A compartmentalized mathematical model of the β1-adrenergic signaling system in mouse ventricular myocytes

Abstract

The β1-adrenergic signaling system plays an important role in the functioning of cardiac cells. Experimental data shows that the activation of this system produces inotropy, lusitropy, and chronotropy in the heart, such as increased magnitude and relaxation rates of [Ca(2+)]i transients and contraction force, and increased heart rhythm. However, excessive stimulation of β1-adrenergic receptors leads to heart dysfunction and heart failure. In this paper, a comprehensive, experimentally based mathematical model of the β1-adrenergic signaling system for mouse ventricular myocytes is developed, which includes major subcellular functional compartments (caveolae, extracaveolae, and cytosol). The model describes biochemical reactions that occur during stimulation of β1-adrenoceptors, changes in ionic currents, and modifications of Ca(2+) handling system. Simulations describe the dynamics of major signaling molecules, such as cyclic AMP and protein kinase A, in different subcellular compartments; the effects of inhibition of phosphodiesterases on cAMP production; kinetics and magnitudes of phosphorylation of ion channels, transporters, and Ca(2+) handling proteins; modifications of action potential shape and duration; magnitudes and relaxation rates of [Ca(2+)]i transients; changes in intracellular and transmembrane Ca(2+) fluxes; and [Na(+)]i fluxes and dynamics. The model elucidates complex interactions of ionic currents upon activation of β1-adrenoceptors at different stimulation frequencies, which ultimately lead to a relatively modest increase in action potential duration and significant increase in [Ca(2+)]i transients. In particular, the model includes two subpopulations of the L-type Ca(2+) channels, in caveolae and extracaveolae compartments, and their effects on the action potential and [Ca(2+)]i transients are investigated. The presented model can be used by researchers for the interpretation of experimental data and for the developments of mathematical models for other species or for pathological conditions.

Figure 1. Schematic representation of the β₁-adrenergic signaling system in mouse ventricular myocytes.

Transmembrane currents are the fast Na⁺ current (I_Na), the two components of the L-type Ca²⁺ current in caveolae and extracaveolae compartments (I_CaL,cav and I_CaL,ecav, respectively), the sarcolemmal Ca²⁺ pump (I_p(Ca)), the Na⁺/Ca²⁺ exchanger (I_NaCa), the rapidly recovering transient outward K⁺ current (I_Kto,f), the rapid delayed rectifier K⁺ current (I_Kr), the ultrarapidly activating delayed rectifier K⁺ current (I_Kur), the noninactivating steady-state voltage activated K⁺ current (I_Kss), the time-independent K⁺ current (I_K1), the Na⁺/K⁺ pump (I_NaK, is regulated by phospholemman, PLM), the Ca²⁺ and Na⁺ background currents (I_Cab and I_Nab). The Ca²⁺ fluxes are uptake of Ca²⁺ from the cytosol to the network sarcoplasmic reticulum (NSR) (J_up) by the SERCA pump and Ca²⁺ release from the junctional sarcoplasmic reticulum (JSR) (J_rel) through the ryanodine receptors (RyRs). There are three intracellular compartments in the β₁-adrenergic signaling system: caveolae, extracaveolae, and cytosol. The subspace volume (V_ss) is located in the extracaveolae domain. Components of the β₁-adrenergic signaling system are the β₁-adrenergic receptors (β1-AR), the α-subunit of stimulatory G-protein (G_sα), the βγ-subunit of stimulatory G-protein (G_βγ), the adenylyl cyclases of type 5/6 or 4/7 (AC5/6 or AC4/7, respectively), the phosphodiesterases of type 2, 3, or 4 (PDE2, PDE3, or PDE4, respectively), the cyclic AMP (cAMP), regulatory (R) and catalytic (C) subunits of protein kinase A holoenzyme, the protein kinase A inhibitor (PKI), the G-protein-coupled receptor kinase of type 2 (GRK2), the protein phosphatases of type 1 or 2A (PP1 or PP2A, respectively), the inhibitor-1 (I-1). The cytosolic proteins which are the substrates of the β₁-adrenergic signaling system are the phospholamban (PLB) and troponin I (TnI). Stimulatory links are shown by black arrows and inhibitory links are shown by red dashed lines with balls. [Ca²⁺]_i, [Na⁺]_i, and [K⁺]_i are the intracellular Ca²⁺, Na⁺, and K⁺ concentrations in the caveolae, extracaveolae, and cytosol; [Ca²⁺]_o, [Na⁺]_o, and [K⁺]_o are the extracellular Ca²⁺, Na⁺, and K⁺ concentrations.

Figure 2. β₁-adrenoceptors phosphorylation.

β₁-adrenoceptors phosphorylation above basal level after 15-minute application of 10 µM isoproterenol or 10 µM isoproterenol+PKA inhibitor H-89. Experimental data from Freedman et al. are shown with black bars with errors, simulation data are shown with gray bars. Effect of H-89 was simulated by setting [PKA]_tot = 0 µM.

Figure 3. Normalized activity of adenylyl cyclases as functions of G_sα and G_sβγ.

Panel A: Experimental normalized activity of AC5 (filled circles) and AC6 (unfilled circles) as functions of G_sα . Simulated data for normalized activity of AC5/6 is shown by a solid line. Panel B: Experimental normalized activity of AC4 (filled circles from and unfilled circles from [38]) as functions of G_sα. Simulated data for normalized activity of AC4/7 is shown by a solid line. Panel C: Experimental normalized activity of AC5 (filled circles) as functions of G_sβγ. Simulated data for normalized activity of AC5/6 is shown by a solid line. Panel D: Experimental normalized activity of AC4 (filled circles) as functions of G_sβγ. Simulated data for normalized activity of AC4/7 is shown by a solid line.

Figure 4. Adenylyl cyclase activity as a function of isoproterenol.

Panel A: Experimental data on AC activity (in pmol/mg/min) in mouse hearts and ventricular myocytes obtained after 10-minutes exposure to isoproterenol are shown by unfilled circles and filled circles . The solid line shows corresponding simulated AC activities at different concentrations of isoproterenol. Panel B: Desensitization of β₁-ARs. Increase in adenylyl cyclase activities above basal level (in %) are measured at maximum (from 50^th to 75^th seconds, control, filled circles) and at two time moments (5 min and 30 min, unfilled circles and unfilled squares, respectively) after exposure to different concentrations of isoproterenol . Corresponding simulated data for the maximum, 5-minute, and 30-minute delays are shown by solid, dashed, and dash-dotted lines, respectively.

Figure 5. The effects of β₁-adrenoceptor stimulation on PDE activities.

Panel A: Absolute total PDE activities and the activities of PDE2, PDE3, and PDE4 obtained experimentally from the mouse and rat hearts (in pmol/min/mg protein, bars with errors [44], [49]) and corresponding simulation data (bars without errors). Panel B: Fractional activities of PDE2, PDE3, and PDE4 obtained experimentally from the mouse and rat hearts (in %, bars with errors [44], [49]) and corresponding simulation data (bars without errors). Panel C: Partial activities of PDE2, PDE3, and PDE4 obtained experimentally from the particulate fraction of the rat hearts (in %, bars with errors [48]) and corresponding simulation data (bars without errors). In the experiments and simulations, PDE activities are obtained at fixed cAMP concentrations of 1 µM. Panel D: The effects of PDE inhibitor IBMX on cAMP levels in ventricular myocytes. Experimental data for rat and canine ventricular myocytes are shown by bars with errors, simulation data are shown by bars without errors. In the experiments with rat ventricular myocytes measurements were performed after 3-minute stimulations (5 µM isoproterenol, or 100 µM IBMX, or both) . In the experiments with canine ventricular myocytes measurements were performed after 5-minute stimulations (10 µM isoproterenol, or 10 µM IBMX, or both) . Simulated cAMP level is determined at the 3^rd minute upon application of 5 µM isoproterenol, or 100 µM IBMX, or both.

Figure 6. The effects of β₁-adrenoceptor stimulation on PKA activities.

Panel A: PKA I and PKA II activities as functions of cAMP. Experimental data for PKA I obtained by two methods are shown by filled and unfilled circles ; data for PKA II obtained by Beavo et al. . Corresponding simulated data are shown by a solid (PKA I) and a dashed (PKA II) line. Panel B: PKA activity ratio. Experimental data were obtained without (−cAMP) and with (+cAMP) externally applied 3 µM cAMP, both without and with 1 µM isoproterenol (black bars [57]). We also performed four simulations: no isoproterenol/basic level cAMP (−cAMP), no isoproterenol/3 µM cAMP (+cAMP), 1 µM isoproterenol/no externally applied cAMP (−cAMP), and 1 µM isoproterenol/3 µM cAMP (+cAMP). Then, the corresponding PKA ratios were calculated. Panel C: Protein kinase A inhibition by a heat-stable protein kinase inhibitor PKI. Experimentally, PKA activities were measured with and without PKI at different concentrations of cAMP, then their ratio was calculated (in %) and subtracted from 100% (filled circles, [54]). Corresponding simulation data for PKA I and PKA II are shown by solid and dashed lines. Concentration of [PKI]_tot = 2·0.2·[PKA]_tot.

Figure 7. The effects of β₁-adrenoceptor stimulation on activities of I-1 and PP1.

Panel A: Relative I-1 activity in ventricular myocytes in control (left bars) and upon stimulation with 1 µM isoproterenol (right bars). Experimental data for guinea pig hearts are shown by black bars, simulation data with our model are shown by gray bars. Panel B: Relative PP1 activity in WT and I-1 knockout mouse hearts. Experimental data are shown by black bars, our simulations – by gray bars. Experimental PP1 activity from I-1 knockout mouse hearts is normalized to 100%.

Figure 8. cAMP and PKA dynamics in mouse ventricular myocytes.

Panel A: cAMP dynamics in ventricular myocytes. Experimental data of normalized cAMP in mouse and rabbit ventricular myocytes are shown by unfilled and filled circles, respectively; simulation data is shown by a solid line. Panel B: PKA dynamics in ventricular myocytes. Experimental data of normalized PKA activity in rabbit ventricular myocytes are shown by unfilled circles; simulation data is shown by a solid line. Data in Panels A and B was obtained upon application of 1 µM isoproterenol.

Figure 9. The effects of β₁-adrenoceptor stimulation on the L-type Ca²⁺ current.

Panel A: Markov model of the L-type Ca²⁺ channel. State diagram consists of two similar sub-diagrams for non-phosphorylated (upper sub-diagram) and phosphorylated states (lower sub-diagram). C₁, C₂, C₃, C₄, and C_P are closed states; O is the open state; I₁, I₂, and I₃ are inactivated states; C_1p, C_2p, C_3p, C_4p, and C_Pp are closed phosphorylated states; O_p is the open phosphorylated state; and I_1p, I_2p, and I_3p are phosphorylated inactivated states. The rate constants α, α_p, and β are voltage-dependent; γ is calcium dependent; k_co, k_oc, k_cop, K_pcf and K_pcb, are voltage-insensitive; and k_phos and k_dephos are the phosphorylation and dephosphorylation rates, respectively (see Appendix S1). Panel B: Time course of the peak values of L-type Ca²⁺ current and L-type Ca²⁺ channel phosphorylation level upon stimulation with 1 µM isoproterenol. Experimental data for peak I_CaL is obtained by a series of pulses to 0 mV for 200 ms from a holding potential of −80 mV with a frequency 0.2 Hz . Modeling data are obtained by a series of pulses to 0 mV for 50 ms from a holding potential of −80 mV with a frequency 0.04 Hz. Increase in phosphorylation level is determined as a fractional increase related to the total increase in phosphorylation of L-type Ca²⁺ channels at 150^th s after application of isoproterenol. Panel C: Peak L-type Ca²⁺ current as a function of isoproterenol concentration. Experimental data are obtained by Kim et al. (filled circles), Sako et al. (unfilled circles), and Mitarai et al. (filled squares). Simulation data is obtained by a voltage pulse to 0 mV from a holding potential of −80 mV after a 600-second exposure to different concentrations of isoproterenol. Simulation data for the total cellular I_CaL,tot, the caveolae-localized I_CaL,cav, and the extracaveolae-localized I_CaL,ecav are shown by black, green, and red solid lines, respectively. Panel D: Simulated time course of the L-type Ca²⁺ currents in control (solid line), after 1000-s exposure to PP1/PP2A inhibitor Calyculin A (65% of PP1/PP2A activity inhibition, dashed line), and after 600-s exposure to 1 µM isoproterenol (dotted line). Currents are obtained by a voltage pulses to 0 mV from a holding potential of −80 mV and without Ca²⁺-induced Ca²⁺ release to account for heavy buffer conditions. Insert in Panel D: Same simulations performed with intact Ca²⁺-induced Ca²⁺ release. Panel E: Peak current-voltage (I–V) relationships for I_CaL in control (filled circles) and after exposure to Calyculin A (unfilled circles) and isoproterenol (filled squares). Panel F: Steady-state inactivation relationships for I_CaL in control (filled circles) and after exposure to Calyculin A (unfilled circles) and isoproterenol (filled squares). Panel G: Normalized maximum conductance (G/G_max) for I_CaL as functions of voltage in control (filled circles) and after exposure to Calyculin A (unfilled circles) and isoproterenol (filled squares). In Panels E, F, and G, currents are obtained by the two-pulse protocols: a 500-ms depolarizing first pulse to between −70 and +50 mV (in 10-mV increment) is applied from a holding potential of −80 mV; this is followed by a second 500-ms pulse to +10 mV. Simulations are performed without Ca²⁺-induced Ca²⁺ release to account for heavy buffer conditions.

Figure 10. The effects of β₁-adrenoceptor stimulation on the fast Na⁺ current.

Panel A: Markov model of the fast Na⁺ channel. State diagram consists of two similar sub-diagrams for non-phosphorylated (upper sub-diagram) and phosphorylated-trafficked states (lower sub-diagram). C_Na1, C_Na2, and C_Na3 are closed states; O_Na is the open state; IF_Na, I1_Na, and I2_Na are the fast, intermediate, and slow inactivated states, respectively; IC_Na2 and IC_Na3 are closed-inactivated states; C_Na1p, C_Na2p, and C_Na3p are closed phosphorylated states; O_Nap is the open phosphorylated state; and IF_Nap, I1_Nap, I2_Nap, IC_Na2p, and IC_Na3p are phosphorylated inactivated states. The rate constants for activation, deactivation, inactivation, phosphorylation-trafficking, and dephosphorylation are given in Appendix S1. Panel B: Time course of the activation of the fast Na⁺ current upon application 0.1 µM isoproterenol. Experimental data of Matsuda et al. obtained for the normalized peak I_Na in rabbit ventricular myocytes is shown by closed circles. Data is obtained with 40-ms pulses from a holding potential of −100 mV to −30 mV at stimulation frequency 0.2 Hz. A solid line shows the time course of simulated data on relative I_Na phosphorylation upon application of 0.1 µM isoproterenol. A dashed line shows the time course of the simulated normalized peak I_Na after application of 0.1 µM isoproterenol. The simulated currents are obtained with 20-ms pulses from a holding potential of −140 mV to −30 mV at stimulation frequency 0.04 Hz. Panel C: An increase in peak I_Na availability upon application of different concentrations of isoproterenol (in %). Experimental data by Kirstein et al. obtained from rat ventricular myocytes are shown by black bars with errors; corresponding simulation data are shown by gray bars. Peak current-voltage (Panel D) and steady-state inactivation (Panel E) relationships for the fast Na⁺ current in ventricular myocytes upon stimulation with 0.1 µM isoproterenol. Experimental data for rats in the absence (unfilled circles) and presence (filled circles) of 0.1 µM isoproterenol are obtained by Kirstein et al. (holding potential is −100 mV, conditioning pulse duration is 2,500 ms; isoproterenol data is obtained after 10 min of application). Simulated data are shown by solid (no isoproterenol) and dashed (10 min after application of 0.1 µM isoproterenol) lines (data are obtained by two-pulse protocol, holding potential is −140 mV, first pulse duration is 500 ms for voltages from −140 to +40 mV in 10 mV steps, second pulse duration is 50 ms at voltage −20 mV). Isoproterenol increases I_Na availability, but does not affect gating properties.

Figure 11. The effects of β₁-adrenoceptor stimulation on ryanodine receptors.

Panel A: Markov model of ryanodine receptors. State diagram consists of two similar sub-diagrams for non-phosphorylated (upper sub-diagram) and phosphorylated states (lower sub-diagram). C₁ and C₂ are closed states; O₁ and O₂ are open states; C_1p and C_2p are closed phosphorylated states; O_1p and O_2p are open phosphorylated states. The rate constants from C₁ to O₁, from O₁ to O₂, from C_1p to O_1p, and from O_1p to O_2p are Ca²⁺-dependent; and k_phos and k_deph are the phosphorylation and dephosphorylation rates, respectively (see Appendix S1). Panel B: Time course of the relative phosphorylation level of RyRs upon activation of the β₁-adrenergic signaling system. Experimental data of Xiao et al. (filled circles) are obtained upon stimulation of rat ventricular myocytes with 1 µM isoproterenol; experimental data of Takasago et al. (filled squares) are obtained from canine ventricular myocytes with endogenous PKA by application of [γ-³²P]ATP. Modeling data are obtained by application of 1 µM isoproterenol. Increase in phosphorylation level is determined as a fractional increase related to the maximum increase in phosphorylation of RyRs after activation of the β₁-adrenergic signaling system. Panel C: Relative phosphorylation level of RyRs at different concentrations of isoproterenol. Experimental data of Xiao et al. (filled circles) obtained upon stimulation of rat ventricular myocytes for 15 minutes. Simulation data are obtained after 10-minute exposure to different concentrations of isoproterenol. Panel D: Effects of PKA on opening probability of RyRs as function of cytosolic Ca²⁺ concentration. Experimental data of Xiao et al. are obtained for cardiac RyRs upon application of active (filled circles) and boiled (unfilled circles) PKA at relatively small luminal Ca²⁺ of 45 nM; experimental data of Xiao et al. (filled squares) for RyRs are obtained from mouse ventricular myocytes. Simulation data for mouse ventricular myocytes in the absence and presence of PKA are obtained at intracellular [Ca²⁺]_i concentration ranged from 0.01 to 10 µM and are shown by dashed and solid lines. Experimental data obtained at small luminal Ca²⁺ concentration shows larger sensitivity of RyRs to cytosolic Ca²⁺.

Figure 12. The effects of β₁-adrenoceptor stimulation on the Na⁺-K⁺ pump.

Panel A: Experimental time course of a relative decrease in the intracellular [Na⁺]_i concentration (filled circles) obtained by Despa et al. from mouse ventricular myocytes after application of 1 µM isoproterenol and corresponding simulated time course of an increase in relative phosphorylation level of phospholemman obtained using our model (solid line). Panel B: Experimental data on a relative increase in I_NaK current (filled circles) obtained by Gao et al. from guinea pig ventricular myocytes at different concentrations of isoproterenol. Corresponding simulation data with our model on a relative increase in phosphorylation of phospholemman is shown by a solid line. Panel C: Experimental (black bars with errors [100]) and simulated (gray bars) data on intracellular [Na⁺]_i concentration before (control) and after 10-minutes application of 1 µM isoproterenol (Iso).

Figure 13. The effects of β₁-adrenoceptor stimulation on I_Kur and I_Kto,f.

Panel A: Experimental time course of the relative increase in the ultra-rapidly activating delayed-rectifier K⁺ current I_Kur obtained by Li et al. from human atrial myocytes after application of 1 µM isoproterenol (line with filled circles) and corresponding simulated time course of an increase in I_Kur obtained with our model (solid line). Simulation data is obtained with 200-ms pulses from a holding potential of −100 mV to +40 mV at stimulation frequency 0.02 Hz. Panel B: Experimental data on a relative increase in I_Kur current (filled circles) obtained by Yue et al. from canine atrial myocytes at different concentrations of isoproterenol. Corresponding simulation data with our model on relative increase in I_Kur are shown by a solid line. Simulation data is obtained with 4.5-s pulses from a holding potential of −90 mV to 0 mV after 800-s exposure to different concentrations of isoproterenol. Panel C: Experimental (human atrial myocytes , solid lines with circles) and simulated (solid and dashed lines) data on current-voltage relationships for I_Kur current. Experimental data for control conditions and those obtained after application of 1 µM isoproterenol are shown by unfilled and filled circles, respectively. Simulated data for control conditions are shown by a solid line, and 1 µM isoproterenol simulations are shown by a dashed line. Simulated currents are obtained by 4.5-s depolarizing pulses to between −80 and +50 mV (in 10-mV increment) from a holding potential of −90 mV. Panel D: Experimental time course of the relative decrease in the rapidly recovering transient outward K⁺ current I_Kto,f obtained by Zhang et al. from mouse Schwann cells after application of 1 µM forskolin (filled circles) or 10 µM db-cAMP (unfilled circles), and corresponding simulated time course of a relative decrease in I_Kto,f phosphorylation obtained with our model after stimulation with 10 µM isoproterenol (solid line). Panel E: Simulated time course of I_Kto,f traces obtained by depolarization pulses to −5 mV from a holding potential of −100 mV for control (solid line) and after stimulation with 10 µM isoproterenol (dashed line). Panel F: Simulated data for G/G_max (black lines) and steady-state inactivation relationships (gray lines) obtained for I_Kto,f with two-pulse protocol (P1 stimuli from −100 to +50 mV in 10 mV intervals for 500 ms, following P2 pulse to +50 mV for 500 ms; holding potential is −100 mV) in control (solid lines) and after application of 10 µM isoproterenol (dashed lines).

Figure 14. The effects of β₁-adrenoceptor stimulation on phospholamban and troponin I.

Panel A: Experimental time courses of the phospholamban phosphorylation (in %) obtained from rat hearts , and mouse ventricular myocytes . Data Karczewski et al. are obtained with application of 0.5 µM PKA catalytic subunit; data Kuschel et al. and Li et al. are obtained with application of 1 µM isoproterenol. Simulated time course of PLB phosphorylation is shown by a solid line. Panel B: PLB phosphorylation (in %) as functions of isoproterenol concentration. Experimental data from rat hearts , were obtained after 2-minute exposure to isoproterenol; data from guinea pig hearts were obtained after 1-minute exposure to isoproterenol; and data from rat ventricular myocytes were obtained after 3-minute exposure to isoproterenol. Simulated data after 2-minute exposure to different concentrations of isoproterenol is shown by a solid line. Panel C: Experimental time courses of the troponin I phosphorylation (in %) obtained from mouse ventricular myocytes after application of 1 µM isoproterenol. Simulated time course of troponin I phosphorylation is shown by a solid line. Panel D: Troponin I phosphorylation (in %) as functions of isoproterenol concentration. Experimental data from rat ventricular myocytes were obtained after 3-minute exposure to isoproterenol. Simulated data after 2-minute exposure to different concentrations of isoproterenol is shown by a solid line.

Figure 15. The effects of β₁-adrenoceptor stimulation on the dynamics of cAMP and PKA.

Panel A: Simulated time courses of cAMP concentrations in caveolae (thin solid line), extracaveolae (dashed line), and cytosolic compartments (dotted line), and in the whole cell volume (bold solid line) after application of 1 µM isoproterenol. Panel B: Simulated time courses of PKA catalytic subunit concentrations in caveolae (thin solid line), extracaveolae (dashed line), and cytosolic compartments (dotted line), and in the whole cell volume (bold solid line) after application of 1 µM isoproterenol. Panel C: Simulated time courses of cAMP production rate by adenylyl cyclases (solid line) and cAMP degradation rate by phosphodiesterases (dashed line) after application of 1 µM isoproterenol. Panel D: Simulated time course of cAMP fluxes between caveolae and extracaveolae (solid line), caveolae and cytosol (dashed line), and extracaveolae and cytosol (dotted line) after application of 1 µM isoproterenol. Fluxes are normalized to the cell volume V^cell.

Figure 16. The effects of PDE3 and PDE4 inhibition on cAMP dynamics upon activation of β₁-adrenergic receptors.

Panel A: Simulated time courses of cellular cAMP concentrations for control conditions (solid line), upon inhibition of PDE3 (dashed line), and upon inhibition of PDE4 (dotted line) after sustained application of 0.1 µM isoproterenol at time moment t = 0 s. Activities of PDE3 or PDE4 are inhibited by 90% to simulate the effects of corresponding selective inhibitors, cilostamide or milrione for PDE3, or rolipam or Ro 20–1724 for PDE4. Panel B: Simulated time courses of cellular cAMP concentrations for control conditions (solid line), upon inhibition of PDE3 (dashed line), and upon inhibition of PDE4 (dotted line) after pulsed application of 0.1 µM isoproterenol at time moment t = 200 s for 30 s (thick solid line). The degrees of inhibition of PDE3 and PDE4 are the same as in Panel A.

Figure 17. Simulated action potentials and underlying ionic currents of the isolated ventricular cell model.

Simulated action potentials and underlying ionic currents are shown for control conditions and after application of 1 µM isoproterenol at two pacing frequencies 1 Hz (Panels A–E) and 5 Hz (Panels F–J) (I_stim = 80 pA/pF, τ_stim = 1.0 ms). Panels A and F: Simulated action potentials for control conditions (solid line) and after isoproterenol application (dashed line). Panels B, D, G, and I: Currents underlying the AP for control condition. Panel C, E, H, and J: Currents underlying the AP after isoproterenol application. The scales for the relatively large fast Na⁺ current, I_Na, are given on the right axis in Panels B, C, G and H. APs and ionic currents are shown after 300 s stimulation. In Panels A, C, E, F, H, and J 1 µM isoproterenol is applied at time t = 0 s.

Figure 18. Simulated and experimental [Ca²⁺]_i transients and their characteristics in an isolated ventricular cell for control conditions and after application of isoproterenol.

Panel A: Simulated [Ca²⁺]_i transients for control conditions (solid line) and after 1 µM isoproterenol application (dashed line) obtained with 1 Hz-stimulation. Panel B: Simulated [Ca²⁺]_i transients for control conditions (solid line) and after 1 µM isoproterenol application (dashed line) obtained with 5 Hz-stimulation. Panel C: Simulated diastolic (filled symbols) and systolic (unfilled symbols) [Ca²⁺]_i magnitudes for control conditions (circles) and after 1 µM isoproterenol application (triangles) as functions of stimulation frequency. Panel D: Decay time constants for intracellular [Ca²⁺]_i transients for control conditions and after 1 µM isoproterenol application as functions of stimulation frequency. Simulated data are shown by solid (control) and dashed (1 µM isoproterenol) lines. The lines with symbols show experimental data from Benkunsky et al. ; experimental data from Knollmann et al. are shown by squares. Panel E: Increase in [Ca²⁺]_i transient amplitudes after application of isoproterenol (folds). Model data and data of Wang et al. and Liu et al. are obtained at 1 Hz and 1 µM isoproterenol; data of Despa et al. are obtained at 2 Hz and 1 µM isoproterenol; data of Knollmann et al. are obtained at 1 Hz and 0.5 µM isoproterenol. All simulation data in this figure are obtained after 300-s stimulations and exposure to 1 µM isoproterenol.

Figure 19. Ca²⁺ fluxes and the SR Ca²⁺ load.

Simulated major integral Ca²⁺ fluxes during one cardiac cycle in the isolated ventricular cell model for control conditions (Panels A and C) and after application of 1 µM isoproterenol (Panels B and D). Pacing frequencies are 1 Hz (Panels A and B) and 5 Hz (Panels C and D). Major integral Ca²⁺ fluxes are shown after 300 s of stimulation. In Panels B and D 1 µM isoproterenol is applied at time t = 0 s. Here, J_CaL is the Ca²⁺ entering the cell 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⁺ outflux from the cytosol through the Na⁺/Ca²⁺ exchanger; and J_pCa is the Ca²⁺ outflux through the sarcolemmal Ca²⁺ pump. Panel E: Increase in the SR Ca²⁺ load after 300-s application of isoproterenol. Model data are shown for stimulation frequencies 1, 2, 3, 4, and 5 Hz; experimental data of Liu et al. and Fernandez-Velasco et al. are obtained at 1 µM isoproterenol; data of Knollmann et al. are obtained at 0.5 µM isoproterenol.

Figure 20. Dynamics of [Ca²⁺]_i transients and [Na⁺]_i concentrations upon stimulation of β₁-adrenoceptors.

Panel A: Simulated peak values of [Ca²⁺]_i transient as function of time for control conditions (solid line) and after 1 µM isoproterenol application at t = 0 s (dashed line). Panel B: Simulated intracellular [Na⁺]_i concentrations as function of time for control conditions (solid line) and after 1 µM isoproterenol application at t = 0 s (dashed line). In both panels, stimulation frequency is 1 Hz.

Figure 21. The effects of isoproterenol on negative staircase of [Ca²⁺]_i transients.

Panel A: Simulated [Ca²⁺]_i transients as function of time for control conditions show clear negative staircase effect. Panel B: Simulated [Ca²⁺]_i transients as function of time after 1 µM isoproterenol application at t = 0 s does not show negative staircase effect. In both panels, stimulation frequency is 1 Hz.

Figure 22. The effects of isoproterenol on integral Na⁺ fluxes.

Simulated major integral Na⁺ fluxes during one cardiac cycle in the isolated ventricular cell model are shown for control conditions (Panels A and C) and after application of 1 µM isoproterenol (Panels B and D). Pacing frequencies are 1 Hz (Panels A and B) and 5 Hz (Panels C and D). Major integral Na⁺ fluxes are shown after 300 s of stimulation. In Panels B and D 1 µM isoproterenol is applied at time t = 0 s. Here, J_Nav is the Na⁺ entering the caveolae compartment through the fast Na⁺ channels; J_Nab is the background Na⁺ influx; J_NaCa is the Na⁺ influx to the cytosol through the Na⁺/Ca²⁺ exchanger; and J_NaK is the Na⁺ outflux through the Na⁺-K⁺ pump.

Figure 23. The effects of two subpopulations of the L-type Ca²⁺ current block on the action potentials and [Ca²⁺]_i transients.

Simulated effects of two subpopulations of the L-type Ca²⁺ current block on the action potentials and [Ca²⁺]_i transients in the isolated ventricular cell model are shown for control conditions and after application of 1 µM isoproterenol at pacing frequency 1 Hz (I_stim = 80 pA/pF, τ_stim = 1.0 ms). Panels A–C: Control conditions; Panels D–F: Isoproterenol is applied. Panels A and D: Simulated action potentials for control conditions (solid black line), after I_CaL,ecav block (green line), and after I_CaL,cav block (red line). Panels B and E: Simulated I_CaL currents. Panel C and F: Simulated [Ca²⁺]_i transients. APs and ionic currents are shown after 300 s stimulation. 1 µM isoproterenol is applied at time t = 0 s.