectively. H-L RMS was significantly smaller than m-E RMS (p < 0.001).

DISCUSSION

These results demonstrate that the h-L function curve-fits CaTII more accurately than the m-E function in isolated aequorin-injected LV papillary muscle of mice. Therefore, we suggest that the h-L function model can be used to more reliably characterize the magnitude and time course of CaTII in cardiac muscle.

Ca²⁺ handling

During a cardiac cycle, the contraction process causes Ca²⁺ inflow into the cytoplasm through voltage-dependent sarcolemmal L-type Ca²⁺ channels and release of Ca²⁺ from SR induced by Ca²⁺ influx through sarcolemmal L-type Ca²⁺ channels13,14 and the sensitivity of myofilaments to Ca²⁺. Most Ca²⁺ flows into the cytoplasm from SR by CICR, with only a small amount of Ca²⁺ flowing into the cytoplasm through L-type Ca²⁺ channels. During each heart beat, a transient increase in the myocardial [Ca²⁺]_i induces a transient increase in the binding of Ca²⁺ to a major Ca²⁺-binding protein, troponin C (TnC), and a transient increase in attachment of the thick filament myosin cross-bridge to thin filament actin, which precedes and initiates myocardial contraction and increase in LV pressure. Because myocardial contraction lags behind the change in [Ca²⁺]_i, the magnitude of the contraction is determined not only by the magnitude of [Ca²⁺]_i but also by its time duration. The relaxation process involves Ca²⁺ dissociation from TnC, Ca²⁺ sequestration into SR,15 and Ca²⁺ removal from the cytoplasm to the extracellular space through the Na⁺/Ca²⁺ exchanger.16

Waveform analysis

The [Ca²⁺]_i transient curve is expected to change in tandem with the magnitudes and time courses of the myocardial tension and LV pressure curves. We have reported that the h-L function outperforms m-E function in curve-fitting the following: (i) the second half of the ascending phase of the isometric myocardial tension curve during the time duration from the point corresponding to the maximum of the first-order time derivative of tension (dF/dt_max) to the point corresponding to the peak tension in the myocardial muscle4 and (ii) the second half of the ascending phase of the isovolumic LV pressure curve during the time duration from the point corresponding to the maximum of the first-order time derivative of LV pressure (dP/dt_max) to the point corresponding to the peak LV pressure in the cross-circulated dog heart.17

The calculated function parameters were used to summarize the waveform of each [Ca²⁺]_i transient phase with almost no loss of information on the characteristics. The m-E function has been used for curve-fitting CaTI shown in Figure 1, in the papillary muscles of mice2 and rabbits,2 and CaTIV shown in Figure 1, in the isovolumic LV in excised whole hearts of rats,18-20 and in myocytes of mice,3,21,22 rabbits,3,22,23 rats,23-25 dogs22 and humans.22 For example, there are significant differences in the m-E time constant of CaTIV (Caτ_4E), reflecting activities of the SR Ca²⁺-ATPase, with values for myocytes of humans > dogs > rabbits > mice.22 In previous and present studies, we found that reliable CaTI,2 CaTII, and CaTIV3 can be predicted from the best-fitted h-L function curves.

Magnitude of [Ca²⁺]_i transient phase

The calculated Ca_2A and Ca_2B have physiological and mathematical meanings. The absolute value of Ca_2A represents the difference between Ca²⁺ concentration at the point corresponding to the maximum rate of increase in the cytoplasmic Ca²⁺ concentration, i.e., dCa/dt_max, and the peak Ca²⁺ concentration, as shown in Figure 2A. Ca_2B represents Ca²⁺ concentration at the point corresponding to the peak Ca²⁺ concentration.

There were significant relationships between the h-L amplitude constant of CaTI (Ca_1A) and the absolute value of Ca_2A, as shown in Figure 4A, and between the m-E amplitude constant of CaTI (Ca₁₀) and the absolute value of Ca₂₀, as shown in Figure 4B (see APPENDIX). Similarly, there were significant relationships between the h-L amplitude constant of CaTIV (Ca_4A) and the absolute value of Ca_2A, as shown in Figure 4C, and between the m-E amplitude constant of CaTIV (Ca₄₀) and the absolute value of Ca₂₀, as shown in Figure 4D. These results show that the magnitude of CaTII relates to that of CaTI and CaTII. Therefore, the proportion of the magnitude of CaTII to CaTI and that of CaTII to CaTIV is constant.

Figure 4.

Relationship between the amplitudes best-fitted for the first half of the ascending phase of the [Ca²⁺]_i transient curve (CaTI) data and for the second half of the ascending phase of the [Ca²⁺]_i transient curve (CaTII) data as well as the relationship between the amplitudes best-fitted for the second half of the descending phase of the [Ca²⁺]_i transient curve (CaTIV) data and for the CaTII data in the 15 isolated aequorin-injected left ventricular (LV) papillary muscles of mice. (A) Points of intersection (open circles) between the half-logistic (h-L) amplitude constant of the CaTI data during the time duration from the point corresponding to the beginning of twitch stimulation to the point corresponding to the maximum of the first-order time derivative of Ca²⁺ concentration (dCa/dt_max) (Ca_1A) and the absolute values of the h-L amplitude constant of the CaTII data (Ca_2A). (B) Points of intersection (open circles) between the mono-exponential (m-E) amplitude constant of the CaTI data (Ca₁₀) and the absolute values of the m-E amplitude constant of the CaTII data (Ca₂₀). (C) Points of intersection (open circles) between the h-L amplitude constant of the CaTIV data during the time duration from the point corresponding to the minimum of the first-order time derivative of Ca²⁺ concentration (dCa/dt_min) to the point corresponding to the resting Ca²⁺ concentration (Ca_4A) and the absolute values of Ca_2A. (D) Points of intersection (open circles) between the m-E amplitude constant of the CaTIV data (Ca₄₀) and the absolute values of Ca₂₀. r, correlation coefficient.

Time constant of [Ca²⁺]_i transient phase

The time constant for each [Ca²⁺]_i transient phase is used for comparison of the time duration. The h-L time constant of CaTI (Caτ_1L) represents the time duration from the point where Ca = 2Ca_1A/(1+e) + Ca_1B to the point where Ca = Ca_1A + Ca_1B2 (see APPENDIX). The h-L time constant of CaTIV (Caτ_4L) represents the time duration from the point where Ca = Ca_4A + Ca_4B to the point where Ca = 2Ca_4A/(1+e) + Ca_4B.3 The m-E time constant of CaTI (Caτ_1E) represents the time duration from the point where Ca = Ca₁₀/e + Ca_1∞ to the point where Ca = Ca₁₀ + Ca_1∞2 and Caτ_4E represents the time duration from the point where Ca = Ca₄₀ + Ca_4∞ to the point where Ca = Ca₄₀/e + Ca_4∞.3 From statistical analyses, Caτ_1L and Caτ_4L are more reliable indices of the time durations of the changes in the increase and decrease in [Ca²⁺]_i than Caτ_1E and Caτ_4E, respectively.2,3 In the present study, compared with Caτ_2E, Caτ_2L is also a more reliable index of the time duration of the change in [Ca²⁺]_l during CaTII.

There were no significant relationships between Caτ_1L and Caτ_2L, as shown in Figure 5A, and between Caτ_1E and Caτ_2E, as shown in Figure 5B. Caτ_2L and Caτ_2E were significantly larger than Caτ_1L and Caτ_1E, respectively, tested using Student’s paired t test (p < 0.0001). Similarly, there were no significant relationships between Caτ_4L and Caτ_2L, as shown in Figure 5C, and between Caτ_4E and Caτ_2E, as shown in Figure 5D. Caτ_2L and Caτ_2E were significantly smaller than Caτ_4L and Caτ_4E (p < 0.0001), respectively. Caτ_2L is independent of Caτ_1L and Caτ_4L. Taken together, these data indicate that the time change in [Ca²⁺]_i of CaTII is different from that of CaTI and CaIV. Therefore, these results indicate that each [Ca²⁺]_i transient phase has its own time constant, and there may be individual differences in Ca²⁺ handling such as Ca²⁺ flow into the cytoplasm from SR by CICR during the contraction process, Ca²⁺ sequestration into SR, and Ca²⁺ removal from the cytoplasm to the extracellular space through the Na⁺/Ca²⁺ exchanger during the relaxation process in isolated papillary muscle.

Figure 5.

Relationship between the time constants best-fitted for the first half of the ascending phase of the [Ca²⁺]_i transient curve (CaTI) data and for the second half of the ascending phase of the [Ca²⁺]_i transient curve (CaTII) data as well as the relationship between the time constants best-fitted for the second half of the descending phase of the [Ca²⁺]_i transient curve (CaTIV) data and for the CaTII data in the 15 isolated aequorin-injected left ventricular (LV) papillary muscles of mice. (A) Points of intersection (open circles) between the half-logistic (h-L) time constant of the CaTI data during the time duration from the point corresponding to the beginning of twitch stimulation to the point corresponding to the maximum of the first-order time derivative of Ca²⁺ concentration (dCa/dt_max) (Caτ_1L) and the h-L time constant of the CaTII data (Caτ_2L). (B) Points of intersection (open circles) between the mono-exponential (m-E) time constant of the CaTI data (Caτ_1E) and the m-E time constant of the CaTII data (Caτ_2E). (C) Points of intersection (open circles) between the h-L time constant of the CaTIV data during the time duration from the point corresponding to the minimum of the first-order time derivative of Ca²⁺ concentration (dCa/dt_min) to the point corresponding to the resting Ca²⁺ concentration (Caτ_4L) and Caτ_2L. (D) Points of intersection (open circles) between the m-E time constant of the CaTIV data (Caτ_4E) and Caτ_2E. r, correlation coefficient.

CaTI encompasses the beginning phase of the increase in Ca²⁺ concentration from SR by CICR in myocardial Ca²⁺ handling, where Caτ_1L evaluates the time duration of the change in the increase in [Ca²⁺]_i during the early part of the contraction process.2 On the other hand, CaTIV refers to the ending phase of the decrease in Ca²⁺ concentration by Ca²⁺ sequestration into SR and Ca²⁺ removal from the cytoplasm to the extracellular space through the Na⁺/Ca²⁺ exchanger in myocardial Ca²⁺ handling. Caτ_4L evaluates the time duration of the change in the decrease in [Ca²⁺]_i during the last part of the relaxation process.3 CaTII corresponds to the combination of the holding phase of the increase in Ca²⁺ concentration from SR by CICR and the beginning phase of the decrease in Ca²⁺ concentration by Ca²⁺ sequestration into SR and Ca²⁺ removal from the cytoplasm to the extracellular space through the Na⁺/Ca²⁺ exchanger.5 Furthermore, CaTII includes both the middle part of the contraction process and the early part of the relaxation process and represents the change in the difference between the increase and decrease in Ca²⁺ concentration, although the degree of increase in Ca²⁺ concentration is larger than that of decrease in Ca²⁺ concentration. Accordingly, Caτ_2L evaluates the time duration of the change in the increase in [Ca²⁺]_i during the combination of the middle part of the contraction process and the early part of the relaxation process. As can be seen in Figure 5C and D, there was a tendency of a slight negative correlation between the time constants of CaTII and CaTIV. We suggest that the change in [Ca²⁺]_i of CaTII with the contraction and relaxation processes relates to the change in [Ca²⁺]_i of CaTIV with the simple relaxation process. It may be possible to obtain significant insight into each process in myocardial Ca²⁺ handling and E-C coupling by using Caτ_1L, Caτ_2L, and Caτ_4L.

The calculated h-L function parameters can enable the comparison between the magnitudes and time constants of the observed CaTII. Curve-fitting for the observed CaTI, CaTII, and CaTIV by the h-L function would be useful for studying the effect of [Ca²⁺]_i during each process in the myocardial Ca²⁺ handling.

Sigmoid logistic function model

The logistic function has been widely used to study rising phenomena in many fields of bioscience and to intuitively describe symmetrical sigmoid curves whose inflection point corresponds to a boundary, such as those found in mortality data26 and growth curves.27,28 The logistic nature of these phenomena predicts that the process is initially minimal, increases gradually, and maximizes to an upper asymptote: conversely, is initially maximal, decreases gradually, and minimizes to a lower asymptote. The logistic function is limited in its ability to represent the symmetrical sigmoid function, especially as it reaches the upper or lower asymptote. In a previous study, we reported that the sigmoid logistic function satisfactorily curve-fits well the entire ascending phase during the time duration from the point corresponding to the beginning of twitch stimulation to the point corresponding to the peak Ca²⁺ concentration and the entire descending phase during the time duration from the point corresponding to the peak Ca²⁺ concentration to the point corresponding to the resting Ca²⁺ concentration.5 Computer simulation of the [Ca²⁺]_i transient also provided a characterization of the sigmoid logistic function.29

Curve-fitting for the entire ascending phase of the [Ca²⁺]_i transient data by the sigmoid logistic function would be useful for evaluating dCa/dt_max. In the present study, however, we found that the entire ascending phase of the [Ca²⁺]_i transient curve is asymmetrical at the point corresponding to dCa/dt_max. SDs of Caτ_2L and Caτ_2E were larger than those of Caτ_1L and Caτ_1E. The asymptote of CaTII, i.e., dCa/dt = 0, is only one point, and CaTI has a longer asymptote.

Study limitations

There are several limitations of this study. First, the experimental condition used to measure the [Ca²⁺]_i transient was not physiological, i.e., 100% L_max with 2 mM extracellular Ca²⁺ at 30 °C and a stimulation frequency of 0.2 Hz. Further examination of these concepts is needed under different physiological conditions of preloads, Ca²⁺ concentrations, temperatures, stimulation frequencies, and pharmacological conditions with varying degrees of blocking or stimulating agents of channels, pumps, exchangers, α- and β-adrenergic receptors, and anesthetic agents. Moreover, it will be beneficial if the future studies include a pathological model, such as dilated cardiomyopathy, hypertrophic cardiomyopathy, and ischemic heart failure to demonstrate correlations of h-L function parameters with changes in Ca²⁺ regulation proteins. Second, we used the Ca²⁺-sensitive photoprotein aequorin, which is a good indicator of the change in [Ca²⁺]_i. Additional investigations with other Ca²⁺-sensitive fluorescent indicators, such as indo-1, rhod-2, fura-2, and fluo-3, would be useful. Third, we curve-fitted CaTII using the h-L function model with three variable parameters, i.e., amplitude constant, time constant, and non-zero asymptote. We intend to develop the h-L function with an additional variable parameter or a new function to curve-fit it more accurately.

The first half of the descending phase of the [Ca²⁺]_i transient curve (CaTIII) shown in Figure 1 corresponds to the combination of the ending phase of the increase in Ca²⁺ concentration from SR by CICR and the holding phase of the decrease in Ca²⁺ concentration by Ca²⁺ sequestration into SR and Ca²⁺ removal from the cytoplasm to the extracellular space through the Na⁺/Ca²⁺ exchanger.5 Moreover, CaTIII also encompasses both the last part of the contraction process and the middle part of the relaxation process and represents the change in the difference between the increase and decrease in [Ca²⁺]_i, although the degree of decrease in [Ca²⁺]_i is larger than that of increase in [Ca²⁺]_i. The h-L time constant of CaTIII (Caτ_3L) evaluates the time duration of the change in the decrease in [Ca²⁺]_i during the combination of the last part of the contraction process and the middle part of the relaxation process. However, CaTIII may not be curve-fitted by h-L and m-E functions. Because CaTIII may be straight rather than convex, the inflexion point of the entire descending phase could not be identified. Further examination is needed to determine the location of the inflexion point of the entire descending phase.

CONCLUSIONS

The h-L functions can describe the magnitudes and time courses of not only CaTI and CaTIV but also CaTII more accurately than the m-E functions in isolated aequorin-injected mouse LV papillary muscle. Compared with Caτ_2E, Caτ_2L is a more reliable index for evaluating the time duration of the change in the increase in [Ca²⁺]_i during the combination of the middle part of the contraction process and the early part of the relaxation process. CaTII can be assessed by the h-L function model in cardiac muscles. The time constant of CaTII is independent of that of CaTI and CaTIV, although the magnitude of CaTII relates to that of CaTI and CaTIV. The h-L approach may provide a more useful model for studying individual processes in myocardial Ca²⁺ handling.

APPENDIX

The following h-L function is used to curve-fit CaTI data by the least-squares method:

Ca(t) = 2Ca_1A/{1 + exp[-(t - t_aC)/Caτ_1L]} + Ca_1B (Eq. 3)

where t is the time from the point corresponding to the beginning of twitch stimulation to the point corresponding to dCa/dt_max, Ca_1A is the h-L amplitude constant of CaTI, Caτ_1L is the h-L time constant of CaTI, Ca_1B is the h-L non-zero asymptote of CaTI, and t_aC is a constant which represents the time at the point corresponding to dCa/dt_max.2 It is noted that t_aC is determined before the h-L curve fitting. The h-L function curve given by Eq. 3 increases monotonically from Ca(0) (= Ca_1B) to Ca(t_aC) [= (Ca_1A + Ca_1B)]. The Caτ_1L value corresponds to the time duration for the curves to increase from Ca(t_aC - Caτ_1L) {= [2Ca_1A/(1 + e) + Ca_1B]~(0.54 Ca_1A + Ca_1B)} to Ca(t_aC) [= (Ca_1A + Ca_1B)]. The Ca_1A, Caτ_1L, and Ca_1B values for the 15 isolated mouse LV papillary muscles were 869 ± 199 nmol/l, 2.22 ± 0.16 ms, and 1.50 ± 5.80 nmol/l, respectively.

The following m-E function is used to curve-fit CaTI data by the least-squares method:

Ca(t) = Ca₁₀exp[(t - t_aC)/Caτ_1E] + Ca_1∞ (Eq. 4)

where t is the time from the point corresponding to twitch stimulation to the point corresponding to dCa/dt_max, Ca₁₀ is the m-E amplitude constant of CaTI, Caτ_1E is the m-E time constant of CaTI, Ca_1∞ is the m-E non-zero asymptote of CaTI, and t_aC is a constant which represents the time at the point corresponding to dCa/dt_max. It is noted that t_aC is determined before the m-E curve fitting. The m-E function curve given by Eq. 4 increases monotonically from Ca(0) (= Ca_1∞) to Ca(t_aC) [= (Ca₁₀ + Ca_1∞)]. The Caτ_1E value corresponds to the time duration for the curve to increase from Ca(t_aC - Caτ_1E) [= (Ca₁₀/e + Ca_1∞)~(0.37Ca₁₀ + Ca_1∞)] to Ca(t_aC) [= (Ca₁₀ + Ca_1∞)]. The Ca₁₀, Caτ_1E, and Ca_1∞ values for the 15 isolated mouse LV papillary muscles were 899 ± 206 nmol/l, 3.08 ± 0.23 ms, and -0.25 ± 5.93 nmol/l, respectively.

The relationships between Ca_1A with Eq. 3 and the absolute value of Ca_2A with Eq. 1, and between Ca₁₀ with Eq. 4 and the absolute value of Ca₂₀ with Eq. 2 are shown in Figure 4A and B, respectively.

The relationships between Caτ_1L with Eq. 3 and Caτ_2L with Eq. 1, and between Caτ_1E with Eq. 4 and Caτ_2E with Eq. 2 are shown in Figure 5A and B, respectively.

The following h-L function is used to curve-fit CaTIV data by the least-squares method:

Ca(t) = 2Ca_4A/{1 + exp[(t - t_dC)/Caτ_4L]} + Ca_4B (Eq. 5)

where t is the time from the point corresponding to twitch stimulation to the point corresponding to the resting Ca²⁺ concentration, Ca_4A is the h-L amplitude constant of CaTIV, Caτ_4L is the h-L time constant of CaTIV, Ca_4B is the h-L non-zero asymptote of CaTIV, and t_dC is a constant which represents the time at the point corresponding to dCa/dt_min.3 It is noted that t_dC is determined before the h-L curve fitting. The h-L function curve given by Eq. 5 decreases monotonically from Ca(t_dC) [= (Ca_4A + Ca_4B)] to Ca(∞) [= Ca_4B]. The Caτ_4L value corresponds to the time duration for the curves to decrease from Ca(t_dC) [= (Ca_4A + Ca_4B)] to Ca(t_dC + Caτ_4L) {= [2Ca_4A/(1 + e) + Ca_4B]~(0.54 Ca_4A + Ca_4B)}. The Ca_4A, Caτ_4L, and Ca_4B values for the 15 isolated mouse LV papillary muscles were 1,252 ± 438 nmol/l, 38.04 ± 4.83 ms, and -0.59 ± 10.83 nmol/l, respectively.

The following m-E function is used to curve-fit CaTIV data by the least-squares method:

Ca(t) = Ca₄₀exp[-(t - t_dC)/Caτ_4E] + Ca_4∞ (Eq. 6)

where t is the time from the point corresponding to twitch stimulation to the point corresponding to the resting Ca²⁺ concentration, Ca₄₀ is the m-E amplitude constant of CaTIV, Caτ_4E is the m-E time constant of CaTIV, Ca_4∞ is the m-E non-zero asymptote of CaTIV, and t_dC is a constant which represents the time at the point corresponding to dCa/dt_min. It is noted that tdC is determined before the m-E curve fitting. The m-E function curve given by Eq. 6 decreases monotonically from Ca(t_dC) [= (Ca₄₀ + Ca_4∞)] to Ca(∞) = Ca_4∞. The Caτ_4E value corresponds to the time duration for the curve to decrease from Ca(t_dC) [= (Ca₄₀ + Ca_4∞)] to Ca(t_dC + Caτ_4E) [= (Ca₄₀/e + Ca_4∞)~(0.37Ca₄₀ + Ca_4∞)]. The Ca₄₀, Caτ_4E, and Ca_4∞ values for the 15 isolated mouse LV papillary muscles were 1,347 ± 472 nmol/l, 52.2 ± 6.9 ms, and -12.06 ± 13.45 nmol/l, respectively.

The relationships between Ca_4A with Eq. 5 and the absolute value of Ca_2A with Eq. 1, and between Ca₄₀ with Eq. 6 and the absolute value of Ca₂₀ with Eq. 2 are shown in Figure 4C and D, respectively.

The relationships between Caτ_4L with Eq. 5 and Caτ_2L with Eq. 1, and between Caτ_4E with Eq. 6 and Caτ_2E with Eq. 2 are shown in Figure 5C and D, respectively.