INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX

SHORT-TERM INTERVAL-FORCE (I-F) relations, as defined by Johnson (23), describe the dependence of contraction strength for short interbeat intervals (3 s) in the physiological range for active mammals. Two phenomena that characterize the short-term I-F relations are restitution and postextrasystolic potentiation. Examples of experimentally obtained I-F relations from Wier and Yue (42) are shown in Fig. 1. The pacing protocol is a priming period, which is followed by variable extrasystolic intervals (ESIs) and then a fixed 3,000-ms postextrasystolic interval (PESI). The family of traces demonstrates restitution by the rise in twitch force (F), rate of force onset (+dF/dt), and aequorin luminescence (L/L_max) as ESI increases (cf. a and b in Fig. 1). Twitch force in response to a second stimulus applied at a fixed PESI shows the opposite behavior. For example, a small extrasystolic force leads to a potentiated postextrasystolic beat (a and a' in Fig. 1). Similarly, as extrasystolic force rises with restitution, postextrasystolic force declines (b and b' in Fig. 1). These data, along with other experimental studies (11, 46), have shown that restitution and postextrasystolic potentiation can be fit to exponentials with similar time constants, suggesting that a common mechanism may underlie both phenomena.

View larger version (34K): [in this window] [in a new window]   Fig. 1.   Original records illustrating restitution and postextrasystolic potentiation from Wier and Yue (42). Top: stimulation protocol. SSI, ESI, and PESI, steady-state, extrasystolic, and postextrasystolic intervals. A: isometric force (F); B: rate of force onset (dF/dt); C: aequorin luminescence (L/Lmax). Traces are averages of 16-32 sweeps, with records aligned to superimpose steady-state responses. Responses a and a' are from an extrasystole and accompanying postextrasystole obtained with ESI of 450 ms; b and b' correspond to ESI of 3,000 ms. In all cases, PESI is fixed at 3,000 ms. Steady-state priming interval is 1,500 ms, and external Ca2+ concentration ([Ca2+]o) is 0.7 mM.

A relatively simple model has been developed to address the general behavior of restitution and postextrasystolic potentiation (12, 42). This model explains these behaviors in terms of the interplay of two features: 1) the total amount of Ca²⁺ loaded in the sarcoplasmic reticulum (SR) and 2) a feature in which Ca²⁺ loaded in the SR slowly becomes available for the next release. More specifically, restitution is explained by more Ca²⁺ becoming available for release as ESI increases. Given a sufficiently long ESI, the saturating plateau level of Ca²⁺ release at full restitution is assumed to reflect the total amount of Ca²⁺ in the SR. This plateau level has been shown to increase with the frequency of stimulation during the priming period. Presumably, this increase occurs because of a greater time-averaged influx of Ca²⁺ that loads the SR to a greater extent. Furthermore, after a small extrasystolic release, postextrasystolic potentiation occurs because 1) there is more Ca²⁺ remaining in the SR for the next release and 2) additional Ca²⁺ influx into the cell increases the SR Ca²⁺ load for the next release. The proposed mechanism for the additional Ca²⁺ influx is a reduction of negative feedback of the SR Ca²⁺ release on influx of Ca²⁺ through L-type Ca²⁺ channels (42).

Although this model makes accurate predictions of the general behaviors of restitution and postextrasystolic potentiation, it lacks mechanistic detail. The purpose of this study is to develop new models that improve on the results of the previous modeling efforts described above. The first phase of this work entails the development of a discrete-time model of excitation-contraction (E-C) coupling to address how systematic parameter variation affects restitution/postextrasystolic potentiation behaviors. The approach of using discrete-time models has been explored previously (1, 35). In these models, all E-C coupling events are represented only at a discrete-time point for each beat, although the interval between beats is allowed to vary. In general, this modeling approach [referred to elsewhere as a "minimum model" (35)] seeks to capture the most salient features of I-F relations in a low-order system. The discrete-time model developed here is of even lower order than in previously published models, because the model developed here focuses only on extrasystolic restitution and postextrasystolic potentiation and not on other I-F phenomena. Also the model output is the isometric contraction force, not the Ca²⁺ transient, as in previous models, to allow for a more direct comparison with experimental results.

The second phase of modeling entails the development of a "detailed" model [referred to elsewhere as a "maximum model" (35)]. The assumption inherent in the development of these models is that I-F relations will be reproduced if the underlying biophysical mechanisms are reproduced. A detailed model is developed here by combining a single ventricular cell model simulating action potentials (APs) and Ca²⁺-handling mechanisms (20) with a model of the myofilaments simulating isometric force generation (32). The single cardiac cell model incorporates recent experimental findings on the mechanisms of E-C coupling in cardiac cells, including 1) adaptation of the ryanodine receptor (RyR), the SR release channel; 2) a model of the L-type Ca²⁺ channel with Ca²⁺-induced inactivation based on mode switching (18); and 3) a restricted subspace thought to exist between the RyR and L-type Ca²⁺ channels, where the local Ca²⁺ concentration ([Ca²⁺]) is thought to rise to 10 times the bulk myoplasmic [Ca²⁺] ([Ca²⁺]_i) (19, 33). The myofilament model used in this model can reproduce isometric force responses seen experimentally and allows for direct comparison of model output with experimental data on I-F relations.

The "minimum" and "maximum" approaches to model I-F relations in this study are complementary. The low-order discrete-time model is suitable for systematic parameter variation studies. However, this model lacks mechanistic detail, and the physiological plausibility of any given parameter choice is difficult to assess on the basis of simple model results alone. The detailed model can better address such issues. However, the complexity of this model precludes exhaustive exploration of the complete parameter space.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX

Simple Model Construction

View larger version (31K): [in this window] [in a new window]   Fig. 2.   Schematic diagram of simple, discrete-time model of Ca2+ handling in restitution and postextrasystolic potentiation. InSC and OutSC, influx and efflux across sarcolemma; InSR and OutSR, sarcoplasmic reticulum (SR) release and uptake. Restitution results from an exponential increase in InSR as governed by (t). Important model parameters are recirculation fraction (r), releasable fraction () of SR Ca2+, and feedback of SR release on sarcolemmal influx (h).

The amount of Ca²⁺ released from the SR during each AP is assumed to depend on the amount of Ca²⁺ in the SR and the state of restitution. The governing equation for the SR release is given by

(1)

where Load_SR(n  1) is the total Ca²⁺ load of the SR computed from the last iteration,  is the fraction of the total load that can be released with the assumption of full restitution, and (n) is the restitution function that assumes a value between 0 and 1. The restitution function is

(2)

where t(n) is the time difference between the last AP and the present AP and  is the time constant of restitution.

The simple model assumes that transarcolemmal influx (In_SC) does not contribute directly to the Ca²⁺ transient but is, instead, sequestered by the SR and is made available for release only on the subsequent beat. Hence, the total Ca²⁺ transient is produced by the SR release alone

(3)

A similar formulation has been used previously (46). Clearly, this is an approximation; influx through L-type channels initiates the SR release and, therefore, must contribute to the Ca²⁺ transient to some extent. However, the initial influx that triggers the SR release may be small, and the subsequent Ca²⁺ influx may play a larger role in the SR loading (9, 16).

In the simple model the SR uptake [Out_SR(n)] contains a component from the recirculation fraction (r) of the Ca²⁺ transient and a component from the sarcolemmal influx

(4)

The fraction of Ca²⁺ transient that is not resequestered into the SR is assumed to be extruded from the cell across the sarcolemma. Therefore

(5)

The load of Ca²⁺ in the SR is updated at each iteration by the governing equation

(6)

The sarcolemmal influx (In_SC) is assumed to be a decreasing function of the Ca²⁺ transient to account for Ca²⁺-induced inactivation of L-type channels. The effect of Ca²⁺-induced inactivation is that a small Ca²⁺ release from the SR on the current beat produces a greater influx of Ca²⁺, which in turn increases the SR load for the next beat. The input across the sarcolemma is given by

(7)

where  is a constant, (n) is defined in Eq. 2, and h is a constant between 0 and 1 that corresponds to the amount of Ca²⁺-induced inactivation. A value of h near 1 produces a large dependence of sarcolemmal influx on the Ca²⁺ transient, whereas a value of 0 corresponds to a constant sarcolemmal influx with no dependence on the Ca²⁺ transient. Although the formulation of Eq. 7 does not appear to depend on the Ca²⁺ transient, there is an implicit dependence through the term (n) [SR release and Ca²⁺ transient are directly proportional to (n), see Eqs. 1 and 7].

The model just described produces Ca²⁺ transients for each beat. The Ca²⁺ transients must be converted to force transients for every beat to compare the model predictions more directly with experimental results. This is accomplished with the following conversion

(8)

where  and  are empirically determined constants. This relation follows directly from the work of Wier and Yue (42) that demonstrates that peak force is linearly correlated with peak [Ca²⁺]. The average values from this study are  = 0.03 N · mm² · µM¹ and  = 0.51 µM. Empirical data also determine the time constant of restitution (in Eq. 2) to be equal to 765 ms. With these parameters constrained, the simple model is run with the following free parameters: r, , and h.

Simple Model Simulation Protocol

The initial conditions are chosen to yield a force close to the standard value of 12.5 ± 0.1 mN/mm². The initial conditions are

(9)

(10)

(11)

(12)

(13)

Detailed Model Construction

The full set of equations for the detailed model is provided in the APPENDIX. The description here will focus on a number of critical differences between the new formulation and the previous model (20) from which it was derived.

RyRs. The original formulation of RyRs was modified to increase the forward rate to and reverse rate from the adapted state (P_{C 2} in Eqs. A71-A74). The forward rate was increased to 100 s¹ to make the time constant of adaptation ~10 ms. A similar value of 15 ms is found experimentally for the rat at room temperature (45). The model assumes a slightly higher rate at a physiological temperature. The increase in forward rate makes adaptation a more important modulatory factor in shaping SR release than in the original model (20). These roles can justified by recent experimental findings that suggest an important role of RyR inactivation in terminating release (36).

Network SR-to-junctional SR transfer rate. The SR is assumed to consist of two compartments: the uptake compartment [network SR (NSR)] and the release compartment [junctional SR (JSR)]. The transfer rate between these compartments is set by J_TR. In the original model (20), J_TR was a relatively small value, resulting in the depletion of JSR during each AP, hastening the termination of the SR release. In the present model, J_TR is made sufficiently large so that NSR and JSR vary by only a small amount. A large value of J_TR is consistent with estimates that Ca²⁺ diffusion between these compartments should be quite rapid, requiring only a few milliseconds at most (9).

Ca²⁺-ATPase pump. The formulation of the SR Ca²⁺-ATPase pump was modified to be similar to a model proposed by Shannon et al. (38) that includes both forward and reverse modes, each with its own binding constant and maximum rate. The forward mode exhibits slight cooperativity, with an experimentally determined value of 1.2 (38). In the original Ca²⁺-handling model (20), forward cooperativity of 2 was assumed, which is consistent with other estimates (28). In the present model, a value of 1.4 is chosen as a compromise between these two conflicting estimates. In general, the choice of this parameter was limited by two extremes. Low cooperativity values cause the Ca²⁺-ATPase pump to run at relatively high rates during diastole, producing unphysiologically low [Ca²⁺]_i. High cooperativity values cause a rapid increase in pump rate as [Ca²⁺]_i increases. This leads to Ca²⁺ transients with narrow peaks. This is a consequence of the large SR uptake at peak [Ca²⁺]_i.

Detailed Model Simulation Protocol

Similar to the experimental work of Wier and Yue (42), a priming period of 30 s is sufficient to bring the detailed model to approximately steady state. The priming beats are delivered at a fixed interval of 1,500 ms in the simulations, unless otherwise noted in RESULTS. The ESI is varied between 236.6 and 2,842.1 ms in 236.6-ms increments. The PESI is fixed at 3,000 ms. It is emphasized that the detailed model simulation involves only changes in the stimulus pattern, and otherwise no parameters are varied to alter the restitution/postextrasystolic behavior.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX

Simple Model

View larger version (29K): [in this window] [in a new window]   Fig. 3.   Sample data for model 3. A: myoplasmic Ca2+ concentration ([Ca2+]i) transients; B: corresponding force transients. Parameter choices for this run are r = 0.75,  = 0.5, and h = 0.27. Last 2 priming beats are shown at left and demonstrate that model reached steady output levels. Plots show a composite of 10 runs with ESI ranging from 300 ms (a) to 3,000 ms (b). Restitution behavior is shown by rise in Ca2+ transients and force between a and b. After each extrasystole, there is a postextrasystole at a delay of 3,000 ms (i.e., a' is 3,000 ms after a). When a' is compared with last steady-state beat (ss), this model shows a small degree of postextrasystolic potentiation.

The postextrasystolic beats show behavior that is opposite from that of the extrasystolic beats; i.e., the least amount of restitution (a in Fig. 3) corresponds to the greatest level of potentiation (a' in Fig. 3). In contrast, the force at full restitution (b in Fig. 3) corresponds to the lowest level of postextrasystolic potentiation (b' in Fig. 3). This behavior is qualitatively consistent with experimental data from a number of studies in whole heart (11) and muscle preparations (24, 29, 42). However, the simulated and experimental results differ quantitatively in the degree of postextrasystolic potentiation. In the experimental data, postextrasystolic force can be two or more times greater than the steady-state force from the last priming beat (29, 42, 44, 46). The level of postextrasystolic potentiation is smaller in this simulation. A potentiation ratio can be computed as the ratio of force at the greatest level of potentiation (a' in Fig. 3 corresponding to the shortest ESI) to the force in response to the last priming beat (ss in Fig. 3). The potentiation ratio is 1.53, which falls short of experimental data (2), at least for the parameters chosen for this simulation.

The next set of simulations addresses the question of whether another choice of r, , and h can produce a high level of postextrasystolic potentiation. The simulation protocol used in Fig. 3 is repeated for r in the range 0.5-0.975 and h in the range 0.0-1.0;  is fixed at 0.25, 0.5, and 0.75 in Fig. 4, A, B, and C, respectively. The resulting potentiation ratios are plotted as a surface plot and as a contour plot. The contour plots show the isoclines for parameter combinations that produce potentiation ratios of 1.5, 2, 3, and 4, as labeled.

View larger version (27K): [in this window] [in a new window]   Fig. 4.   Simple model potentiation ratios are computed for r in range 0.5-0.975 and h in range 0.0-1.0;  is fixed at 0.25 in A, 0.5 in B, and 0.75 in C. Resulting potentiation ratios are plotted as surface (bottom) and contour (top) plots. Contour plots show isoclines for parameter combinations that produce potentiation ratios of 1.5, 2, 3, and 4, as labeled. For any given choice of r and h, potentiation ratios increase as  increases (cf. A, B, and C). For any given , potentiation ratios increase as r increases and h decreases.

In a comparison of Fig. 4, A-C, the potentiation ratio increases as  increases for any given choice of r and h. A small  means that only a small fraction of the total SR Ca²⁺ is released (and resequestered) on each AP. The net effect is that the SR loading shows little beat-to-beat variation. In contrast, a high level of postextrasystolic potentiation requires a large change in the SR load after the extrasystole. Recall that the SR release is a function of restitution and the SR load (see Eq. 1). For the simulations, the PESI is long (3,000 ms), so there is full restitution ( ~ 1). Hence, the SR load is the only variable that changes the level of postextrasystolic potentiation. For example, the difference between a' and b' in Fig. 3A can be attributed solely to differences in the SR loading.

The potentiation ratio increases for decreasing r at any given choice of  and h. Intuitively, decreasing r means that less of the Ca²⁺ released is being recycled back into the SR. The net effect is to increase the beat-to-beat variation in the SR load that is crucial for high levels of postextrasystolic potentiation. In contrast, a large r means that most of the Ca²⁺ released is being recycled back into the SR. Hence, each beat has almost the same SR load, a feature incompatible with a large degree of postextrasystolic potentiation.

The potentiation ratio increases with h at any given choice of  and r. The feedback parameter h controls the degree to which the Ca²⁺ transient has a negative-feedback effect on the sarcolemmal influx. Recall that sarcolemmal influx goes directly to loading the SR for the next beat. Therefore, the negative feedback increases the beat-to-beat variation in sarcolemmal influx and also the SR loading. For example, a short ESI produces a small extrasystolic Ca²⁺ transient, which in turn produces little negative feedback. With a large h, the result is a relatively larger sarcolemmal influx that increases SR load and potentiation.

In Fig. 4, the potentiation ratio is <2 for small h. Moreover, if h = 0, then the potentiation ratio is <2 for all choices of r (range 0.5-0.95) and  (0.05-1.0, all data are not shown). Hence, the simulation results suggest that negative feedback on sarcolemmal influx plays a critical role in producing the large degree of postextrasystolic potentiation observed in the cardiac muscle.

Detailed Model

(14)

View larger version (37K): [in this window] [in a new window]   Fig. 5.   Simulation results for detailed model with use of experimental protocol of Wier and Yue (42) (Fig. 1). A family of traces is shown for different ESIs, each of which is accompanied by a postextrasystolic beat 3,000 ms later. A: [Ca2+]i; B: isometric twitch force. Only force results can be directly compared with experimental results in Fig. 1.

View larger version (32K): [in this window] [in a new window]   Fig. 6.   A: simulated force data from detailed model plotted as rate of force onset (+dF/dt). B: simulated L/Lmax. This signal is derived from [Ca2+]i according to fit to calibration data in Wier and Yue (42). These results can be directly compared with experimental results in Fig. 1.

When corresponding traces are compared, the experimental and model results are similar in extrasystolic restitution and postextrasystolic potentiation. The potentiation ratio is slightly >2 for force (a' = 0.0326 and ss = 0.0157 in Fig. 5B). The relative changes in peak [Ca²⁺]_i levels (Fig. 5A) are considerably smaller than those for peak force, similar to the experimental results. The peak [Ca²⁺]_i of the last priming beat is ~0.825 µM, whereas that of the maximally potentiated beat is 1.11 µM.

We hypothesize that the short-term I-F relations are produced from an interplay of RyR adaptation and the SR loading. This hypothesis is investigated using the detailed model. In Fig. 7A, RyR open probabilities are plotted for the same sequence as in Fig. 5. As ESI increases, the peak open probability increases from 0.269 for the shortest ESI (a in Fig. 7) to 0.853 for the longest ESI (b in Fig. 7). In contrast, the peak open probability is a constant value of 0.869 for each of the postextrasystolic beats. In this case, the SR release depends mainly on the SR load. This feature is shown in Fig. 7B by plotting the [Ca²⁺] in the NSR, where uptake occurs ([Ca²⁺]_NSR), and in the junctional SR, where release occurs ([Ca²⁺]_JSR). In this model, these compartments are coupled with a small time constant so that [Ca²⁺]_NSR and [Ca²⁺]_JSR are very close, except during the SR release, where [Ca²⁺]_JSR is transiently lower than [Ca²⁺]_NSR. During each AP, the transient decreases in the SR load closely mirror the Ca²⁺ transients in Fig. 5A, because Ca²⁺ release from JSR makes the primary contribution to the Ca²⁺ transient.

View larger version (43K): [in this window] [in a new window]   Fig. 7.   A: ryanodine receptor (RyR) open probabilities for detailed model for same sequence as in Fig. 5. Peak open probability increases during restitution phase, whereas peak open probability is a constant value for postextrasystolic beats. B: SR Ca2+ load. Ca2+ concentration is plotted in network SR ([Ca2+]NSR, top traces), where uptake occurs, and in junctional SR ([Ca2+]JSR, bottom traces), where release occurs. In this model, these compartments are coupled with a short time constant so that [Ca2+]NSR and [Ca2+]JSR are very close, except during SR release, where [Ca2+]JSR is transiently lower than [Ca2+]NSR. A short ESI (a) leads to an increased SR [Ca2+] that produces potentiation of postextrasystolic beat (a').

For postextrasystolic potentiation, the important variable is the degree of the SR loading after the restitution beat. For the shortest ESI (a in Fig. 7B), there is a relatively small decrease in [Ca²⁺]_JSR that recovers to a higher value. The small release occurs because most of the RyRs are in the adapted state, and not from a slow transfer from NSR to JSR. [Ca²⁺]_JSR has reached its peak value for a in Fig. 7 compared with larger ESIs so that a lack of available Ca²⁺ for release is not an issue in restitution for this model (i.e., there is no component of restitution from a slow transfer of Ca²⁺ from NSR to JSR). Also there is a slow decrease in [Ca²⁺]_NSR and [Ca²⁺]_JSR when no APs are occurring (i.e., in the period between a and a' in Fig. 7). The slow decrease is produced by two features. First, after the AP, there is still release from the SR, because the RyR open probability is small but nonzero. Consider that, after an AP, most of the RyRs are in the adapted state (P_{C 2}). The RyRs revert back to the resting closed state (P_{C 1}), but this transition requires a sojourn through an open state (P_{O 1}). This transition through the open state causes the nonzero open probability. A second feature that produces a slow unloading of the SR is that the Ca²⁺-ATPase pump can run in a reverse mode. The balance between the forward mode (that fills NSR) and the reverse mode (that empties NSR) depends on [Ca²⁺]_i and [Ca²⁺]_NSR (see Eqs. A76-A78). During diastole, when [Ca²⁺]_i is small and [Ca²⁺]_NSR is larger, the reverse mode is more favored, so some degree of NSR emptying occurs.

Figure 7B shows that the potentiation at a' occurs because the SR [Ca²⁺] recovers to a higher level than the steady-state value achieved after the priming period (ss in Fig. 7B). The small release (a in Fig. 7B) leaves more residual Ca²⁺ in the SR to help potentiate the next beat. Recall from the simple model results that a strong feedback on sarcolemmal influx is required to achieve a high level of potentiation. This effect is shown in Fig. 8A, where L-type Ca²⁺ currents (top traces, right axis) are plotted for the same sequence as in Fig. 5. The L-type Ca²⁺ current is largest at the shortest ESI (a in Fig. 8). The reasons are twofold. First, the small SR release produces less Ca²⁺-induced inactivation of the L-type Ca²⁺ current. Second, the AP (Fig. 8A, bottom traces, left axis) has reduced amplitude, because the time-dependent K⁺ membrane current has not yet fully recovered and is still partially activated. The result is that the peak membrane potential is substantially lower than the reversal potential for the L-type Ca²⁺ current (~50 mV, see Fig. 3 in Ref. 20) that increases the net driving force for Ca²⁺ entry. Hence, the low AP amplitude produces increased Ca²⁺ influx.

View larger version (37K): [in this window] [in a new window]   Fig. 8.   A: membrane potentials (bottom, left axis) and L-type Ca2+ currents (top, right axis) plotted for same sequence as in Fig. 5. Current is largest at shortest ESI (a). B: total flux of Ca2+ computed for release from RyRs (top traces) and influx through L-type Ca2+ channels (bottom traces). Time-integrated fluxes are shown starting at time 0, corresponding to initiation of last priming beat. Time-integrated fluxes are presented as concentrations that are computed with respect to volume of myoplasm to facilitate direct comparisons of a membrane current (bottom traces) and an intracellular flux (top traces). Flux into myoplasm is positive, and extrusion from myoplasm is negative. Top traces, SR release; middle traces, L-type Ca2+ channel flux; bottom traces, sum of other membrane Ca2+ currents (Na+/Ca2+ exchanger, sarcolemmal Ca2+ pump, and background Ca2+ leak). For shortest ESI (a), total SR Ca2+ release is smaller than that for last priming beat (ss). Corresponding postextrasystolic beat (a') produces largest release. Other traces show data for longest ESI (b) and an intermediate ESI (c) and correspond to labeled transients in A.

The increase in L-type Ca²⁺ current at short ESI helps produce a high level of potentiation. This effect is explored further in Fig. 8B, where the time-integrated flux of Ca²⁺ is computed for the influx through L-type channels (middle traces), SR release through RyRs (top traces), and the sum of the other membrane Ca²⁺ currents (bottom traces). The combined fluxes of other membrane Ca²⁺ currents (Na⁺/Ca²⁺ exchanger, sarcolemmal Ca²⁺ pump, and background Ca²⁺ leak) can be added to the influx through L-type channels to compute the net change in cellular Ca²⁺ load. The integrated fluxes are presented as concentrations that are computed with respect to the volume of the myoplasm. This method facilitates direct comparisons of membrane currents (both sets of bottom traces) and intracellular fluxes (top traces). Also the fluxes into the myoplasm are positive, whereas extrusion from the myoplasm is negative.

For the shortest ESI (a in Fig. 8), the total Ca²⁺ release is smaller than for the last priming beat (ss in Fig. 8). The corresponding postextrasystolic beat (a' in Fig. 8) produces the largest release. The other traces show data for the longest ESI (b in Fig. 8) and an intermediate value (c in Fig. 8) and correspond to the labeled transients in Fig. 8A. Figure 8B also illustrates some key characteristics of Ca²⁺ handling. First, the time-integrated SR release flux has a fairly stairlike waveform, indicating that the majority of Ca²⁺ is released over a short period of time. In contrast, the integrated L-type Ca²⁺ flux has a sloping appearance that indicates a smaller-magnitude flux over a longer time interval. The contribution of the cellular Ca²⁺ extrusion mechanisms is shown by the sum of the Na⁺/Ca²⁺ exchanger, sarcolemmal Ca²⁺ pump, and background Ca²⁺ leak fluxes. The contribution of the background Ca²⁺ leak flux is included, because this current counterbalances the other currents so that the net cellular Ca²⁺ load does not deplete too quickly during diastole. The summed flux (bottom traces in Fig. 8B) first has a positive-going deflection after an AP as a result of the Na⁺/Ca²⁺ exchanger running in reverse mode. The positive-going deflection is slightly larger for the shortest ESI (a in Fig. 8), indicating that more Ca²⁺ enters the cell during the small Ca²⁺ transient (favoring the reverse Na⁺/Ca²⁺ exchanger mode). This suggests some contribution of extrusion mechanisms to increased SR loading and potentiation. However, by the postextrasystolic release, the total Ca²⁺ extruded from the cell (indicated by the downward deflection of the trace) is very similar for a', b', and c'. This shows that total extrusion Ca²⁺ is fairly independent of the stimulus pattern and suggests a fairly minor role in beat-to-beat changes in cellular (and hence SR) Ca²⁺ load, at least for the pattern tested.

Recirculation fraction can be estimated from Fig. 8B. One way to estimate this quantity is

(15)

The uptake and release from the SR must balance to maintain no net change after each beat. With use of this fact, recirculation fraction can be estimated by setting

(16)

From the RyR release trace (ss in Fig. 8B), integrated Ca²⁺ released from SR is 8.9 × 10⁷ M. The total extrusion is computed from the combined fluxes of other membrane Ca²⁺ currents. The amount of extrusion in the bottom trace (c in Fig. 8B), just before the extrasystole, shows that integrated Ca²⁺ extruded from the cell is 2.3 × 10⁷ M. With use of these values and Eq. 15, r is calculated to be 0.79.

In summary, the detailed model has been shown to reproduce experimental findings that show slow recovery of force during restitution, with similar results found for the rate of force onset (+dF/dt) and the Ca²⁺ transient. There has not yet been any examination of the temporal details of restitution. Many experimental findings show that restitution can be fit by an exponential (2, 11, 42, 46). For example, the data in Fig. 1 have time constants on the order of 700-800 ms (42). Other sets of experimental findings show that as the priming rate increases, the plateau at full restitution increases while the time constant of the exponential rise is unchanged (11).

The ability of the detailed model to reproduce these experimental findings is now examined in Fig. 9A, where the priming rate is increased from the original basic cycle length (BCL) of 1,500 ms to 1,250 and 1,000 ms. Here the peak rates of force onset (peak dF/dt) are shown for the range of ESIs in Fig. 5, and an exponential fit is shown with  as labeled. Consistent with experimental observations, the higher priming rate increases the plateau level but produces little change in the time constant of restitution. Although only data for peak rate of force onset are shown, similar results are obtained for force, a result also consistent with experimental findings (22, 42). The lowest ESI produced peak force or dF/dt values that are not well fit by the exponentials; hence, they are not included in the fits shown. These points are larger than predicted by an exponential curve fit and, if included, would give the restitution curves a sigmoidal shape. However, this deviation should not be considered to be significant, because this trend can also be seen in some experimentally determined restitution curves (see Fig. 3A in Ref. 12).

View larger version (25K): [in this window] [in a new window]   Fig. 9.   Study of temporal details of restitution and postextrasystolic potentiation when priming rate is increased from original basic cycle length (BCL) of 1,500 ms to 1,250 and 1,000 ms. A: peak rates of force onset (peak dF/dt) for same range of ESIs as in Fig. 5. An exponential fit is shown with time constant () as labeled. Consistent with experimental observations, higher priming rate increases plateau level while producing little change in time constant of restitution. B: postextrasystolic data fit by exponential curves with  as labeled.

The peak rate of force onset for postextrasystolic data is plotted in Fig. 9B. These results show that postextrasystolic data are also well fit by exponential curves with time constants on the order of 700 ms, consistent with experimental findings (42). Very similar exponential fits and time constants are obtained for force data. The close similarity between time constants for restitution and postextrasystolic potentiation is seen in experimental results and has been interpreted to suggest a common mechanism (42, 46).

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES APPENDIX

The simple, discrete-time model produces restitution but is only able to generate high levels of potentiation with appropriate parameter choices. Specifically, potentiation ratio increases with increase in the release fraction (), decrease in the recirculation fraction (r), and increase in the feedback parameter (h). The effects of varying these three parameters are now considered separately.

Release Fraction

The discrepancy between the modeling results and the experimental work may result from the release fraction being an increasing function of the SR load (5, 39). This experimental finding invalidates the constant release fraction assumption in the simple model. A likely consequence of the variable release fraction would be an increase in potentiation, because any increase in the SR loading (from an extrasystole at a short ESI) will also produce release of a greater fraction of the increased load. An alternative explanation for the discrepancy could be that experimental methods underestimate the release fraction. The experiments compared Ca²⁺ released from the SR during an AP with Ca²⁺ released during caffeine application. However, the possibility exists that some Ca²⁺ in the SR may not be involved in normal E-C coupling but is only released with caffeine (14), leading to underestimation of the release fraction.

Recirculation Fraction

(17)

(18)

(19)

(20)

Estimates from experimental studies indicate that sarcolemmal influx provides only 20% of the Ca²⁺ necessary to produce a typical transient (8, 10, 13) (these estimates are from rat and are most likely lower than that for guinea pig, which has a less-developed SR). Now consider the steady-state assumptions, developed above, where sarcolemmal influx equals the SR release [In_SC(n) = In_SR(n)]. In the simple model the total Ca²⁺ in the transient is assumed to be the SR release alone, which would be 20% of the necessary Ca²⁺. Hence, these results suggest that a recirculation fraction of 0.5 (or less) is incompatible with the experimental estimates of sarcolemmal influx per AP and the Ca²⁺ necessary to produce a typical myplasmic Ca²⁺ transient.

Feedback on the Sarcolemmal Influx

Although the simple model allows for complete freedom in setting h, there is an upper limit for plausibility. The simple model assumes that influx does not contribute directly to the Ca²⁺ transient. This is only an approximation, because the Ca²⁺ influx through the L-type channels must contribute to the Ca²⁺ transient, at least to some degree. With preliminary versions of the simple model, the Ca²⁺ transient was assumed to be the sum of the SR release and influx through the L-type channels. However, this construction produced blunted restitution because of a negative-feedback effect. Consider that a small SR release at a short ESI would be bolstered by a large L-type influx. Hence, when these two components are summed, the resulting Ca²⁺ transient is increased at short ESI, effectively reducing restitution behavior. Similarly, the detailed model results indicate that L-type current does produce a noticeable increase in the Ca²⁺ transient, especially when the SR release is small at short ESIs (a in Fig. 5A). Hence, there is likely a practical limit on how large h can be without counteracting the restitution behavior produced by the slow recovery of the SR release.

Previously Published Detailed Models

Modifications to Improve Short-Term I-F Relations

In contrast, our previous model (20) incorporated a long time constant for Ca²⁺ transfer from NSR to JSR to produce greater depletion of JSR. This long time constant is not maintained in the new version of the model, because it produces a number of undesirable behaviors. First, the restitution curves are very sigmoidal instead of exponential (the present model produces only slightly sigmoidal restitution curves, see leftmost points in Fig. 9A). Second, a long time constant, coupled with a high stimulus rate, could produce situations where [Ca²⁺]_NSR far exceeds [Ca²⁺]_JSR. This often leads to spontaneous SR release and ectopic beating on return to a lower stimulus rate. Finally, a long time constant allows JSR to empty while NSR simultaneously fills. This effectively limits the releasable fraction of Ca²⁺ from the SR [the previous model has a releasable fraction of ~0.3 (20)]. In the present version the decreased time constant produces greater emptying of the SR and effectively gives a higher releasable fraction of Ca²⁺. Recall from the simple model results that a high releasable fraction () improves the potentiation ratio by increasing the beat-to-beat variation in the SR load.

Another change in the present model, as suggested by the simple model, is to decrease the recirculation fraction. Recall that a high recirculation fraction decreased the potentiation ratio by decreasing the beat-to-beat variation in the SR Ca²⁺ load. In the detailed model the recirculation fraction is reduced mainly by increasing the Na⁺/Ca²⁺ exchanger rate, which favors extrusion of Ca²⁺ across the sarcolemma rather than resequestration into the SR. However, the precise value of the recirculation fraction is constrained by a number of biophysical factors. As argued earlier, the influx of Ca²⁺ across the sarcolemma must be in net balance with the efflux to be in steady state. The first quantity can be estimated from experimental measurements of L-type Ca²⁺ current, whereas the second quantity is mainly the contribution from the Na⁺/Ca²⁺ exchanger current. This effectively limits efflux from the cell. Likewise, there must be a net balance of influx and efflux from the SR in the steady state. Given that enough Ca²⁺ must be released to activate the myofilaments, then efflux from and influx to the SR must be relatively large.

In the detailed model the recirculation fraction was found to be ~0.79. This value is consistent with experimental estimates for recirculation fraction as computed by comparing efflux of the myoplasmic Ca²⁺ via the competing pathways of the SR Ca²⁺-ATPase and Na⁺/Ca²⁺ exchange. These pathways are considered to be the major pathways, with the SR Ca²⁺-ATPase removing ~70% and the Na⁺/Ca²⁺ exchangers ~20-30% of Ca²⁺ released from SR (9). With the assumption of such a ratio, the recirculation fraction is ~0.7. Other researchers have suggested smaller values. Calculations using the beat-to-beat decay of potentiation (i.e., negative staircase) place the lower value of the recirculation fraction at 0.45 (41). However, this value contained a component from feedback on sarcolemmal input (analogous to h), which could potentially introduce considerable error.

An important finding from the simple model is that a large value of h is required to produce potentiation ratios of 2, as shown in experimental results. The feedback is found to be relatively large in the detailed model. In Fig. 8A, the L-type channel Ca²⁺ current trace for a is roughly twice as large has that for a'. The corresponding h can then be estimated to be on the order of 0.5. This value is actually larger that what might be estimated by comparing experimentally measured L-type currents with and without the SR releases (17). These data show an ~25% increase in L-type current when Ca²⁺-induced inactivation is effectively removed by depleting the SR. However, these data were collected with an AP clamp. In contrast, the detailed model, running without an AP clamp, predicts that a small change in AP shape at short ESIs may increase influx and effectively increase h. Recall from Fig. 8A that the short ESI produced APs with reduced amplitude, which in turn increased the driving force for Ca²⁺ influx. Finally, this is an example in which interaction between the membrane currents and the Ca²⁺ handling required a detailed model approach and could not have been predicted by the simple model alone.

Effects of Low External [Ca²⁺] on Potentiation

Contribution of Myofilaments

In the development of the force generation model, the contribution of end-to-end troponin-tropomyosin interactions is assumed to produce the steady-state F-Ca²⁺ relations with high apparent cooperativity, as observed in real myocardium (32). For twitches, the highly cooperative behavior produces a steep dependence of peak force on the peak of the [Ca²⁺] transient. Another feature of the myofilament model is that troponin/tropomyosin is assumed to shift rapidly to a permissive state, whereas slower cross-bridge dynamics determine the rise of force. Such a construction makes the time to peak force relatively independent of level of peak force, consistent with experimental data (see data in Refs. 6, 21, and 41). Considered another way, the rate of rise of force is proportional to the peak level of force. This leads to the behavior seen in the detailed model, where peak force and the peak rate of force onset behave in a similar fashion (cf. Figs. 5B and Fig. 6A). These results are consistent with experimental findings that I-F relations are virtually identical for peak force and the peak rate of force onset in a variety of isolated muscle preparations (42, 44). Similarly, restitution behavior is equivalent for peak pressure and rate of pressure rise in the whole ventricle (12).

Limitations of the Detailed Model

Another limitation of the detailed model involves species- and tissue-based differences in membrane currents and E-C coupling mechanisms. For example, the experimental data most used in this study (Fig. 1) are from ferret papillary muscle with low [Ca²⁺]_o. Such a choice is justified given the completeness of this data set and the similarity to restitution/postextrasystolic potentiation behavior in other species. However, the detailed model was developed to correspond most closely to the guinea pig under physiological [Ca²⁺]_o. Hence, one could question the validity of such an approach.

In defense of the study, we note that the most important behaviors we have sought to replicate in Fig. 1 are similar to those measured in guinea pig tissue. For example, restitution follows an exponential time course (2, 29, 44), although the rate of recovery may be somewhat faster than that in Fig. 1. Also similar to the data in Fig. 1, restitution and postextrasystolic potentiation follow similar time courses in the guinea pig (29). In this study the targeted level of the potentiation ratio is 2, a reasonable value for guinea pig ventricular (29) and papillary muscle (44). Even larger potentiation ratios are measured in other species, such as dog (46) and rabbit (43), so the mechanisms explored in this study may prove useful in generating models of these tissues.

Finally, difference in AP morphologies should be considered. The guinea pig has a relatively depolarized plateau phase. With such an AP morphology, residual activation of the delayed rectifier current could lower the AP amplitude at short ESI, thereby increasing the Ca²⁺ influx, SR loading, and potentiation. However, this finding may not carry over to other tissues with different AP morphologies or electrical restitution properties. For example, rabbit ventricle shows a less depolarized AP that increases in amplitude and duration at short ESIs (43), presumably as a result of reduced L-type channel inactivation. This AP change may contribute to increased Ca²⁺ influx and the high potentiation ratio in this tissue. Different behaviors are also likely to occur in tissues with a prominent transient outward current (I<