We hypothesize that slow inactivation of Ca2+/calmodulin-dependent kinase II (CaMKII) and its modulatory effect on sarcoplasmic reticulum (SR) Ca2+ handling are important for various interval–force (I–F) relations, in particular for the beat interval dependency in transient alternans during the decay of post-extrasystolic potentiation. We have developed a mathematical model of a single cardiomyocyte to integrate various I–F relations, including alternans, by incorporating a conceptual CaMKII kinetics model into the SR Ca2+ handling model. Our model integrates I–F relations, such as the beat interval-dependent twitch force duration, restitution and potentiation, positive staircase phenomenon and alternans. We found that CaMKII affects more or less all I–F relations, and it is a key factor for integration of the various I–F relations in our model. Alternans arises, in the model, out of a steep relation between SR Ca2+ load and release, owing to SR load-dependent changes in the releasability of Ca2+ via the ryanodine receptor. Beat interval-dependent CaMKII activity, owing to its kinetic properties and amplifying effect on SR Ca2+ load dependency of Ca2+ release, replicated the beat interval dependency of alternans, as observed experimentally. Additionally, our model enabled reproduction of the effects of various interventions on alternans, such as the slowing or accelerating of Ca2+ release and/or uptake. We conclude that a slow time-dependent factor, represented in the model by CaMKII, is important for the integration of I–F relations, including alternans, and that our model offers a useful tool for further analysis of the roles of integrative Ca2+ handling in myocardial I–F relations.
The mechanical characteristics of extrasystolic (ES) beat and post-extrasystolic potentiation (PESP) are strongly linked to beat intervals: the shorter the ES interval (ESI), the smaller the ES beat, and the larger the subsequent PESP. This relation has been well described in the context of restitution and potentiation, conventionally presented by interval–force (I–F) relations (Wier & Yue 1986).
The dynamics of PESP decay is also affected by the beat interval (Shimizu et al. 2000). Rapid pacing, for example, may lead to pronounced mechanical alternans of the decay pattern, with a positive correlation between stimulation rates and alternans amplitude (figure 1; Kihara & Morgan 1991; Laurita et al. 2003). Owing to this beat interval dependency, transient alternans during the decay of PESP may be interpreted as an integral constituent of I–F relations. It is this ‘broad conceptual approach’ that underlies the concept of I–F relations in this paper (i.e. any force change induced by beat interval variations).
Although transient mechanical alternans may be brought about by either haemodynamic or cellular contractile mechanisms (Lab & Seed 1993; Iribe et al. 2004), recent studies suggest transient alternation of the contractile state of cardiomyocytes as a primary mechanism, based in particular on periodic alterations in intracellular Ca2+ cycling, rather than membrane ionic currents (Diaz et al. 2004; Pruvot et al. 2004). Although the detailed mechanisms that give rise to mechanical alternans are still not clear, time-dependent changes in sarcoplasmic reticulum (SR) Ca2+ handling may hold a key to understanding this phenomenon (Diaz et al. 2004).
It has been revealed that the Ca2+/calmodulin-dependent kinase II (CaMKII) plays an important role in regulating Ca2+ handling by phosphorylating several Ca2+ transporting proteins, such as ryanodine receptors (RyR) (Witcher et al. 1991) and phospholamban (Davis et al. 1983). CaMKII is activated upon increased cytosolic Ca2+ concentration ([Ca2+]i) as a result of formation of Ca2+-calmodulin (Ca2+-CaM) complexes. Once CaMKII binds Ca2+-CaM, the regulatory domain of the enzyme (Thr-286) is auto-phosphorylated. Auto-phosphorylation increases the affinity of CaM and CaMKII. This effect traps CaM on CaMKII. Therefore, the enzyme remains active for several seconds even when [Ca2+]i decreases to resting level (Meyer et al. 1992; Maier & Bers 2002). These properties provide CaMKII with relatively slow inactivation kinetics, of the order of several seconds (Braun & Schulman 1995). The CaMKII activity levels that result from the increase in [Ca2+]i during one beat may affect Ca2+ handling, and mechanical output, during the subsequent beats. The extent to which this occurs is determined by the beat interval, which renders the slow inactivation of CaMKII as potentially important for I–F relations.
In the present study, we investigate the hypothesis that the beat interval dependency of transient alternans during the decay of PESP may be explained by CaMKII-dependent changes in SR Ca2+ handling. To test the viability of this hypothesis, we developed a mathematical model of SR Ca2+ handling that takes into account CaMKII dynamics. We incorporated this SR model into the OxSoft ‘electric’ cardiomyocyte model (i.e. the description of membrane currents, transporters and compartments) of Noble et al. (1991), combined with the ‘mechanics’ model of Rice et al. (1999). We also aimed to make the model as complex as necessary and as simple as possible, to help identification of causally linked events, and for computing efficiency. Using this model, we investigate the Ca2+ dynamics during various protocols and I–F relations, and assess the involvement of CaMKII in determining the beat interval dependency of transient alternans during the decay of PESP. Our model shows that CaMKII involvement in Ca2+ handling not only successfully reproduces the beat interval dependency of alternans, but integrates alternans and various other I–F relations at the whole cell level.
(a) Model description
(i) CaMKII kinetics model
Since the detail of dynamic beat-to-beat changes in CaMKII activity is still not fully resolved, we added a conceptual CaMKII kinetics term to describe its involvement in I–F relations. To mimic known properties of CaMKII kinetics, the enzyme activity of the Ca2+-CaM-CaMKII complex can be expressed as a function of Ca2+-CaM, with a rapid binding rate and slow dissociating rate. We calculate an enzyme activity factor (arbitrary units) using equation (A 49) (see appendix A for a complete listing of equations underlying the present model; all further reference to equation numbers relates to this appendix). Figure 2 illustrates how Ca2+-CaM and Ca2+-CaM-CaMKII change in our model when the pacing rate is altered from 1 to 2 Hz. The enzyme activity term dampens changes in Ca2+-CaM. This ‘slow inactivation’ results in a higher CaMKII activity during shorter beat interval, compared with regular ones.
(ii) SR Ca2+ release model
The three-state model of the RyR used in this model is based on the formulations by Hilgemann & Noble (1987). It describes inactivation and recovery processes of RyR via precursor (F1 in equation (A 42)), activator (F2 in equation (A 43)) and product fraction (F3 in equation (A 44)). RyR activation and inactivation rates are based on the Kyoto model (Matsuoka et al. 2003). The Ca2+-induced activation rate (k1 in equation (A 45)) is defined as a function of [Ca2+]i and L-type Ca2+ channel current (ICaL). The inactivation rate (k2 in equation (A 46)) is defined as a function of the SR Ca2+ concentration [Ca2+]SR. The rate of recovery of RyR (k3 in equation (A 47)) is a function of [Ca2+]SR (to express the SR load dependency of the open probability), and the modulating effect of CaMKII on Ca2+ release (Li et al. 1997) is included in it. Additionally, we incorporate an SR load dependency of Ca2+ conduction through RyR (Gyorke & Gyorke 1998; Ching et al. 2000) into our model (A 39).
(iii) SR Ca2+ uptake model
CaMKII regulates phosphorylation of phospholamban, which enhances SR Ca2+-ATPase (SERCA) activity and SR Ca2+ uptake. CaMKII involvement in SR Ca2+ uptake contributes to the interval dependence of twitch duration (Schouten 1990; Bassani et al. 1995), and it may be expected to affect I–F relations as well. To describe this CaMKII involvement, we use the SERCA model of Shannon et al. (2000) and added a CaMKII dependency in the forward (uptake) flux of SERCA (A 51).
(iv) SR and cytosolic compartments
Many of the recent cardiac cell models assume two compartments for the SR: one Ca2+ uptake site, and one release site, with considerable delay in Ca2+ transfer from uptake to release site. However, some recent data suggest that Ca2+ diffusion should be quick enough to cause very little difference in Ca2+ concentration between sites (Shannon et al. 2003). Therefore, the present model uses a single compartment for the SR (for simplicity and lack of better experimental data). Furthermore, many recent models assume the dyadic space as a uniform site for Ca2+ entry into the cytosol via L-type Ca2+ channels and RyR (Nordin 1993; Rice et al. 2000). Since modelling of the dyadic space is not essential for studying beat-to-beat integrative Ca2+ dynamics in I–F relations, we use only one compartment for the cytosolic space to calculate [Ca2+]i (A 58).
(v) Contraction model
To evaluate I–F relations taking into account Ca2+ dynamics, the force computation model describes interactions between force and Ca2+ in relative detail. We use the model of contraction and cooperativity mechanisms of Ca2+, troponin, tropomyosin and crossbridge formation by Rice et al. (1999). Not only a feed-forward pathway from changes in [Ca2+]i to force production, but also a feedback pathway from developed force to Ca2+ handling and troponin binding (A 55) are included.
(vi) Other components
As mentioned earlier, many recent models assume a dyadic space (Nordin 1993; Rice et al. 2000), thus L-type Ca2+ channel formulations in such models are optimized to account for the presence of such a space. Our present model does not separate the dyadic space; therefore, for membrane currents, we incorporate a previously published single cardiac cell model, which does not have a dyadic space, by Noble et al. (1991) (as opposed to Noble et al. (1998), for instance). Appendix A gives the full set of equations. The CellML version of the model will be available on http://www.cellml.org/models/iribe_kohl_noble_2006_version01.
(b) Experimental protocols
The model was validated against published experimental information on Ca2+ dynamics in relation to I–F relations, including the interval dependence of twitch duration, restitution and potentiation, and the staircase phenomenon. On this basis, we investigate the behaviour of alternant decay of PESP, and suggest experimentally testable hypotheses regarding the mechanisms that may underlie this phenomenon. The detail of each pacing protocol is explained in §3.
The simulations were run on MATLAB, using ode15s (Runge–Kutta) as the solver for integrating the ordinary differential equations.
(a) Interval dependence of twitch duration
It has previously been reported that twitch duration decreases when the preceding beat interval is shortened (Schouten 1990; Bassani et al. 1995). The mechanism proposed here is as follows: after a short interval, SERCA is still activated because of the slow inactivation of CaMKII (figure 2); therefore, SR Ca2+ uptake is enhanced, leading to more rapid relaxation.
Figure 3a illustrates the approach to assessing the interval dependence of twitch duration in the model. After pacing the model for 20 s at 1 Hz (to obtain a sufficiently steady control state), an ES beat is introduced at an ESI of either 0.5 or 2 s (figure 3a). Time from stimulus to peak force (PFT) and twitch duration at relaxation to 50% of the peak force (TD50) are compared for the ES beats (figure 3b). PFT and TD50 are reduced at short ESI (0.069 and 0.138 s, respectively) and increased at long ESI (0.099 and 0.167 s, respectively), compared to control activity (0.084 and 0.154 s, respectively), as seen in experiments (Schouten 1990; Bassani et al. 1995). Figure 3c,d shows superimposed SR Ca2+ uptake flux curves and CaMKII activities for both ES beats (ESI=0.5 and 2 s). The uptake flux is larger at the shorter ESI, because of elevated CaMKII activity (peak flux of 0.181 and 0.158 mM s−1 in ESI at 0.5 and 2 s, respectively). These simulations reproduce, in principle, the phenomena and mechanism highlighted in the experiments of Schouten and Bassani (Schouten 1990; Bassani et al. 1995).
(b) Mechanical restitution and potentiation
Figure 4 shows the results of experiments to characterize mechanical restitution and potentiation in our model. After conditioning cells to a steady state by fixed rate pacing (0.5 Hz), ES beats are introduced at variable ESI (from 0.4 to 2 s), followed by a PES beat at a fixed interval of 2 s (figure 4a). As shown in figure 4a, ES amplitude increases with ESI, while PESP shows the opposite trend. This is consistent with experimental observations by Wier & Yue (1986).
The key mechanism underlying restitution of ES beat amplitude is recovery of RyR releasability (defined in the model as the product of RyR flux factor and release fraction of RyR, Krel×Frel; see appendix A for more detail, and figure 4b), while PESP dynamics are mainly linked to SR Ca2+ content (figure 4c), as previously modelled by Rice et al. (2000).
Our model further reveals a possible contribution of CaMKII to these phenomena. The shorter the ESI, the greater the CaMKII activity (figure 4d). Therefore, although Ca2+ release from RyR is small after a short ESI, SR Ca2+ uptake is increased (compared to control; figure 4e). This biases Ca2+ handling towards sequestration, so that less Ca2+ reaches troponin C, thereby reducing force generation. CaMKII also affects PESP. The increased SR load after a shorter ESI (main mechanism for PESP) is not only caused by preservation of Ca2+ in the SR (reduced release via non-recovered RyR), but also by an enhanced Ca2+ uptake into the SR via CaMKII effects. Additionally, RyR releasability during the PES period is also enhanced because of its SR load dependency and the amplifying effect of CaMKII on this load dependency (figure 4b,d). These effects lead to more prominent potentiation during the PES period compared to the model without CaMKII involvement (Nordin 1993; Rice et al. 2000).
(c) Positive staircase phenomenon
Increasing stimulation frequency gives rise to a positive inotropic effect via increasing SR Ca2+ content and release in many species (Kurihara & Sakai 1985; Maier et al. 2000). This phenomenon can be reproduced by the model. Figure 5 illustrates the effects of changing stimulation rate from 1 to 2 Hz, and then back to 1 Hz. The initial drop in twitch force amplitude at the onset of 2 Hz stimulation (figure 5a) reflects the reduced fraction of recovered RyR channels at the faster pacing rate (figure 5d). Thereafter, SR Ca2+ content is gradually raised by the enhanced SR Ca2+ uptake, caused in the model by increased CaMKII activity (figure 5b,c). SR Ca2+ release is additionally enhanced via SR Ca2+ load dependency of Ca2+ releasability. This is also related to the increased CaMKII activity (figure 5c,d).
Conversely, reduction of pacing rate from 2 to 1 Hz causes a transient increase in twitch force via additional recovery of RyR channels. Thereafter, SR Ca2+ gradually returns to control levels, as the reduced CaMKII activity decreases SR Ca2+ uptake and Ca2+ releasability.
(d) Mechanisms of transient alternans of PESP decay
To examine Ca2+ handling during the transient alternans of the PESP decay in our model, we use the pacing protocol illustrated in figure 6a. After a control pacing cycle at a steady state regular interval (RI) of between 0.4 and 0.75 s, ES beats are introduced at an ESI that is shorter than RI (see figure 6 legend for detail). After one ES beat, pacing rate is restored to record PESP and its decay dynamics (see figure 1 for experimental and figure 6 for modelling findings).
Depending on the background pacing rate, alternans of the PESP decay pattern has been observed in experiments of mammalian myocardium, where alternans is most pronounced at faster pacing rates (Kihara & Morgan 1991; Laurita et al. 2003). The model reproduces this beat interval dependency of alternans in PESP decay (compare figures 1 and 6a). At RI of 0.75 s, PESP decays monotonically, both in experiment and model. Increasing RI (to 0.4 s in experiments and to 0.5 s in the model) gives rise to minor oscillations in PESP amplitudes. Clear mechanical alternans can be observed at even shorter RI (0.33 s in experiments, 0.4 s in the model).
Diaz et al. (2004) showed that mechanical alternans is caused by beat-to-beat alternation in [Ca2+]i, which in turn is mainly due to the alternation in SR Ca2+ content. In particular, the steepness of the relation between SR Ca2+ content and Ca2+ release is critical in the induction of mechanical alternans. Our results show that mechanical alternans during PESP decay is linked to SR Ca2+ content fluctuations (figure 6a, right-hand side).
Figure 6b summarizes the relation between SR Ca2+ load and Ca2+ release of several PES beats (same cases as in figure 6a). SR Ca2+ load is equal to the value of [Ca2+]SR prior to the stimulation, while SR Ca2+ release is the difference between SR Ca2+ load and the smallest [Ca2+]SR reached during any given beat. At an RI of 0.75 s, the slope of this relation is shallow, which means that SR Ca2+ content decreases gradually, which results in a monotonic decay of PESP. At RI of 0.4 s, the slope of the relation is steeper, supporting large SR Ca2+ releases during the first PES beat (because of the high SR load). This large release reduces SR Ca2+ load; therefore, the next SR release (and resultant PES beat) are small. This small SR Ca2+ release increases SR Ca2+ content in relative terms (but not to the same level as observed before the first beat), giving rise to transient alternans of the PESP decay pattern, as seen experimentally (Diaz et al. 2004). Accordingly, the ‘beat interval dependency of alternans’ could also be referred to as a ‘beat interval-dependent change in SR Ca2+ load dependency of Ca2+ release’. In the present model, this feature is brought about by an enhancing effect of CaMKII on the SR Ca2+ load-dependent RyR recovery rate (k3; (A 47)). More rapid stimulation results in higher CaMKII activity, which steepens the relations between SR Ca2+ content and Ca2+ release, and causes increasingly pronounced mechanical alternans of the PESP decay.
In addition to its beat interval dependency, mechanical alternans has also been observed upon slowing of either RyR or SERCA (Diaz et al. 2002; Kameyama et al. 2003; Kockskamper et al. 2005). Figure 7a shows the effects of slowing or accelerating the transition between the three functional states of RyR, presented in the model, on alternans during PESP decay. This is based on the ‘intermediate’ scenario in figure 6a (RI=0.5 s), for comparison. Slowing all transition rates by 20% significantly enhanced alternans, while an equally pronounced acceleration eliminates alternans. This result is consistent with previous experimental findings (Diaz et al. 2002). Figure 7b shows superimposed traces of Ca2+ releasability during both slowed and accelerated RyR dynamics in the first and second PES beat (note the fluctuation in amplitude of Ca2+ releasability when RyR function is slowed, also present when accelerated, though to a lesser extent). During the first PES beat, slowed RyR give rise to a slightly smaller peak releasability compared to the accelerated RyR. However, slowed RyR stay open for longer, so that overall more Ca2+ is released. This causes a more pronounced Ca2+ depletion of the SR and contributes to reduced Ca2+ release during the subsequent beat. During the second PES beat, slow RyR are still open for longer, but peak releasability is considerably smaller than in the presence of accelerated RyR (because of the SR Ca2+ load dependency of RyR releasability), which also contributes to the overall reduction in SR Ca2+ release during the second beat with slowed RyR. This behaviour makes the slope of the relationship between SR Ca2+ load and Ca2+ release steeper, as summarized in figure 7c, and leads to more pronounced alternans in the presence of slowed RyR.
Thus, although both rapid pacing and slowed RyR dynamics introduce/enhance alternans by increasing the steepness of the relationship between SR Ca2+ load and Ca2+ release, the primary mechanisms underlying these effects are different. In the case of rapid pacing, the steep relation is induced by increased CaMKII activity; while in the case of slowed RyR it is caused by prolonged RyR opening.
To investigate the experimentally observed effects of SR Ca2+ uptake rates on alternans, we slowed or accelerated the Ca2+ uptake rate parameter (Vmaxf). Figure 8a illustrates the enhancement of alternans in the PESP decay pattern (compared to the ‘intermediate’ control scenario illustrated in figure 6a; RI=0.5 s), that is caused by a 20% slowing of Ca2+ uptake rate. A matching increase in SERCA activity eliminates alternans. This result is consistent with previous experimental findings (Kameyama et al. 2003; Kockskamper et al. 2005).
In the model, slowed SERCA activity reduces Ca2+ sequestration into the SR during RyR release, thereby causing an increase in the first PES beat compared to control (which makes the slope of the relation between SR Ca2+ content and Ca2+ release steep). The reduced SERCA activity shifts the balance of Ca2+ handling from preservation in the SR to extrusion through the sarcolemma, which reduces the second PES beat.
With accelerated SERCA activity, Ca2+ that is being released via RyR is partially sequestered back into the SR, even while RyR release is still ongoing. This reduces the overall amount of SR Ca2+ release during the first PES beat and the amplitude of PESP. In addition, the enhanced SERCA activity increases Ca2+ re-uptake into the SR, raising Ca2+ content for the second beat. From there on, PES beat amplitude decays monotonically.
Figure 8b summarizes the relationship between SR Ca2+ content and Ca2+ release with slowed and accelerated Ca2+ uptake dynamics. Although the properties of RyR in both cases are identical, the slope of the relationship is somewhat shallower in the case with accelerated uptake.
CaMKII effects on SR Ca2+ uptake may play a role in the genesis of alternans. Therefore, we tested transient alternans of the PESP decay in the presence and absence of CaMKII fluctuations, as shown in figure 9. Using the ‘intermediate’ case of figure 6a as a control (RI=0.5 s; figure 9, left-hand side), we set CaMKII activity factor (FCaMK) in the Ca2+ uptake formula (A 51) to its average value in control conditions (figure 9, right-hand side). As shown in figure 9a, CaMKII modulation of Ca2+ uptake reduces alternans. Figure 9b,c shows CaMKII activity and SR Ca2+ uptake flux, respectively. CaMKII activity in the PES period is higher than that in steady state. Therefore, Ca2+ uptake is higher during the PES period in the presence of functional CaMKII. Especially, after the first PES beat, enhanced Ca2+ uptake prevents depletion of SR Ca2+ content at the next beat. As a result, the second PES beat becomes larger than it would have been without CaMKII involvement, thereby reducing alternans.
This result may be seen as somewhat counter-intuitive, as one might have expected that an increased CaMKII activity at the beginning of the second PES beat would sequester more Ca2+ and reduce the amplitude of the second PES beat over and above the level seen without CaMKII involvement. However, the model predicts that the effect of CaMKII on preserving SR Ca2+ content (enhancing uptake) after the first PES beat dominates over the reducing effect on the second PES beat.
Several myocardial Ca2+ handling models have been proposed to describe and study alternans (Adler et al. 1985; Snyder et al. 2000; Tameyasu 2002; Shiferaw et al. 2003). Each of them features Ca2+ handling in different detail. However, a common and apparently fundamental mechanism underlying alternans is a steep relation between SR Ca2+ load and Ca2+ release, which is also supported by experimental evidence (Diaz et al. 2002).
Some of the existing models allow one to replicate the beat interval dependency of alternans. For example, Snyder et al. (2000) describe this phenomenon as instability in the bi-directional feedback system between RyR and calsequestrin during rapid pacing, while Shiferaw et al. (2003) explain it as a result of variable Ca2+-induced inactivation of L-type Ca2+ channels, due to increasing diastolic [Ca2+]i during rapid pacing. Our model proposes an alternative concept, where CaMKII is a major contributor to the beat interval dependency of alternans. This is in keeping with data from Li et al. (1997), who reported that CaMKII increased the amount of SR Ca2+ release for a given SR Ca2+ content and ICaL trigger in intact voltage clamped ventricular myocytes. They showed a shallower relationship between SR Ca2+ load and twitch [Ca2+]i under CaMKII inhibition than in control. This suggests that CaMKII modulation of the relation between SR Ca2+ load and Ca2+ release, and its beat interval-dependent activity, may be reasonable candidate mechanisms underlying the beat interval dependency of alternans.
Indeed, using this approach, our model not only successfully reproduces the beat interval dependency of alternans, but it integrates alternans and various other I–F relations at the whole cell level via CaMKII involvement. Although it is intuitively clear that the role of CaMKII in Ca2+ handling and its time dependency are potentially important contributors to various I–F relations, no previous myocardial cell model has incorporated this, largely because of uncertainties in kinetic features. As a conceptual guide, however, the model has shown that it is not inconceivable for CaMKII to be an important contributor to alternans and it has advanced our integrative understanding of various I–F relations.
The mechanisms that underlie the various I–F relations consist of several components, including time-dependent recovery of RyR, SR Ca2+ load dependency of Ca2+ release, and probably time-dependent modification on Ca2+ release and uptake by CaMKII. These mechanisms are interdependent. In the model, CaMKII involvement in SR Ca2+ uptake is the causative mechanism of interval dependence of twitch duration, which enhances SR Ca2+ accumulation which, in turn, contributes to post-rest potentiation and the staircase phenomenon. SR Ca2+ load dependency of Ca2+ release is a key mechanism of alternans, and CaMKII involvement in SR Ca2+ load regulation enhances its beat interval dependency.
Another interesting point of the present study is that our model enables us to evaluate the effect of modulation of each Ca2+ handling pathway on alternans independently. This is useful when re-investigating possible mechanisms that may underlie the effects of various experimental interventions on alternans. For example, Diaz et al. (2002) reported that decreasing RyR open probability by applying tetracaine or via acidification produces alternans, and resulting Ca2+ transients are characterized by a slowing in rise and decay phases. Our investigation reproduced this behaviour, in principle, by slowing RyR dynamics, rather than decreasing their open probability.
Alternans is an integrated output of various Ca2+ handling components, each of which may affect it in different (partially counter-intuitive) ways. Even similar alternans patterns may be caused by different, even opposing, mechanisms. For example, our model can reproduce sustained alternans by any combination of alternans enhancing factors, such as increasing CaMKII involvement on RyR, slowing RyR and slowing Ca2+ uptake. It is quite difficult to experimentally control these factors separately and to identify the dominant factors. In that respect, the present model is useful for integrative studies into Ca2+ handling in alternans.
Limitations of this model are mainly related to its simplicity. The CaMKII kinetics model we use is conceptual, and not based on a detailed description of chemical reactions. Dupont et al. (2003) reported a more detailed biophysical model of CaMKII kinetics. However, we found that short-term responses of CaMKII activity to frequency changes in our model (figure 2) are consistent with the results obtained using the more complex models. Therefore, in terms of beat-by-beat short-term I–F relation, there is no significant difference between results with detailed and simple models.
Many of the recent more detailed Ca2+ dynamics models describe RyR as a cluster of unitary channels (Greenstein & Winslow 2002; Shiferaw et al. 2003) to address spatially heterogenic effects within a cardiac cell (e.g. Ca2+ sparks or waves). Diaz et al. (2002) showed that subcellular alternans occurs randomly, so that the averaged whole cell signal would be quite uniform. Our point model does not address spatial heterogeneity, since we intended to make the model as simple as possible to clarify the effects of alterations in SR Ca2+ handling on I–F relation, especially on alternans. From the viewpoint of our aim, our phenomenological model offers a reasonable tool to describe underlying mechanisms.
Although altered SR Ca2+ handling is understood to be a primary mechanism underlying alternans, there is a possibility that the primary fluctuation of transmembrane Ca2+ fluxes plays an important role (Fox et al. 2002; Shiferaw et al. 2003). The fact, however, that rapid pacing causes alternans, even under action potential clamp conditions, strongly suggests that SR Ca2+ handling is the dominant causative mechanism (Chudin et al. 1999). Still, the modelling study of alternans by Shiferaw et al. (2003) suggested that inactivation properties of the L-type Ca2+ channel may determine alternans, even during action potential clamp. Furthermore, it has been reported that ICaL is affected by CaMKII (Xiao et al. 1994; Yuan & Bers 1994), and that it shows beating interval-dependent behaviour (Tseng 1988; Hryshko & Bers 1990). Therefore, the lack of CaMKII involvement in membrane currents is a limitation of the present model. Although the presented SR involvement is sufficient to explain a host of experimental observations involving I–F relations and alternans, further investigations will benefit from incorporation into the model of a more detailed ICaL model, with CaMKII and Ca2+-dependent properties, such as proposed by Hund & Rudy (2004).
Thus, we present a cardiac cell model to simulate transient alternans during the decay of PESP, and its beat interval dependency. A key feature of this model is the involvement of CaMKII in Ca2+ handling. A conceptual CaMKII kinetics model that includes its slow inactivating properties was incorporated. The slow inactivation causes relatively increased CaMKII activity at shorter beat intervals. The SR load dependency of RyR Ca2+ release and SR Ca2+ uptake are modulated by CaMKII activity. Shorter beat intervals show alternans because of the steeper relation between SR load and release, due to the higher CaMKII activity that is a consequence of the shorter beat interval. The model suggests that CaMKII effects on Ca2+ handling are important not only for reproducing the beat interval dependency of alternans, but also for the integration of alternans and various other I–F relations at the whole cell level. Furthermore, the effects on alternans of various interventions, such as slowing or accelerating Ca2+ release and/or uptake, could be reproduced. We conclude that a slow time-dependent factor, represented in the model by CaMKII, is important for the integration of I–F relations, including alternans, and that our model offers a useful tool for analysing the roles of integrative Ca2+ handling in myocardial I–F relations.
The authors thank Dr Alan Garny and Dr Jeremy Rice for helpful comments on the manuscript. This study was supported by the British Heart Foundation. G.I. holds a PhD training grant from Eisai Co., Ltd.
One contribution of 13 to a Theme Issue ‘Biomathematical modelling I’.
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