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Am J Physiol Heart Circ Physiol 279: H779-H790, 2000;
0363-6135/00 $5.00
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Vol. 279, Issue 2, H779-H790, August 2000

Cross-bridge kinetics in rat myocardium: effect of sarcomere length and calcium activation

Thomas Wannenburg, Gerardus H. Heijne, Jeroen H. Geerdink, Hendrik W. Van den Dool, Paul M. L. Janssen, and Pieter P. De Tombe

Section on Cardiology, Wake Forest University School of Medicine, Winston-Salem, North Carolina 27157


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES

We tested the hypotheses that Ca2+ concentration ([Ca2+]) and sarcomere length (SL) modulate force development via graded effects on cross-bridge kinetics in chemically permeabilized rat cardiac trabeculae. Using sinusoidal length perturbations, we derived the transfer functions of stiffness over a range of [Ca2+] at a constant SL of 2.1 µm (n = 8) and at SL of 2.0, 2.1, and 2.2 µm (n = 4). We found that changes in SL affected only the magnitude of stiffness, whereas [Ca2+] affected the magnitude and phase-frequency relations. The data were fit to complex functions of two exponential processes. The characteristic frequencies (b and c) of these processes are indexes of cross-bridge kinetics, with b relating to cross-bridge attachment to and c to detachment from certain non-force-generating states. Both were significantly affected by [Ca2+], with an increase in b and c of 140 and 44%, respectively, over the range of [Ca2+] studied (P < 0.01). In contrast, SL had no effect on the characteristic frequencies (P > 0.6). We conclude that Ca2+ activation modulates force development in rat myocardium, at least in part, via a graded effect on cross-bridge kinetics, whereas SL effects are mediated mainly by recruitment of cross bridges.

sinusoidal perturbation; dynamic transfer function of stiffness; myocardial cross-bridge mechanics; skinned fibers


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES

ON THE BASIS OF the Huxley model of muscle contraction, it is generally accepted that the level of force generated by a muscle fiber is proportional to the number of cross bridges that have accumulated in the force-generating state (15). Although calcium concentration and sarcomere length (SL) are known to modulate force generation, the mechanisms involved at the myofilament level remain uncertain. Two general hypotheses have been proposed. First, in the "cross-bridge recruitment" hypothesis, calcium (or SL) functions purely as an on-off switch (33). In this model, cross-bridge kinetics are fixed and only the pool of available force-producing cross bridges is modulated. An alternative hypothesis is that alterations in calcium concentration (or SL) influence the kinetics of the cross-bridge cycle by modulating the rate of transition between passive and force-generating states (17). It is also possible that both mechanisms are operative, simultaneously or separately.

In cardiac and skeletal muscle, these hypotheses have been tested by use of transient analyses of the rate of tension redevelopment after a step length release (Ktr) (4, 11, 30, 31, 42) and the maximum unloaded velocity of shortening (7, 13, 16, 33). Controversy persists, however, because various studies support either hypothesis. For example, two groups of investigators (2, 42) have demonstrated an increase in Ktr in rat myocardium induced by increasing calcium concentrations, consistent with a graded effect of calcium on cross-bridge kinetics. These findings are in agreement with similar studies performed on skeletal muscle (4, 30, 31). However, in support of the switch hypothesis, Hancock et al. (11, 12) found no effect of calcium on Ktr in rat or ferret cardiac muscle. The diverse conclusions of these studies may relate to differences in experimental approach, the impact of muscle loading conditions, or possible muscle type and species differences.

As a complement to step-transient analysis, sinusoidal perturbation has been used to study cross-bridge kinetics in insect flight muscle (28, 40), skeletal muscle (19-21, 25, 26), and cardiac muscle (5, 23, 36-39). The information obtained via this approach should in theory be equivalent to transient analysis, as in the determination of Ktr (20), except the analysis can be performed during steady-state contraction without involving large changes in external loads. This aspect of the sinusoidal approach may make it less subject to possible confounding influences of developed force per se, especially if the analyses were to be performed at various levels of activation. We therefore applied this approach to the issue of force regulation in cardiac muscle.

We hypothesized that changes in SL and calcium activation affect myofilament force generation in cardiac muscle via changes in cross-bridge kinetics. We therefore used sinusoidal perturbation over a range of frequencies to derive the dynamic transfer function of stiffness of skinned rat trabecular muscles at various SL and levels of calcium activation.

We found that the dynamic transfer function of stiffness was fundamentally altered by changes in calcium activation but not by changes in SL. Increases in calcium-activated force generation induced alterations in the stiffness-frequency and the phase-frequency spectra of dynamic stiffness. In contrast, changes in SL resulted in changes in the stiffness-frequency relation alone. These disparate effects of SL and calcium activation on the dynamic transfer function of stiffness suggest fundamentally different mechanisms of myofilament force regulation. Subject to present cross-bridge models of muscle contraction, our findings are best explained by an effect of calcium on cross-bridge kinetics and an effect of SL on cross-bridge recruitment.


    MATERIALS AND METHODS
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES

Muscle preparation and experimental apparatus. All studies were conducted in accordance with institutional guidelines for the care and use of laboratory animals. Trabecular muscles were dissected from the hearts of rats (Harlan LBN-F1). The hearts were rapidly excised under halothane inhalation anesthesia and immediately perfused with a cardioplegic, modified Krebs-Henseleit solution (see Solutions), as previously described (6). Under a binocular microscope, thin unbranched trabecular muscles between the atrioventricular ring and right ventricular free wall were carefully excised with a portion of tricuspid valve and right ventricular free wall on either end. Muscle dimensions were determined via an ocular micrometer mounted in the dissection microscope (~10-µm resolution). On average, the muscles were 1.2 ± 0.2 (SD) mm long, 100 ± 40 µm thick, and 190 ± 20 µm wide. The muscles were incubated for >= 1 h in a "skinning solution" containing 1% Triton X-100, which served to remove cell membranes and sarcoplasmic reticula, thereby isolating the contractile filaments. Custom-made aluminum foil "T" clips were gently attached to both ends of the muscles to serve as handles for mounting the muscle to the experimental apparatus (10). The muscles were then mounted in a plastic chamber (150 µl volume) located on the stage of an inverted microscope (Nikon). The T clip on one end was hooked onto a servomotor (model 6350, Cambridge Technology, Watertown, MA; ~250-µs 90% step response), and the clip on the other end was attached to a modified semiconductor strain gauge (model AE801, Sensonor; resonance frequency ~3 kHz). The muscles were activated by exposure to free calcium in the bathing solution. Sinusoidal muscle length (ML) perturbations were input via the servomotor to determine stiffness. The temperature of the bath was monitored continuously: it averaged 23 ± 0.6°C and did not vary >0.2°C during the course of an individual experiment.

Solutions. In every experiment the muscle was dissected while the heart was perfused with a low-calcium Krebs-Henseleit solution containing (in mmol/l) 140.5 Na+, 5.0 K+, 127.5 Cl-, 1.2 Mg2+, 2.0 H2PO4-, 1.2 SO42-, 19 HCO3-, 10.0 D-glucose, and 0.1 Ca2+. In addition, a calcium-desensitizing agent, 2,3-butanedione monoxime (20 mmol/l), was added to minimize damage to the ends of the trabeculae during dissection (32). The Krebs-Henseleit solution was gassed with 95% O2-5% CO2. The trabecular muscles were then bathed in a skinning solution that contained 1% Triton X-100 for >= 1 h to dissolve lipid membranes. After this "skinning period" the muscles were bathed in a physiological solution that simulated intracellular conditions. Calcium in this solution was highly buffered so that calcium concentration could be strictly controlled. Three types of solution were used: "relaxing solution," "preactivation solution," and "activating solution." The compositions of these solutions are shown in Table 1. The solute concentrations were determined using an iterative computer program as described by Fabiato and Fabiato (8) with use of dissociation constants of Godt and Lindley (9). The level of free calcium was varied in the activating solution as outlined in Experimental protocol.

                              
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  		Table 1.   Ionic composition of skinned fiber solutions

Measurement of SL. SL was measured online by laser diffraction, as previously described (6). Briefly, a beam of laser light (632 nm), perpendicular to the longitudinal axis of the muscle, was directed onto the center of the specimen. The resulting first-order diffraction band was projected onto a 512-element photo-diode array (Reticon), which was scanned electronically every 0.5 ms. An analog circuit converted the intensity distribution of the diffraction band to a voltage proportional to median SL. Glass gratings of known spacing were used to calibrate the system. De Tombe and ter Keurs (6) found that errors due to muscle inhomogeneity and Bragg angle reflection artifacts were <4% with this approach. The length of the muscle was scanned by moving the microscope stage, and SL homogeneity was confirmed. Muscles were discarded if SL was not homogenous during contractions up to ~70% of maximal force development (Fmax).

Measurements of complex stiffness. The amplitude of the force oscillation induced by the sinusoidal ML perturbation was measured as follows. The crude force signal, consisting of a baseline steady-state force level with a small oscillatory component, was fed into a dual-phase lock-in amplifier (model SR 830, Stanford Research Instruments, Stanford, CA). Under computer control, the lock-in amplifier "locked in" on the force signal at the frequency of perturbation. Thus the frequency-specific component of the force signal was amplified while signals outside a very narrow bandwidth were filtered. This arrangement allows for accurate measurements of the magnitude and phase of small force oscillations, superimposed on a large background signal. Data were collected over a range of perturbation frequencies from 0.5 to 70 Hz to derive the muscle transfer function of stiffness. Data were saved to disk for off-line analysis.

Sinusoidal length perturbations. Sinusoidal ML perturbations were input to the servomotor by a dual-phase lock-in amplifier (model SR830, Stanford Research Instruments) under computer control with use of custom-designed software (LabVIEW, National Instruments). We confirmed that the frequency responses of the servomotor and the SL diffractometer were flat at frequencies of perturbation up to 100 Hz.

We wished to determine the dynamic transfer function of stiffness using SL perturbations of 2.5 nm per one-half sarcomere to minimize any potential effects on cross-bridge kinetics. However, such small perturbations could not be resolved by our photo-diode array. Although we could still determine the transfer function of stiffness as a function of the ML perturbation, we needed to confirm that the muscle perturbation resulted in a perturbation of 2.5 nm per one-half sarcomere at all frequencies. As a first step, we confirmed that the ML-SL transfer function was flat over the frequencies used in the study. We used a perturbation of 10 nm per one-half sarcomere (~1% ML perturbation) at levels of activation from the passive state up to 50% of Fmax to determine the ML-SL transfer function in four trabeculae. Because of the limitations of our SL apparatus, we were unable to obtain a narrow enough SL diffraction pattern >50% of Fmax to accurately track the SL perturbation at high frequencies, but the ML-SL was flat up to this level of force development. These results confirmed that the damaged end compliance of our preparations, by use of the T-clip mounting procedure, was not calcium or frequency dependent at levels of activation up to 50% of Fmax. Having confirmed a flat ML-SL transfer function, we wished to convert our ML perturbation to a perturbation of 2.5 nm per one-half sarcomere. We adjusted ML perturbation amplitudes at low frequencies in each muscle to produce a modulation of 10 nm per one-half sarcomere (~1% ML perturbation). We then reduced the ML perturbation to 25% of this value to produce 2.5 nm per one-half sarcomere perturbations, and this magnitude of muscle perturbation was then used throughout the experiment. We therefore made the assumption that the compliance of the muscle was minimally length dependent. To confirm this in five preliminary experiments, we repeatedly derived complex stiffness at the same level of activation, by use of five different perturbation amplitudes, and found no difference in the characteristic frequencies of the derived transfer function. Therefore, although we actually measured the ML-force transfer function during the course of the experiments, we are confident that the ML-SL transfer function was flat and that the same SL perturbation was input at all frequencies. Furthermore, even if there was a small conversion error from ML to SL, this would not affect our derived characteristic frequencies and would merely introduce a small but consistent scaling factor into the magnitude parameters. Steady-state SL was still strictly controlled during activation; only the small sinusoidal perturbation amplitude was assumed from the ML perturbation. In all our experiments, length perturbations were input after SL and force measurements reached steady state (~20 s) after activation.

Experimental protocol. The muscles were mounted on the experimental apparatus, superfused by relaxing solution. Before activation, the solution was exchanged for a preactivating solution to wash out the strong calcium buffers present in the relaxing solution. To activate the muscle, the superfusate was exchanged for activating solution. After data were collected, the superfusate was again exchanged for relaxing solution.

In eight muscles, stiffness parameters and developed force were measured at nine calcium activation levels at a constant SL of 2.1 µm. In a typical experiment the muscle was first superfused with relaxing solution, and resting SL was set to 2.1 µm. A maximal activation at pCa 4.32 was then sustained for 30 s to determine Fmax. The muscle was then relaxed, and SL was again adjusted to 2.1 µm. During subsequent activations, magnitude and phase of stiffness were measured over perturbation frequencies of 0.5-70 Hz. This process was repeated at nine superfusate calcium concentrations between pCa 7.7 and 4.32. The passive state measurements represent the dynamic transfer function of the resting muscle at that SL (24). Magnitude and phase of stiffness in the passive state were very close to zero over the entire frequency range but were subtracted in complex space from the corresponding values in the activated state to isolate the contribution of cross-bridge cycling to the dynamic transfer function. The overall time for data collection at each level of activation was ~2 min. This included time taken to reach steady state at each frequency and time to save the data to disk. At each level of activation, ML was adjusted manually during the contraction, to keep SL constant at 2.1 µm, and muscle homogeneity was confirmed by checking for variation in mean SL across the length of the muscle. Muscles were discarded if internal shortening exceeded 0.05 µm per sarcomere at 2.1 µm (±2.5%) or if SL was inhomogeneous at 70% activation. At the higher levels of activation (pCa >= 5.83), the laser diffraction pattern was lost and SL could not be accurately measured, nor could homogeneity be confirmed. ML was empirically adjusted by the same amount as for pCa 5.92, which yielded forces that were 69 ± 9% (SD) of Fmax. We cannot be sure that SL was maintained at pCa >5.92.

In four muscles the magnitude and phase of complex stiffness were determined at three SL (2.0, 2.1, and 2.2 µm) for each of three levels of activation (pCa 6.06, 5.98, and 5.92). The procedure was otherwise identical to that described above.

Data analysis. Steady-state isometric force was expressed as millinewtons per square millimeter. Stiffness was expressed as dynamic stress divide  strain. The oscillation in force with perturbation was divided by the muscle cross-sectional area to derive stress, which was expressed as meganewtons per square meter. Strain was calculated as baseline SL divided by the amplitude of oscillation (5 nm/sarcomere).

For purposes of statistical analysis and measurement, the stiffness magnitude and phase data were fit to the following transfer function (20)
Y<SUB>M</SUB>(<IT>f</IT>)<IT>=H+</IT><FR><NU><IT>B·i·f</IT></NU><DE>(<IT>b+i·f</IT>)</DE></FR><IT>+</IT><FR><NU><IT>C·i·f</IT></NU><DE>(<IT>c+i·f</IT>)</DE></FR> (1)
where YM(f) is the dynamic transfer function of stiffness fit to a complex function of the sum of two exponential processes, named process B and process C (20). B and C are magnitude parameters, H is the elastic modulus at 0 Hz, b and c are characteristic frequencies of processes B and C, respectively, f is the perturbation frequency, and i is <RAD><RCD>−1</RCD></RAD>. Note that 2pi b and 2pi c represent the rate constants of processes B and C. This approach resulted in a good fit to the dynamic transfer function of stiffness. Processes B and C have been proposed to correlate with steps or groups of steps involved in the transitions between non-force-generating and force-generating states in the cross-bridge cycle (23). A slower process A, which has been found in skeletal muscle (20), was not detectable, as was the case in previous studies on cardiac muscle (23, 36). The range of perturbation frequencies used in this study did not allow us to resolve a faster process D, which has been described in ferret cardiac muscle (23). The fitting procedure was performed using a custom-designed computer program. We adopted the approach described by Kawai and Brandt (20) and used an iterative least-squares method to minimize the sum of modulus squared deviations for the fitting procedure.

Statistical analysis. To determine whether SL or calcium concentration affected the parameters of the complex exponential fit, we applied multiple linear regression analysis using the following model
b=K+&agr;·N+&bgr;·pCa<IT>+&ggr;·</IT>SL<IT>+&dgr;·</IT>pCa<IT>·</IT>SL (2)
where b is the characteristic frequency of process B, K is a constant, and N is a categorical variable coding for the different trabeculae used in the study. A similar analysis was performed for process C. In addition, we applied a paired t-test to determine whether the characteristic frequencies for processes B and C were different at the lowest vs. the highest levels of activation at a constant SL. Unless otherwise stated, values are means ± SE; P < 0.05 was considered significant. All statistical analyses were performed using commercially available software (SYSTAT, Evanston, IL).

Stability of the preparation. To test the stability of the preparation within a given experiment and to rule out possible confounding effects of time or activation sequence on the dynamic transfer function, we performed the following control experiments in five trabeculae. The muscles were prepared and mounted as described above. The muscles were activated to Fmax (pCa 4.32) for 30 s to determine Fmax at SL 2.0 µm. The muscles were then activated 13 times, with alternation between 2 levels of calcium activation, which resulted in levels of force development that were 65 ± 10 and 16 ± 4% of Fmax. The dynamic transfer function of stiffness was determined, and rate constants were derived as described above. The experiments were performed to document the possible effects of force deterioration during an experiment.

A second concern was the possibility of phosphate buildup in the preparation during contractions at higher levels of activation. An increased phosphate buildup during higher levels of activation could theoretically influence cross-bridge kinetics and falsely simulate a calcium effect. To rule this out, we determined the dynamic transfer function twice in a single contraction over perturbation frequencies of 0-70 Hz and then back to 0 Hz.


    RESULTS
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES

Stability of the preparation. Five control experiments were conducted. In each experiment, muscles were activated sequentially 13 times at 2 calcium concentrations (pCa 6.11 and 5.92). In these muscles, mean forces were 65 ± 10% (SD) of Fmax at the high calcium concentration and 16 ± 4% of Fmax at the lower concentration. On average, force deterioration was 2.5% per contraction at the high activation levels and 0.6% per contraction at the low levels. There was a trend for the derived characteristic frequency of process B (b) to diminish by ~5% per contraction on average (r2 = 0.6) in the high activation group. However, the difference between b for the first and last measurements was not statistically different (P = 0.16), and the values of b at each high calcium activation were not statistically different from the others (ANOVA, P = 0.36). Similarly, the values for c at each activation were not different from each other (ANOVA, P = 0.7). The mean values for b in the high and low calcium activations [5.1 ± 0.8 and 1.4 ± 0.3 (SD) Hz, respectively] were significantly different (Mann-Whitney test, P < 0.01), whereas the mean values for c in the high and low calcium activations (17.4 ± 1.2 and 16 ± 1.2 Hz, respectively) were not different (P = 0.07). These results confirm that the dynamic transfer function in an individual muscle at a given level of calcium activation is reproducible and that deterioration of force development by the preparation during an experiment would not explain any increase in the derived characteristic frequency for processes B and C. In addition, the studies suggest that the fitted rate constant of process B is consistently and reproducibly increased at higher levels of activation.

We also performed experiments to determine whether deterioration or phosphate buildup during a contraction at the higher levels of activation could affect cross-bridge kinetics. Figure 1 shows the Nyquist plot from a representative experiment in which the dynamic transfer function was determined twice during a single contraction at a steady-state isometric force of 72 mN/mm2, which was 70% of Fmax in this example. This plot represents the stiffness data as a vector, with magnitude and phase plotted from the zero intercept in complex space. In this representation the x-axis represents the real component of complex stiffness (elastic modulus), and the y-axis represents the imaginary component (viscous modulus). There was no significant difference in the two transfer functions. If phosphate accumulation or deterioration of the preparation during a single contraction affected cross-bridge kinetics, we would expect the second frequency run to yield a different transfer function. The characteristic frequencies fitted to these functions were 3.6 and 3.8 Hz for process B and 24.3 and 25.8 Hz for process C. Figure 1B is an expanded view of the eight lowest frequencies to show that the fit was adequate for these frequencies. In the example shown, the frequency of perturbation was varied from 0.5 to 79 Hz for the first frequency run, and then the same order was repeated for the second run. We performed similar experiments in which the second set of frequencies was varied from high to low, with identical results. These experiments were performed in 4 different muscle preparations and in 12 contractions in each muscle. In all cases, the second data recording was identical to the first.


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  		Fig. 1.   A: Nyquist plot from a representative experiment in which the dynamic transfer function was determined twice during a single contraction at 70% of maximal force (72 mN/mm2). Frequency of perturbation was varied from 0.5 to 79 Hz for the 1st frequency run (black-triangle), and then the same order was repeated for the 2nd run (open circle ). There was no difference in the 2 transfer functions. Characteristic frequencies fitted to these functions for the 2 runs were 3.6 and 3.8 Hz for process B and 24.3 and 25.8 Hz for process C. B: expanded view of the 8 lowest frequencies (area enclosed in rectangle in A) to show that the fit was adequate for these frequencies. Moduli are expressed as MN/m2.

These experiments confirm that deterioration or phosphate buildup during a contraction could not account for activation-related differences in the dynamic transfer function and that the sequence of variation of perturbation frequency had no effect on the transfer function.

Effect of calcium activation. We found that calcium activation level had a significant effect on the dynamic transfer function. An example from a representative muscle is shown in Fig. 2. The dynamic transfer functions of stiffness are plotted at two levels of calcium activation in a single muscle (constant SL of 2.1 µm). The two calcium concentrations resulted in force generation of 76% (open circles) and 10% (solid circles) of Fmax, respectively. The higher calcium concentration in this case was chosen for display, because this was the highest calcium concentration (pCa 5.92) where high-quality SL measurement and control were possible.


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  		Fig. 2.   Results from a representative experiment. Dynamic transfer function of stiffness is shown at pCa 6.06 () and pCa 5.92 (open circle ). Regression lines represent the fit to a complex function of 2 exponential processes obtained using an iterative computer program. A: effect of perturbation frequency on the magnitude of stiffness. Low-frequency stiffness represents the slope of the force-length relation at the indicated level of activation; high-frequency stiffness is a function of the number of attached cross bridges. The frequency at which stiffness is at a minimum is termed the "dip frequency" (Fmin). B: effect of perturbation frequency on the force-muscle length phase shift. Maximal phase shift occurs at a higher frequency when calcium activation is increased. C: Nyquist plot of the imaginary component (viscous modulus) vs. the real component (elastic modulus) of complex stiffness. Moduli are in MN/m2.

Figure 2A shows the stiffness-frequency relation. Certain similarities between the low and high activation states are apparent. In both cases, a relatively low level of stiffness at the low frequency of perturbation decreases to a minimum as the frequency of perturbation increases. The frequency at which this minimum stiffness occurs is termed "dip frequency" or Fmin. From this minimum, stiffness increases dramatically at the high frequencies of perturbation. The stiffness at low frequencies of perturbation represents the slope of the force-SL relations at the given level of activation. This magnitude of stiffness declines as the perturbation frequency increases, because the speed of perturbation outpaces the force-length mechanism. At the higher perturbation frequencies, stiffness increased sharply because the attached cross bridges do not have the opportunity to complete a full cycle during perturbation. The stiffness in the higher frequencies, therefore, increasingly represents the number of cross-bridge attachments as well as the stiffness of individual cross bridges.

A number of differences are apparent between the two levels of activation. The absolute magnitude of stiffness at any frequency is higher when the muscle is in a higher level of calcium activation. This is intuitively compatible with the higher steady-state force development and the increase in the number of force-generating cross bridges. The slightly higher low-frequency stiffness is empirically attributable to a slightly steeper force-length relation, whereas the markedly increased high-frequency stiffness is attributable to an increase in the number of attached cross bridges. It is also apparent that Fmin is shifted to a higher frequency in the higher level of activation. This is indicative of a more rapid translation of a change in ML to a change in force.

Figure 2B represents the phase-frequency relation for the same muscle. At both levels of activation, there are again similarities. In both cases, a small difference in phase between stress and strain is noted at low frequencies, followed by a large shift in phase that was maximal in the range 4-10 Hz and then diminished at higher frequencies of perturbation as the muscle behaved in a more elastic fashion. The lower level of activation was associated with a much earlier (lower-frequency) shift in phase, again suggesting that increases in calcium activation result in a more rapid translation of ML change to force generation.

Figure 2C represents the Nyquist plot (the imaginary vs. the real component) of complex stiffness. This plot most completely describes the data related to the dynamic transfer function, but its complexity makes quantitative analysis difficult. Therefore, to measure objective differences in the dynamic transfer function at different activation levels and for statistical analysis, we fit the data to Eq. 1.

The effects of calcium activation on force development, the fitted characteristic frequencies, and Fmin (from the experiment depicted in Fig. 2) are shown in Fig. 3. Figure 3A represents the force-calcium relation, fitted to a modified Hill equation. The pCa50 was 5.96, with a Hill coefficient of 6.32. The data at the three highest levels of activation are represented with × symbols. This is to indicate that, at these levels of activation, the sarcomere diffraction pattern was lost, and we are unable to confirm SL control or homogeneity. However, as shown in Fig. 3A, we were able to confirm SL at calcium concentrations that resulted in force generation that was very close to Fmax. Figure 3C shows that the characteristic frequency (b) of process B increases with calcium activation up to a level corresponding to 70% of Fmax. The stiffness data at the highest activation levels (× symbols) seem to indicate a trend for the characteristic frequencies to decrease again. This is difficult to interpret, but we believe this indicates a degree of rigor stiffness at these high calcium concentrations. This is also compatible with the loss of the SL diffraction pattern. We also could not confirm SL control or homogeneity at these calcium concentrations (see DISCUSSION) and did not include these data points in the Hill fit. We chose the Hill fit, inasmuch as we believed that if calcium activation modulated force by an effect on cross-bridge kinetics, this relation would be similar to the force-calcium relation. A similar, although smaller, effect of calcium activation was found for the characteristic frequency (c) of process C (Fig. 3B) and for Fmin (Fig. 3D).


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  		Fig. 3.   Plot of the derived data for the experiment in Fig. 2. Effects of calcium activation on the developed force and on the characteristic frequencies b (cross-bridge attachment rate) and c (cross-bridge detachment rate) of complex stiffness, as well as Fmin, are shown. ×, Data at levels of activation where sarcomere length diffraction patterns could not be resolved. Regression lines represent the data fit to modified Hill equations. A: force-pCa relation. Force was close to saturated at the highest level of activation (pCa 5.92) where sarcomere length could still be controlled. B: effect of calcium on characteristic frequency c. C: effect of calcium activation on characteristic frequency b. D: effect of calcium on Fmin.

The pooled data for eight muscles showing the effect of calcium activation at a constant SL of 2.1 µm are summarized in Fig. 4. Fmax was, on average, 87 ± 13 mN/mm2 at full activation. For comparison, forces at different levels of activation in individual trabeculae were normalized as a percentage of Fmax for each muscle. The force-calcium relation for the pooled data (Fig. 4A) was well fit by a modified Hill equation with a pCa50 of 6.03. Figure 4 also shows the effect of calcium on the characteristic frequencies b (Fig. 4C) and c (Fig. 4B) and Fmin (Fig. 4D). Data are shown up to a calcium level where the SL diffraction pattern was preserved (pCa 5.92). On average, at this level of activation, the force development in individual muscles was 69 ± 9% (SD) of Fmax at saturated calcium levels. On average, the characteristic frequencies b and c (Fig. 4, C and B, respectively) increased by 140 and 44%, respectively, and Fmin increased by 49% (Fig. 4D) over a range of calcium concentrations sufficient to result in force generation from 10 to 69% of maximum saturated force (cf. Table 2).


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  		Fig. 4.   Pooled data from 8 trabeculae illustrating the effect of calcium on the developed force and the frequency parameters of complex stiffness. Data are shown for the levels of activation up to the maximum where sarcomere length could still be accurately controlled. Values (means ± SE) are fit to modified Hill equations. A: force-pCa relation. Force was normalized as a percentage of maximal force development at full activation. B: effect of calcium on characteristic frequency c (rate of cross-bridge detachment). C: effect of calcium activation on characteristic frequency b (rate of cross-bridge attachment). D: effect of calcium on Fmin.


                              
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  		Table 2.   Effect of calcium activation on characteristic frequencies

When tested by multiple linear regression analysis, the effect of calcium activation on parameters b and c, as well as Fmin, was found to be significant (P < 0.001 for b, c, and Fmin). When tested by paired t-test for difference between the lowest and highest levels of activation, the effects on b and c were significant (P = 0.0009 and 0.0026 for b and c, respectively), whereas the effect on Fmin was of borderline significance (P = 0.05).

Effect of SL. In a separate series of experiments, we studied the effects of calcium activation at different SL on cross-bridge kinetics in four muscles. The transfer function of stiffness was determined at three SL (2.0, 2.1, and 2.0 µm) for each of three levels of activation (pCa 6.06, 5.98, and 5.92). Data from a representative muscle are shown in Fig. 5. For simplicity, only the data at 2.0 and 2.2 µm SL at pCa 5.92 are shown. Figure 5A shows that the magnitude of stiffness, as expected, increased with SL, compatible with an increase in the number of force-generating cross bridges. However, unlike calcium, an increase in SL did not have a demonstrable effect on Fmin. Figure 5B shows that changes in SL did not have a significant effect on the frequency at which phase shift was maximal. This is in contrast to the effect of calcium concentration. Figure 5C shows the Nyquist plot of the same data. We fit the data to Eq. 1 for statistical analysis of the pooled data, which are summarized in Fig. 6. Figure 6, A and C, shows the effects of SL (2.0, 2.1, and 2.2 µm) at three levels of calcium activation (pCa 6.06, 5.98, and 5.92) on the characteristic frequencies b and c, respectively; Fig. 6, B and D, shows the effect of SL on force development and Fmin. Although force development consistently increased with SL and calcium activation, there was no effect of SL on the characteristic frequencies or Fmin. By multiple linear regression analysis, the effect of SL on the characteristic frequencies was not significant (P = 0.6 and 0.914 for b and c, respectively), although the effect of calcium activation, independent of SL, was significant (P < 0.001 for b and c).


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  		Fig. 5.   Results from a representative experiment. Dynamic transfer function of stiffness is shown at 2 sarcomere lengths: 2.0 µm () and 2.2 µm (open circle ). Regression lines represent the fit to a complex function of 2 exponential processes obtained using an iterative computer program. A: effect of perturbation frequency on the magnitude of stiffness. Low-frequency stiffness represents the slope of the force-length relation at the indicated level of activation; high-frequency stiffness is a function of the number of attached cross bridges. B: effect of perturbation frequency on the force-muscle length phase shift. C: Nyquist plot of the imaginary component (viscous modulus) vs. the real component (elastic modulus) of complex stiffness. Moduli are in MN/m2.



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  		Fig. 6.   Effect of sarcomere length (2.0, 2.1, and 2.2 µm) on the developed force and frequency parameters of complex stiffness in 4 trabeculae at 3 levels of activation: pCa 6.06 (), 5.98 (open circle ), and 5.92 (triangle ). Values are means ± SE. A: effect of sarcomere length on characteristic frequency b. B: force-sarcomere length relation. Force was normalized as a percentage of maximal force. C: effect of sarcomere length on characteristic frequency c. D: effect of sarcomere length on Fmin.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES

Using the technique of sinusoidal analysis of the dynamic transfer function of stiffness, we studied the effects of calcium activation and SL on cross-bridge kinetics in chemically skinned rat myocardium. We found that increases in calcium activation and SL resulted in increases in the magnitude of stiffness, especially in the high perturbation frequencies (Figs. 2 and 5). This is compatible with an increase in the number of force-generating cross bridges and overall force development. However, we also found that increases in calcium concentration altered the dynamic transfer function in ways that were not seen with increases in SL; notably, there was an increase in the frequency at which stiffness is at a minimum (Fmin) as well as the frequency at which the SL-stiffness phase shift was maximal. This suggests that an increase in calcium activation increases the speed with which a change in length can be translated into a change in muscle force generation. Because the rate-limiting step of such a process is likely to be the rate at which cross bridges can transfer from passive states to force-generating states, the data support the hypothesis that increases in calcium activation increase the apparent rate of cross-bridge attachment.

To quantitate and statistically test our findings, we fit the transfer function data in complex space to two parallel exponential processes (B and C) according to Eq. 1, as described by Kawai and Brandt (20). The rate constants of these processes have been proposed to represent the rate constants of groups of steps involved in the transition between non-force-generating and force-generating cross-bridge states (23). The derived curves fit the data well enough for complete reconstruction of the transfer functions from the fitted parameters (Figs. 2 and 5); therefore, we believe that the derived parameters were good measures for statistical analysis. We found that an increase in calcium activation up to pCa 5.92, which resulted in 0-70% Fmax, resulted in an increase in the magnitudes and the derived rate constants of processes B and C. In contrast, changes in SL resulted in increases in the magnitudes but not the rate constants of processes B and C. Using multiple linear regression analysis, we were able to show that the effect on the rate constants was specific to calcium activation. This confirms our intuitive analysis of the magnitude and phase spectra and suggests that calcium concentration had an effect on cross-bridge kinetics.

Apart from reproducibly fitting the data and describing the dynamic transfer function for statistical analysis, the complex fit parameters have been investigated as a tool for measuring specific aspects of cross-bridge kinetics. Kawai and co-workers determined the sensitivity of processes B and C to phosphate, ADP, and ATP concentrations (18, 19, 22, 23) and correlated their findings with the cross-bridge cycle deduced from biochemical studies of extracted proteins (27) to deduce that process B represents transitions to force-generating states, whereas process C represents reversible transitions to non-force-generating states. According to this interpretation, our results suggest an increase in the rate of cross-bridge attachment with calcium. Naturally, this type of interpretive analysis is somewhat dependent on the biochemical cross-bridge model. Different, empirical analyses by Berman et al. (3) and Campbell et al. (5) involve minimal model-dependent assumptions and confirm homology between muscle and chamber stiffness spectra to provide strong arguments that the stiffness spectra represent the processes of cross-bridge kinetics.

The graded cross-bridge kinetics and the recruitment hypotheses have been previously investigated in cardiac and skeletal muscle by time domain analyses of velocity of shortening, step length perturbations, and frequency domain analyses of the effects of sinusoidal perturbations in activated muscles at steady state. The answers have remained controversial, because evidence has been advanced in support of both mechanisms. Using the force-velocity relation as an index of cross-bridge kinetics in very similar experiments, Podolsky and Teichholz (33) found evidence in favor of an on-off calcium switch, whereas Julian (16) presented evidence in favor of changes in cross-bridge kinetics. Hofmann and Moss (13) found that increasing levels of calcium activation resulted in an increase in shortening velocity in rat ventricular myocardium, favoring an effect on cross-bridge kinetics. In cardiac and skeletal muscle, the rate constant of isometric force redevelopment after a step length change at steady-state activation (Ktr) has been proposed to reflect the apparent rate of transition from passive to force-generating states (4). Application of this technique to studies of cross-bridge kinetics has also returned conflicting results. For example, Hancock et al. (11, 12) found in rat and ferret myocardium that Ktr was not affected by calcium activation, whereas Wolff et al. (42) and Baker et al. (2) in rat myocardium and Brenner (4) and Metzger and Moss (31) in skeletal muscle fibers found that Ktr was accelerated at higher levels of activation. Although some of these differences in findings may reflect differences in species or muscle type, the findings of Hancock et al. in favor of the recruitment hypothesis in rat and ferret myocardium suggest that some of the controversy may relate to differences in loading conditions, experimental technique, and analysis.

The conflicting conclusions of the above studies illustrate the complexity involved in interpretation of studies on cross-bridge kinetics. All our present approaches to this issue involve essentially indirect measurements, and the confounding effects of possible internal viscous loads or force sensitivity of the measurements are somewhat unpredictable. Furthermore, interpretations of findings are, by definition, model dependent. Sinusoidal analysis is, in theory, equivalent to transient analysis (as in the determination of Ktr) and subject to the same assumptions and model dependency. However, we believe that sinusoidal analysis has the advantage that the muscle is studied in steady-state conditions rather than during large changes in loading conditions. The complexity and possible load sensitivity of the activation process in muscle make this an attractive feature, because the activation process is perturbed only at the myofilament level with very small length perturbations.

Using sinusoidal analysis, Kawai et al. (21) found in skinned skeletal muscle fibers that the complex stiffness of the activated muscle changed with calcium in a way that suggested a possible change in a rate constant of at least one step in the process of cross-bridge attachment, resulting in an apparent increase in the rate of process B. However, in this study, the lower levels of activation were associated with the appearance of a phase change in the phase-frequency relation, which suggested the presence of an additional process with a rate constant close to that of process B. The authors therefore concluded that the apparent change in the rate of attachment may reflect a separate force-generating state rather than a calcium dependence of the intrinsic rate constant. In our study we found no evidence for an additional process in cardiac muscle, although we cannot rule out this possibility. Using pseudorandom binary noise length modulation, Rossmanith et al. (35) also examined the effect of calcium activation in rat papillary muscle. These investigators found no effect of calcium on Fmin but did not analyze their data in complex space. Their data do seem to suggest a phase effect of calcium, which would likely affect the complex fit. Analysis of the stiffness spectrum by determination of Fmin is not as sensitive as the overall complex fit, because it is determined only from the frequency-stiffness relation, with exclusion of data provided by the frequency-phase relation. This did appear to be the case in our experiments also, because the effect of calcium on Fmin was smaller than the effect on characteristic frequency b.

In intact rabbit papillary muscles in barium contracture, Shibata et al. (38) found no effect of level of activation on the phase-frequency relation, although the characteristic frequencies were not determined. This finding is somewhat in conflict with our findings. This discrepancy suggests that the changes in cross-bridge kinetics we identified are mediated by calcium but not barium, possibly via a calcium-specific binding site, and also serves as confirmation that complex stiffness is not affected by force per se. Findings reported by Metzger and Moss (31) showing that Ktr remained calcium sensitive in skeletal muscle after removal of troponin subunits or myosin light chain-2 suggest that the calcium effect may be partly independent of these proteins.

In our studies the effect of SL on the dynamic transfer function of stiffness was clearly different from the effect of calcium. Although force increased at least twofold over the range of SL tested, there was no change in the characteristic frequencies. This suggests that SL modulates force generation via recruitment of cross bridges and also serves as an internal control for the observed calcium effect. The ranges of forces were similar in both groups. We cannot determine the exact mechanism for the SL recruitment effect. It could involve an increase in calcium sensitivity of troponin C, it may be modulated by other contractile subunits, or it may involve changes in the structural relations of actin and myosin molecules. Our findings are somewhat in conflict with a study by McDonald et al. (29), who found that Ktr varied with SL in rat soleus fibers, suggesting possible fiber-specific mechanisms.

By definition, steady-state force divided by high-frequency stiffness returns the calculated step length release expected to abolish force generation. From our high-frequency stiffness data, we predict that a step length release of 12.9 nm would abolish force development in an activated cardiac fiber. This is close to, and not statistically different from, the value of 12 nm previously reported from analysis of T1 curves (1) and represents a 1.2% SL change, which is also close to previous estimates of an overall ML change of 1.6-1.85% (23, 34, 37). A likely explanation for the higher value we found, compared with T1 curves, is that we did not measure stiffness at very high frequencies (>100 Hz) and, therefore, probably underestimate high-frequency stiffness somewhat. The percent length change we predict is smaller than that found by others, possibly because of differences in fiber end compliance and SL control.

Certain limitations regarding our studies need to be considered. First, the highest level of activation at which we are confident about our stiffness data is pCa 5.92, which resulted in force generation of 61 ± 8 mN/mm2, which was 69 ± 9% (SD) of Fmax. Above this, we lost SL measurement and we could not confirm muscle homogeneity. The force-high-frequency stiffness relation for our pooled data was linear up to the level of activation where SL could be confirmed. With saturated activation the data had more scatter and fell off the linear relation slightly. We also noted some scatter in our characteristic frequencies in this activation range, with a tendency for these frequencies to decrease again. This scatter may represent true properties of cardiac muscle, possibly a switch to a predominant cross-bridge recruitment mechanism or, as we suspect, an element of rigorlike stiffness at supraphysiological calcium concentrations. Alternatively, this phenomenon may represent an artifact due to insensitivity of the sinusoidal length perturbation technique at levels of activation where the force-length relation is very shallow. Because we could not resolve a SL diffraction pattern at these activation levels, we cannot be certain that we are not seeing the effects of internal shortening during data collection. Accordingly, we are not certain whether a real transition in the calcium-force mechanism occurs at these calcium concentrations or whether these changes are an artifact of our preparation. The loss of SL diffraction pattern at high levels of activation is a well-described phenomenon in skinned cardiac muscle and is present in all but the thinnest trabecular preparations.

A second limitation of our study is that we did not uniformly collect data at perturbation frequencies >70 Hz and, therefore, were not able to resolve a third process (process D) identified by Kawai (36) in ferret myocardium. We do not believe that this significantly impacts on our interpretation of the effects of calcium on cross-bridge kinetics or our ability to resolve processes B and C for statistical analysis, because the major features of the dynamic transfer function fell well within the range of perturbation frequencies used. In initial experiments we collected data to 100 Hz but were unable to consistently, satisfactorily fit process D. We noted, however, that whereas the magnitude parameters were affected somewhat, the values for characteristic frequencies b and c were not significantly altered by the addition of a third process to the fitting procedure. Therefore, we subsequently limited our data collection to a maximum frequency of 70 Hz.

The use of chemically skinned fibers may also introduce certain limitations, because length-dependent phenomena may be altered. Alternatives to the use of skinned fibers include the use of barium contractures (36-38) or high-caffeine tetanic contractions (14). Saeki et al. (36) showed that, compared with barium contracture, there is no effect of chemical skinning on myocardial cross-bridge kinetics. In addition, a study by Baker et al. (2) using intact rat myocardium supports our finding that calcium affects cross-bridge kinetics. The use of skinned fibers has the advantage of anatomically isolating the myofilaments. This allows for the study of specific myofilament kinetics, but caution must be used in extrapolation of findings to the intact state.

The sinusoidal analysis approach does not allow us to study the rate of cross-bridge detachment at the end of the cross-bridge cycle, because this process is too slow to be resolved by this method. We have, however, studied this previously by measuring the rate of ATP consumption during contraction and found no effect of SL or calcium activation on the rate of cross-bridge detachment (41). Our approach has been to look for evidence of calcium- or length-mediated changes in cross-bridge kinetics. Our methods do not allow us to confirm or rule out the recruitment hypothesis. Although we believe that we have found evidence of calcium-mediated changes in cross-bridge kinetics, we cannot rule out the possibility that both mechanisms are active in cardiac muscle.

In conclusion, we studied the mechanisms of modulation of force development of chemically skinned rat myocardium. We found that calcium activation induced specific changes in the dynamic transfer function of stiffness. These effects on the transfer function were not found with changes in SL, despite similar levels of force development. We interpret our findings to suggest that calcium activation modulates force in part via an increase in the rate of cross-bridge attachment.


    ACKNOWLEDGEMENTS

This study was supported by American Heart Association, North Carolina Affiliate, Grant NC-94-GS-42 (to T. Wannenburg) and National Center Grants 95-012390 (to T. Wannenburg) and 94-006380 and 95-001050 (to P. P. de Tombe); National Heart, Lung, and Blood Institute Grants HL-03255 (to T. Wannenburg) and HL-52322 (to P. P. de Tombe); and the Whitaker Foundation for Biomedical Research (P. P. de Tombe). P. P. de Tombe is an Established Investigator of the American Heart Association.


    FOOTNOTES

A preliminary report of this work has been presented in abstract form (see Ref. 40a).

Present addresses: P. P. de Tombe, Dept. of Physiology and Biophysics (M/C 902), University of Illinois at Chicago, 900 S. Ashland Ave., Chicago, IL 60607-7171; P. M. L. Janssen, Dept. of Cardiology and Pneumology, University of Göttingen, Robert-Koch Str. 40, D37075 Göttingen, Germany.

Address for reprint requests and other correspondence: T. Wannenburg, Section on Cardiology, Wake Forest University School of Medicine, Medical Center Blvd., Winston-Salem, NC 27157-1045 (E-mail: twannen{at}wfubmc.edu).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

Received 26 April 1999; accepted in final form 31 January 2000.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES

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