Mathematical modeling mechanisms of arrhythmias in transgenic mouse heart overexpressing TNF-α

Abstract

Transgenic mice overexpressing tumor necrosis factor-α (TNF-α mice) possess many of the features of human heart failure, such as dilated cardiomyopathy, impaired Ca²⁺ handling, arrhythmias, and decreased survival. Although TNF-α mice have been studied extensively with a number of experimental methods, the mechanisms of heart failure are not completely understood. We created a mathematical model that reproduced experimentally observed changes in the action potential (AP) and Ca²⁺ handling of isolated TNF-α mice ventricular myocytes. To study the contribution of the differences in ion currents, AP, Ca²⁺ handling, and intercellular coupling to the development of arrhythmias in TNF-α mice, we further created several multicellular model tissues with combinations of wild-type (WT)/reduced gap junction conductance, WT/prolonged AP, and WT/decreased Na⁺ current (I_Na) amplitude. All model tissues were examined for susceptibility to Ca²⁺ alternans, AP propagation block, and reentry. Our modeling results demonstrated that, similar to experimental data in TNF-α mice, Ca²⁺ alternans in TNF-α tissues developed at longer basic cycle lengths. The greater susceptibility to Ca²⁺ alternans was attributed to the prolonged AP, resulting in larger inactivation of I_Na, and to the decreased SR Ca²⁺ uptake and corresponding smaller SR Ca²⁺ load. Simulations demonstrated that AP prolongation induces an increased susceptibility to AP propagation block. Programmed stimulation of the model tissues with a premature impulse showed that reduced gap junction conduction increased the vulnerable window for initiation reentry, supporting the idea that reduced intercellular coupling is the major factor for reentrant arrhythmias in TNF-α mice.

tumor necrosis factor-α (TNF-α) is significantly (10-fold) elevated in the serum of patients with end-stage congestive heart failure (15, 36) and in the myocardium of patients with dilated cardiomyopathy and ischemic heart disease (61). These facts suggest that the inflammatory cytokine TNF-α may play a significant role in the promotion of cardiac arrhythmias. To investigate the role of the cytokine, transgenic (TG) mice were generated with cardiac-specific overexpression of TNF-α (33). These mice demonstrated dilated cardiomyopathy, impaired Ca²⁺ dynamics, and increased mortality (24, 33).

More recent investigations have shown that overexpression of TNF-α in mouse hearts leads to electrical remodeling, larger susceptibility to Ca²⁺ alternans, and reentrant arrhythmias (38, 50). These genetically modified hearts demonstrated significantly prolonged action potential (AP) duration, smaller intracellular Ca²⁺ concentration ([Ca²⁺]_i) transients, and significantly smaller magnitudes of two major repolarization currents, the rapidly inactivating and rapidly recovering transient outward K⁺ current, I_Kto,f, and the ultrarapidly activating delayed-rectifier K⁺ current, I_Kur [I_K,slow1 and I_K,slow2 in notation of Petkova-Kirova et al. (50)]. It is interesting to note that AP prolongation would tend to suppress reentrant arrhythmias, yet TG hearts were more susceptible to this type of arrhythmia. London et al. (38) suggested that such arrhythmia was due to a slowing of conduction velocity in TNF-α-overexpressing mouse hearts.

We developed an experimentally based mathematical model that was focused on simulating the changes in the AP and Ca²⁺ handling in single TNF-α ventricular myocytes and on modifying the intercellular conductance in TNF-α multicellular tissues to elucidate the mechanisms underlying the enhanced susceptibility of TNF-α mice to arrhythmias. The single cell model reproduced experimentally observed differences in AP shape and duration between wild-type (WT) and TNF-α mice myocytes that were predominantly due to reduced expression of two major K⁺ repolarization currents: I_Kto,f and I_Kur (50). The model also reproduced experimentally observed differences in Ca²⁺ handling (24, 25, 38, 69), which included smaller and longer [Ca²⁺]_i transients in TNF-α myocytes and a slower Ca²⁺ sequestration rate into the sarcoplasmic reticulum (SR) due to a decreased sarco(endo)plasmic reticulum Ca²⁺-ATPase (SERCA) function, and possibly to an increased Ca²⁺ extrusion through the Na⁺/Ca²⁺ exchanger. To study further the potential contribution of AP prolongation, reduced gap junction conductance, and reduced magnitude of the fast Na⁺ currents to the development of arrhythmias in TNF-α mice, we investigated multicellular model tissues using reduced/WT gap junction conductance, prolonged/WT AP, and decreased/WT Na⁺ current (I_Na). All model tissues were examined for susceptibility to Ca²⁺ alternans, AP propagation block, and reentrant arrhythmias.

Modeling showed that, similar to experimental data in TNF-α mice (38), Ca²⁺ alternans in TNF-α tissues developed at longer basic cycle lengths. The greater susceptibility to Ca²⁺ alternans was predominantly due to the prolonged AP resulting in larger inactivation of I_Na, and to the decreased SR Ca²⁺ uptake and corresponding smaller SR Ca²⁺ load. Simulations of our model tissues showed that prolongation of AP increased the possibility for AP propagation block. Application of programmed stimulation to the model tissues using an S1–S2 protocol similar to the protocol applied in the experiments of London et al. (38) showed that a reduction in gap junction conduction increases the interval for initiation reentry, pointing to a decrease in gap junction conductance as the major mechanism of reentrant arrhythmias in TNF-α-overexpressing mice. Ca²⁺ alternans and AP prolongation do not appear to be particularly connected to the inducibility of reentry at least as assessed through the size of the vulnerable window.

METHODS

Mathematical modeling.

Mathematical models of isolated myocytes from WT and TNF-α-overexpressing TG mice were based on our previously published model for apical mouse ventricular myocytes developed for room temperature T – 298°K (+25°C) (9). Four major modifications were made to our previously published model for the WT cell (Table 1 and Fig. 1). First, the voltage-dependent modulation factor for ryanodine receptors (RyRs) $e^{-{(V-5.0)}^2/648.0}$ in Eq. A15 from Ref. 5 was replaced by $e^{-{(V+5.0)}^2/648.0}$ to improve graded SR Ca²⁺ release (V is the voltage; initial increase in normalized [Ca²⁺]_i in Fig. 2D is less steep than in Fig. 6B from Ref. 9). Second, the nonmonotonic complex voltage dependence of activation rate constant α(V) (Eq. A31 in Ref. 9) was replaced by a monotonic dependence α(V) – 0.4e^{(V+15.0)/15.0}, with corresponding adjustments of other parameters in the Markov model for L-type Ca²⁺ current, I_CaL (Table 1). Third, the maximal pumping rate of the sarcolemmal Ca²⁺ pumps, Ip(Ca)max, was significantly decreased by a factor of 12, according to observations of Li et al. (37), to account for the smaller fraction of Ca²⁺ extrusion from the cytosol by the slow mechanism (I_p(Ca) − I_Cab) [<1% of total released Ca²⁺; here I_p(Ca) is the sacrolemmal Ca²⁺ pump and I_Cab is the Ca²⁺ background current] compared with the fraction given by the Na⁺/Ca²⁺ exchanger, I_NaCa. Fourth, in the equations for the corresponding activation time constants, deactivation of two repolarization currents, the ultrarapidly activating delayed-rectifier K⁺ current, I_Kur, and the noninactivating steady-state K⁺ current, I_Kss, was taken into account. Other changes were also applied to improve the mouse ventricular cell model (Table 1). All changes made to the model (9) are highlighted in gray on the scheme of the mouse ventricular myocyte (Fig. 1). Some of these changes were also implemented in our mathematical model for epicardial and endocardial mouse ventricular myocytes (8).

Table 1.

Differences between current model and the model for apical cell from Bondarenko et al. (9)

See text for definitions.

Fig. 1.

Schematic diagram of the mouse model ionic currents and Ca²⁺ fluxes. Transmembrane currents are as follows: I_Na, the fast Na⁺ current; I_CaL, the L-type Ca²⁺ current; I_p(Ca), the sarcolemmal Ca²⁺ pump; I_NaCa, the Na⁺/Ca²⁺ exchanger; I_Kto,f, the rapidly recovering transient outward K⁺ current; I_Kto,s, the slowly recovering transient outward K⁺ current; I_Kr, the rapid delayed-rectifier K⁺ current; I_Kur, the ultrarapidly activating delayed-rectifier K⁺ current; I_Kss, the noninactivating steady-state voltage-activated K⁺ current; I_K1, the time-independent K⁺ current; I_Ks, the slow delayed-rectifier K⁺ current; I_NaK, the Na⁺-K⁺ pump; I_Cl,Ca, the Ca²⁺-activated chloride current; I_Cab and I_Nab, the Ca²⁺ and Na⁺ background currents; I_stim, the external stimulation current. The Ca²⁺ fluxes within the cell are as follows: J_up, uptake Ca²⁺ from the cytosol to the network sarcoplasmic reticulum (NSR); J_rel, Ca²⁺ release from the junctional sarcoplasmic reticulum (JSR); J_tr, Ca²⁺ flux from the NSR to the JSR; J_leak, Ca²⁺ leak from the sarcoplasmic reticulum (SR) to the cytosol; J_xfer, Ca²⁺ flux from the subspace volume to the bulk myoplasm; J_trpn, Ca²⁺ flux to troponin. The model includes Ca²⁺ buffering by troponin and calmodulin in the cytosol and by calsequestrin in the SR. Current systems changed from the original model of Bondarenko et al. (9) are highlighted in gray.

Fig. 2.

I_CaL and intracellular Ca²⁺ transients ([Ca²⁺]_i) in wild-type (WT) and tumor necrosis factor-α (TNF-α)-overexpressing ventricular myocytes. A: a family of simulated current traces for WT myocytes. A 5-s depolarizing first pulse to between −70 and +50 mV (in 10-mV increments) was applied from a holding potential of −80 mV. This was followed by a second 250-ms pulse to 0 mV. Current simulations were performed to match the experimental conditions of Petkova-Kirova et al. (50) in which [Ca²⁺]_i was buffered with 10 mM EGTA. For TNF-α-overexpressing ventricular myocytes, the current traces are almost identical to those for WT cells. Only the first 250 ms are shown to demonstrate details of activation and inactivation. B: current-voltage relationships for peak I_CaL for simulated and experimental data. The solid and dashed lines show data from simulations for WT and TNF-α-overexpressing ventricular myocytes, respectively, under the same physiological conditions as in A. Filled and unfilled circles are our experimental data for WT and TNF-α-overexpressing ventricular myocytes, respectively (50). Experimental results from other groups on WT mice are shown by unfilled triangles (68), unfilled squares (28), and filled diamonds (30). C: normalized channel conductance G/G_max and steady-state inactivation relationships for WT cells for 250-ms P1 pulse simulations without Ca²⁺ buffer (dashed lines) and 5-s P1 pulse simulations with buffered Ca²⁺ (solid lines). Filled circles are our simulation data for 5-s P1 pulses and [Ca²⁺]_i buffered with 5 mM EGTA (9). Unfilled triangles are from the experimental measurements of Yatani et al. (68) where Ca²⁺ was buffered with 5–10 mM EGTA (BAPTA). D: voltage dependence of simulated normalized I_CaL (solid line with filled triangles) and normalized (norm) [Ca²⁺]_i (dashed line with filled circles) for WT myocytes. Simulations clearly show graded release of Ca²⁺.

Fig. 6.

Integral Ca²⁺ fluxes during one cardiac cycle. Simulations are shown for a 1-s cycle after 10 stimuli at 1 Hz for WT (A) and TNF-α-overexpressing (B) cells. Here, J_CaL is the Ca²⁺ entering the cytosol through L-type Ca²⁺ channels; J_up − J_leak is the uptake Ca²⁺ from the cytosol to the network SR with subtracted Ca²⁺ leak from the SR to the cytosol; J_NaCa is the Ca²⁺ flowing through the Na⁺/Ca²⁺ exchanger; and J_pCa − J_Cab is the Ca²⁺ efflux through the slow mechanism.

The model was improved to better describe Ca²⁺ handling in mouse ventricular myocytes. In a Markov model of I_CaL the voltage dependence of two rate constants [α(V) and β(V)] was simplified and adjusted, resulting in a model that fits experimental data (Fig. 2). Experimental data did not show changes in I_CaL from WT and TG myocytes (50), therefore, both models included the same formulation of this current. Simulated I_CaL traces obtained by depolarizations for 5 s to between −70 and +50 mV (in 10-mV increments) from a holding potential of −80 mV and under heavy buffer conditions (10 mM EGTA) are shown in Fig. 2A. The improved description of I_CaL allowed for simulation of the voltage dependence of the peak current (Fig. 2B), inactivation and normalized conductance G/G_max (Fig. 2C), and graded SR Ca²⁺ release (Fig. 2D). Small differences were observed in the inactivation properties of I_CaL obtained with and without Ca²⁺ buffer (dashed and solid lines in Fig. 2C, respectively).

The differences between the models for WT and TNF-α-overexpressing mouse ventricular myocytes are shown in Table 2. Consistent with experimental data (50), the models of ventricular myocytes for TG mice have smaller maximum conductances of two major K⁺ repolarization currents, I_Kto,f and I_Kur (Table 2). Smaller values of the maximum conductance were obtained from the experiments with TNF-α-overexpressing cardiac myocytes compared with WT cells. The maximum conductances of the two currents were normalized to fit the experimental data (50). Figure 3, A–D, shows experimental and simulation voltage-clamp data for depolarization-activated K⁺ currents from WT and TNF-α-overexpressing ventricular myocytes. Modeling data include the sum of three currents, I_Kto,f, I_Kur, and I_Kss. As seen from Fig. 3, there is a good agreement between experimental and simulation voltage-clamp data. In addition to inactivation, in the model we also included deactivation of I_Kur and I_Kss. Deactivation of these currents was not included in our previously published model (9). In the new model, deactivation was included by modification of the τ_aur and τ_Kss gating variables (Table 1). For I_Kur, the activation time constants compared well with the corresponding data from mice (71), whereas deactivation time constants were verified by the experimental data from rats (10, 56) (Fig. 3E). Because of difficulties in the identification of the molecular basis and properties of I_Kss, we used a symmetric bell-shaped extension of the voltage dependence of the activation rate constants to more negative potentials about a voltage of −40 mV. The voltage dependence of the activation time constants for this current compares well with the corresponding experimental data from mouse ventricular myocytes (67).

Table 2.

Differences between WT and TNF-α-overexpressing cell models

See text for definitions.

Fig. 3.

Total depolarization-activated K⁺ currents from WT and TNF-α-overexpressing mouse ventricular myocytes. A and C: experimental whole cell K⁺ currents from WT and TNF-α-overexpressing mouse ventricular myocytes (50). B and D: corresponding simulated currents. Experimental currents were recorded in response to 4.5-s voltage steps to test potential between −40 and +50 mV from a holding potential of −90 mV in 10-mV increments (50). Simulated currents were elicited by a 4.5-s depolarizing step to between −90 and +50 mV in 10-mV increments from a holding potential of −90 mV. E: activation and deactivation time constants. Simulated data are shown by solid and dashed lines for the I_Kur and I_Kss, respectively. Experimental data for activation time constants of I_Kur (71) and I_Kss (67) are shown by filled and unfilled circles, respectively; experimental data for deactivation time constants of I_Kur are shown by filled squares (10) and a triangle (56). Voltage dependence for deactivation time constant of I_Kss was chosen to be symmetric to those for activation in respect to voltage V – −40 mV.

The myocytes from the two groups differed in the magnitude of their [Ca²⁺]_i transients. [Ca²⁺]_i transients were smaller and longer in TNF-α-overexpressing ventricular myocytes compared with WT myocytes (38). In the new model, this difference was achieved by a decrease in the SR Ca²⁺ sequestration rate via the SERCA pump and by an increase in the pumping rate of the Na⁺/Ca²⁺ exchanger (Table 2). Although there was direct experimental evidence for decreased Ca²⁺ sequestration rate in TG cells, which was evaluated from the decay of the [Ca²⁺]_i transients (38), we did not find experimental data on Ca²⁺ extrusion through the Na⁺/Ca²⁺ exchanger. However, during continuous pacing, only a change in the SERCA pumping rate was not sufficient to achieve smaller [Ca²⁺]_i transients in TG compared with WT model cells. Additionally, while an increase in the SR Ca²⁺ release rate through RyRs together with decreased SERCA pumping decreased SR Ca²⁺ load, such a modification again did not produce decreased [Ca²⁺]_i transients in the TG model. We needed a mechanism for significant transsarcolemmal Ca²⁺ outflux in the TG model cell. We considered that such a large Ca²⁺ outflow could occur due to upregulation of the Na⁺/Ca²⁺ exchanger, which was often observed in failing hearts and TNF-α-treated cardiac cells (17, 22, 26, 35, 59, 60). Small differences between WT and TG cells were found experimentally and were implemented in the new models for the time-independent K⁺ current, I_K1 (Fig. 4 and Tables 1 and 2).

Fig. 4.

The time-independent K⁺ current I_K1. Solid and dashed lines show simulated voltage dependences for WT and TNF-α-overexpressing ventricular myocytes, respectively; filled and unfilled circles correspond to our experimental data on WT and TNF-α-overexpressing ventricular myocytes, respectively. Experimental currents were recorded in response to 500-ms voltage steps to test potentials between −130 and −50 mV from a holding potential of −80 mV in 10-mV increments. For simulated currents, we used the same voltage-clamp protocol as for the experiment.

The one-dimensional (1D) model of mouse ventricular tissue consists of 190 cardiac cells, both for WT and TG mice. Two configurations of the tissue were simulated: a 1D cable and a ring of N – 190 model cells connected end to end with gap junction conductances g_gap (7). Cell length was 100 μm (7, 41); therefore, the length of the cable and the ring was 19 mm, which is approximately equal to the circumference of a mouse ventricle with a diameter of 6 mm (54).

Each model myocyte was described by the equations from Ref. 9 with the corresponding modifications (Tables 1 and 2) and initial conditions plus intercellular currents:

$$\begin{array}{cc}\frac{\text{d}{V}_i}{\text{d}t}=\frac{1}{C_{\text{m},i}}(I_{\text{memb},i}-I_{\text{stim},i}+I_{\text{gap},i})& i=1,.\; .\; .,N\end{array}$$

where i is the cell number, C_m,i is the membrane capacitance of the ith cell, V_i is the membrane potential, I_memb,i is the total membrane ion current, I_stim,i is the stimulus current, I_gap,i – g_gap(−V_i-1 + 2V_i − V_i+1). In this paper, we used values of g_gap – 25 nS/pF (WT g_gap) and g_gap – 10 nS/pF (TNF-α g_gap) to reproduce experimentally observed conduction velocities 0.62 m/s (20, 62) and 0.40 m/s (38) for WT and TNF-α mouse tissues, respectively.

We employed different stimulation protocols to investigate the frequency dependence of AP amplitudes, AP durations (APDs), and [Ca²⁺]_i transients for isolated myocyte models and 1D model tissues. Isolated cardiac myocytes were stimulated at basic cycle lengths (BCLs) ranging from 30 to 1,000 ms with the stimulus current I_stim – 60–80 pA/pF (which was ∼120% of the threshold value) and pulse duration τ_stim – 0.5 ms. 1D tissue models were stimulated by applying stimulus currents to single cells with BCLs from 50 to 1,000 ms with τ_stim – 0.5 ms and I_stim – 800–900 pA/pF, to be equal to ∼150% of the threshold value. A significant increase in I_stim for 1D tissue is due to the effects of electrotonic currents, which significantly increase a threshold for initiation of AP. Stimulation protocols are described in more detail in the text or in figure legends. AP amplitude was measured as the difference between the peak and minimum values of the transmembrane potential over the specified time interval. [Ca²⁺]_i transient amplitude was also measured as the difference between the peak and minimum values of [Ca²⁺]_i over the specified time interval. In the case of the S1–S2 restitution protocol (see Fig. 8), the maximum and minimum values of the transmembrane potential and the [Ca²⁺]_i transient were determined over the 400-ms time interval following the S2 stimulus. Once the AP amplitude was determined, we evaluated the threshold transmembrane potential corresponding to 50% of repolarization, and APD at 50% of repolarization (APD₅₀) was defined as the difference between the times when AP crosses the threshold value during repolarization and depolarization phases. The decay time constant τ_Cai of the [Ca²⁺]_i transient was determined over the specified time interval by fitting the decaying part of the transient with a monoexponential function from the inflection point to the end of the time interval.

Fig. 8.

S1–S2 restitution. Model cells are paced with constant basic cycle length of 200 ms for 10 s (S1 stimuli); then, the test stimulus is applied at variable S1–S2 intervals from 30 to 200 ms after the last pacing stimulus S1. The dependences (bifurcation diagrams) of AP amplitude (A), AP duration at 50% of repolarization (APD₅₀, B), and [Ca²⁺]_i transients (C) for WT (black squares) and TNF-α-overexpressing (gray squares) ventricular myocytes are shown. AP amplitude, AP durations, and [Ca²⁺]_i transients are calculated during the variable S1–S2 interval from 30 to 200 ms. The bifurcation diagrams show a steady decrease in AP amplitude and increase in AP duration at 50% of repolarization. [Ca²⁺]_i transients are significantly larger in WT model cells compared with those in TNF-α-overexpressing cells. D and E show the time series for APs and [Ca²⁺]_i transients for two characteristic values of the S1–S2 interval (170 and 50 ms, respectively) for both WT and TNF-α-overexpressing cells.

Model equations were solved by a fourth-order Runge-Kutta method, with a time step of 0.0001 ms. Initial conditions were obtained by running a program code without stimulation for at least 1,000 s. AP of single ventricular myocytes, voltage-clamp experiments, bifurcation diagrams, and simulations of 1D tissue models were performed on a personal computer using Intel FORTRAN 90 [processor Intel Core2 Quad Q9550 (2.83 GHz)].

Myocyte isolation and measurements of AP.

The investigation conforms with the Guide for the Care and Use of Laboratory Animals published by the United States National Institutes of Health (NIH Publication No. 85–23, revised 1996) and was approved by the Institutional Animal Care and Use Committee at the University of Pittsburgh. Mouse ventricular myocytes were isolated using the same technique, as previously described (50). Briefly, mice were anesthetized with pentobarbital sodium (35 mg/kg ip) and injected with heparin (50 U ip). Under deep anesthesia, hearts were rapidly excised, cannulated, and perfused at 35–36°C with Tyrode solution containing also 0.5 mg/ml collagenase, 0.02 mg/ml protease, and 0.025 mM Ca²⁺ using the Langendorff perfusion method. After a 50–100% increase of flow rate, single cardiac cells were obtained by gentle trituration of the left ventricle. Isolated myocytes were resuspended in Tyrode solution containing 0.1 mM Ca²⁺ with 1 mg/ml BSA and further used for electrophysiological recordings.

To measure inwardly rectifying K⁺ currents (I_K1), the pipette solution contained (in mM) 135 KCl, 1 MgCl₂, 10 EGTA, 10 HEPES, 5 glucose, and 3 Mg₂ATP, pH adjusted to 7.2 with KOH, and the external solution contained (in mM) 136 NaCl, 4 KCl, 1 CaCl₂, 2 MgCl₂, 10 HEPES, and 10 glucose, pH adjusted to 7.35 with NaOH. CoCl₂ (5 mM) and tetrodotoxin (20 μM) were added to the external solution to block Ca²⁺ and Na⁺ currents, respectively. Inwardly rectifying K⁺ currents were recorded in response to 500-ms voltage steps to test potentials between −130 and −50 mV from a holding potential of −80 mV. All current amplitudes were normalized to the cell capacitances and expressed as densities (in pA/pF).

APs were recorded in the current clamp configuration of the patch-clamp method and elicited by brief, 3-ms, depolarizing current injections delivered at 1 Hz. The pipette solution contained (in mM) 135 KCl, 1 MgCl₂, 10 EGTA, 10 HEPES, 5 glucose, and 3 Mg₂ATP, pH adjusted to 7.2 with KOH, and the external solution contained (in mM) 136 NaCl, 4 KCl, 1 CaCl₂, 2 MgCl₂, 10 HEPES, and 10 glucose, pH adjusted to 7.35 with NaOH. Electrophysiological measurements were performed at room temperature (22–24°C).

Statistics.

Data are expressed as means ± SE. Differences between membrane currents in TG and WT mice were evaluated using Student's t-test and considered statistically significant at P < 0.05.

RESULTS

AP and Ca²⁺ handling in TNF-α-overexpressing ventricular myocytes.

Simulated APs from WT and TNF-α-overexpressing myocytes and the time behavior of [Ca²⁺]_i transients and the major repolarization currents are shown in Fig. 5. As expected from the voltage-clamp data of K⁺ currents, the APD for WT cells was shorter than that for TG cells (APD₅₀ are equal to 3.0 and 4.3 ms, respectively; Table 3). These values are close to the corresponding experimental durations, 2.37 ± 0.13 and 4.64 ± 0.27 ms for WT and TG cells, respectively. Experimental resting potentials are similar for WT and TG cells, −76.25 ± 0.78 and −76.03 ± 0.62 mV, respectively. The values are close to our simulated equilibrium membrane potentials, −76.25 and −75.37 mV for WT and TG cell models, respectively. The WT APD₅₀ fit reasonably well to the corresponding experimental values obtained by others that range from 2.6 to 6.2 ms (3, 12, 29). The data can vary due to different experimental conditions and mice strains. Figure 5B shows the simulated time behavior of [Ca²⁺]_i transients, which are smaller and more prolonged for TG compared with WT cells, as observed experimentally (38). The slower [Ca²⁺]_i transient decay is due to a smaller Ca²⁺ sequestration rate of the SERCA pump. Simulations of the time course of the major repolarization currents (Fig. 5, C and D) demonstrate larger outward K⁺ currents in WT compared with TG cells, which underlies shorter AP. The magnitude of I_Kto,f is only slightly larger than that of I_Kur in WT cells, but they are approximately equal in TG cells. I_Kur declines more slowly in both cell types and affects repolarization at a longer time scale. These data are different from our previously published model, where the magnitude of I_Kur is ∼2.5 times smaller than that of I_Kto,f (9). I_Kss has a relatively small amplitude in both cell types and does not affect repolarization significantly. Such differences in K⁺ conductances may reflect the differences in mice strains on which ventricular myocyte models were based [C57BL6 for the model in Ref. 9 and FVB for the model in this study; the two mice strains have different ratios in the expression of transient outward K⁺ current (I_to), I_K,slow1, and I_K,slow2 (50, 67)].

Fig. 5.

Simulated action potentials (APs), [Ca²⁺]_i transients, and underlying currents of the isolated ventricular myocyte models from WT and TNF-α-overexpressing mice. A: APs in WT (solid line) and TNF-α-overexpressing (dashed line) myocytes. B: [Ca²⁺]_i transients from WT (solid line) and TNF-α-overexpressing (dashed line) mouse ventricular myocytes. C: currents underlying the WT AP. D: currents underlying the AP for TNF-α-overexpressing myocytes. The scale for the relatively large Na⁺ current, I_Na, is given on the right axis in C and D. All other currents are scaled to the left axis. Pacing frequency was 1 Hz. APs, [Ca²⁺]_i, and ionic currents are shown after the 10th stimulus.

Table 3.

Comparison of simulated and experimental APDs from adult WT and TNF-α-overexpressing ventricular myocytes

	APD25, ms	APD50, ms	APD75, ms
		Experiment
WT	1.51 ± 0.09 (8)	2.37 ± 0.13 (8)	4.48 ± 0.28 (8)
TNF-α	2.11 ± 0.12 (9)	4.64 ± 0.27 (9)	12.05 ± 1.31 (9)
		Simulation
WT	1.8	3.0	5.8
TNF-α	2.3	4.3	8.6

Values are means ± SE; no. of myocytes is shown in parentheses. APD₂₅, action potential duration (APD) at 25% of repolarization; APD₅₀, APD at 50% of repolarization; APD₇₅, APD at 75% of repolarization; WT, wild type; TNF-α, tumor necrosis factor-α.

Figure 6 demonstrates the simulated relationships between the different integral Ca²⁺ fluxes in WT and TG cells during one cardiac cycle. About 0.95 μM Ca²⁺ enters the WT cell through the L-type Ca²⁺ channels during a single AP and triggers significantly larger (32.5 μM) Ca²⁺ release from the SR. The latter value is close to the experimental data of 39 ± 3 μM (42). About 90% of the total Ca²⁺ that enters the cytosol is pumped back to the SR (31.9 μM); only ∼8% (2.6 μM) and ∼1% (0.23 μM) are extruded by the Na⁺/Ca²⁺ exchanger and the slow mechanism [I_p(Ca) - I_Cab]. These percent fractions agree well with the corresponding experimental data of Li et al. (Ref. 37, 90.3, 9.2, and 0.5% by the SR pump, the Na⁺/Ca²⁺ exchanger, and the slow mechanism, respectively). Similar integral flux calculations for TNF-α-overexpressing cells show an increase in Ca²⁺ entry through L-type Ca²⁺ channels (1.24 μM) due to the AP prolongation, a decrease in Ca²⁺ release (17.7 μM) due to a smaller SR Ca²⁺ content (1,000 vs. 390 μM for WT and TG cells, respectively), and an increased fraction of Ca²⁺ extruded by the Na⁺/Ca²⁺ exchanger (3.6 μM or 19% of the total Ca²⁺ that enters the cytosol). Simulation results correlate with experimental data in TNF-α-treated cardiomyocytes [∼20% decrease in the SR Ca²⁺ load compared with control (35)] and in mouse ventricular myocytes, which have a decreased SERCA function [40–60% or more of a decrease in the SR Ca²⁺ content and ∼40% increase in I_NaCa (26, 40)].

WT and TNF-α-overexpressing ventricular myocytes also show a difference in the response to stimulation rates. Figure 7 shows experimental and simulation data for the frequency dependence of [Ca²⁺]_i transient amplitudes and decay times in isolated cells. For WT cells, both experiments (solid lines with symbols) and simulations (solid lines) show an increase in the peak and basal [Ca²⁺]_i magnitudes at stimulation frequencies larger than 2 Hz (Fig. 7A). Simulations for TG cells (dashed lines in Fig. 7A) demonstrate smaller peak and larger basal values compared with WT cells, similar to the experimental data (38). Simulated and experimental data show a similar behavior and close values for the relaxation time constant τ_Cai as a function of stimulation frequency (Fig. 7B). For example, simulated values of τ_Cai for WT and TG myocytes at BCL – 200 ms, 68 and 84 ms, are close to the corresponding experimental values, 54 ± 5 and 72 ± 6 ms, respectively (38).

Fig. 7.

A: experimental peak (unfilled symbols) and basal (filled symbols) [Ca²⁺]_i transients during mouse AP excitation as functions of stimulation frequency. Solid and dashed lines show simulation data for WT and TNF-α-overexpressing ventricular myocytes, respectively. Squares and circles show experimental data from Refs. 1 and 23, respectively. B: frequency dependence of [Ca²⁺]_i transient decay. Simulation data were fit with a monoexponential function; solid and dashed lines refer to WT and TNF-α-overexpressing cells, respectively. Filled circles and a square show experimental data for [Ca²⁺]_i transient relaxation in WT cells (Refs. 65 and 38, respectively); an unfilled square corresponds to data for TNF-α-overexpressing cells (38).

We also studied two different restitutions, S1–S2 and steady-state restitutions. Figure 8 shows S1–S2 restitution curves for AP amplitudes, APD₅₀, and peak [Ca²⁺]_i transients for WT and TG myocytes. The stimulus protocol included stimulation of the model cell with 51 pulses (I_stim – 60.0 pA/pF, τ_stim – 0.5 ms) with BCL – 200 ms (pacing), and the 52nd stimulus was applied with a variable time interval from 30 to 200 ms (S1–S2 interval). The stimulus protocol was similar to that used by London et al. (38) in experiments on WT and TNF-α-overexpressing mouse hearts. Simulation data show a decrease in AP amplitude with reduction of the S1–S2 interval, with TG cells showing a steeper decrease (Fig. 8A). Similar behavior was observed experimentally (38). In contrast, the APD₅₀ demonstrates an increase for WT cells from 3.3 to 7.75 ms and for TG cells from 4.6 to 15.5 ms when the S1–S2 interval is reduced from 200 to 30 ms (Fig. 8B). An increase in APD₅₀ with decreasing BCL is an experimentally observed peculiarity of mice (31). Peak [Ca²⁺]_i transients demonstrate decreasing and increasing parts as the S1–S2 interval decreases (Fig. 8C). The increasing part of restitution in peak [Ca²⁺]_i transients at very small S1–S2 intervals is due to an incomplete [Ca²⁺]_i transient relaxation, when the second peak appears on the top of the relatively large [Ca²⁺]_i transient from the previous stimulation (Fig. 8E). Note that [Ca²⁺]_i transients from TG cells are significantly smaller than those from WT cells with most S1–S2 intervals. Figure 8, D and E, shows the time series for APs and [Ca²⁺]_i transients for two S1–S2 intervals. At the length of the S1–S2 interval of 170 ms, the shape of both APs and [Ca²⁺]_i transients was not much different from those obtained with the S1 stimuli (Fig. 8D). However, APs and [Ca²⁺]_i transients were reduced significantly at S1–S2 – 50 ms (Fig. 8E).

Steady-state restitution curves for AP amplitudes, APD₅₀, and peak [Ca²⁺]_i transients are shown in Fig. 9. They were obtained by stimulation with constant BCL in the range from 30 to 200 ms with I_stim – 60.0 pA/pF and τ_stim – 0.5 ms. AP amplitudes, APD₅₀, and peak [Ca²⁺]_i transients were calculated on a 2-s time interval between the 48th and 50th s when an approximate steady state was reached. APs of both WT and TG cells show a decrease in amplitude when stimulus BCL decreases from 200 to 30 ms (see also time series in Fig. 9, D and E). At relatively fast pacing rates (∼40 ms for WT cells and ∼55 ms for TG cells), a steep decrease in AP amplitudes is observed (Fig. 9A); further decreasing BCL elicited relatively small-amplitude APs (Fig. 9E). The steep decrease in AP amplitudes correlates with a steep increase in APD₅₀ (Fig. 9B) at the same pacing rates. In contrast to S1–S2 restitution, steady-state restitution curves demonstrate intervals of BCLs where APD₅₀ decreases together with the decrease in the BCL; however, the BCL interval for such a behavior is smaller for WT cells compared with TG cells. The rate dependence of [Ca²⁺]_i transients for steady-state pacing is also different from that for dynamic restitution (Fig. 9C). First, the peak [Ca²⁺]_i transients decrease in amplitude, when BCL changes from 200 to 48 ms for WT cells and 62 ms for TG cells. Second, at smaller BCLs, an irregular behavior of [Ca²⁺]_i transients is observed until BCL – 44 ms for WT and BCL – 56 ms for TG cells (Fig. 9E and Table 4). At even smaller BCLs, no [Ca²⁺]_i transients are observed, indicating the absence of Ca²⁺-induced Ca²⁺ release at very fast pacing rates (Fig. 9E).

Fig. 9.

Steady-state restitution. Simulated frequency dependence (bifurcation diagrams) of AP amplitude (A), AP duration at 50% of repolarization (B), and [Ca²⁺]_i transients (C) for WT (black squares) and TNF-α-overexpressing (gray squares) ventricular myocytes. Basic cycle length (BCL) as a bifurcation parameter was varied from 30 to 200 ms. AP amplitude, AP durations, and [Ca²⁺]_i transients are calculated in the time interval from 48 to 50 s. The bifurcation diagrams for AP amplitude and AP duration at 50% of repolarization show regular behavior, but the bifurcation diagrams for [Ca²⁺]_i transients show irregular behavior when pacing frequency increases. [Ca²⁺]_i transients lose their stability at more rapid pacing in WT cells compared with those in TNF-α-overexpressing cells. D and E show time series for APs and [Ca²⁺]_i transients for two characteristic values of BCL (170 and 47 ms, respectively) for both WT and TNF-α-overexpressing cells.

Table 4.

Threshold basic cycle lengths for instability of AP propagation and [Ca²⁺]_i transients in WT and TNF-α-overexpressing isolated mouse ventricular myocytes and model cardiac tissues

	AP Instability Type and Threshold				[Ca2+]i Instability Type and Threshold
	Isolated ventricular myocyte*	Cardiac tissue (pacing), ms	Cardiac tissue (time window for reentry), ms	Cardiac tissue (width of the time window for reentry), ms	Isolated ventricular myocyte,* ms	Cardiac tissue (pacing), ms
WT AP	Stable	72 (conduction block)	52.41–52.53	0.12	48 (irregular)	121 (alternans)
WT ggap
TNF-α AP	Stable	98 (conduction block)	63.29–64.78	1.49	62 (irregular)	141 (alternans)
TNF-α ggap
WT AP		72 (conduction block)	56.84–58.86	2.02		123 (alternans)
TNF-α ggap
TNF-α AP		97 (conduction block)	59.93–59.94	0.01		139 (alternans)
WT ggap
TNF-α AP		100 (conduction block)	73.48–73.51	0.03		206 (alternans)
WT ggap 70% INa
TNF-α AP		130 (conduction block)	112.67–112.73	0.06		Not determined; very small [Ca2+]i
WT ggap 35% INa

AP, action potential; [Ca²⁺]_i, intracellular Ca²⁺ concentration; g_gap, gap junction conductance; I_Na, Na⁺ current.

^*For isolated ventricular myocyte models neither g_gap must be taken into account.

Stability of [Ca²⁺]_i transients in WT and TG mouse ventricular tissues.

The stability of AP propagation and [Ca²⁺]_i transients was studied by pacing the 1D model of WT and TG tissues with different BCLs. Each tissue contained 190 cells connected in a cable with a different intercellular conductance g_gap. Because two major factors determine AP propagation velocity in cardiac tissue, the magnitude of intercellular conductance g_gap and the magnitude of the fast Na⁺ current I_Na, we investigated velocity of AP propagation in 1D model tissues where g_gap and I_Na were changed. AP propagation through the quiescent tissue was initiated by a single stimulus (τ_stim – 0.5 ms and I_stim – 900 pA/pF). Figure 10, A and B, shows the dependence of AP propagation velocity as a function of g_gap and I_Na magnitudes. AP propagation velocity increases when g_gap (Fig. 10A) and the amplitude of I_Na (Fig. 10B) increase. The dependence of AP velocity on g_gap and I_Na is not related to tissue type and is virtually the same for WT and TG tissues (Fig. 10, A and B). Of interest is to note that, at values for I_Na as low as 20% of its original (maximum) value, no APs are propagated (Fig. 10B) and that the reduction of I_Na also leads to a decrease in AP amplitude (Fig. 10C).

Fig. 10.

A: AP propagation velocities as functions of intercellular conductance [gap junction conductance (g_gap)] for 1-dimensional (1D) cardiac tissues consisting of WT (filled circles) and TNF-α-overexpressing (unfilled circles) cells. Each model tissue consists of 190 cells connected in a cable. Stimulation I_stim (900 pA/pF, τ_stim – 0.5 ms) is applied to the 190th cell. B: AP propagation velocities as functions of the amplitude of I_Na for 1D cardiac tissues consisting of WT (filled circles) and TNF-α-overexpressing (unfilled circles) cells. The amplitude of I_Na is presented in %fraction of its original value. Intercellular conductance g_gap – 25 nS/pF. AP failed to propagate at the amplitudes of I_Na less than 20% of its original value. C: AP amplitude as a function of the amplitude of I_Na for a 1D cardiac tissue consisting of TNF-α-overexpressing cells. Intercellular conductance g_gap – 25 nS/pF. Note that the simulated AP velocities for WT and TG mice are very close to each other and overlap in A and B.

To investigate the contribution of g_gap and/or I_Na to the reduction in AP propagation velocity in TNF-α-overexpressing tissue to the stability of AP propagation and [Ca²⁺]_i transients, we studied six model tissues composed of 1) a WT cell and WT g_gap – 25 nS/pF (WT tissue); 2) a TG cell and TG g_gap – 10 nS/pF (TG tissue); 3) a WT cell and TG g_gap; 4) a TG cell and WT g_gap; 5) a TG cell with 70% of original I_Na amplitude and WT g_gap; and 6) a TG cell with 35% of original I_Na amplitude and WT g_gap. 1D tissues, similar to isolated cardiac myocytes, demonstrate different behavior at rapid pacing (Table 4). As described above, isolated WT and TG myocytes show a transition to irregular behavior of [Ca²⁺]_i transients at different threshold BCLs. For 1D tissues, the threshold BCLs for instability of [Ca²⁺]_i transients become longer compared with isolated myocytes, and the model tissues lose stability through a period-doubling bifurcation (or alternans). For WT tissue, the threshold BCL for alternans is 121 ms, whereas, for TNF-α-overexpressing tissue, it is 141 ms (Table 4). Figure 11 shows simulated propagation of [Ca²⁺]_i transients for WT and TG mouse tissues, paced at the 190th cell with two BCLs, 170 and 130 ms. The propagation velocity of [Ca²⁺]_i transients is smaller in TG mouse tissue due to a smaller intercellular conductance g_gap, and they are more prolonged compared with WT tissue. At BCL – 170 ms, none of the tissues show [Ca²⁺]_i alternans (Fig. 11, A and C). When BCL is reduced to 130 ms, WT tissue still demonstrates periodic [Ca²⁺]_i transients, whereas TG tissue shows [Ca²⁺]_i alternans that changes in time and space (Fig. 11D).

Fig. 11.

Three-dimensional (3D) plots of [Ca²⁺]_i transients in 1D mouse model tissues. Model tissues consist of 190 cells connected in a cable. Stimulation I_stim (900 pA/pF, τ_stim – 0.5 ms) is applied to the 190th cell with two BCLs (170 and 130 ms). Data are shown for the time interval from 3.5 to 4.0 s. A: WT tissue model (BCL – 170 ms). B: WT tissue model (BCL – 130 ms). C: TNF-α-overexpressing tissue model (BCL – 170 ms). D: TNF-α-overexpressing tissue model (BCL – 130 ms). Conductances for WT and TNF-α-overexpressing tissue models are g_gap – 25 nS/pF (A and B) and g_gap – 10 nS/pF (C and D), respectively. TNF-α-overexpressing tissue model is more susceptible to [Ca²⁺]_i alternans than WT tissue.

Four other model tissues show both similarities and differences with WT and TG tissues in respect to [Ca²⁺]_i alternans development. The tissue consisting of WT AP and TG g_gap has a threshold value for alternans that is equal to 123 ms, a value close to the threshold value for alternans in WT tissue, 121 ms. Similarly, the tissue consisting of TG AP with WT g_gap has a threshold value of 139 ms, which is close to the TG tissue value of 141 ms. This leads to a conclusion that the threshold for [Ca²⁺]_i alternans is determined predominantly by cellular properties rather than by the intercellular conductance. Furthermore, a reduction of I_Na to 70% of its original value in the tissue consisting of TG AP with WT g_gap increased the threshold BCL for alternans to 206 ms.

An example of the cellular properties of propagated behavior in multicellular cardiac tissues is shown by APs and [Ca²⁺]_i transients for the 170th cell in WT (trace on top in the two panels for AP and [Ca²⁺]_i transients in Fig. 12) and TG (trace on bottom in the two panels for AP and [Ca²⁺]_i transients in Fig. 12) tissue models at BCLs 170 and 130 ms (Fig. 12, A and B, respectively). [Ca²⁺]_i transients in WT cells have larger peak amplitudes and a smaller diastolic [Ca²⁺]_i level compared with those in TG cells as observed experimentally (38). Ca²⁺ handling of WT and TG cells differs predominantly in the Ca²⁺ sequestration rate set by the different activity of the SERCA pump in the two cell types. The reduced activity of the SERCA pump in TG cells results in a significantly smaller Ca²⁺ load in the SR and a smaller SR Ca²⁺ release in TG mouse cells. Clear alternans is seen in the lower trace for TNF-α-overexpressing cells (Fig. 12B, bottom). Analysis of the gating properties of the L-type Ca²⁺ channel and RyRs suggests that the slower RyR opening rate due to a smaller amplitude of Ca²⁺ transients in the subspace region between Ca²⁺ channels and RyRs in TG cardiac cells is the major cause for [Ca²⁺]_i alternans (7). It is remarkable that Ca²⁺ alternans does not significantly affect APs (Fig. 12), unlike in larger species where Ca²⁺ alternans is coupled to AP alternans (for review, see Ref. 19).

Fig. 12.

Representative APs and [Ca²⁺]_i transients obtained from the 170th cell of the same WT and TNF-α-overexpressing model tissues, as in Fig. 11. A: BCL – 170 ms. B: BCL – 130 ms. Top and bottom traces in the subpanels (APs and [Ca²⁺]_i) of each panel correspond to WT and TNF-α-overexpressing models, respectively. The cell from TNF-α-overexpressing tissue shows clear [Ca²⁺]_i alternans at BCL – 130 ms, whereas APs are periodic.

Stability of AP propagation in WT and TG mouse ventricular tissues.

AP propagation instability in all six simulated 1D cardiac tissues represents AP propagation block (Table 4). For each cardiac tissue, the conduction block occurs at smaller BCLs compared with the threshold BCLs for Ca²⁺ alternans. However, as in the case of Ca²⁺ alternans, the threshold for AP propagation block is predominantly determined by cellular properties. For example, WT tissue has the same threshold BCL – 72 ms for conduction block as the tissue composed of WT cells connected with TG g_gap. For TG tissue, the threshold BCL – 98 ms, which is close to the threshold BCL – 97 ms for the tissue composed of TG cells connected with WT g_gap (Table 4). The model tissues composed of TG cells with reduced magnitude of I_Na (70 and 35% of its original value) and WT g_gap have even longer BCLs for AP propagation block, 100 and 130 ms, respectively.

Figure 13 shows AP propagation through WT and TG model tissues at different BCLs, 130 and 80 ms. At BCL – 130 ms, both tissues demonstrate stable AP propagation, with a slower propagation velocity in TG tissue. However, when the BCL decreases to 80 ms, AP propagation in TG tissue shows multiple blocks (Fig. 13D). Stimulated with the same BCL – 80 ms, WT tissue demonstrates stable AP propagation (Fig. 13C).

Fig. 13.

Stability of AP propagation in 1D mouse model tissues. The model tissue parameters are the same as in Fig. 11. A: WT tissue model (BCL – 130 ms). V, voltage. B: WT tissue model (BCL – 80 ms). C: TNF-α-overexpressing tissue model (BCL – 130 ms). D: TNF-α-overexpressing tissue model (BCL – 80 ms). Arrows show time moments at which stimuli were applied. TNF-α-overexpressing tissue model shows clear conduction block at BCL – 80 ms.

Simulation of reentry in WT and TG mouse ventricular tissues.

TNF-α-overexpressing mouse hearts have been demonstrated experimentally to be more susceptible to reentrant arrhythmias than WT hearts (38). Reentrant arrhythmias were triggered experimentally in Langendorff perfused mouse hearts using programmed stimulation: a train of 10 stimuli with a BCL of 200 ms was followed by a premature stimulus at a variable time interval. Reentrant arrhythmias were observed in 14 out of 17 TG hearts and only in 1 out of 7 WT hearts (38). During reentry, the AP propagated around the perimeter of the mouse heart.

To simulate this susceptibility to reentry, we used a 1D ring mouse ventricular tissue model with 190 cells and a stimulation protocol similar to that used by London et al. (38). After a train of 11 S1 stimuli with BCL – 200 ms (I_stim – 900 pA/pF, τ_stim – 0.5 ms) to cell 1, a premature stimulus S2 (I_stim – 800 pA/pF, τ_stim – 1.0 ms) was applied to cell 50 with a variable time interval from 40 to 70 ms (Fig. 14). Note that the duration of the S2 stimulus is doubled to facilitate reentry. To elucidate contributing factors in reentrant arrhythmias, we investigated six model tissues described above (Table 4). In the WT model tissue, reentry could be triggered only in a very narrow time window for S2 (∼0.12 ms), from 52.41 to 52.53 ms (Fig. 14, A–C). Even when excited, reentry was not sustained for more than one cycle (Fig. 14B). At smaller time windows for S2, AP was not propagated. At larger time windows for S2, a bidirectional AP propagation was observed. In TG model tissue, reentry was triggered in a much wider time window for S2 (∼1.49 ms) compared with WT tissue, from 63.29 to 64.78 ms (Fig. 14, D–F), and reentry was sustained (Fig. 14E). In the model tissue with TG AP and WT conductance, reentry was triggered only in an even smaller time window for S2 (∼0.01 ms) compared with WT tissue, from 59.93 to 59.94 ms (Fig. 14, G–I), and the resulting reentry was not sustained (Fig. 14H). In the model tissue with WT AP and TG conductance, reentry was initiated within the largest time window, from 56.84 to 58.86 ms (2.02 ms; Fig. 14, J–L), and reentry was sustained (Fig. 14K). Finally, reentry was studied in the tissue with TG AP, WT conductance, and reduced magnitude of I_Na. The reduction of I_Na yielded a time window for reentry of 0.03 and 0.06 ms for 70 and 35% of control I_Na, respectively, and significantly prolonged the refractory time interval to 73.48 and 112.67 ms for 70 and 35% of control I_Na, respectively. In both cases, reentry was not sustained. These results suggest that the smaller intercellular conductance in TG mouse tissue is the major substrate for reentry in TG mouse hearts.

Fig. 14.

Reentry of APs in 1D mouse tissue models. Model tissues consist of 190 cells connected in a ring. Stimulation with S1 pulses (I_stim – 900 pA/pF, τ_stim – 0.5 ms) is applied to the first cell. The tissue first was stimulated with BCL – 200 ms (11 S1 stimuli), then S2 pulse (I_stim – 800 pA/pF, τ_stim – 1.0 ms) was applied to cell 50 at the indicated time moment T_stim. The 11th S1 and S2 stimuli are shown by red arrows. A–C: APs for WT tissue model (g_gap – 25 nS/pF). D–F: APs for TNF-α-overexpressing tissue model (g_gap – 10 nS/pF). G–I: APs for TNF-α-overexpressing tissue model (g_gap – 25 nS/pF). J–L: APs for WT tissue model (g_gap – 10 nS/pF). Note different time scales in D–F and J–L to show sustained reentry. Nonsustained reentry is seen in B and H for WT and TNF-α-overexpressing tissue models (g_gap – 25 nS/pF) while sustained reentry is developed in WT and TNF-α-overexpressing tissue models at g_gap – 10 nS/pF (E and K).

DISCUSSION

TG mice overexpressing TNF-α were generated as an animal model of human heart failure (33, 38, 50). They possess many of the features of the disease observed in patients, such as dilated cardiomyopathy, impaired Ca²⁺ handling, arrhythmias, and decreased survival. Our mathematical model reproduced the electrical properties and Ca²⁺ handling in isolated ventricular myocytes and cardiac tissues from TG mice. Additional multicellular tissue simulations examined the relative role of AP prolongation, reduced gap junction conductance, and reduced amplitudes of the fast Na⁺ currents in the development of Ca²⁺ alternans, AP propagation block, and reentrant arrhythmias. Our models showed that a decrease in SR Ca²⁺ uptake due to decreased SERCA function and an increase in the function of the Na⁺/Ca²⁺ exchanger in TNF-α multicellular tissues reduced SR Ca²⁺ load and together with prolonged AP promoted Ca²⁺ alternans. Prolongation of APD is the major contributing factor for the greater susceptibility to AP conduction block, and the reduced tissue conductance is the predominant determinant of the wider vulnerable window for reentrant arrhythmias.

Differences in APs and Ca²⁺ handling in isolated ventricular myocytes from WT and TNF-α-overexpressing mouse hearts.

Mouse ventricular myocytes overexpressing TNF-α have significantly longer APs than nontransgenic cells (38). The longer APs correlate with significantly reduced amplitudes of two major repolarization K⁺ currents, the rapidly inactivating transient outward K⁺ current, I_Kto,f, and the ultrarapidly activating delayed-rectifier K⁺ current, I_Kur (50). These two currents have significantly larger amplitudes in mice compared with larger species that ensure very short mouse AP, with APD₅₀ – 2.37 ± 0.13 ms in WT cells. Reduction of these currents in TG mice prolonged their APDs (APD₅₀ – 4.64 ± 0.27 ms). Our mathematical model reproduced AP prolongation in TG mice compared with WT mice (Fig. 5 and Table 3). Downregulation of a major K⁺ repolarization current, I_to, and the corresponding AP prolongation were also observed in rat cells that were exposed to different concentrations of TNF-α (16). In larger species, ventricular myocytes have longer APs resulting from a different set of K⁺ currents. In these animals, pacing-induced heart failure also leads to downregulation of K⁺ currents (predominantly I_to and the slow delayed-rectifier K⁺ current) and associated prolongation of APs (45). The pacing-induced remodeling of K⁺ currents was directly related to activation of TNF-α (11, 43).

Despite significant effects on K⁺ currents, activation of TNF-α does not affect I_CaL in different species [mouse (50), rat (16), and dog (27)]. However, failing hearts consistently demonstrate significant changes in the function of two other Ca²⁺-handling proteins, the SERCA2 pump and the Na⁺/Ca²⁺ exchanger (45, 59). A decrease in SERCA function together with a trans-sarcolemmal increase of Ca²⁺ loss due to an increased extrusion by the Na⁺/Ca²⁺ exchanger lead to reduced SR Ca²⁺ load, increased diastolic Ca²⁺ level, a reduced amount of released Ca²⁺ during Ca²⁺ cycling, and reduced [Ca²⁺]_i transients. Such abnormalities in Ca²⁺ handling were reproduced by our model of mouse ventricular myocytes overexpressing TNF-α (Figs. 5, 6, 8, 9, 11, and 12). The reduction in the SR Ca²⁺ sequestration rate and the enhancement of I_NaCa in our mathematical model resulted in a decreased SR Ca²⁺ load, smaller and prolonged [Ca²⁺]_i transients, and a higher level of diastolic Ca²⁺. Elevation of the diastolic level of Ca²⁺ appears to occur as a result of the slowing of Ca²⁺ sequestration into the SR. This effect overwhelms the increased function of the Na⁺/Ca²⁺ exchanger, since the SR Ca²⁺ flux through SERCA is much larger (by a factor ∼10) than that through the Na⁺/Ca²⁺ exchanger (37), resulting in elevation of the diastolic Ca²⁺ concentration.

Stability of [Ca²⁺]_i transients in TNF-α-overexpressing myocytes and tissues.

Overexpression of TNF-α in mouse hearts results in an increased susceptibility to [Ca²⁺]_i alternans compared with WT hearts (38), and our simulations showed a similar susceptibility in both isolated myocytes and ventricular tissue models of TG mice. One of the major causes of [Ca²⁺]_i transient instability in isolated ventricular mouse cells at fast pacing rates was a decrease in AP amplitude. A decrease in AP amplitude decreases the time window for activation of I_CaL and leads to smaller Ca²⁺ entry, a smaller amount of Ca²⁺ released from the SR, and eventually lower Ca²⁺ concentration in the subspace between L-type Ca²⁺ channels and RyRs. The second cause of [Ca²⁺]_i transient instability is decreased SR Ca²⁺ load, which decreases Ca²⁺ release from the SR, and, as a consequence, decreases the Ca²⁺ concentration in the subspace volume between L-type Ca²⁺ channels and RyRs. Because the opening rate of RyRs is proportional to the fourth power of the Ca²⁺ concentration (9), this leads to their slower opening. A smaller Ca²⁺ release in one Ca²⁺ cycle results in a relatively large Ca²⁺ concentration in the junctional SR before the next cycle. The amount of released Ca²⁺ during the next stimulus cycle is dependent on several factors, such as Ca²⁺ transfer from the network to the junctional SR, the closing rate of RyRs, and the number of RyRs that remain in the open state upon arrival of the stimulus. Such multiple factors act on a portion of the frequency dependence curve with significantly reduced AP amplitude and lead to irregular behavior of [Ca²⁺]_i transients in isolated ventricular myocytes (Fig. 9).

Irregular behavior of [Ca²⁺]_i transients is observed at longer BCLs (BCL – 62 ms) in TG cells compared with WT cells (BCL – 48 ms) because of a larger decrease in AP amplitude with BCL in TG cells and a smaller SR Ca²⁺ load (Fig. 9). The larger decrease in AP amplitude in TG cells is a result of the larger I_Na inactivation during AP and smaller I_Na availability, which is a consequence of the longer APD. Note that the irregular behavior of [Ca²⁺]_i transients is observed in both WT and TNF-α-overexpressing mouse hearts at very rapid pacing with BCL – 80 ms (38).

When TG cells were connected into a tissue model, [Ca²⁺]_i became unstable at a considerably longer BCL – 141 ms compared with isolated cells (BCL – 62 ms), resulting in [Ca²⁺]_i alternans. The simulated value of BCL for the alternans in TG tissue is close to the corresponding experimental value of 140 ms obtained when pacing TG mouse hearts (38). At these BCLs, the peak values of [Ca²⁺]_i transients in isolated ventricular cells have a less steep dependence on BCL (Fig. 9) that results in period doubling bifurcations. Note that the type of [Ca²⁺]_i transient instability in cardiac tissue (period-doubling bifurcation, or alternans) is different from that for isolated myocytes (transition to irregular behavior).

The mechanism of [Ca²⁺]_i alternans in rodent cardiac tissues has been studied both experimentally and using computer simulations (7, 14, 38). Diaz et al. (14) showed that pharmacological depression of RyRs with tetracaine led to alternans in rat cardiac myocytes. This mechanism was supported by later simulation studies (7) and in this study, where rapid pacing rates resulted in small opening probabilities of RyRs and [Ca²⁺]_i alternans. We believe this is the mechanism of [Ca²⁺]_i alternans observed with increased frequency of stimulation for both WT and TG mice. The increased susceptibility to [Ca²⁺]_i alternans in TG mice (in the tissue model, [Ca²⁺]_i alternans appear at longer BCLs in TG mice compared with WT mice, BCL – 141 ms vs. BCL – 121 ms, respectively) is explained with a steeper AP decrease with rapid pacing, as it follows from model simulations. Such a steeper decrease is mostly due to larger inactivation of I_Na during more prolonged APs in TG mice. The larger inactivation results in decreased I_Na, which in turn underlies smaller AP amplitude. Additionally, the smaller SR Ca²⁺ load due to decreased SERCA function in TG tissue compared with WT tissue is another major cause for larger susceptibility to [Ca²⁺]_i alternans of TG mice. In larger species, with significantly longer APs, [Ca²⁺]_i alternans is generally accompanied by AP alternans, where Ca²⁺-dependent inactivation of I_CaL plays an important role (18, 52). However, in rabbit ventricular myocytes, Chudin et al. (13) demonstrated that the genesis of [Ca²⁺]_i alternans can appear solely from instability in Ca²⁺ cycling (57).

Proarrhythmic activity in TNF-α-overexpressing mouse ventricular tissue: Reentry.

Reentry is a well-established mechanism of tachycardia in the hearts of large mammals and humans (see Refs. 44 and 47 for recent reviews). During reentry, AP circulates in a small part of the ventricle rather than propagating from the apex to the base of the ventricle. However, unlike large mammals, it is difficult to generate reentry in healthy normal WT mouse hearts (62). Even if reentrant AP propagation is initiated in the WT mouse heart, it is not sustained (46). Nonetheless, reentry is considered to be a major mechanism of arrhythmia in TG and failing mouse hearts (4, 38, 46).

Our model was able to simulate reentry in a ring approximating the perimeter of the mouse heart consistent with the observations of Baker et al. (4). Reentry in WT tissue was initiated with an S1–S2 protocol in a very narrow time window between the two stimuli, but it was not sustained (Fig. 14B). However, reentry was sustained in our TNF-α-overexpressing model tissue in which APD was increased, but AP propagation velocity was reduced (Fig. 14E).

Although AP prolongation underlying prolongation of the QT interval could promote reentrant arrhythmias through early afterdepolarizations (EADs) [as in Torsade de Pointes (53)], mice studies reveal that QT prolongation alone is not sufficient to create a proarrhythmic substrate and that gradients of refractoriness across the wall of the ventricle (epicardium to endocardium) as well as along the wall (base to apex) reflecting differential expression of ion channels (2, 12, 39) are critical to assess arrhythmia vulnerability (58). A most conspicuous example are Kv4.2DN dominant-negative TG mice, lacking a major repolarization current, the rapidly recovering component of I_to. These mice have a significant prolongation of AP in isolated myocytes and QT intervals in electrocardiogram recordings (5) but show an anti-arrhythmic phenotype due to uniform refractory periods along the epicardium (39). Our simulations confirmed a protective role for AP prolongation in TNF-α-overexpressing mice: in the model tissue with TG AP and WT conductance, reentry was triggered in a smaller time window for S2 (∼0.01 ms) compared with WT tissue with normal AP and conduction (∼0.12 ms), and the resulting reentry was not sustained. Even combining a prolonged TG AP and WT g_gap with a significant I_Na decrease (35 and 70% of control value) in two other simulated model tissues, which significantly increases the time window for reentry to 0.06 and 0.03 ms, respectively, vs. 0.01 ms for a model where I_Na was not changed (TNF-α AP, WT g_gap, and 100% I_Na), the time windows were still smaller than those for reentry in WT tissue (WT AP and WT g_gap, 0.12 ms), again pointing to a protective role for AP prolongation. This conclusion is further supported by experimentally observed lack of EADs and lack of increased dispersion of refractoriness between apex and base in TNF-α-overexpressing mice (38). Regardless of the above discussion, our cardiac tissue models with prolonged TG APs are more susceptible to conduction block compared with tissues with WT AP (Table 4). This increased susceptibility to conduction block could be partly attributed to modeled decrease in I_Na, since threshold BCLs for conduction block were larger for models with decreased I_Na (Table 4). However, even if I_Na was not decreased, models with prolonged TG APs still had larger threshold BCLs for conduction block compared with models with WT AP (Table 4). Therefore, we do not consider AP prolongation to be a major factor contributing to reentry in TNF-α-overexpressing mice.

Conduction slowing is another major proarrhythmic factor that can lead to reentry. It could be either due to changes in the magnitude of I_Na or reduced conduction of gap junctions. As discussed above, model cardiac tissues with decreased I_Na have the largest threshold BCLs for conduction block compared with other model tissues; it is unlikely that changes in I_Na play a role in inducing reentry in this system. The most important result in support of this conclusion is that the time windows for reentry in model tissues with reduced I_Na (up to 35% of its original value; smaller values of I_Na produced too small AP amplitudes that were not observed experimentally or completely blocked AP propagation) are smaller than those of WT tissue. Based on our modeling studies, we believe that the reduction in propagation velocity due to reduced conduction of gap junctions is the major factor in generating sustainable reentry in TG mouse tissue.

Conduction between ventricular myocytes depends on the excitability of cardiomyocytes and the expression and distribution of connexin43 (Cx43), which is the predominant connexin type in the mouse ventricles (55). Western blot analysis and immunohistochemistry studies in mouse hearts overexpressing TNF-α do not show decreased Cx43 expression in ventricles (25, 55). Immunohistochemical studies, however, show that, while in WT mice Cx43 is primarily located in the intercalated discs of cardiac myocytes, in TG mice, with cardiac-specific overexpression of TNF-α, Cx43 is dispersed throughout the sarcolemma and in intercalated disks (55). According to Peters et al. (48) disturbed Cx43 gap junction distribution correlates with the location of reentrant circuits in the epicardial border zone of healing canine infarcts that cause ventricular tachycardia, which points to disorganization of Cx43 in TG mice as a possible cause for a decrease in conduction velocity. Another fact is that TNF-α-overexpressing mice used in experiments (25, 55) are quite young: 7–12 wk (25) and 2–4 mo (55). For comparison, mice used to measure K⁺, Ca²⁺, and inward-rectifier K⁺ currents as well as APs in isolated cells were 5 mo old (50), whereas mice used to assess [Ca²⁺]_i transients and susceptibility to arrhythmias were 3–9 mo old (38). There is a possibility that the expression and function of Cx43 change at older age, with more severe heart failure. Another factor that can change Cx43 conductance is phosphorylation. Although experimental data on mouse hearts overexpressing TNF-α do not show decreased Cx43 expression in ventricles (25, 55), TNF-α could increase phosphorylation of Cx43 through activation of the Ca²⁺-dependent protein kinase C (PKC) (34, 63), which reduces channel conductance and could result in slower AP conduction velocity in TG ventricles.

[Ca²⁺]_i alternans could predispose to arrhythmias, especially if it is accompanied by AP alternans. It is believed that repolarization alternans could arise from primary changes in various ionic currents like I_Na, I_CaL, or the Na⁺/Ca²⁺ exchanger current, or from secondary alterations in sarcolemmal ion channels due to alterations in intracellular Ca²⁺ cycling (64). Figure 12 shows that [Ca²⁺]_i alternans does not contribute significantly to the shape of AP and, therefore, to the gating properties of ionic currents, in particular, to the inactivation of the fast Na⁺ current, I_Na. Recovery from inactivation of I_Na importantly contributes to refractory time after AP propagation, which determines the interval for reentry initiation (46). This is why Ca²⁺ alternans does not significantly affect the vulnerable window for reentry. We do not exclude though that impaired Ca²⁺ handling could contribute to reducing g_gap, since internal Ca²⁺ has been shown to have an effect on this parameter (66).

Thus, according to our modeling studies, in the case of TNF-α-overexpressing tissues, the crucial parameter for initiation reentry is reduced g_gap, since reduced Na⁺ current amplitude is not sufficient to promote reentry.

A mouse model of human heart failure.

Mice with cardiac-specific overexpression of TNF-α were created as an animal model for heart failure (15). TG mice demonstrate properties similar to those in patients with heart failure, such as dilated cardiomyopathy, extracellular matrix remodeling with fibrosis, a reduced β-adrenergic response, reduced I_to, reduced [Ca²⁺]_i transients, downregulation of SERCA expression, and larger susceptibility to arrhythmias. Despite such similarities, there are debates whether mice in general are a suitable species for modeling human heart failure due to a smaller heart size and different mechanisms of AP generation and Ca²⁺ handling (54). Both experimental data (15, 33, 38, 50) and our simulations demonstrate that the TG mouse model is very useful for studying electrophysiological modifications and their effects on AP generation, AP propagation, Ca²⁺ dynamics, and arrhythmias. Both human and TG mouse ventricular myocytes show prolonged APD, and the mechanism of AP prolongation can be simulated by our model. Our model also reproduces smaller [Ca²⁺]_i transients and smaller SR Ca²⁺ load, and the role of reduced SERCA pump activity and upregulation of the Na⁺/Ca²⁺ exchanger in such modifications that are also observed in human heart failure (21) [note, however, the study of Piacentino et al. (51), where in the failing human heart I_NaCa is not upregulated]. The current model was able to elucidate a substrate for reentrant arrhythmias, a reduced AP propagation velocity due to reduced conductance of gap junctions. Reduced expression of connexins and reduced AP propagation velocity are also observed in the human failing hearts (49, 55).

There are also differences in the molecular origin of the pathological modifications in mouse hearts overexpressing TNF-α and failing human hearts. For our modeling purposes, we modified the magnitude of the SR Ca²⁺-ATPase maximum pump rate, the conductances of the I_Kto,f and the I_Kur, the scaling factor for the Na⁺/Ca²⁺ exchanger, and the intercellular coupling strength. Experimental data show that the decreased magnitudes of the K⁺ currents, I_Kto,f and I_Kur, correlate with reduced expression of the proteins Kv4.2, Kv4.3, and Kv1.5, which are the molecular substrates for these currents (50). Experimental data also demonstrate decreased levels of SERCA and phospholamban (PLB) gene expression in TNF-α-overexpressing mouse hearts (24, 32), however, both PLB and SERCA protein levels were not altered in TG mice (24). Similarly, no differences were found between WT and TG mice at the level of protein expression of Cx43 and the Na⁺/Ca²⁺ exchanger (25). However, protein expression level does not always correlate with an increased or decreased protein function. Electrophysiological studies show that, despite a nonaltered level of SERCA and Cx43, TG mice do show decreased Ca²⁺ sequestration rates and slower AP propagation (38). Such changes in protein function could be due to posttranslational modifications, e.g., protein phosphorylation. Experimental data show that conductance of Cx43 could be reduced by phosphorylation by the Ca²⁺-dependent PKC (34, 63), and the Na⁺/Ca²⁺ exchanger function could be enhanced by PKC (70). SERCA pump function is also affected by several phosphorylation mechanisms and an interaction with PLB (25). The failing human heart also demonstrates altered expression of several proteins involved in this study. Most of the experimental data on human heart failure show decreased expression of the SERCA pump and increased expression of the Na⁺/Ca²⁺ exchanger (21, 59), a significantly decreased magnitude of the I_to (6), and a reduced amount of the Cx43 gap junction protein (49, 55).

In conclusion, our mathematical modeling results support the idea that a decrease in SR Ca²⁺ uptake and an increase in the function of the Na⁺/Ca²⁺ exchanger are plausible mechanisms of abnormal [Ca²⁺]_i transients in single ventricular myocytes from mice with cardiac-specific overexpression of TNF-α. The greater susceptibility to Ca²⁺ alternans of TG mouse ventricular tissue could result from a smaller Ca²⁺ release from the SR and slower opening of RyRs as well as AP prolongation as a contributing factor. A marked decrease in the conduction velocity due to decreased g_gap in TNF-α mouse tissue is suggested as a major mechanism of experimentally observed reentrant arrhythmias. Our mathematical model provides a useful tool for predicting the behavior of TNF-α cardiac cells and multicellular cardiac tissues. In future studies, it might be helpful for understanding the influence of different metabolic and pharmacological manipulations that could contribute to the in silico development of treatments for heart failure.

Model limitations.

Despite that the presented model reproduces a significant number of experimental data on mouse ventricular myocytes and mouse cardiac tissues, and provides plausible mechanisms for the stability of AP propagation and Ca²⁺ dynamics, it has several limitations. One of them is that model tissues do not take into account heterogeneity of mice hearts, which could be proarrhythmic. Another limitation is that we modeled 1D cardiac tissues while real tissues are three-dimensional. Nevertheless, most mechanisms described in this study have either a cellular basis or rely on intensive parameters, such as g_gap. Additionally, the cellular model does not include signaling pathways, which limits interpretation of some of the observed experimental phenomena, e.g., identical expression levels of Cx43 in WT and TNF-α-overexpressing hearts and reduced conduction velocity in TG mouse tissue. Further investigations are necessary to extend experimental and modeling efforts for understanding the role of TNF-α in the development of heart failure.

GRANTS

This work was supported by grants from the National Heart, Lung, and Blood Institute (R01HL-66096 to B. London, R01HL-59614 and R01HL-70722 to G. Salama, and R01HL-59526 to R. L. Rasmusson), the National Science Foundation (DBI9873173 to R. L. Rasmusson), and the American Heart Association (10GRNT4720012 to V. E. Bondarenko) and by a Georgia State University Seed Grant from the Brain and Behavior Program and a Research Initiation Grant to V. E. Bondarenko.

DISCLOSURES

No conflicts of interest are declared by the authors.