INTRODUCTION

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PARMLEY AND CHUCK (17) first showed that the immediate increase in cardiac muscle tension due to a step increase in length is followed by a slow rise in stress that continues for at least 10 min in cat papillary muscle. Similar slow changes in stress (SCS) in response to step changes in length or volume have been observed in a variety of preparations and species, including isolated rat and cat trabeculae and papillary muscles (1), single guinea pig cardiac ventricular myocytes (21), and the intact left ventricle in dog (12) and rat (20). These changes are of large magnitude. In rabbit right ventricular papillary muscles (4), SCS accounted for about one-third of the total change in active stress after a step change in length.

The slow increase in active stress after a sustained length increase is accompanied by a parallel slow increase in the magnitude of the intracellular Ca²⁺ transient (1). However, recent experiments (4) showed that SCS in rabbit right ventricular papillary muscles is not dependent on sarcoplasmic reticulum (SR) Ca²⁺ content and is not significantly altered by SR Ca²⁺ depletion or inhibition of SR Ca²⁺ reuptake. If then SCS is due to a slow increase in myoplasmic Ca²⁺ but does not depend on the SR, altered sarcolemmal ion fluxes are a possible mechanism of the SCS phenomenon.

Much of the extensive experimental data on the ionic kinetics of the cardiac myocyte are summarized within various mathematical models of the cardiac action potential that have included an increasing number of sarcolemmal ion channels (3, 5, 9, 14). The dynamics of excitation-contraction coupling in general and intracellular Ca²⁺ transients in particular have also been investigated in detail. Therefore, Luo and Rudy (15) published a refined version of their earlier action potential model that includes the SR with Ca²⁺ uptake and release and several intracellular Ca²⁺ buffers. Sarcolemmal Ca²⁺ currents are more accurately described than by previous models, making it possible to account for changes in intracellular Ca²⁺ concentration ([Ca²⁺]_i).

Because myocardial force development is directly related to intracellular Ca²⁺, we used the Luo-Rudy model (15) to investigate potential contributions of sarcolemmal ion fluxes to SCS. By analyzing the effects of a step change in each parameter of individual ion fluxes on the subsequent force development, we used the model to test whether a stepwise change in a sarcolemmal ion current accompanying stretch could provide a potential mechanism for SCS.

Glossary

ACap	Capacitive membrane area
[B]	Concentration of ion B
Ca50	Ca2+ concentration for half-maximum force generation
Cm	Membrane capacitance
fNa,K	Voltage dependence parameter of Na+-K+ pump
F	Faraday constant
	Maximum permeability for current x
	Maximum current for flux x
ICa,b	Ca2+ background leakage current
ICa	Ca2+ current through L-type Ca2+ channel
ICa,K	K+ current through L-type Ca2+ channel
ICa,Na	Na+ current through L-type Ca2+ channel
Ii	Sum of all ionic currents
IK	Time-dependent K+ current
IK1	Time-independent K+ current
IKp	Plateau K+ current
Ileak	Ca2+ leakage from network SR to myoplasm
INa	Fast Na+ current
INa,b	Na+ background leakage current
INa,Ca	Current through Na+/Ca2+ exchanger
INa,K	Current through Na+-K+ pump
Ins	Nonspecific Ca2+-activated current
Ip(Ca)	Sarcolemmal Ca2+ pump current
Irel	Ca2+ release from junctional SR to myoplasm
Ist	Stimulus current
Itr	Ca2+ transfer from network SR to junctional SR
Iup	Ca2+ uptake from myoplasm into network SR
Km,x	Half-saturation concentration for Ix
n	Hill's coefficient of force-pCa relation
PB	Membrane permeability for ion B
R	Gas constant
V	Membrane potential
VC	Compartment volume
zB	Valency of ion B
	Extracellular Na+ concentration dependence factor of fNa,K

	METHODS

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The definitions of the slow change in stress (SCS) and its half time (t_1/2) are shown in Fig. 1.

View larger version (11K): [in this window] [in a new window]   Fig. 1.   Definition of slow change in stress (SCS). In this example from an experimental protocol (4), muscle length was increased from 85 to 95% of Lmax (muscle length at maximal isometric active stress). Active stress before length change (T1) immediately increased after length change (T2) and then further increased to a new steady state (T3) after ~15-20 min. SCS was defined as (T3  T2)/(T3  T1); i.e., SCS expressed slow increase in active stress (T3  T2) as a fraction of total increase in active stress between steady states (T3  T1). t1/2, Time after which 50% of slow change in stress is completed. [Adapted from Bluhm and Lew (4).]

The Luo-Rudy model (15) is based on a numerical reconstruction of the action potential using the following equation where V is the membrane potential, C_m is the membrane capacitance, I_st is a stimulus current, and I_i is the sum of 13 ionic currents through sarcolemmal ion channels, pumps, and exchangers, as summarized schematically in Fig. 2. The ionic currents are determined by voltage-dependent ion gates that are obtained as the solution to a system of coupled differential equations. The equations for the Na⁺-K⁺ pump are provided in the APPENDIX, and the original publication (15) gives the equations for all ionic fluxes.

View larger version (21K): [in this window] [in a new window]   Fig. 2.   Schematic of ionic model. Ionic model by Luo and Rudy (15) was used for computations of this study. Sarcoplasmic reticulum (SR) is represented as 2 distinct compartments: network SR (NSR) and junctional SR (JSR). Model calculates dynamic changes in intracellular concentrations of Ca2+, Na+, and K+. Ca2+ ions are buffered by troponin (TRPN) and calmodulin (CM) in myoplasm and by calsequestrin (CS) in JSR. Force was calculated from a sigmoidal function of intracellular free Ca2+ concentration ([Ca2+]i). We simulated two different muscle lengths by assuming myofilament sensitivities to Ca2+ (Ca50) of 1 µM (95% of Lmax) and 2 µM (85% of Lmax). A Hill coefficient of 2.5 was chosen for either force-Ca2+ relation. See Glossary for definition of other abbreviations. [Reproduced with modifications from Luo and Rudy (15) with permission of the American Heart Association].

The model accounts for changes in [Ca²⁺]_i and intracellular concentrations of K⁺ ([K⁺]_i) and Na⁺ ([Na⁺]_i) using the following equation where [B] is the concentration of ion B, I_B is the sum of ionic currents carrying ion B, A_Cap is the capacitive membrane area, V_C is the volume of the compartment where [B] is calculated, z_B is the valence of ion B, and F is the Faraday constant.

Force was computed as an output variable of the model from a sigmoidal function of [Ca²⁺]_i with a Hill coefficient of 2.5, representing the binding of Ca²⁺ to troponin C. The Hill coefficient chosen was similar to published values (7, 8). Force generation at 85 or 95% of L_max (muscle length at maximal isometric active stress) was simulated using low or high myofilament sensitivities to Ca²⁺ (Ca₅₀, i.e., Ca²⁺ concentration for half-maximum force generation). Ca₅₀ equal to 1 µM was chosen at 95% of L_max by assuming a 50% saturation of the myofilaments by the peak of the Ca²⁺ transient. At 85% of L_max, Ca₅₀ equal to 2 µM was chosen to reflect the decrease in myofilament sensitivity (10). With these values for the Hill coefficient and the myofilament sensitivity, simulated step changes from 85 to 95% of L_max produced force changes similar to those observed experimentally (4). All other model parameters were used as published previously (15), except for _Na,K, the maximum current through the Na⁺-K⁺ pump, which was adjusted from 1.5 to 1.7 µA/µF for immediate stability of [Na⁺]_i and [K⁺]_i. Muscle contractions were simulated to occur at 5-s intervals.

The iterative integration algorithm (14) is based on an analytic solution to some of the differential equations (dealing with the kinetics of the gates), which assumes the rate constants to be unchanged for sufficiently small time increments. A fourth-order Runge-Kutta scheme with adaptive time stepping was used to integrate the other equations. The equations were programmed in C on a Silicon Graphics R4400 workstation (Mountain View, CA) using double precision.

	RESULTS

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SCS. A step in myofilament sensitivity from 2 to 1 µM generated a step increase in force from 0.143 · F_max to 0.485 · F_max (where F_max is maximum generated force), and the peak value of the Ca²⁺ transient was 0.976 µM. Model-simulated muscle contractions were stable over long periods of time, with changes of <0.1% over 15 min.

View larger version (8K): [in this window] [in a new window]   Fig. 3.   Step change in L-type Ca2+ current. In addition to step change in Ca50, permeability of L-type channel to Ca2+ (PCa) was doubled. Force increased from 0.143 · Fmax (where Fmax is maximum generated force) before step to 0.649 · Fmax immediately after step. Force further increased to 0.675 · Fmax at 5th beat after step and to 0.692 · Fmax after 15 min. Therefore, additional increase in force after length step is too small and occurs too fast to resemble slow changes in stress seen experimentally.

View larger version (9K): [in this window] [in a new window]   Fig. 4.   Step change in Na+-K+ pump current. In addition to step change in Ca50, maximum current through Na+-K+ pump (Na,K) was decreased by 20%. Force increased from 0.143 · Fmax immediately before step to 0.483 · Fmax immediately after length step and continued to increase to 0.658 · Fmax over following 15 min, amounting to a 34% slow change in stress (SCS); t1/2 of this simulated SCS was 2.6 min.

Mechanism of simulated SCS. The mechanisms for the simulated SCS are illustrated in Fig. 5. The decrease in _Na,K resulted in a decrease in I_Na,K. Peak systolic pump activity dropped from 0.599 to 0.479 µA/µF, and diastolic activity decreased from 0.315 to 0.252 µA/µF. The diminished Na⁺ extrusion caused [Na⁺]_i to rise from 10 to 12.5 mM 15 min after the step. This led to a recovery of the Na⁺-K⁺ pump, a systolic pump activity of 0.559 µA/µF, and a diastolic pump activity of 0.294 µA/µF, allowing [Na⁺]_i to stabilize at the increased concentration. The increase in [Na⁺]_i was accompanied by an increase in systolic Ca²⁺ entry through the Na⁺/Ca²⁺ exchanger. Peak systolic I_Na,Ca doubled from 0.578 to 1.145 µA/µF, with little change in diastolic I_Na,Ca. The increase in Ca²⁺ entry caused the Ca²⁺ transient to increase from 0.976 to 1.299 µM and, therefore, the increase in force.

View larger version (12K): [in this window] [in a new window]   Fig. 5.   Mechanisms of simulated SCS. Na,K was decreased by 20% concurrently with change in Ca50 (shown in Fig. 4). Consequently, actual current through Na+-K+ pump decreased. Decreased pump activity caused intracellular Na+ concentration ([Na+]i) to increase. Because of resulting partial recovery of Na+-K+ pump activity, [Na+]i restabilized after 15 min. Increase in [Na+]i resulted in an increase in systolic activity of Na+/Ca2+ exchanger and, hence, an increase in Ca2+ entry into cell during systolic phase of exchanger. This increased Ca2+ entry yielded increase in intracellular Ca2+ transients (bottom) and slow increase in force shown in Fig. 4. Right: last beats of INa,K, INa,Ca, and [Ca2+]i shown on an expanded time scale to distinguish between systolic and diastolic phases.

Simulated experimental interventions. By use of the slow force response to the step changes in Ca₅₀ and _Na,K as the control, several experimental interventions were simulated. SR Ca²⁺ uptake was inhibited by setting _up, Ca²⁺ uptake from myoplasm into network SR, equal to 0 (Fig. 6, left of double hatch mark). This led to SR Ca²⁺ depletion, a decline of the peak Ca²⁺ transient to 0.52 µM, and a decline of peak force to 0.033 · F_max. When the step changes in Ca₅₀ and _Na,K were imposed, the peak Ca²⁺ transient increased to 0.64 µM, and force rose from 0.162 · F_max to 0.244 · F_max after 15 min (39% SCS). Therefore, the simulated SCS did not depend on the SR, similar to experimental observations (4). However, with the SR intact, the 34% model SCS (Fig. 4) was accompanied by a 10% increase in SR Ca²⁺ content (not shown). Experimentally (4), a 33 ± 12% SCS was accompanied by a 13 ± 9% increase in SR Ca²⁺ content as measured by rapid cooling contractures (n = 19).

View larger version (11K): [in this window] [in a new window]   Fig. 6.   Simulated SR Ca2+ depletion. SR was depleted of Ca2+ by inhibition of Ca2+ uptake into SR. Because of SR Ca2+ depletion, force and intracellular Ca2+ transients declined over first 10 beats. Ca50 and Na,K were then changed as in Figs. 4 and 5. Force increased from 0.033 · Fmax before step to 0.162 · Fmax immediately after step and then continued to increase to 0.244 · Fmax over 15 min; SCS was 39%. Hence, simulated SCS was not dependent on SR.

View larger version (11K): [in this window] [in a new window]   Fig. 7.   Simulated length steps during diastolic phases of contraction. Protocol is illustrated in top panels, resulting in force shown in bottom panels. Changes in Ca50 and Na+-K+ pump current were maintained only during 4-s periods between contractions. Similar to experiments performed by Nichols (16) and Allen et al. (2), a slow increase in force was observed, although systolic muscle contraction never occurred at an increased muscle length. * Same time in all panels.

View larger version (11K): [in this window] [in a new window]   Fig. 8.   Simulated length steps during systolic phases of contraction. Protocol is illustrated in top panels, resulting in force shown in bottom panels. Changes in Ca50 and Na+-K+ pump current were maintained only during 1-s periods immediately after stimuli. Immediate increase in force from 0.143 · Fmax to 0.483 · Fmax was identical to results of maintained length step (Fig. 4); however, slow increase in stress (SCS) was only 8% (to 0.514 · Fmax after 15 min) compared with 34% SCS obtained with maintained length step. * Same time in all panels.

View larger version (14K): [in this window] [in a new window]   Fig. 9.   Effect of extracellular Ca2+ concentration ([Ca2+]o). [Ca2+]o was varied as indicated. Ca50 and Na+-K+ currents were changed as in Fig. 4. SCS decreased with increasing Ca2+ concentration. A similar trend was observed experimentally (17).

Parameter sensitivity analysis. We examined the sensitivity of the model results to changes in the parameters of the force-Ca²⁺ relations (Table 2). All simulations used the same 20% step decrease in _Na,K as in the simulations in Figs. 4 and 5. Although there was only a small increase in the magnitude of SCS with an increase in the Hill coefficient (condition 2), SCS increased significantly with several changes in myofilament sensitivity (conditions 3-5). In the last two cases (conditions 4 and 5), the changes in myofilament sensitivity simulated a smaller length step. The increased magnitude of SCS in these cases indicates that a smaller step change in _Na,K would have been sufficient to model the same magnitude of SCS as with the original step (condition 1). The t_1/2 of SCS, in contrast to its magnitude, was relatively insensitive to parameter changes.

	DISCUSSION

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In this study, a theoretical ionic model was used to investigate whether SCS could be caused by length-induced step changes in individual sarcolemmal ion currents. Step changes in the model parameters of sarcolemmal Ca²⁺ or K⁺ fluxes did not yield force changes that could resemble the experimentally observed SCS. SCS, however, could be reproduced by step changes in the parameters of I_Na,K or I_Na,b. The increase in [Na⁺]_i concurred with an increase in systolic Ca²⁺ entry through the Na⁺/Ca²⁺ exchanger, which caused an increase in Ca²⁺ transients and thus force. This simulated SCS compared well with experimental observations in its response to several simulated interventions. Therefore, the mechanisms of SCS may involve primary changes in sarcolemmal Na⁺ fluxes.

The model chosen for this study (15) was primarily designed as a model of the mammalian ventricular action potential. However, the model is also able to account for dynamic changes in intracellular ion concentrations, which distinguishes it from several previous action potential models (3, 5, 9, 14). This can be attributed to several factors. A considerable number of sarcolemmal ion channels, exchangers, and pumps are included, and their currents could be more accurately described than by the previous models because of a substantial amount of recent single cell and single channel patch-clamp data. Accurate intracellular Ca²⁺ transients, which are clearly important for the model to be applied to study myocyte contraction, were obtained by including an SR with Ca²⁺ uptake and release as well as several intracellular Ca²⁺ buffers.

The description of sarcolemmal K⁺ and Na⁺ fluxes proved sufficient to maintain intracellular K⁺ and Na⁺ homeostasis, since [K⁺]_i and [Na⁺]_i remained stable over >15 min. This is important, because changes in [K⁺]_i and [Na⁺]_i led to secondary changes in sarcolemmal Ca²⁺ fluxes and, hence, in myocardial contraction. On the basis of more recent experimental work, the delayed rectifier current was later replaced by two distinct components (25). Most recently, the ATP-sensitive K⁺ current was also introduced into the model (19). Although these changes present important refinements of the version used in our simulations (15), they should not affect the present conclusions.

We introduced force into the model by means of two sigmoidal force-pCa relations with a Hill exponent of 2.5 and myofilament sensitivities (Ca₅₀) of 2 and 1 µM to resemble myocardial contraction at 85 and 95% of L_max. These parameters fall within the wide range of published experimental data. Reported myofilament sensitivities at maximum muscle length or sarcomere length range from Ca₅₀ of 1.45 µM (10) to Ca₅₀ of 3.39 µM (8) in skinned preparations, whereas estimates in intact muscle are much lower, ranging from Ca₅₀ of 0.50 µM (24) to Ca₅₀ of 0.62 µM (7). Hill coefficients of 1.67 (10) to 2.72 (7) have been reported in skinned preparations, although estimates in intact muscle are higher, ranging from 6.08 (24) to 4.87 (7). Some of this variation can probably be attributed to differences in species (13) and temperature (8). The Hill coefficient and myofilament sensitivity chosen in this model were within the range of reported values and produced force responses to simulated length steps that were similar to typical experimental observations (4). We found that the time course of SCS was relatively insensitive to changes in Hill coefficient and myofilament sensitivities (Table 2). The main conclusion of this study, i.e., identifying I_Na,K and I_Na,b as the most likely candidates for a mechanism of the slow tension response to length changes, is therefore independent of these parameter choices.

Although the current parameters of the model were based on experimental data from guinea pig ventricular myocytes at 37°C (15), we compared the model results with our experimental data obtained from rabbit papillary muscles at 23°C (4). The temperature difference might in part explain why the model-predicted t_1/2 of SCS of 2.6 min underestimated the experimental t_1/2 of 4.3 min (4), since Parmley and Chuck (17) stated that SCS was more rapid at higher temperatures.

One limitation of the force-pCa model relations is the unrealistic time course of force development. Maximum force reached its peak value within ~15 ms after stimulation, which is faster than that observed experimentally (typically ~100 ms). Furthermore, there was no force plateau, and force decline was relatively fast. This discrepancy of time courses is related to the instantaneous force-Ca²⁺ relation used in the model. In contrast, experimental data show that the intracellular Ca²⁺ transient is significantly faster than the time course of force development (23). The model did not include actin-myosin kinetics and, therefore, could not account for the fact that cross bridges can stay attached even after the Ca²⁺ transient declines (18). Furthermore, Yue (23) suggests that the intracellular Ca²⁺ transient might be related more closely to the rate of force development than to force itself. These limitations regarding the time course of force development do not affect the conclusions of this study, however, since the slow changes in stress were shown to be accompanied by slow changes in the peak intracellular Ca²⁺ transient (1).

The model showed that a step change in myofilament sensitivity alone produced only an immediate step in active force, but no additional slow component. By design, the model does not include time-dependent changes in myofilament sensitivity. However, the model does include Ca²⁺ buffering by troponin, calmodulin, and calsequestrin. The lack of an SCS after a step change in myofilament sensitivity, therefore, suggests that Ca²⁺ buffering is not changed in such a way as to contribute to the SCS. The model results do not rule out the possibility that SCS might be related to length-induced slow changes in myofilament sensitivity. However, this would be an unlikely sole mechanism of SCS, since SCS was shown to be accompanied by an increase in the magnitude of the intracellular Ca²⁺ transient (1).

Further simulations tested the effects of step changes in individual parameters governing the various sarcolemmal ion fluxes on the subsequent force development in the model. SCS has been shown to be accompanied by an increase in the magnitude of the intracellular Ca²⁺ transient (1, 2) and to be independent of the SR (4). Because Ca²⁺ not supplied by the SR enters the cell mainly through the L-type Ca²⁺ channel, SCS might be mediated by stretch-induced changes in I_Ca. Although the model results do not support such a mechanism, Ca²⁺ entry might be slowly increased secondary to other sarcolemmal or intracellular processes. Only 2 of the 13 ionic fluxes included in the model produced changes in force that resembled the experimentally observed SCS. These currents are I_Na,K and I_Na,b.

To evaluate whether the experimentally observed SCS might be mediated by this mechanism, several other experiments from the literature were simulated. A model SCS was simulated by a step increase in myofilament sensitivity and a concurrent step decrease in _Na,K. This model SCS was then subjected to several simulated experimental interventions.

First, we examined whether the model SCS depended on the SR for comparison with experimental results (4). As in the experiments, the model SCS was SR independent. After simulated SR uptake inhibition and resulting SR depletion, the model SCS was not abolished but, in fact, slightly increased compared with control, thus showing that SR Ca²⁺ was not a necessary requirement. However, in the presence of a functional SR, the slow increase in stress was accompanied by a 10% increase in SR Ca²⁺ content, similar to the 13% increase observed experimentally (4). This confirms the suggestion that these two results, concomitant increases in SR Ca²⁺ content and active stress, on the one hand, and the SR independence of SCS, on the other hand, are only seemingly contradictory. Rather than being a primary cause for SCS, the increase in SR Ca²⁺ content was secondary to increased sarcolemmal Ca²⁺ entry in the model.

The model also offered a potential explanation for the observations by Nichols (16) and Allen et al. (2) that the slow increase in active force development can be triggered by diastolic length changes alone. They found that increasing muscle length only during the diastolic periods and returning it to the original length before each contraction still produced a slow increase in stress. The mechanism suggested by the model reproduced these experimental findings, since the Na⁺-K⁺ pump is constantly active, i.e., during systole as well as during diastole. A simulated length change (step changes in Ca₅₀ and _Na,K) that was maintained only during 4-s diastolic periods produced a considerable slow increase in force. Conversely, when the length changes were simulated only during a 1-s systolic period, the immediate step increase was followed by a much smaller slow increase than in the case of a maintained step change.

We also subjected the model SCS to changes in [Ca²⁺]_o. Similar to experimental data (17), the model SCS was attenuated with increasing [Ca²⁺]_o. This simulation provides a simple example in which the proposed mechanism of SCS is sensitive to inotropic interventions.

In the simulations in the second half of this study, the change in the Na⁺-K⁺ pump was adopted as a simulated mechanism for SCS. However, changes in Na⁺ homeostasis may not necessarily be mediated through the Na⁺-K⁺ pump. In the model, SCS was most sensitive to changes in _Na,K, which had to be decreased only by 20% to produce a magnitude of SCS as seen in the experiments. However, a change in I_Na,b also resulted in a similar SCS, although a larger change was required (+100%). To the best of our knowledge, a length dependence has not been demonstrated for any of the above ion fluxes.

The Na⁺-K⁺ pump plays an important role in the regulation of myocardial contraction in the heart. Several processes alter force development through changes in the activity of the Na⁺-K⁺ pump, which is the major mechanism for the inotropic action of cardiac glycosides (11). The Na⁺ pump lag hypothesis for the force-frequency relation of cardiac muscle states that the greater Na⁺ entry due to more frequent stimulation is balanced by increased Na⁺ pump activity but only at the cost of elevated [Na⁺]_i and, hence, increased Ca²⁺ entry. Mechanical restitution may be mediated in part through changes in [Na⁺]_i. Wilde and Kléber (22) observed in guinea pig ventricular trabeculae that a transition from regular stimulation to quiescence was followed by a decline in intracellular Na⁺ activity with a time course of ~1.5 min. In sheep cardiac Purkinje fibers, Eisner et al. (6) found that Na⁺-K⁺ pump inhibition with strophanthidin resulted in parallel increases in tension and intracellular Na⁺ activity with a t_1/2 of ~5 min.

In summary, step changes in the parameters of sarcolemmal ion currents were analyzed with a theoretical ionic model. The results suggested that SCS may be caused by length-induced step changes in sarcolemmal Na⁺ fluxes, leading to an increase in [Na⁺]_i and a concurrent increase in systolic Ca²⁺ entry through the Na⁺/Ca²⁺ exchanger. Future studies need to determine whether this proposed mechanism is indeed responsible for SCS and how sarcolemmal Na⁺ flux might be altered by changes in muscle length.