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Author's personal copy Journal of Molecular and Cellular Cardiology 46 (2009) 332–342
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Journal of Molecular and Cellular Cardiology j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / y j m c c
Original article
Theoretical investigation of action potential duration dependence on extracellular Ca2+ in human cardiomyocytes Eleonora Grandi a,b, Francesco S. Pasqualini a, Chiara Pes a, Cristiana Corsi a, Antonio Zaza c, Stefano Severi a,⁎ a b c
Biomedical Engineering Laboratory – D.E.I.S., University of Bologna, Cesena, Italy Department of Pharmacology, University of California, Davis, CA, USA Department of Biotechnology and Bioscience, University of Milano Bicocca, Milano, Italy
a r t i c l e
i n f o
Article history: Received 30 August 2008 Received in revised form 10 November 2008 Accepted 3 December 2008 Available online 11 December 2008 Keywords: Extracellular calcium Computer modeling Action potential duration Calcium current inactivation
a b s t r a c t Reduction in [Ca2+]o prolongs the AP in ventricular cardiomyocytes and the QTc interval in patients. Although this phenomenon is relevant to arrhythmogenesis in the clinical setting, its mechanisms are counterintuitive and incompletely understood. To evaluate in silico the mechanisms of APD modulation by [Ca2+]o in human cardiomyocytes. We implemented the Ten Tusscher-Noble-Noble-Panfilov model of the human ventricular myocyte and modified the formulations of the rapidly and slowly activating delayed rectifier K+ currents (IKr and IKs) and L-type Ca2+ current (ICaL) to incorporate their known sensitivity to intra- or extracellular Ca2+. Simulations were run with the original and modified models at variable [Ca2+]o in the clinically relevant 1 to 3 mM range. The original model responds with APD shortening to decrease in [Ca2+]o, i.e. opposite to the experimental observations. Incorporation of Ca2+ dependency of K+ currents cannot reproduce the inverse relation between APD and [Ca2+]o. Only when ICaL inactivation process was modified, by enhancing its dependency on Ca2+, simulations predict APD prolongation at lower [Ca2+]o. Although Ca2+-dependent ICaL inactivation is the primary mechanism, secondary changes in electrogenic Ca2+ transport (by Na+/Ca2+ exchanger and plasmalemmal Ca2+-ATPase) contribute to the reversal of APD dependency on [Ca2+]o. This theoretical investigation points to Ca2+-dependent inactivation of ICaL as a mechanism primarily responsible for the dependency of APD on [Ca2+]o. The modifications implemented here make the model more suitable to analyze repolarization mechanisms when Ca2+ levels are altered. © 2008 Elsevier Inc. All rights reserved.
1. Introduction It is well known that changes in serum calcium influence the cardiac electrical activity particularly affecting ventricular repolarization [1–3]. The primary electrocardiographic manifestation of hypocalcaemia is QTc prolongation [4,5], which is associated with increased risk of early after-depolarizations and triggered arrhythmias. On the other hand, hypercalcaemia exerts an opposite effect on the electrocardiogram with the hallmark of abnormal shortening of the QTc interval [5]. Ca2+-dependency of repolarization is particularly relevant Abbreviations: AP, Action Potential; APD, Action Potential Duration; APDR, Action Potential Duration Restitution; APD90, Action Potential Duration to 90% of repolarization; DI, Diastolic Interval; endo, endocardial; epi, epicardial; GKs, maximal conductance of IKs; ICaL, L-type Ca2+ current; IKr, rapidly activating delayed rectifier K+ current; IKs, slowly activating delayed rectifier K+ current; INaCa, Na+/Ca2+ exchanger current; IpCa, plasmalemmal Ca2+-ATPase current; Ito, transient outward K+ current; PLB, phospholamban; QTc, Heart rate corrected QT interval; SR, sarcoplasmic reticulum; TNNP, Ten Tusscher-Noble-Noble-Panfilov; [Ca2+]i, intracellular calcium concentration; [Ca2+]o, extracellular calcium concentration; [Na+]i, intracellular sodium concentration. ⁎ Corresponding author. Biomedical Engineering Laboratory – D.E.I.S., University of Bologna, Via Venezia 52, I-47023 Cesena – Italy. Tel.: +39 0547 339127; fax: +39 0547 339114. E-mail address:
[email protected] (S. Severi). 0022-2828/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.yjmcc.2008.12.002
in the setting of hemodialysis, when plasma Ca2+ levels may vary widely according to the treatment. During dialysis, QTc was found to inversely correlate with plasma Ca2+ levels by several authors [6–9]. Experimental studies in isolated cells have demonstrated that the duration of the cardiac AP is sensitive to [Ca2+]o. As a general rule, elevated [Ca2+]o shortens the action potential and reduced [Ca2+]o lengthens it. Such behavior has been observed in ventricular muscle [10–12] and Purkinje fibers [13,14] from different species. Data on the effects of hypo- and hypercalcaemia on human APs are not available, however, the consistency between QTc and ventricular APD changes suggests that the same phenomenon may occur in the clinical and experimental settings. Although these effects have been known for some time, and in spite of the clinical relevance, their explanation remains unclear. A subtle balance between inward and outward currents active during the AP plateau determines the APD. In AP phase 2, ICaL is the main inward current and the outward contribution is dominated by the rapidly and slowly activating components (IKr and IKs) of the delayed-rectifier K+ current. An increase in inward current is expected to prolong APD; indeed, blockade of ICaL depresses the plateau phase and shortens APD. The change in the electromotive force resulting from an increment of [Ca2+]o per se should increase
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ICaL, thereby prolonging APD. Nevertheless, the picture is complicated by secondary changes in the cytosolic and SR Ca2+ concentrations and by Ca2+-sensitivity of other membrane currents. Voltage-dependent gating of IKr and IKs conductances is modulated by external and cytosolic Ca2+ respectively [12,15–18] and the latter directly determines an important component of ICaL gating (Ca2+-dependent inactivation) [19–22]. Moreover, intra- and extracellular Ca2+ concentrations affect the rate of electrogenic Ca2+ transport through the Na+/Ca2+ exchanger (INaCa) and the plasmalemmal Ca2+-ATPase (IpCa). Which of these mechanisms prevails in determining Ca2+-dependency of repolarization is a relevant question that cannot be addressed by electrophysiological measurements, due to the complex interaction among currents. An alternative and complementary approach is the use of mathematical AP models, which allow investigating the effects of channel alterations on cardiac repolarization individually and collectively. In recent years, studies on human myocytes have made available a large number of data on the gating properties of many channels, thus setting the bases for the development of computational models of human AP [23–26]. The aim of the present study was to identify the ionic mechanism/s likely to underlie the APD dependency on [Ca2+]o by using an in silico approach. To this purpose, new formulations of the Ca2+-dependency of IKs, IKr and ICaL were incorporated in the TNNP model of the human ventricular AP [25]. 2. Materials and methods 2.1. Action potential model The TNNP model of the human ventricular myocyte [25] provided the basis for the AP simulations in this study. It describes the main membrane currents and active transport mechanisms participating in the AP and the processes that regulate intracellular Ca2+ concentration according to a “common pool” formulation of Ca2+-induced Ca2+ release. This model can reproduce the action potentials of three ventricular cell types, epi, endo and M cells, by changing a few parameters in Ito and IKs formulations. The model has been validated against a wide set of experimental data [25]. Model differential equations were implemented in Simulink (Mathworks Inc., Natick, MA, U.S.A.). A variable order solver based on the numerical differentiation formulas was used to numerically solve the model equations (ode15s) [27]. Pacing at 1 Hz was maintained until a steady AP was reached. APD was measured as the interval between the AP upstroke and the 90% repolarization level (APD90). AP phase 2 was determined by considering the AP second derivative: in detail, phase 2 starts from the local maximum following the AP notch and ends when the rate of change of repolarization velocity (i.e. the AP second derivative) reaches a fixed threshold (10 mV/ms2). The S1-S2 restitution protocol, used to assess APD restitution, consists of 10 S1 stimuli (at 1 Hz pacing rate) followed by a S2 extrastimulus delivered at some diastolic interval after the AP generated by the last S1 stimulus. The APD restitution curve is generated by decreasing DI and plotting APD corresponding to the S2 stimulus against DI. In AP clamp simulations, a train of steady-state APs was obtained at 0.2 Hz and used to pace the digital cells. The model parameters are listed in the TNNP paper [25] and our modifications to the original formulation are described below. 2.2. IKs The slowly activating delayed rectifier K+ current has been shown to be a target of regulation by Ca2+ in guinea pig ventricular myocytes [12,16–18]. Elevation in intracellular Ca2+ concentration enhances IKs and the range of [Ca2+]i that regulates IKs (between 10−2 and 1 μM) corresponds to the range of [Ca2+]i variation during the cycle of contraction and relaxation of cardiomyocytes. Therefore, Ca2+ sensi-
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tivity of IKs seems an important determinant of cardiac repolarization. Changes in [Ca2+]o affect [Ca2+]i thus modulating IKs density [12]. Although there are no available data, a similar modulation by Ca2+ may be assumed in human ventricular myocytes. We incorporated the dependency of the maximal conductance GKs on intracellular Ca2+ as it was formulated by Viswanathan et al. [18] (Fig. 1B), on the basis of the study of Nitta et al. [16]: 0 B B B GKs = α B1 + B @
1 C C C !1:4 C C −5 3:8 10 A 1+ ½Ca2 + i 0:6
with α = 0.157 mS/μF for epi and endo cells, α = 0.04 mS/μF for M cell. Ion concentrations are expressed in mM. 2.3. IKr Human ether-a-go-go-related (HERG) gene K+ channels, responsible for IKr, have been shown to be sensitive to external Ca2+ concentration. An increase in [Ca2+]o shifts the voltage dependence of channel activation to more positive membrane potentials. Such sensitivity has not yet been incorporated into any ventricular myocyte model. The [Ca2+]o dependence of the half maximal voltage of activation (V1/2) was derived from the three-dimensional fit of the voltage-dependent Monod–Wyman–Changeux model as reported by Johnson et al. [15] and shifted in order to have V1/2 = −26 mV when [Ca2+]o = 2 mM, as in the TNNP model: 0 0 h i 14 1 Ca2 + B @ C oA B 1+ C B C K a B C RT logB V1=2 = − h i 14 C 0 B C−0:026 F Q 2+ B C Ca B C oA @@1 + L0 A Kc where F is the Faraday constant, T is the temperature, R is the gas constant. Q is the apparent gating charge that moves through the membrane electrical field during the transitions from closed to activated, Kc is the Ca2+ dissociation constant for the closed channel, Ka is the Ca2+ dissociation constant for the activated (open) channel, and L0 is the closed-open equilibrium constant in the absence of calcium (see Johnson et al. [15] for the derivation of the expression for V1/2 and parameter values). Shifts in the IKr activation curve (V1/2 from −37 to −20 mV) corresponding to [Ca2+]o changes from 1 to 3 mM were tested for their effects on APD (Fig. 1A). 2.4. ICaL After ICaL activation, the L-type Ca2+ channel undergoes voltage and Ca2+ dependent inactivation [22,28–33]. In the TNNP model ICaL kinetics is described by a voltage-dependent activation gate (d), a voltage-dependent inactivation gate (f), and a [Ca2+]i-dependent inactivation gate (fCa):
ICaL = GCaL
h i h i 2VF 1 0 2+ − Ca2þ e RT V F 2 @0:341 Ca o i A: d f fCa 4 2VF RT 1−e RT
Because experimental data on Ca2+-dependence of inactivation in human myocytes are unavailable, we modified the fCa formulation (Fig. 1C) according to the following qualitative observations: a) experimental studies demonstrated that the Ca2+-dependent inactivation of ICaL is dominant over the voltage-dependent process and its maximum inactivation level is higher than the 80% included in the
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Fig. 1. Modulation by Ca2+of IKr, IKs, and ICaL. Curves from the TNNP model and our modifications (modified model) are shown. (A) Effect of extracellular Ca2+ on the voltage dependence of steady-state activation of IKr. Activation curves were obtained by incorporating in the model the Ca2+ dependence of the half maximal voltage (V1/2) of activation of HERG channels (inset). (B) Effect of intracellular Ca2+ on the maximal conductance of IKs. (C) Steady-state Ca2+-dependent inactivation of the L-type calcium current. (D) Effect of extracellular Ca2+ on the voltage-dependent inactivation time constant.
TNNP model [19–21]; b) during AP clamp experiments on guinea pig ventricular myocytes ICaL inactivation ranged from 40% to 95% in control conditions [30]. It seems therefore unlikely that at [Ca2+]i equal to 500–600 nM the Ca2+-dependent inactivation was completely exploited so that an 80% degree of inactivation would be expected for any higher [Ca2+]i, as assumed by the TNNP formulation. Taken together, these observations led to the following fCa formulation, in which the switch shape is slightly smoothed, the threshold is shifted towards higher [Ca 2+ ] i, and the level of inactivation beyond the threshold is higher with respect to the TNNP model (Fig. 1C). fCa; inf −fCa dfCa =k dt τ Nfca k = 0 if fCa, inf N fCa and V N -60 mV;k = 1 otherwise fCa; inf =
α fca =
α fca þβfca þγ fca 1:3156
1 0 h 2 + i 18 Ca i A 1+@ 0:000600
characteristic (Fig. 1D) by adding to the formulation of the voltagedependent inactivation time constant, τf, a linear dependence on [Ca2+]i based on experimental data in rabbit cardiomyocytes [34]: ðV + 27Þ2 165 τf = 1125 e− 240 + + 80 25−V 1 h i + e 10 1 + 1433 Ca2 + −50 10−6 i
if finfNf
We verified the ability of the model to qualitatively reproduce the influence of [Ca2+]i on L-type Ca2+ current and L-type channel inactivation (fCa∙f) during an action potential by comparing simulations with the TNNP and our modified models with data by Linz and Meyer from guinea pig ventricular myocytes [30]. In their study, [Ca2+]i was manipulated, for example, by blocking the efflux of Ca2+ from the myocyte via Na+/Ca2+ exchange. This block leads to ICaL reduction and is accompanied by a reduction of fCa. The TNNP model fails to reproduce the effects of these maneuvers on Ca2+ homeostasis (Fig. 2, left), which are recapitulated by the modified model (Fig. 2, right): under AP clamp conditions, INaCa blockade increases [Ca2+]i, thus leading to a reduction of fCa (e.g. enhancement of Ca2+ dependent inactivation) and consequent decrease of ICaL. 3. Results
βfca =
γ fca =
0:1 ð½Ca2 + i −0:0009Þ 0:0001 1+e 0:3 ð½Ca2 + i −0:00075Þ 0:0008 1+e
Recovery from inactivation of ICaL is also dependent on intracellular Ca 2+ and it slows as [Ca2+]i increases. We incorporated this
3.1. Simulations with TNNP model In the simulations performed with the original TNNP model, APD90 increases linearly with [Ca2+]o (Figs. 3A and C) in all the cell types (endo, epi and M cells). The increase in driving force, resulting from increased Ca2+ gradient, causes a significant increase of ICaL peak (−8.3 vs. −5.6 pA/pF at 3 vs. 1 mM external Ca2+, in epi cell), which augments Ca2+ transient amplitude (from 0.54 to 1.2 μM), and ICaL plateau value (−1.5 vs. −1 pA/pF), thus prolonging repolarization. Prolongation of APD at higher [Ca2+]o is clearly in contrast with experimental data,
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Fig. 2. ICaL characteristics under AP clamp (first row) with and without Na+/Ca2+ exchange blockade (dashed vs. solid lines). Block of INaCa increases [Ca2+]i (second row) in both models; in contrast with experimental data [30], this does not produce changes in Ca2+ current inactivation when simulated with the TNNP model (third and fifth rows, left). With the modified model, INaCa blockade leads to a reduction in ICaL (third row, right). This is accompanied by a reduction of fCa∙f during the action potential plateau (sixth row, right), which is entirely due to a decrease in fCa (fifth row, right), since f is unchanged in AP clamp (fourth row, right).
showing that an increase in [Ca2+]o above the physiological value (1.8 mM) to 2.4 and 3 mM shortens the APD in guinea pig ventricular myocytes [12] (Fig. 3E). On the basis of this divergence, which suggests an incomplete description of [Ca2+] modulation of the ionic currents underlying repolarization, we included the description of Ca2+ dependency of IKr and IKs and the modified formulation of Ca2+-dependent ICaL inactivation in the TNNP model. 3.2. Simulations with the modified model 3.2.1. IKr and IKs dependency on Ca2+ Incorporation of [Ca2+]o modulation of IKr into the TNNP formulation was expected to prolong AP with increased [Ca2+]o. However, no significant change is observed on APD and its dependency on [Ca2+]o (Figs. 3F vs. C). In fact, IKr peak does not significantly change after inclusion of Ca2+-dependency (e.g. 0.51 vs. 0.52 pA/pF when [Ca2+]o = 1 mM, in epi cell). Inclusion of GKs dependency on intracellular Ca2+ has negligible effect on APD. Indeed, the increase in APD observed upon increasing [Ca2+]o is similar to that observed with the original TNNP model version (Fig. 3G). The paucity of effects on APD (Figs. 3G vs. C) is explained by the fact that the IKs activated during the AP is almost insensitive to inclusion of Ca2+-dependency (e.g. in epi cell, peak IKs = 0.26 vs. 0.27 and 0.49 vs. 0.48 pA/pF when [Ca2+]o = 1 and 3 mM, respectively). 3.2.2. ICaL dependency on Ca2+ Modification of ICaL Ca2+-dependent inactivation reverses the APD dependency on [Ca2+]o predicted by the original TNNP model. An
increase in [Ca2+]o leads to APD shortening for all the three cell types (Fig. 3H). Peak ICaL is increased by [Ca2+]o elevation resulting in slowing of AP phase 1 (not shown), whereas ICaL amplitude during the AP plateau remains unchanged. This is the consequence of a marked increase in Ca2+-dependent inactivation of ICaL, which completely offsets the increment in the driving force for the permeating ion. Indeed, with the new formulation, the amount of Ca2+-dependent inactivation of ICaL during an AP (i.e. the maximum percent reduction of the Ca2+-dependent inactivation gate fCa) changes significantly with respect to the original version of the TNNP model, being more pronounced at higher Ca2+ concentrations (61% vs. 72%, 82% vs. 78%, and 87% vs. 79%, when [Ca2+]o = 1, 2 and 3 mM, respectively, in epi cell). The inverse APD-[Ca2+]o relationship is maintained whether Ca2+ dependency of ICaL recovery from inactivation is incorporated or not. Finally, the new ICaL, IKs and IKr formulations were incorporated simultaneously in the model (see representative traces in Fig. 3B). Results obtained are comparable to those obtained by the model in which only ICaL is modified (Figs. 3D vs. H), thus emphasizing the important role of Ca2+-dependent ICaL inactivation in determining APD dependency on [Ca2+]o. In the modified model, [Ca2+]o increment causes APD shortening mainly by decreasing phase 2 duration (171, 153 and 147 ms at 1, 2 and 3 mM external Ca2+ respectively, in epi cell), whereas the duration of repolarization (phase 3) remains unchanged (58 ms, in epi cell). Data obtained after modification of the TNNP model are in agreement with the experimental observations by Bai et al. [12] showing a reduction of APD when increasing [Ca2+]o. Fig. 3E shows that, although APD dependency on [Ca2+]o is underestimated,
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the positive sign of d(APD90)/d[Ca2-]o observed experimentally is correctly predicted by the modified version of the TNNP model over the full range of [Ca2+]o tested. 3.2.2.1. Changes in Ca2+ homeostasis. Ca2+-dependent inactivation of ICaL is regulated by the Ca2+ released from the SR, which in turn depends on the diastolic SR Ca2+ load. We have analyzed the dependency of SR Ca2+ content and peak [Ca2+]i transient on [Ca2+]o in both the TNNP and modified models. As [Ca2+]o increases, the SR Ca2+ load (Fig. 4A) and the peak [Ca2+]i (Fig. 4B) are enhanced in both models. The changes in cytosolic and SR Ca2+ are significantly smaller in our model compared to the TNNP (Figs. 4A and B, squares vs. circles). In fact, in response to an increase in [Ca2+]i (due to a larger Ca2+ influx) our modified Ca2+-dependent inactivation feeds back more strongly to narrow the amount of Ca2+ entering the cell upon depolarization. This limits the trigger for SR Ca2+ release and reduces the amount of Ca2+ available to be re-uptaken by the SR. 3.2.2.2. Istantaneous vs. steady-state response. Since changes in diastolic SR Ca2+ load occur over several beats, we examined whether immediate changes in APD due to increased [Ca2+]o differ from the
steady-state changes previously described. Fig. 5 shows the percentage APD90, systolic [Ca2+]i and diastolic SR Ca2+ changes upon an increase in extracellular Ca2+ from 1.8 to 3 mM. Both models predict a biphasic response of APD, with an instantaneous increase followed by decay (Figs. 5A and B, black lines). The former is due to an increase in the driving force for ICaL, which increases the depolarizing current; the enhanced Ca2+-dependent ICaL inactivation, due to the increase over time of SR (Figs. 5C and D) and cytosolic [Ca2+] (Figs. 5A and B, grey lines), contributes to the latter. This feedback mechanism is much stronger in our formulation and is involved in the steady state AP shortening with increasing [Ca2+]o, whereas in the TNNP model APD90 stays longer. Both models reach the new steady state in approximately the same time. 3.2.2.3. APD changes during SR Ca2+ reloading. In Fig. 6, we studied the changes in APD as the SR Ca2+ content is restored after depletion (e.g. with caffeine) for two different levels of external Ca2+ (1.8 and 3 mM). The AP prolongs upon SR depletion in both models, to a greater extent in our formulation (Figs. 6A vs. B, black lines), and it shortens as the SR starts reloading (Figs. 6C and D) and the [Ca2+] transient is increasing (Figs. 6A and B, grey lines). The time course of changes in APD does not differ with different [Ca2+]o in our modified model
Fig. 3. Modulation of APD by Ca2+ in the TNNP and modified models. (A and B) Example of simulated ventricular APs with different [Ca2+]o: the TNNP model predicted prolonged APs with higher Ca2+ levels (A), whereas our modifications led to AP shortening with high Ca2+ levels (B). (C) Sensitivity analysis of the dependence of epi, endo, and M cell APD90 on [Ca2+]o was performed with the TNNP model. A positive correlation between APD90 and [Ca2+]o was found for all the cell types. (D) The modified model inverted the relation between APD and [Ca2+]o for all the cell types. (E) Comparison of simulation results with experimental data by Bai et al. [12]. Percentage changes of APD are reported with respect to APD with 1.8 mM external Ca2+. Rise in [Ca2+]o to 2.4 and 3.0 mM shortened AP in ventricular myocytes (upper panel). The TNNP model predicted AP prolongation due to rise in external Ca2+, whereas the modified model showed a reduction of APD when [Ca2+]o increases (lower panel). Effects on APD vs. [Ca2+]o relationship of modifications to: (F) IKr, (G) IKs, (H) ICaL only. The modification to ICaL was sufficient to invert the relation between APD and [Ca2+]o for all the cell types. In contrast, the newly introduced dependence on [Ca2+] of IKr and IKs had almost no effects on APD.
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Fig. 4. Changes in SR and cytosolic [Ca2+] upon rise in [Ca2+]o. [Ca2+]SR (A) and peak and diastolic [Ca2+]i (B) increase with external Ca2+ both in the TNNP (black circles) and modified model (grey squares).
(18–22 beats to reach 90% of the steady state value, Fig. 6B, solid vs. dashed line), which predicts a faster restoration of the SR Ca2+ load compared to the TNNP model at physiological concentrations. In the TNNP model, the higher the [Ca2+]o the faster the reloading of the SR (60 and 47 beats to reach 90% of the steady state value at 1.8 and 3 mM respectively, Fig. 6A, solid vs. dashed line).
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Cytosolic Ca2+ levels and the expression of Ca2+-dependent currents (e.g. IKs) may depend on the diastolic interval. Fig. 8A compares the APD restitution curve measured from endocardial monophasic APs in human hearts by Morgan et al. [35] with the TNNP and modified model predictions for endo cell (S1-S2 restitution protocol, see Methods). Although APD values are different, both models appropriately predict the restitution kinetics. A similar verification was also performed for the intracellular calcium and sodium staircase. Fig. 8 compares the changes in systolic calcium (C) and sodium (D) levels when the pacing frequency is increased in a stepwise fashion in the original and modified versions of the TNNP model. Simulated data are compared to the peak Ca2+-frequency relationship obtained experimentally for human myocardial cells by Schmidt et al. [36] (Fig. 8C). With increasing frequencies of stimulation there is an increase in the intracellular systolic calcium concentration (for [Ca2+]i b 2.5, 3, 3.5 μM in experiments, TNNP and our model respectively), which is accompanied by a positive force-frequency relationship in the undiseased human myocardium [37,38]. Our simulated calcium-frequency staircase is similar to the experimental relationship, and it is attenuated in our model compared to the original TNNP model, due to a stronger negative feedback at elevated [Ca2+]i, which limits the entry of Ca2+ into the cell. Intracellular sodium levels increase with the pacing frequency: from an initial value of 7.9 mM and 8.7 mM at 0.25 Hz [Na+]i goes up to 17 mM and 13.4 mM at 3 Hz in the TNNP and modified model, respectively. Simulated percentage [Na+]i increase was compared to the results obtained by Pieske et al. [38] in human cardiomyocytes (Fig. 8D). While the TNNP model overestimates the rate-induced increase in [Na+]i, this is correctly reproduced with the modified model, since the attenuated rise in [Ca2+]i reduces the entry of Na+ through INaCa. We also compared the predictions of our new model formulation to clinical data from patients undergoing significant changes in their plasma [Ca2+] during hemodialysis. The scatter plot and regression line in Fig. 8B show the significant inverse correlation between QTc interval duration and serum [Ca2+] changes measured during hemodialysis sessions (data from [9]). Although QTc values are generally higher in the study by Genovesi et al. [8], shown for comparison, QTc–Ca2+ dependency (e.g. the steepness of the interpolation lines) is substantially similar between simulations and the two experimental studies. 4. Discussion
3.3. Role of Ca2+ transporters in AP shortening In the original TNNP model an increase of [Ca2+]o from 1 to 3 mM also causes an outward shift of both IpCa and INaCa, which only partly offsets the large ICaL increment (Fig. 7, left). In our modified version the increase in external Ca2+ outwardly shifts both IpCa and INaCa (Fig. 7, right), whereas the plateau ICaL level is unmodified (despite the peak current being markedly increased). This contributes to an outward shift of net membrane current and consequently shortens the APD. While the outward shift of IpCa is justified by the increased [Ca2+]i, the latter should shift INaCa inwardly since the inward Na+ flux due to the exchanger is increased. However, INaCa sensitivity to external Ca2+ is dominant and it reduces the amount of Ca2+ that is extruded, thus increasing the net outward current. 3.4. Validation of the model against human data The TNNP model was derived in order to conform to a large number of experimental data. We proposed a modified version that is more suitable to reproduce the effect of changes in extracellular Ca2+. However, to make sure that the modified model matches the data simulated with the original TNNP model, we validated our model against the same pool of single cell human data shown in the paper of Ten Tusscher et al. [25].
We explored the APD dependency on external calcium concentration with a previously published mathematical model of the human ventricular myocyte (TNNP). This model has been validated against a wide range of experimental data [25]. However, it does not succeed in the attempt to replicate the AP shortening observed in vitro and in vivo when extracellular or plasma calcium concentration is increased. In fact, most of the other commonly used ventricular cell models suffer from the same defect, as we have shown in the Online Data Supplement, thus our results may be applied more extensively. We modified the existing formulations of delayed rectifier K+ currents and L-type Ca2+ current to account for their modulation by Ca2+ and assessed how these alterations relatively contribute to the APD dependency on external Ca2+. Extracellular Ca2+ directly modulates IKr gating via association of the ions with closed HERG K+ channels [15], whereas IKs and ICaL kinetics is only indirectly affected by variations in external Ca2+. Indeed, [Ca2+]o changes alter cytosolic [Ca2+], which influences both IKs and ICaL. 4.1. Dependency of IKs and IKr on Ca2+ In guinea pig ventricular myocytes intracellular Ca2+ is known to modulate the cardiac delayed rectifier K+ current IKs [16,17] and
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Fig. 5. Instantaneous vs. steady-state APD90, SR and cytosolic [Ca2+] changes upon increase in [Ca2+]o. In both the TNNP and modified models, the APD exhibits a biphasic response to the rise in external Ca2+ from 1.8 to 3 mM. AP instantaneously prolongs in both the TNNP and modified models (A and B, black), due to the increased driving force for ICaL. The increased Ca2+ influx rapidly loads the SR (C and D) and increases the Ca2+ transient amplitude (A and B, grey), enhancing the Ca2+ dependent ICaL inactivation, which feedbacks to shorten the AP. As already shown, at steady state APD is longer in the TNNP and shorter in the modified model compared to 1.8 mM.
results from Bai et al. [12] suggest that IKs enhancement contributes to the [Ca2+]o-induced APD shortening. However, in our human model IKs dependency on intracellular Ca2+ has negligible effects on
APD. In fact, in our simulations even at low [Ca2+]o (1 mM) the [Ca2+]i “sensed” by the current during the AP is always higher than 0.1 μM (Fig. 7, panel G). Above this concentration the weak dependence of
Fig. 6. Changes in APD90 and [Ca2+]i during SR Ca2+ reloading with 1.8 and 3 mM external [Ca2+]. Upon SR Ca2+ depletion (lower panel), AP prolongs in both models and then recovers to steady state (A, B black traces) with increasing SR (C and D) and cytosolic [Ca2+] (A, B grey traces). Recovery in our model is faster than in the TNNP (18 vs. 60 beats to reach 90% of the steady state value at [Ca2+]o = 1.8 mM).
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GKs on [Ca2+]i (Fig. 1B), as characterized by Nitta et al. [16] in excised patch experiments, fails to justify the APD changes upon external Ca2+ variations. Bai et al. did not show the extent of [Ca2+]o-induced [Ca2+]i changes in their experiments, therefore it is not clear whether the difference with their results is due to different [Ca2+]i transient amplitudes or to a different Ca2+ sensitivity of IKs. The discrepancy between the model and the experimental data of Bai et al. might be seen as a limitation of the present work, since we could have underestimated the sensitivity of IKs to relevant Ca2+ changes. Nevertheless, several reports suggest that IKs has smaller amplitude and faster deactivation in larger mammals [39,40], including humans [41], than in guinea pigs. IKs contribution to repolarization is negligible in human myocytes in physiological conditions and in absence of adrenergic stimulation [42]. We believe that this makes the formulation of IKs dependence on [Ca2+]i less crucial and that our conclusions on the role of IKs stand regardless the uncertainty on this phenomenon. Extracellular Ca2+ is an important modulator of HERG channels. Significant changes in channel gating occur with changes in [Ca2+]o [15]. However, the shift in the activation curve we incorporated into the model does not produce significant changes in APD. In fact, during the plateau phase, IKr is dominated by the kinetics of the inactivation gate, whereas, in spite of the Ca2+-induced activation shift, the activation gate is always fully activated. Thus, IKr contribution to repolarization is not affected by [Ca2+]o.
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4.2. Ca2+ dependence of ICaL inactivation ICaL inactivation is both voltage- and Ca2+-dependent. The intrinsic voltage-dependent process is relatively slow, so that in physiological conditions most of the inactivation is Ca2+-dependent during the cardiac AP [22]. In our simulations the APD regulation is significantly affected by Ca2+-dependent ICaL inactivation. This is consistent with the experimental observations of Alseikhan et al. [43]. They showed that the use of engineered calmodulins to eliminate Ca2+-dependent ICaL inactivation leads to a 4-to 5-fold prolongation of APs in guinea pig cardiomyocytes. Linz and Meyer [30] performed AP clamp experiments showing that after an initial rapid phase, ICaL inactivation proceeds during the AP plateau up to 95%. This scenario is underestimated in the original TNNP model; however, it is approached by the new formulation of ICaL inactivation implemented in the present study (see Fig. 1C). Moreover, in our formulation the degree of current inactivation keeps increasing for [Ca2+]i from 0.54 to 1.2 μM (the values of [Ca2+]i transient peaks corresponding to 1 and 3 mM [Ca2+]o). This is the key feature that allows inverting the relation between APD and [Ca2+]o with respect to the original model. As in the TNNP formulation, the level of current inactivation is constant during phase 2 instead of progressively increasing as found in [30]. A more detailed formulation (e.g. Markov model [44,45]) should be used to account for a complete description of the inactivation kinetics.
Fig. 7. Sarcolemmal Ca2+ transporters during AP. APs at two different [Ca2+]o levels (first row), Ca2+ transients (second row), and the corresponding currents (third to fifth rows) obtained with the TNNP (left) and our modified model (right) are shown. Note that the time scale was normalized with respect to APD90 to allow a direct comparison of the different currents.
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Fig. 8. (A) APD restitution curve obtained before and after model modifications. Experimental data from endocardial monophasic APs in human hearts [35] are included for comparison (circles). (B) Comparison of [Ca2+]o dependence of measured QTc interval and simulated APD (modified model). Scatter plot and regression line show the significant inverse correlation between QTc interval duration and serum [Ca2+] changes measured during hemodialysis sessions (data from [9]). Simulated APD values and polynomial interpolation were derived from data in Fig. 3 for epicardial cell after normalization to the APD value obtained at the average pre-dialysis [Ca2+] (1.2 mM). Data from Genovesi et al. [8] show a similar Ca2+ dependence, even if the regression curve is shifted towards larger QTc changes. (C) Changes in peak Ca2+ and (D) [Na+]i levels when pacing frequency is increased in a stepwise fashion in the original (thin lines) and modified (thick lines) versions of the TNNP model (each frequency was maintained for 10 min). Experimental results from Schmidt et al. [36] and Pieske et al. [38] are also shown for comparison.
Altamirano and Bers [34] showed that ICaL recovery from inactivation is also Ca2+ dependent. This might be relevant at fast pacing rates, when the time for channel recovery is shorter. We incorporated this characteristic (τf dependency on [Ca2+]i, see methods) and assessed whether Ca2+ modulation of APD is rate dependent by stimulating the digital cell at 1, 2 and 3 Hz. However, no difference in the APD is found when τf is purely voltage- or both voltage- and Ca2+-dependent (data not shown). 4.3. Role of Ca2+ transporters in AP shortening Once a role for IKs and IKr in modulating APD dependency on external Ca2+ was ruled out, we investigated the impact of sarcolemmal Ca2+ transporters (Fig. 7). The rise in [Ca2+]o affects ICaL in two opposite directions: it increases the driving force thus allowing a larger Ca2+ influx (peak ICaL −8.3 vs. −5.6 pA/pF at 3 vs. 1 mM external Ca2+, in epi cell) and enhances the Ca2+-dependent inactivation due to a progressive increase in the SR Ca2+ load. These two effects result in negligible current increase during AP phase 2 (Fig. 7H) along with increased Ca2+ transient amplitude and diastolic level (Fig. 7G). Such intra- and extra-cellular alterations lead to an increased Ca2+ efflux via IpCa (Fig. 7J) and decrease Ca2+ extrusion via INaCa (Fig. 7I), both leading to augmented net outward currents which cause AP shortening. Although IKs Ca2+-dependency could also contribute to APD shortening, the results with the complete model and the one with isolated ICaL modification are similar. Thus, one may conclude that the secondary changes in IpCa and INaCa are the main cause of APD
shortening upon [Ca2+]o elevation. If this is the case, APD and its Ca2+-dependency might be affected by the rate and extent of Ca2+ compartmentation in the SR. We show that the decrease in APD with external Ca2+ is accompanied by increased cytosolic and SR [Ca2+] (Fig. 4). On the other hand, suppression of SR uptake by 30% in our model lead to a decreased SR Ca2+ content, which limits the amount of Ca2+ released by the SR and weakens the Ca2+-dependent inactivation of ICaL, thus allowing more depolarizing current which prolongs the AP (not shown). This APD increase is predicted at all [Ca2+]o concentrations, and the dependency of APD on [Ca2+]o remains substantially unaffected. Our simulation results are in agreement with experimental findings showing that PLB overexpression in mouse myocytes slows down ICaL inactivation by inhibiting SR Ca2+ uptake [46] and depletion of the SR (with thapsigargin) prolongs the AP in rat cardiomyocytes [47]. Analogously, overexpression of SR Ca2+ pump has been shown to shorten the AP and increase the SR Ca2+ content in rabbit myocytes [48]. In addition, we monitored the changes in APD after SR depletion and during SR reloading: we have shown (Fig. 6) that the AP prolongs as the store is depleted and shortens as the SR is refilled with Ca2+ in about 20 beats. A similar time course has been shown experimentally [47]: after removal of caffeine (which empties the SR, suppresses the Ca2+ transient and prolongs the AP), the APD is gradually shortened to basal length in parallel with the restoration of the magnitude of the Ca2+ transient during the refilling of the SR [49] as the rat cardiomyocytes are stimulated. Our simulations predicted a similar biphasic behavior in the APD changes when switching from 1.8 to 3 mM in the extracellular solution
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(Fig. 5). Increase in [Ca2+]o initially results in an increased driving force for Ca2+ to enter the cell through the L-type channel. This increases the depolarizing current and leads to AP prolongation, and also enhances the SR Ca2+ content and cytosolic transient. This process augments Ca2+ dependent inactivation, which shortens the AP at steady state. The increase in SR Ca2+ matches experimental results in rat ventricular myocytes and trabeculae, where an increase of [Ca2+]o from 1 to 6–8 mM leads to augmented amplitude of the caffeineevoked increase of [Ca2+]i due to an increase of SR content [50,51]. Measurements of caffeine-induced Ca2+ release in cultured neonatal rat myocardium found that increasing [Ca2+]o (0.25 to 4 mM) increases the SR content [52]. Trafford et al. [49] have shown that when [Ca2+]o is elevated from 1 to 2 mM, the increase in systolic [Ca2+]i occurs without changes in the SR Ca2+ content. Other studies showed that a decrease in [Ca2+]o (2.5–2 to 0.5–0.2 mM) leads to Ca2+ transient depression with slight or no changes in SR Ca2+ [53,54]. The differences in species, preparations and ranges of concentrations explored may explain the divergent outcomes. 4.4. Limitations of the study The incorporated changes on IKr are derived from heterologous expression of the human HERG cDNA, whereas the modifications in GKs and ICaL are based on guinea pig data. Transfer of modulation by Ca2+ of these channels in human myocytes seems reasonable, but might be quantitatively different and have implications on our results. As an example, for the mechanism of Ca2+ dependent inactivation of ICaL, Linz and Meyer [21] found species-specific differences when comparing rabbit, rat and guinea pig myocytes: while L-type channels showed similar Ca2+ dependency in the rabbit and the rat, a 2-times higher Ca2+ influx was necessary to achieve a given degree of inactivation in the guinea-pig. Therefore, it would be ideal to experimentally assess the Ca2+ sensitivity of the L-type channel protein in human myocytes and generate a specie-specific model. Common pool models, such as the TNNP and the modified model presented here, fail to capture the local aspects of ICaL Ca2+ dependent inactivation. In fact, the [Ca2+] sensed by the channels in the dyadic junction is likely to be one to two orders of magnitude greater than the cytosolic [Ca2+]. Thus, the representation of Ca2+-dependent inactivation is purely phenomenological rather than mechanistic, since in the model average cytosolic [Ca2+] is responsible for inactivation. Ten Tusscher and Panfilov have published an extension of the original TNNP model [55]. They added a cleft subspace where the L-type Ca2+ channels and the ryanodine receptors inject calcium, and in turn their dynamics are influenced by the cleft Ca2+ concentration; we tested the model for its APD dependency on [Ca2+]o (see the Supplementary material) and found that it is not significantly changed with respect to the original formulation, whereas inclusion of our proposed modification of ICaL qualitatively leads to the same results we presented here. Nevertheless, a more thorough analysis with this and other models of human AP could further assess the robustness of our results. 4.5. Conclusions Our computational analysis pointed out that the mechanism of Ca2+-dependent ICaL inactivation seems to be a sensible pathway for modulating APD adaptation to variations in [Ca2+]o. We formulated a model of the human ventricular AP, which successfully reproduces AP shortening with increasing [Ca2+]o, and validated it against in vitro and in vivo data. This provides a framework to computationally explore the impact of electrolyte imbalances on the myocyte electrical activity. As a relevant example, in silico analysis proved to be a valuable tool to gain insights into the impact of hemodialysis treatment at the cellular level [9], where APD prolongation might lead to arrhythmias when patients undergo Ca2+ and K+ changes.
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Acknowledgments The authors would like to thank Giulia Callisesi, Davide Bianchini and Pietro Avanzini for their help with the simulations. We are thankful to Dr. Ruchi Patel for editing the manuscript. The work was partially supported by Hospal SpA (Bologna, Italy). Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.yjmcc.2008.12.002. References [1] Braunwald E, Sonnenblick E, Ross J. In: Braunwald E, editor. The ECG and electrolyte abnormalities. Philadelphia: Saunders; 1992. p. 149–51. [2] Fisher NG, Armitage A, McGonigle RJ, Gilbert TJ. Hypocalcaemic Cardiomyopathy; the relationship between myocardial damage, left ventricular function, calcium and ECG changes in a patient with idiopathic hypocalcaemia. Eur J Heart Fail 2001; 3(3):373–6. [3] Diercks DB, Shumaik GM, Harrigan RA, Brady WJ, Chan TC. Electrocardiographic manifestations: electrolyte abnormalities. J Emerg Med 2004;27(2):153–60. [4] Huang TC, Cecchin FC, Mahoney P, Portman MA. Corrected QT interval (QTc) prolongation and syncope associated with pseudohypoparathyroidism and hypocalcemia. J Pediatr 2000;136(3):404–7. [5] Surackwicz B. The interrelationship of electrolyte abnormalities and arrhythmias. In: Mandel WJ, editor. Cardiac Arrhythmias. Their Mechanisms, Diagnosis & Management. Philadelphia: JPLippincott; 1995. p. 89–108. [6] Nappi SE, Virtanen VK, Saha HHT, Mustonen JT, Pasternack AI. QTc dispersion increases during hemodialysis with low-calcium dialysate. Kidney Int 2000;57(5): 2117–22. [7] Covic A, Diaconita M, Gusbeth-Tatomir P, Covic M, Botezan A, Ungureanu G, et al. Haemodialysis increases QTc interval but not QTc dispersion in ESRD patients without manifest cardiac disease. Nephrol Dial Transplant 2002;17(12):2170–7. [8] Genovesi S, Rivera R, Fabbrini P, Dossi C, Bonforte G, Mircoli L, et al. Dynamic QT interval analysis in uraemic patients receiving chronic haemodialysis. J Hypertens 2003;21(10):1921. [9] Severi S, Grandi E, Pes C, Badiali F, Grandi F, Santoro A. Calcium and potassium changes during haemodialysis alter ventricular repolarization duration: in vivo and in silico analysis. Nephrol Dial Transplant 2008;23(4):1378–86. [10] Hoffman BF, Suckling EE. Effect of several cations on transmembrane potentials of cardiac muscle. Am J Physiol 1956;186(2):317–24. [11] Leitch SP, Brown HF. Effect of raised extracellular calcium on characteristics of the guinea-pig ventricular action potential. J Mol Cell Cardiol 1996;28(3): 541–51. [12] Bai CX, Namekata I, Kurokawa J, Tanaka H, Shigenobu K, Furukawa T. Role of nitric oxide in Ca2+ sensitivity of the slowly activating delayed rectifier K+ current in cardiac myocytes. Circ Res 2005;96(1):64–72. [13] Kass RS, Tsien RW. Control of action potential duration by calcium ions in cardiac Purkinje fibers. J Gen Physiol 1976;67(5):599–617. [14] Temte JV, Davis LD. Effect of calcium concentration on the transmembrane potentials of Purkinje fibers. Circ Res 1967;20(1):32–44 Ref Type: Generic. [15] Johnson Jr JP, Balser JR, Bennett PB. A novel extracellular calcium sensing mechanism in voltage-gated potassium ion channels. J Neurosci 2001;21(12): 4143–53. [16] Nitta J, Furukawa T, Marumo F, Sawanobori T, Hiraoka M. Subcellular mechanism for Ca(2+)-dependent enhancement of delayed rectifier K+ current in isolated membrane patches of guinea pig ventricular myocytes. Circ Res 1994;74(1): 96–104. [17] Tohse N. Calcium-sensitive delayed rectifier potassium current in guinea pig ventricular cells. Am J Physiol, Heart Circ Physiol 1990;258(4):H1200–7. [18] Viswanathan PC, Shaw RM, Rudy Y. Effects of IKr and IKs heterogeneity on action potential duration and its rate dependence a simulation study. Circulation 1999;99 (18):2466–74. [19] Peterson BZ, Lee JS, Mulle JG, Wang Y, de Leon M, Yue DT. Critical determinants of Ca2+-dependent inactivation within an EF-hand motif of L-type Ca2+ channels. Biophys J 2000;78(4):1906–20. [20] Peterson BZ, DeMaria CD, Adelman JP, Yue DT. Calmodulin is the Ca2+ sensor for Ca2+ -dependent inactivation of L-type calcium channels. Neuron 1999;22(3): 549–58. [21] Linz KW, Meyer R. Profile and kinetics of L-type calcium current during the cardiac ventricular action potential compared in guinea-pigs, rats and rabbits. Pflufgers Arch Eur J Physiol 2000;439(5):588–99. [22] Bers DM. Excitation-Contraction Coupling and Cardiac Contractile Force. Kluwer Academic Press; 2001. [23] Priebe L, Beuckelmann DJ. Simulation study of cellular electric properties in heart failure. Circ Res 1998;82(11):1206–23. [24] Bernus O, Wilders R, Zemlin CW, Verschelde H, Panfilov AV. A computationally efficient electrophysiological model of human ventricular cells. Am J Physiol, Heart Circ Physiol 2002;282(6):H2296–308. [25] Ten Tusscher KH, Noble D, Noble PJ, Panfilov AV. A model for human ventricular tissue. Am J Physiol, Heart Circ Physiol 2004;286(4):H1573–89.
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