Synergistic Dual Automaticity in Sinoatrial Node Cell and Tissue Models

Abstract

Background

The mechanism of sinoatrial node (SAN) automaticity is traditionally attributed to membrane ion currents. Recent evidence indicates spontaneous sarcoplasmic reticulum (SR) Ca²⁺ cycling also plays an important role.

Methods and Results

We performed computer simulation on SAN cell and 1D tissue model. In the SAN cells, SR Ca²⁺ cycling broadly modulated sinus rate from 1.74Hz to 3.87Hz. Shortening of the junctional SR refilling time and increase of SR Ca²⁺ release were responsible for sinus rate acceleration. However, under the fast SR Ca²⁺ cycling, decreased L-type Ca²⁺ current (I_CaL) resulted in irregular firing. When Ca²⁺ cycling was suppressed, I_f and I_CaT both acted to stabilize the pacemaker rhythm, but I_CaT had less effect than I_f. At the 1D level, the electrical coupling between neighboring cells had little effect on the earliest pacemaker location. The leading pacemaking site always colocalized with the site with the highest SR Ca²⁺ cycling rate, but shifted to the site with less inhibited I_CaL.

Conclusions

The rate of SR Ca²⁺ cycling can effectively and broadly modulate the sinus rate. I_f, I_CaL and I_CaT play integral roles to guarantee SAN cell rhythmic firing. The leading pacemaker site is determined by intracellular Ca²⁺ dynamics and membrane currents, indicating the synergistic dual automaticity not only exists in single SAN cells, but also at the tissue level.

Introduction

The sinoatrial node cells (SANC) are the primary pacemakers of the heart. They spontaneously exhibit slow diastolic depolarization (DD) to a threshold for firing, thereby periodically initiating action potentials to set the rhythm of the heart^{1, 2}. Spontaneous DD has traditionally been attributed to a “membrane clock (M clock)” mechanism ^{3, 4}, in which the membrane potential changes with the activation and inactivation of the membrane ion channel currents. Recently, the “Ca²⁺ clock” has been put forward as a complementary mechanism for SAN automaticity^{5, 6}. Results from single SANC studies suggested that isolated sarcoplasmic reticulum (SR) could operate as a self-sustained Ca²⁺ oscillator and a small spontaneous Ca²⁺ release from the junctional SR to the intracellular subspace occurred first as the initiating event, which in turn activated Na⁺/Ca²⁺ exchanger current (I_NCX) and caused the accelerated DD. With these experimental results, a mathematical model of SANC was recently constructed to include the intracellular Ca²⁺ cycling together with the more traditional ionic channel currents ⁷. On the other hand, at the tissue level, the functional organization of SANC is still not very clear today. This makes evaluation of the interaction between the two clocks especially challenging.

The synergistic action of the M and Ca²⁺ clocks has been demonstrated by the Maltsev-Lakatta model⁷, but previous works did not extensively explore mechanisms of heart rate modulation. Furthermore, it is unclear whether the two clock synergy can be applied to multicelluar SAN structures to determine the dominant pacemaking site. In this paper, by introducing a variety of simulated interventions, such as β-adrenergic stimulation, I_CaL and I_CaT alteration, on a single cell and a constructed tissue model, we evaluated the synergistic effects of two clocks on the rhythmic firing and leading pacemaker site.

Method

The single cell model and its solution

We selected the Maltsev-Lakatta SAN rabbit model ⁷ for the current study because it is the latest and most advanced numerical formulations to address dynamic activities of the SR Ca²⁺ pump and ion buffering in all cell compartments, including Ca²⁺ buffering by calsequestrin in junctional SR (jSR). The model included twenty nine first-order differential equations with the following format to describe action potential of a single SANC:

$$\frac{dV}{dt}=-\frac{1}{C_m}(I_{\mathit{ion}})$$ (1)

where V was the transmembrane potential, C_m was the membrane capacitance, t was the time, I_ion was the total ionic current.

In general I_ion was a function of the voltage V and gating variables Y₁,….Y_i,….Y_M that satisfied the following ordinary differential equations:

$$\begin{array}{l}\frac{{dY}_i}{dt}=\frac{I_{i\_inf}-Y_i}{{\tau }_{Y_i}}i=1,\dots ,\text{M},\\ Y_{i\_inf}=\frac{{\alpha }_i}{{\alpha }_i+{\beta }_i}\\ {\tau }_{Y_i}=\frac{1}{{\alpha }_i+{\beta }_i}\end{array}$$ (2)

where Y_{i_inf} was the steady-state value of the gating variable Y_i. τ_Yi was its time constant. α_i, β_i were opening and closing rates of the channel gating.

Ca²⁺ release from the SR via ryanodine receptors (RyRs)⁸, its intracellular fluxes, buffering, and concentrations in cell compartments satisfied another set of ordinary differential equations.

$$\frac{{dZ}_i}{dt}=f(j,r)$$ (3)

where Z_i, j, r were a variety of introduced variables, such as Ca²⁺ concentrations in different cell compartment or flux coefficients during dynamic processes.

The fourth-order Runge-Kutta method was applied to integrate the single cell model with a time step of 0.005ms. Decreasing the time step to 0.001ms was not found to change the simulation results.

One-dimensional SAN tissue and its numerical integration

Cardiac tissue could be treated as a continuous and excitable medium connected by gap junctions between adjacent cells. If only the intracellular space was taken into account, the tissue was considered as a monodomain system. Then Eq.4 could be used to describe the tissue, together with the no-flux boundary conditions imposed at both ends of the fiber as described by Eq.5.

$$C_m\frac{\partial V}{\partial t}=-(I_{\mathit{ion}})+\frac{1}{{\rho }_x{S}_v}\frac{{\partial }^{2}V}{\partial x^2}$$ (4)

$$(\nabla V){\mid }_{x=0}=(\nabla V){\mid }_{x=l}=0,$$ (5)

where x was the spatial coordinates in the string of tissue, ρ_x was the bulk cytoplasmic resistivity, S_v was the surface-to-volume ratio, and l was the length of the whole fiber.

SAN central area is normally the dominant pacemaker location. To explore the role of SR Ca²⁺ dynamics in determining the leading pacemaker site, we performed simulation around the central SAN region. The fiber consisted of 30 cells with strand length of 0.3cm that was approximately half the length of SA node in the rabbit ⁹, and parallel to the direction of the superior to the inferior vena cava which was consistent with extension direction of the central SANC ¹⁰. The fiber was divided into three parts: cells from the first to the 10^th were designated the inferior part, the 11^th to 20^th cells middle part, and the left 10 cells superior part.

So far, two models (mosaic and gradient model) of the organization of SAN tissue have been put forward¹¹. According to the mosaic model, all SANC had reasonably uniform properties, it was the peppered atrial myocytes that made them show the different morphology. Recently, a mix of atrial and SAN cells in the SAN tissue periphery was reported, but no atrial cells were found in the SAN central area ¹⁰, indicating that our concerned region could be considered homogeneous in the controlled condition. In the gradient model¹², it was suggested that SANCs showed transition of electrophysiological properties from the center to the periphery. An empirical exponential formula was given to describe changes of parameters such as cellular size and current conductances. According to this formula, these parameters almost kept the same within half length of the fiber, and only started to show heterogeneity close to the defined periphery. This implied that the central region was also considered uniform in this organization. Therefore, based on these two models, our concerned region could be set homogeneous to the controlled condition.

It was reported that the leading pacemaker site changed dynamically within the central area because of interventions affecting different regions of the node differentially ^{13, 14, 15}. For β-adrenergic stimulation, there has been little report about heterogeneous sensitivities among primary pacemakers. Our main purpose here is to investigate relationship between the leading pacemaker site and SR Ca²⁺ activity, therefore the exact form of its distribution has less significance on the results. In this case, we hypothesized three kinds of heterogeneity: one was a linear change of SERCA pump rate( P_up) from the superior to the inferior parts along the fiber. The second adopted an exponential change from the putative equation 80 ¹² shown in formula (6). The third was a discontinuous state: the middle cells had higher P_up than the superior and inferior cells. On the basis of such a discontinuity, P_up in different parts of the SAN received different percentage of increase to investigate whether the leading pacemaker site shifted with P_up.

$$F_{\mathit{cell}}=1.07\times 29.0\times \frac{1.0\times (30-n)}{\mathit{cell}\_\mathit{number}}/(30.0\times (1.0+0.7745\times exp(-(29.0\times \frac{1.0\times (30-n)}{\mathit{cell}\_\mathit{number}}-24.5)/1.95)))$$ (6)

where n stands for the numbered cell, cell_number is the total amount of cells in the tissue ( here cell_number = 30), F_cell is a scaling factor of P_up, which is directly based on the location of a cell relative to the firstSAN.

Assuming that all cells were cylindrical and adhered to the dimensions suggested in the Maltsev-Lakatta’s single cell model, a radius of 4.0 μm and length of 70μm were used for all cells in the fiber, resulting in a surface-to-volume ratio S_v = 5300cm⁻¹. According to the experimental data obtained in pairs of rabbit SANC using the double whole-cell patch clamp technique ¹⁶, the mean conductivity value of the intercellular coupling was 7.5nS. Therefore, based on the Eq.86 of ¹² in which at each grid point the conductivity tensor was determined from both the intercellular conductivity and the dimensions of the cell, we used ρ_x = 9.57 × 10⁻⁶ GΩ ··cm at each grid point in our tissue simulation. The model was considered a monodomain and continuum system, thus the geometry of the cells and gap junctions would not significantly affect the results of computation.

The three-point centered difference method with spatial step of Δ x = h = 0.01cm was used in the calculation. Source code was developed by us using C++ programming language and all simulations were performed on a personal computer.

Results

Intracellular SR Ca²⁺cycling

Two main parameters, a SERCA pump rate P_up and a Ca²⁺ release rate K_s, were included in the model to account for the operation of the cardiac SR. In Ref.7, by removing the membrane clock (all ion currents = 0), the investigators analyzed characteristics of the isolated SR Ca²⁺ in simultaneous P_up and K_s changes. We simulated their combined effects on sinus rate regulation with the intact two clocks (Fig. 1). Panel A showed that spontaneous action potential could be generated within a broad frequency ranging from 1.74Hz to 3.87Hz. Increasing P_up accelerated sinus rate whereas decreasing K_s had the opposite effect. Under slow sinus rate the diastolic depolarization was markedly prolonged with slight hump-like elevation during DD comparing (b) with (a)in Panel B. When either P_up or K_s was greatly reduced from their baseline values (P_up =12mM/s, Ks =250/μs), irregular firing and intermittent sinus pause (Fig. 1Bc and Fig. 1Bd) occurred in the grey zone.

Figure 1.

Effects of SR Ca²⁺ cycling on sinus rate. Panel A: P_up (Ca²⁺ pump rate) changed from 0.0mM/s to 24.0mM/s with the step size of 1.0mM/s. K_s (SR Ca²⁺ release rate) changed from 0.0/μs to 500/μs with the step size of 10/μs. Grey area represented the irregular firing while red area corresponded to the rhythmic firing. Panel B: some specific action potentials positioned in Panel A. a* was the baseline point (P_up =12mM/s, Ks =250/μs). a and b were two points which showed regular firings but with different sinus rates. c and d displayed two sites showing irregular firings.

Fig. 2 displayed internal processes of sinus rate regulation by P_up and K_s. According to Ref.7, Ca²⁺ release from the jSR was divided into three stages (Fig. 2G and green arrows in the inset on the right): the first spontaneous primary release (during mid-DD), a secondary larger release (during late DD indicated by the grey bars) and the third AP-induced release (Ca²⁺-induced Ca²⁺ release or CICR¹⁷ due to activation of L-type Ca²⁺ current I_CaL). Therefore, compared with baseline (all solid curves), with elevation of P_up to 20 mM/s (dashed curves), Ca²⁺ concentration in network SR (Fig. 2E) and junctional SR (Fig. 2I) both increased, leading to a faster rate of jSR refilling. Therefore, within a shorter time period that the higher jSR Ca²⁺ loads was attained, a secondary larger Ca²⁺ release during the DD period (red arrow in the inset of Fig. 2G) drove a strong and fast increase of I_NCX (red arrow in Fig. 2B) to bring the membrane potential to threshold and produced an action potential with a shorter cycle time than the baseline.

Figure 2.

Simulation traces of the primary dynamic interactions of the M clock and Ca²⁺ clock during spontaneous production of action potential. Action potentials, major ionic currents and Ca²⁺ behaviors in different compartments were shown. Solid curves denoted the baseline (P_up =12mM/s, K_s =250/μs). Dashed curves were under the high SERCA pump rate (P_up =20mM/s, K_s =250/μs) while dotted curves corresponded to the high SERCA pump but low Ca²⁺ release rate (P_up =20mM/s, K_s =12.5/μs). Grey bars indicated the period of late DD under the baseline conditions. The jSRCarel curves inside the red circle were enlarged at the right side.

When K_s was significantly reduced under the high P_up (all dotted curves), although accumulation of Ca²⁺ in jSR (Fig. 2I) and nSR (network SR, Fig. 2E) increased remarkably, Ca²⁺ release from the jSR (blue arrow in Fig. 2G) during the late DD decreased, resulting in smaller diastolic I_NCX than that during high P_up (about 35% decrease at the time indicated with the blue dashed lines in Fig. 2B), thereby increasing the time to bring the membrane potential to threshold and reducing the sinus rate. Of note, when I_CaL was activated, because of the high Ca²⁺ concentration in the jSR during CICR process, the third release (AP-induced) was strong in this situation, resulting in a higher peak of subspace Ca²⁺ (dotted curve in Fig. 2F) and I_NCX (dotted curve in Fig. 2B) than with the normal Ks.

L-type Ca²⁺current

I_CaL is one of the most important interactions between Ca²⁺ and M clocks because SR is reset and refueled by CICR process due to I_CaL activation. Fig. 3A showed effects of I_CaL on the electrical firing. To facilitate comparison, the baseline was plotted first with instant parameter changes (black bars) simulating various conditions explained with labels above the bars. Fig. 3Aa column corresponded to P_up 67% increase and gCaL 50% increase. As noted simultaneous increase P_up and gCaL resulted in fast sinus rate compared with baseline (15% rate increase). Large P_up led to the large I_NCX (arrow 1) and large Ca²⁺ release (arrow 2) during late DD. Increased gCaL further strengthened the CICR process, resulting in the large peak of Ca²⁺ release and I_NCX during fast firing of AP. However, if gCaL was reduced by 34% (Fig. 3Ab column), a notch (arrow 3) appeared in the phase 0 of the action potential with prolonged duration, implying delayed activation of I_CaL. Although Ca²⁺ release and I_NCX during late DD was still larger than that of the baseline due to the larger P_up (arrows 4 and 5), reduced gCaL made their peak value become small because of the suppressed CICR process. Further reduction of gCaL by 35% (Fig. 3Ac column) resulted in membrane oscillation (arrow 6) and spontaneous Ca²⁺ release (arrow 7) with sinus rhythm halt characterized with delayed afterdepolarization (DAD).

Figure 3.

Effects of I_CaL on rhythmic AP firing. Panel A: Action potentials, major ionic and Ca²⁺ changes during large P_up (67% higher than baseline) and large gCaL (50% increase) (a), or large P_up and small gCaL (34% decrease in (b) and 35% decrease in (c)). Panel B: exploration of AP firing with variations of P_up and gCaL at the same time. The steps for P_up and gCaL were 1.0mM/s and 0.001nS/pF, respectively.

Fig. 3B explored effects of decreased I_CaL on AP firing under different values of P_up. If we define AP of Fig. 3Aa as rhythmic firing, Fig. 3Ab as firing with DAD, and Fig. 3Ac as no firing, Fig. 3B could be divided into three parts. Small gCaL led to failed firing even under the high P_up condition. Rhythmic firing existed only when the SR Ca²⁺ activity and I_CaL dynamically and properly interacted with each other. With decreasing P_up, large I_CaL was needed to sustain the rhythmic firing.

Hyperpolarization-activated “funny” current

I_f has long been considered as the most important ion channel involved in the rate regulation of cardiac pacemaker cells ³. Fig. 4 showed change of sinus rate with gI_f (panel A), and a parametric analysis of sinus rate versus gI_f and P_up (panel B and C). Of note, complete blockade of gI_f made sinus rate fall to 2.86Hz, but without appearance of pause or abnormal firing. The rate modulation range was less than 0.3Hz when gI_f changed from 0.0nS/pF to two times of its baseline (0.30nS/pF). Even if gI_f was elevated to 0.6nS/pF the change of rate was less than 0.5Hz. But in panel C, blocking I_f and down-regulating P_up caused intermittent sinus pause. At the same low P_up, the persistent irregular firing (a) changed to transient irregular firing (b) and finally to rhythmic firing (c) by increasing gI_f. Also in B, increasing gI_f reduced the area of irregular firing at low P_up.

Figure 4.

Role of I_f in the rhythmic firing of the whole system. A: sinus rate modulation by I_f. B: analysis of P_up and gI_f. The step size was 0.01nS/pF for gI_f and 0.2mM/s for P_up. Action potentials of the three specific points within the panel A were depicted in panel C.

T-type Ca²⁺current

Since T-type Ca²⁺ current (I_CaT ) has a low activation threshold (~-60 mV), its activation was suggested as an additional mechanism contributing to the mid-late DD. Fig. 5 showed the role of I_CaT in sinus rate and rhythmic firing. As noted in the panel A, with decreasing of g_CaT, sinus rate gradually reduced. When g_CaT changed from 0.0nS/pF to two times of its baseline (0.3664nS/pF), the firing rate was smoothly regulated within the range of 2.67Hz to 3.33Hz. Four times of g_CaT resulted in a rate of 3.98Hz which was larger than the effect of I_f on sinus rate. Additionally, if the SR Ca²⁺ uptake rate was greatly inhibited as in panel B, a transient firing pause occurred after the suppression of P_up. But with I_CaT increased (from (a) to (c)), the time of sinus pause shortened and the firing recovered to the normal in the end.

Figure 5.

Role of I_CaT in sinus rate and rhythmic firing. Panel A: (a) relationship between g_CaT and sinus rate. (b) Action potentials at g_CaT 0.1832nS/pF (solid lines), 0.09nS/pF (dashed lines) and 0.00 nS/pF (dotted lines), respectively. Panel B: Role of I_CaT at the low Ca²⁺ pump rate.

Leading pacemaker site

We performed simulation on the developed fiber. In the first situation, when P_up increased linearly from 10 mM/s (superior) to 15 mM/s (inferior) along the strand, the leading pacemaker site located at the inferior region (Fig. 6Aa). In the second situation in which P_up increased exponentially, the inferior part was also the leading pacemaker site (Fig. 6Ab). In the third discontinuous situation, when P_up of the middle cells was 13mM/s in contrast to 12mM/s in other parts, the cells in the middle region exhibited the earliest electrical excitation and propagated towards the superior and inferior regions (Fig. 6Ba). Therefore, the above three cases consistently showed colocolization of the leading pacemaker site and the highest SERCA pump activity. To further test this relationship, on the bases of the conditions in Fig. 6Ba, we increased P_up of the superior cells by 30% and 15% for the middle and inferior cells. The results demonstrated that the leading pacemaker site shifted from the 15^th cell (red arrows) to the 30^th cell in the superior side (blue arrows in Fig. 6Bb) where the SR Ca²⁺cycling was most active.

Figure 6.

Role of Ca²⁺clock in determining the leading pacemaker site in a sin us node fiber. Panel A: electrical propagation when Ca²⁺ uptake rate (P_up) was smoothly changed in a linear (a) or exponential (b) way along the strand from 10mM/s (superior) to 15mM/s (inferior). Along the strand the upper 10 cells was defined as the superior site, the lower 10 cells was defined as the inferior site as denoted. Panel B: leading pacemaker site shifted from the middle (red arrow in a,) to the superior (blue arrow in b,) when P_up of the superior cells increased 30% while other cells only had 15% increase. (c) showed relationship of the intercellular conductance with the location of leading pacemaker site and conduction velocity. Similar to the (a) and (b), the earliest excitation always shifted from the 15^th cell in the middle (solid red circles) to the 30^th cell in the superior site (solid blue circles) if P_up of the superior cells became the maximum no matter what value the coupling conductance was. But electrical conduction velocity from the superior to the inferior was greatly increased (solid blue triangles) if the coupling conductance elevated from its baseline (7.5nS) to five times (37.5nS).

Additionally, relationship between the pacemaker site location, conduction velocity and the intercellular coupling conductance was assessed and summarized in Fig. 6Bc, where location of the earliest excitation was designated with the cell number. Leading pacemaker site and velocity both were plotted as a function of the coupling conductance between two neighboring cells, but with regards to the different y-axis. As noted, similar to the Fig. 6Ba and Fig. 6Bb, the leading pacemaker site always shifted from the 15^th cell in the middle (solid red circles) to the 30^th cell in the superior site (solid blue circles) if P_up of the superior cells became the maximum no matter what value the coupling conductance was. However, electrical conduction velocity from the superior to the inferior was greatly increased if the coupling conductance elevated from its baseline (7.5nS) to five times (37.5nS) (solid blue triangles).

Besides SR Ca²⁺ cycling, we found that I_CaL could also in some extent determine location of the earliest excitation. Based on the conditions in Fig. 6Bb, we suddenly decreased gCaL of the superior cells after the time notified with respective labels in Fig. 7. The action potential of the superior cell was shown in the upper panels while the lower panels displayed electrical propagations along the fiber. The leading pacemaker site was observed to shift from the superior (red arrows) towards the middle site (blue arrows) again. Comparing the gCaL effects, more suppressed gCaL caused more obvious shift of the leading pacemaker site.

Figure 7.

Effects of I_CaL on the leading pacemaker site. gCaL of the superior cells was changed after the time indicated with respective labels in each panel. Their action potentials were shown in (a) while (b) displayed electrical propagations along the fiber. Before changing gCaL, action potential propagated from the superior to the inferior as indicated by the red arrows. After decreasing gCaL, leading pacemaker site started to shift towards the middle site indicated by the blue arrows.

Discussion

The present study focuses on the mechanistic investigation of recent experimental findings based on the latest mathematical model. Shortening of the jSR refilling time and increase of Ca²⁺ release from SR during DD were found to be responsible for the isoproterenol-induced sinus rate acceleration. In contrast with I_f and I_CaT, SR Ca²⁺ cycling was a major contributor to the broad sinus rate modulation. However, appearance of DAD-like firing at down-regulated I_CaL, and stabilizing effect of I_f and I_CaT on sinus rhythm at suppressed SR Ca²⁺ cycling suggested the importance of membrane clock. In the simulated tissue, the leading pacemaker site was found to always colocalize with the site having the highest SR Ca²⁺ cycling, but I_CaL down-regulation also showed an obvious effect on the location.

Mechanisms of rate modulation and irregular firing

When the membrane clock was removed (all ion currents = 0), SR was shown to be a release-pump-delay oscillator, but unable to generate high-amplitude firing at higher rates⁷. Within the full system of two clocks, it was demonstrated that the interaction between the secondary releases of Ca²⁺ and I_NCX was responsible for producing rhythmic firing. Based on these findings, we investigated the combined effects of P_up and K_s on firing rate and the mechanisms to explain experimental observations, in which heart rate increased with elevated concentration of isoproterenol and decreased with infusion of ryanodine¹⁸. Fig. 2 suggested that increased SERCA pump activity by isoproterenol could markedly accelerate the rate of jSR refilling and increase the amount of Ca²⁺ release from SR during DD, thus shortening the time of I_NCX to threshold and thereby the cycle length. On the contrary, Ca²⁺ release inhibition by ryanodine would attenuate effects of isoproterenol and delay activation of I_NCX, resulting in sinus rate reduction. When P_upor K_sdramatically decreased, Ca²⁺ release from jSR was too small to activate I_NCX. Then transient sinus pause occurred until accumulated Ca²⁺ was enough to trigger I_NCX and started a new action potential. In this case, the firings would most likely lose their intrinsic rhythm and became irregular as shown in Fig. 1.

We did the study under conditions of high P_up (higher than 12mM/s) and broad change of g_CaL (from 0 to 0.5nS/pF) to investigate their combined effects on automaticity. Area of rhythmic firing was found to extend with P_up elevation. Such results are in line with the finding in Ref.7 and support the importance of Ca²⁺ clock. However, our results additionally suggested that even with higher P_up, lower contribution of I_CaL could lead to DAD-like firing and persistent sinus pause, implying a key role of I_CaL on rhythmic firing as well. Because I_CaL is mainly responsible for refilling the SR through Ca²⁺ influx and triggering CICR process, our results illustrated that the Ca²⁺ cycling could not be sustained when I_CaL was significantly down-regulated. Thus, after a spontaneous rhythmic Ca²⁺ release from the SR, depletion of Ca²⁺ in SR could cause sinus rhythm to completely stop after the DAD as observed in Fig. 3Ac. If down-regulation of I_CaL could be compensated by elevation of SERCA pump activity, which meant Ca²⁺ refueled into SR was just enough to maintain their cycling, then firing with DAD occurred as shown in Fig. 3Ab. In this case, because of decreased Ca²⁺ influx due to reduced I_CaL, increased SERCA pump would transport more Ca²⁺ into SR, thereby resulting in small Ca_sub concentration in submembrane space as shown in the trace of Ca_sub in Fig. 3Ab. This case was suggested to be a delicate balance between Ca²⁺ available for pumping and the speed of the SERCA pump, thus it only existed in a small area as shown in Fig. 3B. The above finding suggested importance of delicate corporation between SR Ca²⁺ cycling and I_CaL in keeping the system robustness.

In addition, we observed different effects of SR Ca²⁺ cycling and I_CaL on the action potential amplitude (APA). As observed in Fig. 1Ba and Fig. 1Bb, with increase of sinus rate, the maximum diastolic potential became more negative (from −57.5mV in Fig. 1Bb to −62.7mV in Fig. 1Ba), resulting in increase of APA. But 106.7% increase of the rate led to only 9.5% increase of APA. Therefore, the effect was only marginal. Nevertheless, as shown in Fig. 3, changes of I_CaL had obviously greater effects on APA. It might due to their different phases of function: SR Ca²⁺ cycling was mainly responsible for the spontaneous diastolic depolarization after the previous action potential to bring the membrane potential to threshold and activated I_NCX, while I_CaL was the primary current to generate the AP upstroke, thus should be the main factor to affect APA and overshoot.

Stabilization of the membrane currents on rhythmic firing

Activation of I_f and I_CaT were both suggested to be able to trigger a depolarization because of their low threshold ³. In Ref.7, the authors proved that inhibition of I_f resulted in rate reduction and the function of I_f in increasing the system robustness at low firing rates. On the other hand, our simulation in Fig. 4 demonstrated that compared with the rate modulation through SR Ca²⁺ cycling, I_f adjusted sinus rate within a narrow range (less than 0.3Hz) and without occurrence of firing pause at its blockade, suggesting a limited role of I_f in controlling pacemaker cell automaticity. Besides, as shown in Fig. 5 when g_CaT changed from 0.0nS/pF to 0.3664nS/pF, sinus rate was modulated within a range of 0.66Hz which was larger than that of I_f, indicating the more important effect of I_CaT than I_f on rate adjustment. As noted in Fig. 4 and Fig. 5, similar to I_f, increase of I_CaT could stabilize the firing rhythm. But in contrast to I_f, complete elimination of I_CaT could not result in persistent sinus pause. Therefore, as far as keeping the rhythm stabilization was concerned, I_f was suggested to be more important than I_CaT. This might due to their different phases of activation, because I_f was mainly responsible for the earlier spontaneous DD while I_CaT was the primary current during the mid-late DD.

Effects of SR Ca²⁺ cycling and I_CaL on the leading pacemaker site

With the observation that the maximum LDCAE (late diastolic Ca_i elevation) slope always colocalized with the leading pacemaker site and that LDCAE could faithfully represent Ca²⁺ clock activity, the leading pacemaker site was proposed to be the site exhibiting the highest Ca²⁺ activity¹⁸. To test this proposal, we modified the distribution of P_up in various ways along the developed fiber. In either the linear, exponential or discontinuous situation, leading pacemaker site showed good consistency with the highest SR Ca²⁺cycling.

In an optical mapping experiment, after isoproterenol infusion, the leading pacemaker site was found to shift to the superior SAN¹⁸. The SERCA2a/phospholamban molar ratio was reported lower in the superior SAN than the inferior SAN and right atrial tissue. The key regulator of SR Ca²⁺ uptake was phospholamban that inhibited SERCA2a in its dephosphorylated state, thereby superior site was supposed to have more phospholamban molecules available to regulate SERCA2a molecules and more robust Ca²⁺ uptake. We simulated this response by increasing P_up more in the superior than in other regions of the 1D model based on the third situation. The results in Fig. 6B confirmed the shift of the leading pacemaker site with increase of P_up. The site with the highest SERCA pump activity was characterized with the fastest and earliest spontaneous firing, driving the surrounding cells to fire and resulting in shift of the dominant pacemaker. Additionally, the results showed that the coupling conductance between neighboring cells had little effect on the location of the earliest excitation but velocity of the electrical propagation increased with the conductance elevation.

Besides, we observed the role of I_CaL in determining location of the earliest excitation. If I_CaL was suppressed as we did in Fig. 7, even with the superior cells showed the highest P_up, the leading pacemaker site still shifted to the site with less inhibited I_CaL. Note that in Fig. 7Aa and Fig. 7Ba, decreasing gCaL in the superior cells led to reduced amplitude of action potential, longer cycle length between the two neighboring action potentials and the appearance of DAD, thereby might suppress their dominant status in spontaneous activity. Consequently, in response to an intervention or diseased condition that suppresses the SAN spontaneous activity, the leading pacemaker is expected to shift to the site with the least suppressed intrinsic spontaneous activity, forming a hierarchical structure of pacemaker dominance.

Limitations

Because little is known about the heterogeneous distribution of the Ca²⁺ cycling in the SAN tissue, especially during β-adrenergic stimulation, we tested three cases and confirmed good consistency between the leading pacemaker site and the highest SR Ca²⁺ activity, and the site shift with change of P_up as well. This tissue model provided us with an opportunity to exclusively investigate role of Ca²⁺ and I_CaL in determining the earliest pacemaker location. However, in reality, two-dimensional tissue would be more ideal and realistic in the study. Heterogeneities, such as cellular size, ionic currents and SR Ca²⁺cycling among cells are neededfor further consideration.

The interaction of intracellular calcium dynamics and membrane current is a complicated process. We provided theoretical analysis of the SR Ca²⁺ cycling and effects of several main currents I_CaL, I_f and I_CaT on automaticity. However, other ionic channels, such as non-selective sustained current (I_st), was also suggested as additional mechanisms contributing to DD ¹⁹. But due to its exhibition of many properties of I_CaL and I_NCX, molecular origin and specific blockers of I_st have not been found yet, the identity of I_stremains unclear.