Effect of stimulation rate, sarcomere length and Ca²⁺ on force generation by mouse cardiac muscle

Abstract

The relations between stress, stimulation rate and sarcomere length (SL) were investigated in 24 cardiac trabeculae isolated from right ventricles of mice (CF-1 males, 25-30 g) and superfused with Hepes solution ([Ca²⁺]_o = 1 mm, pH 7.4, 25 °C). Stress and SL were measured by a strain gauge transducer and laser diffraction technique, respectively. Stress versus stimulation frequency formed a biphasic relation (25 °C, [Ca²⁺]_o = 2 mm) with a minimum at 0.7-1 Hz (≈15 mN mm⁻²), a 150 % decrease from 0.1 to 1 Hz (descending limb) and a 75 % increase from 1 to 5 Hz (ascending limb). Ryanodine (0.1 μm) inhibited specifically the descending limb, while nifedipine (0.1 μm) affected specifically the ascending limb. This result suggests two separate sources of Ca²⁺ for stress development: (1) net Ca²⁺ influx during action potentials (AP); and (2) Ca²⁺ entry into the cytosol from the extracellular space during diastolic intervals; Ca²⁺ from both (1) and (2) is sequestered by the SR between beats. Raising the temperature to 37 °C lowered the stress-frequency relation (SFR) by ≈0-15 mN mm⁻² at each frequency. Because the amount of Ca²⁺ carried by I_Ca,L showed a ≈3-fold increase under the same conditions, we conclude that reduced Ca²⁺ loading of the SR was probably responsible for this temperature effect. A simple model of Ca²⁺ fluxes addressed the mechanisms underlying the SFR. Simulation of the effect of inorganic phosphates (P_i) on force production was incorporated into the model. The results suggested that O₂ diffusion limits force production at stimulation rates >3 Hz. The stress-SL relations from slack length (≈1.75 μm) to 2.25 μm showed that the passive stress-SL curve of mouse cardiac trabeculae is exponential with a steep increase at SL >2.1 μm. Active stress (at 1 Hz) increased with SL, following a curved relation with convexity toward the abscissa at [Ca²⁺] = 2 mm. At [Ca²⁺] from 4 to 12 mm, the stress-SL curves superimposed and the relation became linear, which revealed a saturation step in the activation of force production. EC coupling in mouse cardiac muscle is similar to that observed previously in the rat, although important differences exist in the Ca²⁺ dependence of force development. These results may suggest a lower capacity of the SR for buffering Ca²⁺, which makes the generation of force in mouse cardiac ventricle more dependent on Ca²⁺ entering during action potentials, particularly at high heart rate.

Genetically modified mice provide unique models for investigating the roles of specific proteins in cardiac excitation-contraction coupling and helping to understand molecular alterations that contribute to cardiac diseases (James et al. 1998). The mouse is, however, a relatively recent model in cardiac physiology and data published in the literature are sparse and often contradictory. For instance, in most mammalian animal models, ventricular myocardium exhibits a positive relationship between force development and stimulation rate (Braveny & Sumbera, 1970; Edman & Johannsson, 1976; Maylie, 1982; Pieske et al. 1999; Maier et al. 2000). The increase of force with pacing frequency has been correlated with a simultaneous rise in Ca²⁺ content of the SR (for review see Bers, 2000). In mouse and rat heart, however, this aspect is controversial, since some authors have reported a negative force- frequency relation (FFR) (Mitchell et al. 1985; Bouchard & Bose, 1989; Bers, 2000; Maier et al. 2000), whereas others confirmed a strongly positive FFR (Gao et al. 1998; Layland & Kentish, 1999; Kassiri et al. 2000) but with an uncoupling between force and the Ca²⁺ transient suggesting the presence of a frequency-dependent ‘sensitization’ of the myofilaments (Gao et al. 1998).

Length dependence of force generation constitutes another fundamental aspect of cardiac mechanics that has been poorly documented in mice. Although this relationship constitutes the essence of the Frank-Starling law, no measure of the isometric force versus sarcomere length relation has ever been published for this species.

In order to establish a clear database for mouse cardiac physiology, we examined these two fundamental relations that govern the mechanical functioning of the cardiac pump: (1) the force-frequency (FFR); and (2) the force- sarcomere length relations. We studied mouse cardiac muscle using methods and protocols developed previously for the rat with, however, slight modifications for the smaller size of mouse trabeculae. Under our experimental conditions, we found a clear biphasic FFR in mouse trabeculae that matched a simple structural model of Ca²⁺ fluxes. The active stress-sarcomere length relation was characterized at [Ca²⁺]_o from 1 to 12 mm with a stimulation frequency of 1 Hz.

Methods

Preparation of the trabeculae

Twenty-four cardiac trabeculae were dissected from the right ventricles of mouse hearts. Mice (≈120 adult CF-1 males, 25-35 g, Charles River) were anaesthetized deeply with ether and the hearts rapidly removed. The aorta was cannulated and the heart perfused with Hepes buffer solution. The trabeculae extended from the right ventricular free wall to the atrioventricular ring; the mean length, width and thickness were 1.04 ± 0.10, 0.18 ± 0.02 and 0.11 ± 0.01 mm, respectively (mean ± s.e.m.; n = 24); cross-sectional areas were calculated based on an elliptical approximation. Each trabecula was transferred to the experimental chamber and mounted horizontally between a force transducer and a servo-controlled motor arm (Van Heuningen et al. 1982; Stuyvers et al. 1997). The muscle was field stimulated through platinum electrodes positioned along the bath; 2 ms square electrical pulses ≈40 % above the threshold were delivered at a frequency of 1 Hz. Each muscle equilibrated for ≈1 h at 25 °C in the superfusion chamber, i.e. until twitch force and sarcomere shortening reached a steady state. The standard solution used for dissection, equilibration and experiments was a modified Hepes buffer solution (mm): 140 NaCl, 1.2 MgCl₂, 5 or 10 (dissection) KCl, 2.8 sodium acetate, 1 CaCl₂, 10 glucose, 10 Hepes and 10 taurine, equilibrated with O₂; pH 7.4 (adjusted with NaOH); in dissection, equilibration and standard experimental solutions, [Ca²⁺] was 2 mm.

Experimental procedures used in this study have been approved by The Animal Care Committee of the University of Calgary.

Force and sarcomere length measurements

The length of the muscle (ML) was set using a servo-controlled motor (model 318 B, Aurora Scientific Inc., Aurora, ON, Canada) hooked to the tricuspid valve attached to the trabecula. A silicon strain gauge (model X17625, SensoNor, Horten, Norway), extended with a 6 mm long carbon fibre rod, constituted the force transducer so that the resolution of the force measurement system was ≈0.5 μN (see Stuyvers et al. 1997). SL was determined by laser diffraction technique as described before (see, e.g. Van Heuningen et al. 1982; Kentish et al. 1986; Stuyvers et al. 1997); briefly, the beam of a He-Ne laser (λ, 633 nm) was projected vertically onto the muscle. SL was calculated from the position of the median of the resulting first-order of the diffraction pattern. The resolution of SL measurements was ≈4 nm (see Stuyvers et al. 1997). The small size of the mouse trabeculae (length ≤ 1.4 mm) with respect to the width of the laser beam (≈300-500 μm) required a condenser lens (focal distance, 8 cm) mounted together with additional neutral density filters in the path of the incident laser beam; this typically reduced the diameter of the beam to the width of the trabeculae (100-200 μm), which still generated a crisp first-order band with Gaussian intensity distribution, permitting the accurate determination of median SL at 2 kHz.

A digital oscilloscope (model 6000, Data Precision, Danvers, MA, USA) and a chart recorder (Gould Model 2800S, Cleveland, OH, USA) were used to monitor force (F) and SL; data were recorded on a Pentium III computer via a custom Labview program (version 6, National Instrument) interfaced with an I/O board 6040E (National Instrument).

Force, SL and Ca²⁺ relations: experimental protocols

We measured the force response to different stimulation frequencies applied in random order. One of our primary concerns was to limit the incidence of artefacts that could potentially interfere with force-frequency measurements. (1) To avoid history or rundown effects that may occur in a stepwise increase protocol, the different frequencies were tested in a random order. (2) At the end of experiments, two or three frequencies chosen randomly were tested again on the same muscle; the measure of force was compared with the one obtained previously at the same frequency during the initial force- frequency protocol; a difference of ≈5 % was tolerated between the two measures. (3) The small variability observed occasionally in the time taken for force development was overcome by measuring the reference force, i.e. the force at 1 Hz, before each individual frequency was tested; individual measures of force were then normalized to this reference. (4) It appeared from previous work that, compared with passive force, SL constitutes a more accurate estimate of the mechanical state of resting muscles (Stuyvers et al. 1997). In the present study, SL was, therefore, monitored continuously throughout the experiment and maintained between 2 and 2.1 μm. Before each test, we verified that resting SL did not vary more than 2 % so that the muscle contracted from the same basal conditions from test to test. (5) We chose muscles with a thickness < 0.2 mm to limit the metabolic deficiency that arises especially at high frequency in thicker preparations (Schouten & ter Keurs, 1986).

In the second part of the study, we assessed the force response to changes in SL. Different SLs were tested randomly between slack length and 2.3 μm. We measured the resting F, resting SL (SL₀) and the corresponding maximal F developed during the twitch. Stress was computed from the measured force and cross-sectional area (Stress = F/S, where S is the cross-sectional area in mm²). Active stress was calculated from total stress minus passive stress at the same SL. To facilitate comparisons, each individual value of stress was expressed as a percentage of its own reference measured at 1 Hz (Stress_{1 Hz}): [Stress/(Stress_{1 Hz})] × 100. Every time we changed the experimental conditions (stimulation frequency, muscle length, temperature, [Ca²⁺]_o), we started to record the data when amplitudes of twitch force and sarcomere shortening reached their new steady state.

Electrophysiological measurements

Ventricular myocytes were isolated enzymatically from CD-1 mouse hearts as described previously (Wang et al. 1999). Action potential (AP) and whole cell currents were recorded at 25 and 35 °C, using conventional ruptured patch recording techniques. Borosilicate electrodes filled with internal solution had resistances of 3-4 MΩ. AP was recorded under current clamp configuration with the internal solution containing (mm): 110 potassium aspartate, 10 KCl, 5 MgCl₂, 5 ATP-Na₂, 10 EGTA, 10 Hepes and 1 CaCl₂ (pH 7.2). A liquid junction potential of -10 mV was corrected on all the potentials measured. The internal solution, used for recording of L-type Ca²⁺ current (I_Ca,L), contained (mm): 100 CsCl, 20 tetraethylammonium chloride, 5 ATP-Mg, 0.5 GTP-Na₂, 10 Hepes, 10 EGTA and 1 CaCl₂ (pH adjusted to 7.2 with CsOH). The bath solution used for recording I_Ca,L was Hepes-Tyrode buffer containing an extra 5 mm CsCl to block K⁺ currents.

Action potential duration (APD) was measured at 0 mV because it was more directly related to I_Ca,L than traditional relative repolarization level. Conventional I_Ca,L was induced by a depolarization to 0 mV after a conditioning pulse to -40 mV for 100 ms from a holding potential of -80 mV. The I_Ca,L under AP clamp was recorded by using an AP as the command potential in the voltage clamp configuration. The AP clamp activated two distinct inward currents, one that was activated by the rapid depolarization phase of the AP and was sensitive to TTX (20 μm) and insensitive to nisoldipine (0.2 μm) and another that showed a peak at ≈0 mV, was TTX insensitive and nisoldipine sensitive and was, therefore, identified as I_Ca,L. Integration of this second inward current (over APD) was taken as a measure of the amount of Ca²⁺ entering the cell during the action potential. An axopatch 200B amplifier (Axon Instruments) was used. Data acquisition and analysis were carried out with the software pCLAMP 8 (Axon Instruments).

Statistics

Results are expressed as the means ± s.e.m., or means ± s.d. when n ≤ 8. We judged the significance of stress-frequency and stress-SL relations by ANOVA and the correlation coefficients of the regressions. We used paired and unpaired t tests for comparisons of two means; the difference was significant when P < 0.05.

Results

Relation between force and frequency

Figure 1 and Figure 2A show the relation between twitch and stimulation frequency in isolated trabeculae. Resting SL was controlled (SL_o = 2.05 ± 0.03 μm; n = 24) and passive tension measured before each individual test frequency so that measurements were carried out at constant resting conditions. Stimulation frequencies ranging between 0.106 ± 0.003 (n = 12) and 4.6 ± 0.15 Hz (n = 21) were tested. At frequencies above 5 Hz, we observed a partial fusion of twitches, while small and large twitches alternated (see example of Fig. 1A at 10 Hz); this was accompanied by a rise in resting force. This alternans phenomenon made the determination of force amplitude inaccurate at frequencies above 5 Hz. At frequencies lower than 0.1 Hz, spontaneous contractions appeared frequently between stimulated twitches (Fig. 1B: see arrows a); the occurrence of such extra-contractions decreased the amplitude of the subsequent stimulated twitch (arrows b in Fig. 1B). The extent of the decline depended on the number of spontaneous beats that occurred in the preceding interstimulus interval. This phenomenon would decrease the average force and therefore we did not use data collected at frequencies lower than 0.1 Hz, except in one experiment where the muscle responded to stimulation at 0.07 Hz for several minutes without showing spontaneous contractions. On average, over the range of frequencies tested, stress formed a biphasic relation with a minimum of ≈15 mN mm⁻² at 0.7-1 Hz and maxima of 25.4 ± 2.93 and 35.5 ± 6.2 mN mm⁻² at 4.6 and 0.1 Hz, respectively. At 25 °C and a [Ca²⁺] of 2 mm, developed stress increased on average by more than 150 % from 1 to 0.1 Hz and by 75 % from 1 to 5 Hz (Fig. 2A). These changes in stress took place while passive stress remained constant. The rate of relaxation and twitch duration were investigated in five representative trabeculae. At low frequencies, the rate of relaxation (V_relax) and twitch force duration did not change significantly (V_relax: 0.08 ± 0.03 mN mm⁻² ms⁻¹ at 0.2 Hz versus 0.07 ± 0.04 mN mm⁻² ms⁻¹ at 1 Hz; duration: 548 ± 92 ms at 0.2 Hz versus 480 ± 115 ms at 1 Hz). At frequencies above 1 Hz, however, V_relax increased 4-fold (0.28 ± 0.12 mN mm⁻² ms⁻¹ at 5 Hz versus 0.07 ± 0.04 mN mm⁻² ms⁻¹ at 1 Hz; n = 5, P < 0.05) and duration decreased 2-fold (247 ± 73 ms at 5 Hz versus 480 ± 115 ms at 1 Hz; n = 5, P < 0.05; see Fig. 2B).

Figure 1. Typical force and sarcomere length responses to different stimulation rates in mouse cardiac trabeculae.

A, raw F (bottom trace) and SL (upper trace) signals were recorded from a trabecula stimulated at 10, 5, 1, 0.25 and 0.125 Hz. Note the presence of alternans phenomenon at 10 Hz, which limited our investigations to frequencies <10 Hz when temperature was 25 °C. B shows traces obtained at a stimulation frequency of 0.07 Hz. Extra-contractions occurred after the stimulated twitch (see arrows a); as indicated by arrows b, the amplitude of the next stimulated contraction was reduced. Temperature was 25 °C and [Ca²⁺]_o was 2 mm. Stimulations are represented in B (under F trace) in order to mark stimulated twitches.

Figure 2. Relation between stress and stimulation rate in mouse cardiac trabeculae.

A, averaged SFR in mouse cardiac muscle at 25 °C (•; n = 24) and 37 °C (□; n = 10); results are expressed as means ± s.e.m. Data at 25 and 37 °C were fitted through a double sigmoid function. Data were compared individually with stress measured at 1 Hz (* P < 0.05; ** P < 0.001); [Ca²⁺]_o = 2 mm, pH 7.4. B, active stress development and corresponding sarcomere shortening at stimulation rates of 0.2, 1 and 5 Hz and at temperatures of 25 (left part) and 37 °C (right part). Traces were recorded from a typical trabecula with a diameter of 0.09 mm. Vertical bars indicate vertical scales of stress and SL traces and represent 10 mN mm⁻² and 0.1 μm, respectively.

The frequency at which the slope of the stress-frequency curve reversed was denoted as the ‘reversal frequency’ (Freq_Rev ≃ 1 Hz at 25 °C). When stress was expressed as a percentage of the reference stress at 1 Hz, the averaged stress-frequency relation (SFR) flattened above 2.5 Hz. Actually, at frequencies above 1 Hz, 45 % of individual muscles exhibited a continuous increase of stress and a stress-frequency curve above the average curve. The remaining 55 % showed stress-frequency curves superimposed on or lower than the average curve (see examples in Fig. 3A and B and Fig. 7D).

Figure 3. Effects of nifedipine and ryanodine on the SFR of mouse cardiac trabeculae.

Effects of 0.1 μm nifedipine (A, □) and 0.1 μm ryanodine (B, ▿) on SFR were tested on two individual muscles; for each muscle, data were compared with their respective controls (nifedipine: •, ryanodine: ○). C, data from A and B were superimposed on the average SFR of Fig. 2; to facilitate the comparison, the data were expressed as percentage of stress measured at 1 Hz before the frequency test (see Methods); SFR data were described over the full range of frequencies by using two sigmoid functions: (1) function 1 (red curve) fitted the data of the descending limb of the SFR (r = 0.932; P < 0.05); extrapolation of function 1 to high frequencies predicted stress development close to stress determined experimentally in the presence of 0.1 μm nifedipine (□); (2) function 2 (blue curve) described the data of the ascending limb of the SFR (r = 0.886; P < 0.05); extrapolation of function 2 to low frequencies shows that the predicted stress closely matched the experimental data in the presence of 0.1 μm ryanodine (▿). The double sigmoid function used to fit the full set of data (thick line) was modified from the sum of functions 1 and 2 (r = 0.907; P < 0.05); coloured arrows indicate that ryanodine (red arrow) reduced specifically the descending limb while nifedipine (blue arrow) decreased the ascending limb of the SFR. D, effect of temperature on calcium influx through Ca²⁺ channel in mouse cardiac cell. a, typical I_Ca,L induced at 0 mV at 25 and 35 °C; the 0 current level is symbolized by the dashed line. The action potential was recorded at both temperatures (b, upper part) and the corresponding nisoldipine-sensitive currents (I_Ca,L) were sampled under action potential clamp condition (b, lower part); the nisoldipine-sensitive current resulted from the subtraction of current measured in presence of 0.2 μm nisoldipine from the control; the fast component of the action potential clamp was due to a residue of sodium current (I_Na). The last slow inward current corresponded to I_Ca,L; the amount of Ca²⁺ entering the cell per unit of time was estimated by the integration of I_Ca,L.

Figure 7. Simulation of the effect of muscle thickness on the SFR.

This figure shows the simulated effect of limitation in O₂ diffusion related to muscle thickness and proposes an explanation for the discrepancy between ex and in vivo SFR data (see Discussion). In A, SFR predicted (blue curve) by the double sigmoid model described in Fig. 6 is compared with the average SFR determined from the experimental data of Fig. 3 (red curve). In B, force- and frequency-dependent increases of [P_i] were calculated based on varied muscle thickness (10 (bottom curve), 50, 100, 200 and 400 μm (top curve)); see Appendix and Discussion. These calculated variations of [P_i] were incorporated in the double sigmoid model described in Fig. 6, which predicted the effects on the SFR reported in C (top curve corresponds to SFR variation when diameter is 10 μm and bottom curve when diameter is 400 μm). Results predicted by the model in C are then compared with individual SFR measured from nine representative trabeculae as represented in D; to facilitate comparison, ordinate and abscissa scales are identical in C and D. Note the similarity between the data predicted by the model above 1 Hz in C and experimental variability shown in D for the same frequencies. The average SFR determined experimentally is shown in C and D for comparison (red curve).

Figure 8. The major parameters of the SFR model.

A and B represent the theoretical parameters used to predict respectively the ascending and the descending limbs of the SFR (see Results and Appendix).

The same measurements were conducted at a higher temperature in order to test whether force was limited at higher frequencies (>5 Hz) because of the duration of the twitch at 25 °C. The results revealed the existence of a similar biphasic relation between stress and stimulation frequency at 37 °C (Fig. 2A). However, raising temperature lowered the stress at each frequency (Fig. 2B) such that the stress-frequency curve was shifted downward by 10-15 mN mm⁻² (Fig. 2A). At 37 °C, a marked alternans still occurred at 10 Hz and was responsible for the low value of stress represented in Fig. 2A. Increasing temperature from 25 to 37 °C reduced the twitch force duration by 50 % and induced a 1.5- to 2-fold increase of the rate of relaxation at both high and low frequencies (see Fig. 2B).

The effects of nifedipine (0.1 μm), ryanodine (0.1 μm) and caffeine (50 μm) were tested on SFRs at 25 °C. Nifedipine and ryanodine effects on typical trabeculae are represented in Fig. 3A and B, respectively, and compared with the averaged SFR in Fig. 3C. Nifedipine, ryanodine and caffeine substantially altered the SFR at 25 °C; nifedipine (Fig. 3A) decreased the stress at frequencies >1 Hz. Both ryanodine (Fig. 3A) and caffeine (similar to its effect previously shown in rat cardiac trabeculae; Schouten & ter Keurs, 1991, data not shown) reduced the stress specifically below 1 Hz. Ryanodine (0.1 μm) abolished the stress almost completely between 0.1 and 1 Hz (as indicated by red and blue arrows in Fig. 3C). At 5 Hz, the stress was almost nonexistent in the presence of 0.1 μm nifedipine.

Forces below 1 Hz fitted a simple saturating function of time interval (function 1: see Fig. 3C, red curve) that predicted values of stress close to data obtained experimentally at frequencies >1 Hz, when Ca²⁺ current was inhibited by nifedipine. A second saturating function of stimulation rate (function 2: see Fig. 3C, blue curve) described the data above 1 Hz and values of stress predicted at frequencies <1 Hz matched the experimental values measured in the presence of ryanodine. Finally, the sum of the two functions described the SFR observed experimentally at 25 °C (see Fig. 3C). This model was used to fit the SFR data presented throughout this report.

Finally, we tested whether a temperature dependence of I_Ca,L could explain the depression of the SFR when the temperature was raised from 25 to 37 °C (Fig. 3D). It was interesting to notice that action potentials recorded in single myocytes (Fig. 3Db) were nearly identical to monophasic and transmembrane APs measured in the intact heart (Knollmann et al. 2001). APD₅₀ (duration of AP at half-maximal amplitude) occurred at ≈10 ms in both conditions. Intact heart and single cells exhibited APs with the same shape: a clear peak due to I_Na followed by the repolarization phase, probably driven in the mouse by the repolarizing K⁺ current I_to (Wang & Duff, 1997) and the inward current I_Ca,L and, finally, ending with the slow termination of the repolarization that originated in the combined effect of the Na⁺-K⁺ pump and Na⁺-Ca²⁺ exchange (Schouten & ter Keurs, 1985; Schouten et al. 1990).

In agreement with previous studies that reported a Q₁₀ of 2-3 for I_Ca,L (McDonald et al. 1994), the amplitude of I_Ca,L increased ≈2-fold when temperature was raised from 25 to 35 °C (Fig. 3Da). Inactivation of I_Ca,L was slightly faster at higher temperatures. The AP was shorter at 35 versus 25 °C: APD at 0 mV was 1.6 ± 0.4 ms at 35 °C vs. 2.8 ± 0.5 ms at 25 °C (n = 6, P < 0.05). Moreover, in the same conditions, AP overshoot decreased from 71 ± 2.8 mV at 25 °C to 50 ± 9.1 mV at 35 °C (n = 6, P < 0.05) and resting membrane potential decreased from -74 ± 1.6 mV to -77 ± 1.9 mV (n = 6, P < 0.05; Fig. 3Db, upper part). Under AP clamp, the amplitude of the nisoldipine-sensitive current was ≈2-fold greater at 35 versus 25 °C (Fig. 3Db, lower part) and the integrated inward Ca²⁺ current was 81.15 ± 16.98 pA ms pF⁻¹ at 35 °C, compared with 31.35 ± 8.85 pA ms pF⁻¹ (n = 6) at 25 °C.

Relation between force, SL and [Ca²⁺]

Figure 4 shows the stress-SL curves of mouse cardiac muscle at different Ca²⁺ concentrations for a stimulation rate of 1 Hz. As described previously in rat trabeculae (Kentish et al. 1986), the passive stress-SL relation varied between individual trabeculae (Fig. 4A). Data from each preparation were fitted to an exponential function, exp (k_p(SL - SL₀)) - 1, where k_p is the exponent of passive stiffness in mN mm⁻² μm⁻¹ and SL₀ is the intercept on the SL axis, i.e. the zero load sarcomere length. We calculated the average passive stress-SL curve (Fig. 4A, continuous thick line) using the mean values of k_p and SL₀ (respectively 19.8 ± 2.5 mN mm⁻² μm⁻¹ and 2.00 ± 0.03 μm, n = 11). SL₀ was obtained after regression of the data in a range where the stress-SL curve is shallow. This may overestimate the value of SL₀. When the resolution of the sarcomere diffraction pattern allowed accurate measurement of SL and when the resting muscle started to buckle with further compression, SL indeed appeared to be lower than 1.75 μm.

Figure 4. Force-sarcomere length relations of mouse cardiac muscle.

A group of 11 trabeculae was used to study the force-SL relation in mouse cardiac muscle. A shows the passive stress-SL relation of typical trabeculae; for clarity, results from only 7 muscles are represented. Individual stress-SL relations were fitted through the exponential function: Stress = exp(k_p × (SL - SL₀)), where k_p represents the passive stiffness coefficient and SL₀ represents SL at zero load. Average values of k_p and SL₀ were k_p = 19.83 ± 2.45 mN mm⁻² μm⁻¹ and SL₀ = 2.00 ± 0.03 μm (n = 11). The resulting mean passive stress-SL curve was plotted as indicated by the thick line. Curves in A reveal variability in the stress-SL relations. Such variability could be simulated only by a model of variation of SL₀ in the exponential function, as demonstrated in the inset of A, which shows the theoretical passive stress-SL curves according to this exponential function for k_p = 2 mN mm⁻² μm and SL₀ 1.7, 1.8, 1.9, 2.0 and 2.1 μm, respectively. B shows the relation between active stress and SL at five different values of [Ca²⁺]_o: 1, 2, 4, 8 and 12 mm. For each muscle, the active stress was calculated from the total stress subtracted from the corresponding passive stress measured at the same SL. The stimulation frequency was 1 Hz. Data were expressed as means ± s.e.m. Nonlinear regressions were performed on each group of data based on polynomial functions; regression lines were drawn on the corresponding group of data.

The influence of SL on active stress development was tested at 1 Hz randomly in each individual muscle and at [Ca²⁺] varying between 1 and 12 mm. The results are represented in Fig. 4B. Maximal active stress developed at 1 mm [Ca²⁺] was less than 5 mN mm⁻², which represents approximately 7 % of the maximal value measured at higher [Ca²⁺]. At a [Ca²⁺] of 2 mm, active stress exhibited a continuous and exponential increase with increasing SL. Active stress increased almost linearly with [Ca²⁺] above 2 mm. Stress-SL curves obtained at 4, 8 and 12 mm nearly superimposed such that no significant stress increment could be detected at a given SL when [Ca²⁺] was increased. Even though extra-contractions often occurred during the resting periods between stimuli at [Ca²⁺] > 4 mm, the force stayed constant over time and a mild spontaneous contractile activity of small groups of sarcomeres was seen in only four preparations. Figure 5 illustrates the effect of [Ca²⁺]_o on active stress at three different SLs (1.9, 2.05 and 2.15 μm) and confirms that active stress generated at 1 Hz by the mouse cardiac muscles increased steeply with [Ca²⁺] up to ≈4 mm (see Fig. 5B). Above 4 mm, active stress reached a plateau. At a [Ca²⁺] of 2 mm, two trabeculae stimulated at 1 Hz developed maximal active stress up to 70-80 mN mm⁻².

Figure 5. Force-[Ca²⁺] relation at various sarcomere lengths.

In A, results of Fig. 4 were grouped in three different bins based on resting SL: 1.91 ± 0.009 (n = 50; •), 2.05 ± 0.005 (n = 43; ▿) and 2.15 ± 0.004 μm (n = 59; ▪). B shows the results from A after normalization to the maximal stress. Nonlinear regression was performed on each group of data by using the Hill equation: Stress = (Stress_max × [Ca]^nH)/([Ca]₅₀^nH + [Ca]^nH), where Stress_max is maximal stress, [Ca]₅₀ is the Ca²⁺ concentration at half of Stress_max and n_H is the Hill coefficient. The regression lines were drawn through the data in A and B. Results of the regression of data from B are n_H = 5.8 ± 1.3, Stress_max = 0.92 ± 0.03 and [Ca]₅₀ = 1.6 ± 0.1 mm (r² = 0.94; P < 0.0001).

Discussion

The force-frequency relation in mouse trabeculae

In the present study, mouse cardiac trabeculae showed a clear biphasic SFR: stress amplitude decreased by 2- to 3-fold from 0.1 to 1 Hz (negative limb) and increased by 1- to 2-fold from 1 to ≈5 Hz (positive limb). This observation is consistent with the previous report of Schouten & ter Keurs (1991), who described a similar biphasic relation in rat cardiac trabeculae under equivalent experimental conditions (25 °C, [Ca²⁺]_o ≤ 1 mm).

Multiple factors can interfere with the shape of the SFR, which may explain discrepancies and contradictions found in the literature. It was reported, for instance, that elevated [Ca²⁺]_o (>2 mm) renders the SFR negative in the rat (Schouten & ter Keurs, 1991; Layland & Kentish, 1999). This may explain the negative SFR reported in rat cardiac muscle when studies were performed at a [Ca²⁺]_o of 2 or 2.5 mm (Mitchell et al. 1985; Maier et al. 1998, 2000). The diameter of the preparation can also contribute to create a negative SFR because of the metabolic failure that occurs as the stimulation rate increases, particularly in muscles thicker than 0.2 mm (Schouten & ter Keurs, 1986). Conversely, strongly positive SFRs have been reported in rats and mice at physiological [Ca²⁺]_o (Gao et al. 1998; Layland & Kentish, 1999; Kassiri et al. 2000). Freq_Rev in the rat was ≈0.3 Hz at a [Ca²⁺]_o of 1 mm and 25 °C (Schouten & ter Keurs, 1991), which means that SFR was negative below 0.3 Hz and positive above. By using frequencies between 0.2 and 4 Hz, Layland & Kentish (1999) and Kassiri et al. (2000) evidently investigated the positive limb of the SFR.

Possible mechanisms underlying the SFR

Consistent with Schouten & ter Keurs (1991), modulation of the negative segment of the SFR by ryanodine (cf. Fig. 3B and C) indicated that the amount of Ca²⁺ releasable from the SR was a determinant parameter of the SFR at low stimulation rate in the mouse. Also consistent with a previous study of rat cardiac muscle (Schouten & ter Keurs, 1991), blockers of the current I_Ca,L, such as nifedipine, caused a pronounced reduction of force amplitude in mouse trabeculae at higher frequencies (>1 Hz; cf. Fig. 3A and C). This indicates that I_Ca,L is a major contributor to the frequency-dependent potentiation of contraction above 1 Hz.

Hence, our observations support the idea that the SFR depends on the amount of Ca²⁺ accumulated in the SR, which results from the combination of two different sources that exhibit opposite dependence on stimulation rate: (1) the amount of Ca²⁺ sequestered during the diastolic interval, which is inversely related to the stimulation rate; and (2) the net amount of Ca²⁺ entering the cell during the action potential and sequestered during the twitch force, which increases with stimulation rate.

As shown in Fig. 3C, each of the two components of the SFR could be described over the full range of frequencies by two sigmoidal functions that predicted stress values close to the experimental data obtained in the drug-free state or during blockade of I_Ca,L or by inducing Ca²⁺ leak from the SR. These results fitted into an intuitively simple model of excitation-contraction coupling summarized in Fig. 6A. We assumed first that the net Ca²⁺ release from the SR was a function of the Ca²⁺ load of the SR. In turn, the Ca²⁺ load of the SR was a simple saturating function of the product of the stimulation rate and the integral of the net Ca²⁺ influx during the action potentials (due to Ca²⁺ entry via I_Ca,L and removal via forward operation of the Na⁺-Ca²⁺ exchanger). These assumptions generated the ascending limb of the SFR (Fig. 6Ba, green curve). We assumed that at low frequencies Ca²⁺ loading of the SR was determined by the amount of Ca²⁺ entering the cell during diastolic intervals (mainly via spontaneously opening L-type Ca²⁺ channels and reverse operation of the Na⁺-Ca²⁺ exchanger). The net Ca²⁺ load of the SR was a saturating function of this diastolic Ca²⁺ influx diminished by the diastolic release of Ca²⁺ from the SR through opened ryanodine receptors, which leads to the time-dependent decline of force in the presence of ryanodine (Banijamali et al. 1991). These assumptions generated the descending limb of the SFR (Fig. 6Ba, red curve). We assumed that the Ca²⁺ load through the diastolic mechanisms adds to the contribution of net Ca²⁺ entry during the action potential. Accordingly, the sum of the two sigmoid functions described a biphasic relation (Fig. 6Ba, blue curve), which was similar to the experimental SFR. In Fig. 6Bb, the model was tested by simulating the effect of Ca²⁺ release from the SR on the function responsible for the descending limb of the SFR. Increasing intensity of diastolic SR release (red curves) lowered the descending limb and induced changes in the computed SFR (blue curves) similar to those observed experimentally in the presence of ryanodine (cf. Fig. 3C). Conversely, by simulating the effect on the ascending limb of blockade of I_Ca,L with different intensities (Fig. 6Bc, green curves), the model predicted changes in the SFR (blue curves) similar to those obtained experimentally by using nifedipine (cf. Fig. 3C).

Figure 6. A simple model for the SFR of mouse trabeculae.

An intuitively simple model of excitation-contraction coupling (ECC) is proposed to explain the cellular mechanisms underlying the frequency dependence of the force production in mouse cardiac muscle. As illustrated in A, the model considers two pathways for SR Ca²⁺ loading: (1) Ca²⁺ loaded during the excitation process (blue arrows); and (2) Ca²⁺ loaded during resting intervals between contractions (red arrows). The first process relies on Ca²⁺ transport through I_Ca,L and, probably to a lesser extent (see Discussion), Na⁺-Ca²⁺ exchanger during action potentials (dashed blue line), while, in the second process, the Ca²⁺ channel and Na⁺-Ca²⁺ exchanger provide Ca²⁺ also during the diastolic periods separating beats (dashed red line). The amount of Ca²⁺ carried during action potentials is directly related to stimulation frequency and contributes to the ascending limb of the SFR (green curve in Ba). In addition, the amount of Ca²⁺ loaded into the SR during diastole determines the descending limb of the SFR (red curve in Ba); it depends on the duration of the diastolic interval and is, therefore, inversely related to frequency. Furthermore, the net Ca²⁺ load of the SR, i.e. Ca²⁺ available for release into the cytosol and subsequent contraction, is reduced by the degree of diastolic Ca²⁺ release from the SR through the ryanodine receptor (Banijamali et al. 1991). Ba represents the resultant saturation functions fitting the descending (red curve) and the ascending (green curve) limbs of the SFR as described in Fig. 3C. The sum of the two functions evidently leads to a biphasic SFR (double sigmoid function) as represented by the blue curve. Bb shows stepwise changes induced in the descending function (red curves) by reduction of Ca_Dias in the equation in point (2) in the Appendix, and the lumped SFR (blue curves) as predicted by the model when the intensity of Ca²⁺ leakage from the SR is enhanced. Bc shows the effect of stepwise increase of blockade of I_Ca,L on the ascending function (green curves) and the SFR (blue curves).

A limitation of this model is the evident proportionality suggested between force and Ca²⁺ release from the SR, which may not be correct at all frequencies over which the mouse heart operates (Backx et al. 1995). Although Fig. 2 shows that activation and relaxation of the twitch accelerate substantially in response to stimulation at higher frequency, we considered incorporation of frequency dependence of cross-bridges and activation kinetics in the model to be beyond the scope of this study, because the major features of the SFR were already reproduced by this simple approach. The proposed contribution of the I_Ca,L and Na⁺-Ca²⁺ exchange to the SFR can be tested by modifying the open probability of the Ca²⁺ channel pharmacologically or by genetic modification of the channel. The putative role of the forward and reverse mode of operation of the Na⁺-Ca²⁺ exchanger involved in the ascending and descending limbs of the SFR, respectively, may be tested in a similar manner (see for example Elias et al. 2001).

As indicated in Fig. 7A, compared with the predicted biphasic SFR showing a continuous increase with frequency above 1 Hz (blue curve), the average SFR determined experimentally flattened above 2 Hz (red curve). Such flattening was not in agreement with the in situ study of the mouse heart that showed a further increase of 35 % of the ventricular elastance from 6 to 10 Hz (Georgakopoulos & Kass, 2001). Neither did the averaged data reflect observations of approximately 50 % of trabeculae that exhibited a continuous increase of stress above 1 Hz (Figs 3A and B and 7D). As shown in Fig. 7D, the stress-frequency data above 2 Hz showed a more scattered distribution among the muscles. One explanation for this variability may be that force production at high frequencies was limited by O₂ diffusion, which depended upon the thickness of the muscles (Schouten & ter Keurs, 1986); this possibility was incorporated in the model as a force reduction due to the accumulation of inorganic phosphates (P_i; see Fig. 7B and C). O₂ consumption (_O2) by mouse heart has been shown to reach 400 nmol g⁻¹ s⁻¹ in a Langendorff preparation (Gustafson & Van Beek, 2000), equivalent to an O₂ flux of 0.16 nmol cm⁻² s⁻¹ to the core of the cell (e.g. 15 μm diameter). The O₂ flux (F_O2) by diffusion from a bathing solution to the core of a ribbon-shaped trabecula obeys:

where D represents the O₂ diffusion coefficient (10⁻⁵ cm² s⁻¹), C₁ and C₂ the O₂ concentrations and h the thickness of the sheet of muscle and its adjacent unstirred layers (Crank, 1979). Ignoring the contribution of the unstirred layers, it follows that the P_O2 in the centre of the muscle (bathed at a P_O2 of ≈600 mmHg) falls to zero, when F_O2 is 0.16 nmol cm⁻² s⁻¹ and h = 220 μm. The effect of unstirred layers cannot be ignored, however, as they can be several hundred micrometres thick. Therefore, the physical diffusion of O₂ (F_O2) even in thin trabeculae with a thin unstirred layer (e.g. 50 μm) does not match the maximal _O2 of mouse heart. Mitochondrial synthesis of ATP slows at a P_O2 of a few Torr (Loiselle, 1987) and P_i will accumulate when ATP consumption by ECC and contraction proceed. In turn, the rise of [P_i] will inhibit force development (Xiang & Kentish, 1995) so that a further rise in [P_i] is limited. Consequently, we have incorporated the assumption that force development at high rates of stimulation is limited by P_i accumulation as a third term in the lumped model for the SFR of mouse trabeculae (see Appendix).

In Fig. 7B, the increase in [P_i] with frequency was simulated for different muscle thicknesses ranging from 10 to 400 μm. Then the different increases of [P_i] were used as inputs of the model. Corresponding SFR curves were represented in Fig. 7C and compared with the experimental relation (red curve). Comparison of Fig. 7C and D showed that the different responses of the SFR to variations of [P_i] matched the variability observed experimentally among the trabeculae at high frequency. These results suggest that hypoxia might indeed have occurred in some trabeculae, since the stimulation rate increased even though the muscles were probably thin enough to contract adequately below 1-2 Hz. We conclude that increase of stress with increasing frequency above 1-2 Hz is probably underestimated in studies using multicellular preparations even as thin as mouse trabeculae.

Temperature effect on the SFR in mice

An interesting finding of our study is the reduction of force at all frequencies with increasing temperature without alteration of the biphasic shape of the relation. It is well known that Ca²⁺ sensitivity of the myofilaments increases with temperature (Harrison & Bers, 1989) and this cannot, therefore, explain the decline of force observed in our study at 37 °C. However, consistent with our observation, the amplitude and the duration of Ca²⁺ transient decrease with temperature (Puglisi et al. 1996).

As discussed above, two factors participating in the Ca²⁺ transient are probably involved in the control of force in cardiac muscle, namely the net Ca²⁺ influx during the AP and the Ca²⁺ content of the SR. We considered that Ca²⁺ carried by I_Ca,L dominated the net Ca²⁺ influx in mouse cardiac muscle during the AP. Ca²⁺ entry through the Na⁺-Ca²⁺ exchanger may contribute to the net Ca²⁺ influx during the contraction as well, since it is favoured during the action potential plateau (Su et al. 1999). However, because of the short action potential in mouse heart cells, the quantitative contribution of the Na⁺-Ca²⁺ exchanger is probably small compared with I_Ca,L.

We found that integrated I_Ca,L was ≈30 × 10⁻¹⁵ and 80 × 10⁻¹⁵ C pF⁻¹ per cell at 25 and 35 °C, respectively, in myocytes with a volume of ≈15 pl and total capacitance of ≈150 pF. Hence, the amount of Ca²⁺ entering the cell during the AP was 4 μmol l⁻¹ at 37 °C and 1.5 μmol l⁻¹ at 25 °C. Moreover, the stress measured at 1 Hz was ≈9 mN mm⁻² at 37 °C versus 17.5 mN mm⁻² at 25 °C, i.e. approximately 10 and 20 %, respectively, of the maximal stress that cardiac muscle can produce (≈100 mN mm⁻²). This indicated that ≈7 and 13 μmol l⁻¹ of Ca²⁺ activated troponin C (TNC) respectively at 37 and 25 °C ([TNC] in the heart, 70 μm; Solaro et al. 1974). Comparison of these numbers with the amount of Ca²⁺ carried by I_Ca,L during the AP suggested that the SR amplified the Ca²⁺ signal ≈10-fold at 25 °C and only ≈2-fold at 37 °C. Hence, a difference in SR loading or release is, by inference, the most plausible source for the temperature-induced depression of force observed in our study. One mechanism causing reduced Ca²⁺ loading of the SR at increased temperature that would fit in the proposed model (see Fig. 6) would consist of a larger increase of Ca²⁺ extrusion from the cell compared to accelerated reuptake via the Ca²⁺ pumps into the SR, because at increased temperature the [Na⁺]_i would decrease as a result of acceleration of the Na⁺-K⁺ pumps, which would enhance Ca²⁺ extrusion by Na⁺-Ca²⁺ exchange even more than by acceleration of the exchanger alone. The observed increase of the rate of relaxation with increase of the temperature would be consistent with this assumption.

Physiological relevance of the SFR in mice

The fact that we could not investigate the force response above 5 Hz in mouse cardiac trabeculae was at first striking when one considers that normal resting heart rate in mice ranges from 500 to 700 beats min⁻¹ (≈8-12 Hz; Georgakopoulos & Kass, 2001).

Furthermore, Georgakopoulos & Kass (2001) could study the SFR of mouse heart in situ at frequencies up to 14 Hz. Even though increasing temperature shortens the Ca²⁺ transient (Puglisi et al. 1996), a temperature difference between in and ex vivo conditions could not explain why isolated trabeculae failed to contract normally at stimulation rates 3- to 4-fold lower than the heart in situ by developing alternans even at 37 °C. We found that APD₅₀ in single myocytes was ≈10 ms (Fig. 3D), i.e. nearly identical to APD₅₀ of the monophasic AP measured in intact mouse heart (Knollmann et al. 2001). Therefore, the faster response of the muscle to a shorter AP cannot explain the ability of the mouse heart to beat at a high rate in vivo either. Alternatively, accelerated relaxation of the twitch due to circulating catecholamines may also explain the larger ability of cardiac muscle to contract more rapidly in living mouse compared to ex vivo conditions.

Finally, a major difference between ex and in situ experiments is that diffusion operates from external superfusion milieu in isolated trabeculae whereas myocytes are supplied individually over a distance of a few micrometers by capillaries in the intact heart (Krogh, 1919). As seen above, it is difficult to completely rule out the possibility that hypoxia affects the force production in ex vivo conditions when the preparation is stimulated at high frequency.

Stress-SL relation in mouse cardiac trabeculae

To our knowledge, this study constitutes the first report of the stress-SL relations in mouse cardiac muscle. Passive stress varied with SL following a monotonic curvilinear relation (Fig. 4A). The relation is shallow from slack length up to ≈2.1 μm and then rises steeply. The similarity with passive stress-SL relation of the rat (Van Heuningen et al. 1982; Kentish et al. 1986) suggests that the shape of the passive stress-SL curve of the mouse originates in the combination of the respective stress-strain curves of titin and collagen (Granzier & Irving, 1995). Regression of stress-SL data using a single exponential function provided SL₀ ≃ 2 μm, while the slack length measured experimentally was less than 1.75 μm. This may reflect a need for more parameters in the estimation, such as two different exponential functions: the one that we used to describe the steep part of the stress-SL relation and a second one to describe more accurately the shallow part (< 2 μm). Such a requirement for two exponential stress-strain relations would be consistent with a combined contribution of titin (shallow part) and collagen (steep part) to the passive properties of cardiac muscle of the mouse. We also observed variability among passive stress-SL curves, which was reproduced by changing SL₀ in the regression model (see theoretical curves calculated from the exponential function in the inset of Fig. 4A; see also legend of Fig. 4 for more details). A change of stiffness (k_p), however, did not fit the observed variability. Individual passive stress-SL curves vary due to collagen that contributes significantly to stiffness at slightly different SL (Hanley et al. 1999) or, less likely, as a result of different lengths of titin molecules among the muscles (Trombitas et al. 2000).

Under the experimental conditions of this study, the active stress was on average ≈50 mN mm⁻² and reached 70-80 mN mm⁻² in two trabeculae, i.e. active stress was of the same order as in muscles from larger animals. Interestingly, at [Ca²⁺] up to 4 mm, the stress-SL curves in the mouse were of the same shape as curves obtained in the cat (de Tombe & ter Keurs, 1991), the rat (Kentish et al. 1986; ter Keurs & Tyberg, 1987; Banijamali et al. 1991), the ferret and the sheep (see ter Keurs & Tyberg, 1987). Data from skinned mouse cells suggest length-dependent sensitivity of the myofilaments to Ca²⁺ (Cazorla et al. 2001) similar to the length dependence that has been shown for rat cardiac muscle (Kentish et al. 1986). Hence, we expect that similar mechanisms affecting the length dependence of Ca²⁺ sensitivity operate in mice and rats (and other mammals).

A striking feature of our study, however, is the limitation of force at [Ca²⁺] > 4 mm (see Fig. 4B and Fig. 5), which constitutes a major difference of mice compared with other mammals (ter Keurs & Tyberg, 1987). At [Ca²⁺] ≥ 4 mm, active stress increased almost linearly with SL. Raising [Ca²⁺] up to 12 mm did not significantly change the relation and, particularly, no inversion in the curvature of the relation was seen, unlike other species.

These results appeared comparable to those of Gao et al. (1998), which showed an initial steep rise in stress followed by a shallower increase with [Ca²⁺] (see Fig. 3 in Gao et al. 1998). The results of Gao et al. (1998) are even more consistent with our study when one takes into account the free Ca²⁺ concentration, which represents 100 % of the total Ca²⁺ concentration in Hepes buffer solution like the one used in our study, whereas it is only ≈70 % in bicarbonate-based buffer (De Tombe, 1989) like the Krebs-Henseleit used by Gao and collaborators.

A possible reason for a saturation of force development with [Ca²⁺] like that seen in our study is that the amount of Ca²⁺ available for activating the contractile filaments is limited by the SR, consistent with the assumption that the SR Ca²⁺ release function is the limiting factor for stress development at high [Ca²⁺] in rat cardiac muscle (Schouten & ter Keurs, 1991). Nevertheless, the hypothesis implying that SR Ca²⁺ release may be limited more in the mouse than in other mammalian species would require further studies of both Ca²⁺ sensitivity of the myofilaments and the Ca²⁺ transient in intact muscle.

Conclusion

Our observations support the conclusion that excitation- contration coupling in mouse cardiac muscle is similar to what we observed previously in the rat, although important differences exist in the Ca²⁺ dependence of force development. Our results may indeed suggest a lower capacity of the SR for buffering Ca²⁺, which would make the generation of force in mouse cardiac ventricle more dependent on the amount of Ca²⁺ carried by I_Ca,L at high stimulation frequency and, particularly, at beating rates of mouse heart in vivo, which would render extrapolations between the two species, without verification, hazardous.

This study contributes to the groundwork for investigations using genetically modified mice, animal models that certainly will provide important information regarding mechanisms underlying cardiac excitation-contraction coupling in normal and pathological conditions.

APPENDIX

Lumped model for the force- (or stress)-frequency relation

We made the following assumptions.

1. The ascending limb of the force- or stress-frequency relation:

   

   where Ca_AP is the frequency-dependent amount of Ca entering during the AP:

   

   Ca_AP reflects the net calcium accumulated by the SR owing to Ca²⁺ entry during the APs at varied rate (normalized for Ca_control), Rate is the heart stimulation rate and Ca_entry is the net Ca²⁺ influx into the cell carried by I_Ca,L and Na⁺-Ca²⁺ exchange. Ca_entry is sensitive to interventions that affect the L-type Ca²⁺ channel (e.g. nifedipine). Ca_control is the Ca²⁺ influx in the drug-free state and at physiological [Ca²⁺]_o. n_AP and Ca_AP50 are the parameters of the Hill function that determine the ascending limb of the FFR (F_ascversus frequency; see Fig. 3 and Fig. 6).
2. The descending limb of the SFR obeys:

   

   where the Ca_Dias is the net SR Ca²⁺ accumulation during the diastolic interval(TD):

   

   Ca_entryDias is the amount of Ca²⁺ accumulated in the cell during diastole (presumably through Ca²⁺ L-type channels, Na⁺-Ca²⁺ exchanger, see Schouten et al. 1987; Schouten & ter Keurs, 1991). Ca_entryDias is sensitive to external [Ca²⁺]. Ca_leakDias represents the amount of Ca²⁺ leaking from the SR during diastole. Ca_leakDias is sensitive to interventions that influence SR function (e.g. ryanodine, caffeine). Ca_Dias50 and n_Dias are parameters of the Hill function that defines the descending limb of the SFR.
3. Total force (or stress) is:

   
4. Force development in the presence of P_i obeys:

   

   where [P_i]₅₀ and n_Pi represent the parameters of the Hill function fitting data of the force-[P_i] relation (from Xiang & Kentish, 1995); [P_i]₅₀ = 2 mm and n_Pi = 1.8.

5. [P_i] relates to the synthesis of ATP, ADP + P_i ™⇌ ATP, with a synthesis rate (k_Synth) which depends on P_O2 and is inversely dependent on thickness, and with a degradation rate (k_Degrad) which depends on the force-frequency product (FFP; see Discussion).

   Hence, it follows that the average [P_i] is:

   

   where the degradation rate of the P_i,pool ([P_i] from ATP and PCr; P_i,pool = 38 mm, Bittl et al. 1987) is proportional to the force-frequency product of the twitches (ignoring the duration of the twitch):

   

   and the average synthesis rate of ATP is inversely proportional to the effective thickness of the muscle:

   

   where k_O2 is a proportionality constant linking ATP synthesis to F_O2 (the O₂ flux into the muscle), D is the diffusion coefficient for O₂ and th is the muscle thickness + unstirred layers in mm.

   We assumed that, at the normal heart rate of the mouse, [P_i] was 0.7 mm in a myocyte (16 μm diameter) with unlimited O₂ supply (Katz et al. 1988).

6. Coefficients of the SFR when O₂ diffusion is not limited by muscle thickness (e.g. for a muscle with a thickness of 16 μm) are: F_max, 80 mN mm⁻²; n_AP, 2.4; Ca_AP50, 2.7 Hz; n_Dias, 0.9; and Ca_Dias50, 8 s.