The ionic model of the ventricular myocyte developed by Luo and Rudy (Circ. Res. 74: 1071–1096, 1994) was used to investigate potential mechanisms of the slow changes in stress (SCS) that follow step changes in muscle length. A step change in myofilament sensitivity alone caused an immediate increase in active tension, but no SCS. The effects of additional step changes in the parameters of sarcolemmal ion fluxes were examined for each ion flux in the model. Changes in the coefficients of Ca2+ or K+ channels did not produce SCS. SCS was produced by step changes in parameters of the Na+-K+pump or the Na+ leak current. This simulated mechanism was mediated through a slow increase in intracellular Na+ concentration and a resulting increase in systolic Ca2+ entry through the Na+/Ca2+exchanger. The model reproduced the effects of several experimental interventions such as sarcoplasmic reticulum Ca2+ depletion, “diastolic” length changes, and changes in extracellular Ca2+. Thus SCS in cardiac muscle may be caused by length-induced changes in sarcolemmal Na+ fluxes.
- myocardial contraction
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.
- Capacitive membrane area
- Concentration of ion B
- Ca2+ concentration for half-maximum force generation
- Membrane capacitance
- Voltage dependence parameter of Na+-K+pump
- Faraday constant
- Maximum permeability for current x
- Maximum current for flux x
- Ca2+ background leakage current
- Ca2+ current through L-type Ca2+ channel
- K+ current through L-type Ca2+ channel
- Na+ current through L-type Ca2+ channel
- Sum of all ionic currents
- Time-dependent K+ current
- Time-independent K+ current
- Plateau K+ current
- Ca2+ leakage from network SR to myoplasm
- Fast Na+ current
- Na+ background leakage current
- Current through Na+/Ca2+exchanger
- Current through Na+-K+pump
- Nonspecific Ca2+-activated current
- Sarcolemmal Ca2+ pump current
- Ca2+ release from junctional SR to myoplasm
- Stimulus current
- Ca2+ transfer from network SR to junctional SR
- Ca2+ uptake from myoplasm into network SR
- Half-saturation concentration forIx
- Hill’s coefficient of force-pCa relation
- Membrane permeability for ion B
- Gas constant
- Membrane potential
- Compartment volume
- Valency of ion B
- Extracellular Na+ concentration dependence factor offNa,K
The definitions of the slow change in stress (SCS) and its half time (t 1/2) are shown in Fig. 1.
The Luo-Rudy model (15) is based on a numerical reconstruction of the action potential using the following equation whereV is the membrane potential,C m is the membrane capacitance,I st is a stimulus current, and Ii 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 , and the original publication (15) gives the equations for all ionic fluxes.
The model accounts for changes in [Ca2+]iand intracellular concentrations of K+([K+]i) and Na+([Na+]i) using the following equation where [B] is the concentration of ion B,IB is the sum of ionic currents carrying ion B,A Cap is the capacitive membrane area, VC is the volume of the compartment where [B] is calculated,zB is the valence of ion B, andF is the Faraday constant.
Force was computed as an output variable of the model from a sigmoidal function of [Ca2+]iwith 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% ofL 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% ofL 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+]iand [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.
A step in myofilament sensitivity from 2 to 1 μM generated a step increase in force from 0.143 ⋅ Fmax to 0.485 ⋅ Fmax(where Fmax 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.
Superimposing a step change in myofilament sensitivity with a step change in any one of the parameters governing sarcolemmal ion fluxes enabled the potential role of a single ion flux in SCS to be examined. A step increase inI Ca, the Ca2+ current through the L-type Ca2+ channel, is simulated at the time of the length step (Fig. 3). The step increase in myofilament sensitivity was accompanied by a twofold step increase in P Ca, the permeability of the L-type Ca2+ channel to Ca2+. Force increased with the step from 0.143 ⋅ Fmax to 0.649 ⋅ Fmax. The immediate increase in force was therefore larger than with the step change in myofilament sensitivity alone. This larger step increase in force was due to a step increase in the Ca2+ transient from 0.976 to 1.28 μM. The immediate increase in force was followed by an additional increase to 0.692 ⋅ Fmaxafter 15 min. However, more than one-half of this increase occurred during the first five beats. When Ca2+ depletion of the SR was simulated, the immediate increase in force under the above conditions was not followed by an additional increase in force after the step (results not shown). Because of the fast time course and the apparent SR dependence, the force increases after these step changes do not resemble the experimentally observed SCS.
In the same way, we examined step changes in 20 parameters of 13 ionic fluxes included in the model. The results of these simulations are summarized in Table 1. Step changes in 12 parameters of 7 ionic currents produced changes in force of >10% after the steps (boldface values in SCS column). However, for five of these ionic currents the increases in force hadt ½ values between 5 and 15 s, in contrast tot ½ values observed experimentally (4.3 ± 0.4 min,n = 6) (4). The force changes induced by step changes in four parameters (boldface values in SCS andt ½ columns) closely resembled the experiment. These parameters are the amplitude of the Na+ background current (I Na,b) and three parameters describing the Na+-K+pump (I Na,K).
Of these parameters, only a small change in Na,K, the maximum current through the Na+-K+pump, was needed to yield a slow change of force comparable in magnitude to the experimentally observed SCS. This parameter was therefore selected for further analysis. Na,K was decreased by 20% from 1.7 to 1.36 μA/μF (Fig.4). The immediate increase in force from 0.143 ⋅ Fmax to 0.483 ⋅ Fmax was followed by a further increase to 0.658 ⋅ Fmax, amounting to an SCS of 34% with at ½ of 2.6 min. These compare with an SCS of 32 ± 12% (n = 22) and at ½ of 4.3 ± 0.4 min (n = 6) observed experimentally (4).
Mechanism of simulated SCS.
The mechanisms for the simulated SCS are illustrated in Fig.5. The decrease in Na,Kresulted in a decrease inI 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+]ito 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+]ito stabilize at the increased concentration. The increase in [Na+]iwas accompanied by an increase in systolic Ca2+ entry through the Na+/Ca2+exchanger. Peak systolicI Na,Ca doubled from 0.578 to 1.145 μA/μF, with little change in diastolicI 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.
Comparable results were obtained for a 100% step increase in Na,b, maximum permeability for background Na+ current (not shown), instead of a 20% decrease in Na,K.
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 ⋅ Fmax. 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 ⋅ Fmax to 0.244 ⋅ Fmaxafter 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).
Nichols (16) and Allen et al. (2) showed that the muscle does not have to contract at the increased muscle length but that diastolic stretches between stimuli are sufficient to produce SCS. We simulated this length protocol by imposing the step changes in Ca50 and Na,K only during a 4-s diastolic period starting 1 s after each stimulus and returning to the original values at the time of the next stimulus (Fig.7). At the onset of this protocol, there was no immediate increase in force, but there was a slow increase in force from 0.143 ⋅ Fmax to 0.217 ⋅ Fmaxwith a t ½ of 2.9 min. The peak Ca2+ transient increased from 0.976 to 1.198 μM. This increase of 0.222 μM was 68% of the 0.323 μM increase in peak Ca2+ transient with a sustained change in Ca50 and Na,K (Fig.4).
When the changes in Ca50 and Na,K were made only during a systolic 1-s period immediately after each stimulus (Fig. 8), there was an 8% SCS with at ½ of 2.1 min. The increase in peak Ca2+transient was 15% of the increase with maintained changes in Ca50 and Na,K (Figs.4 and 5).
Finally, we examined the effects of changing extracellular Ca2+ concentration ([Ca2+]o) on the simulated SCS (Fig. 9). Maintained changes in Ca50 and Na,K were imposed in the presence of 1.25, 1.8, or 2.5 mM extracellular Ca2+. With 1.25 mM extracellular Ca2+, force increased from 0.050 ⋅ Fmax to 0.229 ⋅ Fmaximmediately with the step and continued to increase to 0.434 ⋅ Fmaxafter 15 min, amounting to 53% SCS. In the presence of 1.8 mM extracellular Ca2+, force increased initially from 0.143 ⋅ Fmax to 0.483 ⋅ Fmax and continued to 0.658 ⋅ Fmax(34% SCS). With 2.5 mM extracellular Ca2+, an immediate increase in force from 0.296 ⋅ Fmax to 0.702 ⋅ Fmax was followed by a further increase to 0.816 ⋅ Fmax(22% SCS).
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 and5), 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). Thet ½ of SCS, in contrast to its magnitude, was relatively insensitive to parameter changes.
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 ofI Na,K orI Na,b. The increase in [Na+]iconcurred 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+]iand [Na+]iremained stable over >15 min. This is important, because changes in [K+]iand [Na+]iled 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% ofL max. These parameters fall within the wide range of published experimental data. Reported myofilament sensitivities at maximum muscle length or sarcomere length range from Ca50of 1.45 μM (10) to Ca50 of 3.39 μM (8) in skinned preparations, whereas estimates in intact muscle are much lower, ranging from Ca50of 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,Kand 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-predictedt ½ of SCS of 2.6 min underestimated the experimentalt ½ 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 inI 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 andI 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 inI 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+]iand, 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 at ½ 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+]iand 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.
We thank Dr. James Weiss and Scott Lamp (University of California, Los Angeles) for kindly providing the source code for an implementation of the Luo-Rudy model.
Address for reprint requests: A. D. McCulloch, Dept. of Bioengineering, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093-0412.
This work was supported by American Heart Association, California Affiliate, Predoctoral Fellowship Award 94-409A (W. F. Bluhm) and Grant-In-Aid 91-147A (W. Y. W. Lew), the Office of Research and Development, Medical Research Service of the Department of Veterans Affairs (W. Y. W. Lew), National Heart, Lung, and Blood Institute Specialized Center of Research in Sudden Cardiac Death P50-HL-52319 (A. Garfinkel), and National Science Foundation Presidential Young Investigator Award BCS-9157961 (A. D. McCulloch). The work was performed during the tenure of an Established Investigatorship from the American Heart Association (W. Y. W. Lew).
- Copyright © 1998 the American Physiological Society
Equations for Na+-K+Pump
with Na,K = 1.7 μA/μF and half-saturation concentrations for intracellular Na+( ) and extracellular K+( ) of 10 mM and 1.5 mM, respectively