Intracellular Na+ Modulates Pacemaking Activity in Murine Sinoatrial Node Myocytes: An In Silico Analysis

Background: The mechanisms underlying dysfunction in the sinoatrial node (SAN), the heart’s primary pacemaker, are incompletely understood. Electrical and Ca2+-handling remodeling have been implicated in SAN dysfunction associated with heart failure, aging, and diabetes. Cardiomyocyte [Na+]i is also elevated in these diseases, where it contributes to arrhythmogenesis. Here, we sought to investigate the largely unexplored role of Na+ homeostasis in SAN pacemaking and test whether [Na+]i dysregulation may contribute to SAN dysfunction. Methods: We developed a dataset-specific computational model of the murine SAN myocyte and simulated alterations in the major processes of Na+ entry (Na+/Ca2+ exchanger, NCX) and removal (Na+/K+ ATPase, NKA). Results: We found that changes in intracellular Na+ homeostatic processes dynamically regulate SAN electrophysiology. Mild reductions in NKA and NCX function increase myocyte firing rate, whereas a stronger reduction causes bursting activity and loss of automaticity. These pathologic phenotypes mimic those observed experimentally in NCX- and ankyrin-B-deficient mice due to altered feedback between the Ca2+ and membrane potential clocks underlying SAN firing. Conclusions: Our study generates new testable predictions and insight linking Na+ homeostasis to Ca2+ handling and membrane potential dynamics in SAN myocytes that may advance our understanding of SAN (dys)function.


Introduction
In a healthy individual, each cardiac beat is initiated by the periodic activation of the sinoatrial node (SAN), the primary pacemaker of the heart [1]. The SAN is a complex and heterogeneous structure [2] that consists of a mix of fibroblasts, atrial myocytes, and a subpopulation of specialized myocytes characterized by the peculiar capability of spontaneously firing an action potential (AP), which is the main determinant of SAN pacemaking activity. It is well known that multiple mechanisms forming a "coupled-clock system" are involved in sustaining the automaticity of the spontaneously beating SAN myocytes (SAMs) [3,4]. The first subsystem, called "membrane clock", encompasses sarcolemmal ion channels and transporters that exhibit voltage-and time-dependent properties and interact nonlinearly to shape AP characteristics. Those include the hyperpolarization-activated cyclic nucleotide-gated channels (carrying the "funny" current I f , the dominant driver of Our study provides new insight into the link between [Na + ] i , [Ca 2+ ] i , and AP dynamics in SAMs and generates new testable predictions that may advance our understanding of cardiac pacemaking function and dysfunction.

A Computational Model of Murine SAMs Well Recapitulates a Broad Experimental Dataset
We simulated the murine SAM AP using the published model developed by Kharche et al. [34], updated to more closely recapitulate our experimental datasets. First, we adjusted the formulation of several ionic currents based on our measurements of I CaL [35], transient and steady-state outward K + currents (I to and I sus ) [36], and I f [37] ( Figure 1A). Then, we applied a global optimization method [38] to scale selected model parameters (Table A1) to match measured AP characteristics ( Figure 1B and Table A2) [39,40]. The resulting optimized model reproduces AP waveform properties that closely resemble our experimental results ( Figure 1C,D and Figure 2A). As an additional validation step, we confirmed that the model reproduces the reported effects of selectively blocking individual ion channels and transporters on SAM firing rate (FR, Figure 2B) [41][42][43][44][45][46]. ulates SAM electrophysiology, whereby Na + changes can lead to an array of arrhythmogenic phenotypes, including tachycardia, bursting behavior, and complete loss of automaticity. Our study provides new insight into the link between [Na + ]i, [Ca 2+ ]i, and AP dynamics in SAMs and generates new testable predictions that may advance our understanding of cardiac pacemaking function and dysfunction.

A Computational Model of Murine SAMs Well Recapitulates a Broad Experimental Dataset
We simulated the murine SAM AP using the published model developed by Kharche et al. [34], updated to more closely recapitulate our experimental datasets. First, we adjusted the formulation of several ionic currents based on our measurements of ICaL [35], transient and steady-state outward K + currents (Ito and Isus) [36], and If [37] ( Figure 1A). Then, we applied a global optimization method [38] to scale selected model parameters (Table A1) to match measured AP characteristics ( Figure 1B and Table A2) [39,40]. The resulting optimized model reproduces AP waveform properties that closely resemble our experimental results ( Figures 1C,D and 2A). As an additional validation step, we confirmed that the model reproduces the reported effects of selectively blocking individual ion channels and transporters on SAM firing rate (FR, Figure 2B) [41][42][43][44][45][46]. and simulated (bottom panels) voltage-dependence of peak ICaL [24], peak Ito [25], and If availability [26]. (B) Parameter scaling factors yielded by the global optimization process aimed at minimizing the differences between average measured and simulated AP characteristics (shown in the schematic in the inset). Definition of both model parameters and AP characteristics is reported in Appendix A. (C) Time course of membrane voltage in a representative experimental trace and in simulations performed with the original Kharche et al. model (orange) and our optimized dataset-specific reparameterization (blue). (D) Comparison between experimentally measured AP characteristics [23] and the outputs predicted with original and optimized models. and simulated (bottom panels) voltage-dependence of peak I CaL [24], peak I to [25], and I f availability [26]. (B) Parameter scaling factors yielded by the global optimization process aimed at minimizing the differences between average measured and simulated AP characteristics (shown in the schematic in the inset). Definition of both model parameters and AP characteristics is reported in Appendix A. (C) Time course of membrane voltage in a representative experimental trace and in simulations performed with the original Kharche et al. model (orange) and our optimized dataset-specific reparameterization (blue). (D) Comparison between experimentally measured AP characteristics [23] and the outputs predicted with original and optimized models.  [28,29], Ito and Isus [29], ICaT [30], ICaL [31], NCX [32], and NKA [33]. "Stop" indicates interruption of spontaneous firing activity.

NKA and NCX Inhibition Can Both Boost and Disrupt SAM Automaticity
Regulation of [Na + ]i depends on the balance between Na + influx and efflux through the sarcolemmal membrane during the beat cycle. Figure 3A depicts the predicted contribution of various mechanisms for Na + removal (NKA) and Na + influx, mediated by the voltage-gated TTX-sensitive, TTX-insensitive, and sustained Na + currents (INa1.1, INa1.5, and Ist, respectively), a passive Na + leak (INaB), If (which carries both Na + and K + ), and NCX [34]. In the model, NCX constitutes the main Na + entry pathway, accounting for ~50% of the total Na + influx, whereas NKA is the only Na + extrusion pathway. Thus, we manipulated their respective maximal transport rates (vNKA and vNCX) to produce substantial changes in Na + homeostasis and investigate the effects induced by Na + accumulation and depletion on SAN automaticity. Interestingly, despite opposite effects on [Na + ]i, we observed similar phenotypes when progressively increasing the degree of the block of each transporter ( Figure 3B,C). Reducing NKA or NCX function up to ~60 and 50%, respectively, has a slight positively chronotropic effect. However, higher degrees of block lead to a decrease in FR and irregularities in pacemaking function, in which the SAM capability of firing APs is periodically lost and restored. Further reductions in NKA (≥70%) or NCX (≥80%) eventually lead to permanent loss of SAM automaticity, in agreement with experimental observations [45,46]. We analyzed these results and performed simulations to dissect the mechanisms underlying the observed phenotypes, as detailed in the following sections.  2+ and Na + concentrations, and main ion currents in the optimized model. (B) Simulated and experimentally observed effects on FR induced by complete block of I Na1.1 and I Na1.5 [28,29], I to and I sus [29], I CaT [30], I CaL [31], NCX [32], and NKA [33]. "Stop" indicates interruption of spontaneous firing activity.

NKA and NCX Inhibition Can Both Boost and Disrupt SAM Automaticity
Regulation of [Na + ] i depends on the balance between Na + influx and efflux through the sarcolemmal membrane during the beat cycle. Figure 3A depicts the predicted contribution of various mechanisms for Na + removal (NKA) and Na + influx, mediated by the voltage-gated TTX-sensitive, TTX-insensitive, and sustained Na + currents (I Na1.1 , I Na1.5 , and I st , respectively), a passive Na + leak (I NaB ), I f (which carries both Na + and K + ), and NCX [34]. In the model, NCX constitutes the main Na + entry pathway, accounting for 50% of the total Na + influx, whereas NKA is the only Na + extrusion pathway. Thus, we manipulated their respective maximal transport rates (v NKA and v NCX ) to produce substantial changes in Na + homeostasis and investigate the effects induced by Na + accumulation and depletion on SAN automaticity. Interestingly, despite opposite effects on [Na + ] i , we observed similar phenotypes when progressively increasing the degree of the block of each transporter ( Figure 3B,C). Reducing NKA or NCX function up to~60 and 50%, respectively, has a slight positively chronotropic effect. However, higher degrees of block lead to a decrease in FR and irregularities in pacemaking function, in which the SAM capability of firing APs is periodically lost and restored. Further reductions in NKA (≥70%) or NCX (≥80%) eventually lead to permanent loss of SAM automaticity, in agreement with experimental observations [45,46]. We analyzed these results and performed

Na + Accumulation Induced by NKA Impairment Reduces SAM Automaticity via Ca 2+ Overload and Excessive Ca 2+ -Dependent ICaL Inactivation
The time course of SAM response to NKA blockade reveals both instantaneous and gradual changes in SAM electrophysiology (Figures 4 and 5). This is due to the abrupt reduction in the electrogenic NKA current that alters voltage dynamics and thus FR instantaneously, and the slower consequent increase in Na + , which modulates Ca 2+ homeo-

Na + Accumulation Induced by NKA Impairment Reduces SAM Automaticity via Ca 2+ Overload and Excessive Ca 2+ -Dependent I CaL Inactivation
The time course of SAM response to NKA blockade reveals both instantaneous and gradual changes in SAM electrophysiology (Figures 4 and 5). This is due to the abrupt reduction in the electrogenic NKA current that alters voltage dynamics and thus FR instantaneously, and the slower consequent increase in Na + , which modulates Ca 2+ homeostasis and AP dynamics. When reducing v NKA by 40%, decreased outward current accelerates diastolic depolarization and suddenly enhances SAM automaticity ( Figure 4; after a transient spike that fades in a few beats). Then, [Na + ] i accumulates over time and leads to further enhancement in SAM automaticity. This further FR increase is due to [Na + ] i -mediated increase in [Ca 2+ ] i and consequently increased NCX activity during diastole. In fact, when we repeated the simulation with [Na + ] i clamped at the initial value to prevent its elevation, the slow FR adaptation phase is prevented. Our simulations also predict a reduced steady-state AP amplitude due to increased I CaL CDI limiting the AP upstroke ( Figure 4). stasis and AP dynamics. When reducing vNKA by 40%, decreased outward current accelerates diastolic depolarization and suddenly enhances SAM automaticity ( Figure 4; after a transient spike that fades in a few beats). Then, [Na + ]i accumulates over time and leads to further enhancement in SAM automaticity. This further FR increase is due to [Na + ]imediated increase in [Ca 2+ ]i and consequently increased NCX activity during diastole. In fact, when we repeated the simulation with [Na + ]i clamped at the initial value to prevent its elevation, the slow FR adaptation phase is prevented. Our simulations also predict a reduced steady-state AP amplitude due to increased ICaL CDI limiting the AP upstroke ( Figure 4).  As shown in Figure 3B, the steady-state response leads to very different outcomes upon a higher degree of NKA block. When reducing vNKA by 60%, the model enters a bistable regime in which the SAM exhibits intermittent firing activity ( Figure 5A). The switch between these two states is determined by slow changes in [Na + ]i that accumulates ulations in which [Na + ]i was clamped to the maximal or minimal Na + levels (~13.5 and ~12 mM, respectively) seen during the oscillations. Indeed, model results predict that the bursting behavior is suppressed if Na + is kept constant, whereby Na + elevation (or depletion) can permanently interrupt (or restore) regular AP firing ( Figure 5A). Permanent loss of automaticity is observed when the NKA block is ≥70%. In this case, Na + levels increase due to limited extrusion, and the SAM membrane potential stabilizes at ~−40 mV, similarly to the value reported in rabbit multicellular SAN preparations [46]. In these conditions, regular firing activity could be restored by clamping [Na + ]i in our model to a lower level (12 vs. ~14 mM, Figure 5B). To identify the mechanism linking Na + overload to loss of AP firing, we analyzed the changes in currents and transporters modulated by [Na + ]i, either directly or indirectly (e.g., via Ca 2+ overload). We found that Na + -dependent Ca 2+ accumulation increases CDI of ICaL to the extent that it hampers a current essential for generating the AP [44,47]. Indeed, our simulations reveal that firing activity can be re-initiated (even with high Na + load) by restoring CDI to the levels seen before perturbing NKA function ( Figure 5B). As shown in Figure 3B, the steady-state response leads to very different outcomes upon a higher degree of NKA block. When reducing v NKA by 60%, the model enters a bistable regime in which the SAM exhibits intermittent firing activity ( Figure 5A). The switch between these two states is determined by slow changes in [Na + ] i that accumulates during the active phase and diminishes during the pauses. To demonstrate that this arrhythmic phenotype is caused by Na + accumulation and depletion, we ran additional simulations in which [Na + ] i was clamped to the maximal or minimal Na + levels (~13.5 and~12 mM, respectively) seen during the oscillations. Indeed, model results predict that the bursting behavior is suppressed if Na + is kept constant, whereby Na + elevation (or depletion) can permanently interrupt (or restore) regular AP firing ( Figure 5A). Permanent loss of automaticity is observed when the NKA block is ≥70%. In this case, Na + levels increase due to limited extrusion, and the SAM membrane potential stabilizes at~−40 mV, similarly to the value reported in rabbit multicellular SAN preparations [46]. In these conditions, regular firing activity could be restored by clamping [Na + ] i in our model to a lower level (12 vs.~14 mM, Figure 5B).
To identify the mechanism linking Na + overload to loss of AP firing, we analyzed the changes in currents and transporters modulated by [Na + ] i , either directly or indirectly (e.g., via Ca 2+ overload). We found that Na + -dependent Ca 2+ accumulation increases CDI of I CaL to the extent that it hampers a current essential for generating the AP [44,47]. Indeed, our simulations reveal that firing activity can be re-initiated (even with high Na + load) by restoring CDI to the levels seen before perturbing NKA function ( Figure 5B).

Ca 2+ Accumulation Induced by NCX Impairment Reduces SAM Automaticity via Excessive Ca 2+ -Dependent I CaL Inactivation
Despite the opposing roles of NKA and NCX in regulating cellular Na + loading, the block of these transporters has similar consequences in terms of changes in FR (Figure 3). The next set of simulations aimed at identifying similarities and differences in the underlying mechanisms. We found that positive chronotropy induced by reducing v NCX up to 60% is primarily mediated by increased [Ca 2+ ] i that enhances NCX activity during diastole and limits I CaL (via CDI), leading to a lower AP peak ( Figure S1). The moderate [Na + ] i decrease predicted in this case weakly counteracts Ca 2+ accumulation and contributes to the chronotropic effect by slightly decreasing outward NKA current and facilitating diastolic depolarization ( Figure S1). Upon 75% reduction of v NCX , the cell enters a bistable regime characterized by periodic transitions between the active (firing) and inactive states ( Figure 6A), accompanied by small oscillations in [Na + ] i . Once again, disruption of regular pacemaking is due to excessive CDI of I CaL induced by intracellular Ca 2+ accumulating during the active phase. When simulating ≥80% v NCX reduction, SAM automaticity is permanently lost (Figure 6B), in agreement with observations in murine SAMs lacking NCX [45]. As previously shown for NKA block, disruption of SAM automaticity is primarily determined by Ca 2+ overload, which hampers I CaL via excessive CDI ( Figure 6B).

Concomitant NKA, NCX, and I CaL Block Increases the Susceptibility to Pacemaking Dysfunction
We demonstrated that individual blocks of NKA and NCX could impair SAM automaticity. Interestingly, however, concomitant downregulation of NKA and NCX has been observed in mice lacking functional ankyrin-B [31,32] and could be a key contributing factor to the emerging irregularities in SAN function. We hypothesized that impairment in both transporters could act synergistically, whereby smaller concomitant reductions in v NKA and v NCX are required to disrupt SAM firing activity compared to those needed when NKA (≥60%) and NCX (≥75%) are altered individually. To test our hypothesis, we simulated various combinations of NKA and NCX block, ranging from normal function to complete block, and assessed the impact on average FR and FR variability ( Figure 7). As hypothesized, our simulations predicted developing irregular pacemaking activity, including bursting and permanent loss of automaticity, with a combined ≥50% reduction of v NKA and v NCX .
We previously described the key role of I CaL in the generation of SAM APs, and the deleterious consequences of increasing CDI (Figures 5 and 6). Notably, I CaL was found reduced in dysfunctional SAMs from both NCX knockout and ankyrin-B-deficient mice [29,32], potentially due to a compensatory mechanism to reduce Ca 2+ load [48]. To assess how changes in I CaL influence SAM response in the face of combined NKA and NCX block, we expanded the parametric analysis in Figure 7 by simulating the effects of variations in I CaL maximal conductance (G CaL ) and voltage-dependent gating (Figure 8). Our results show that the region in which regular SAM firing activity is preserved shrinks as G CaL is reduced, suggesting that even smaller perturbations of NKA and NCX function could result in SAM dysfunction in the case of I CaL downregulation. Similarly, we observed that the region with irregular firing activity expands as voltage-dependence of I CaL is shifted toward more positive potentials (i.e., to increase its voltage threshold of activation). depolarization ( Figure S1). Upon 75% reduction of vNCX, the cell enters a bistable regime characterized by periodic transitions between the active (firing) and inactive states ( Figure  6A), accompanied by small oscillations in [Na + ]i. Once again, disruption of regular pacemaking is due to excessive CDI of ICaL induced by intracellular Ca 2+ accumulating during the active phase. When simulating ≥80% vNCX reduction, SAM automaticity is permanently lost (Figure 6B), in agreement with observations in murine SAMs lacking NCX [45]. As previously shown for NKA block, disruption of SAM automaticity is primarily determined by Ca 2+ overload, which hampers ICaL via excessive CDI ( Figure 6B).

Concomitant NKA, NCX, and ICaL Block Increases the Susceptibility to Pacemaking Dysfunction
We demonstrated that individual blocks of NKA and NCX could impair SAM automaticity. Interestingly, however, concomitant downregulation of NKA and NCX has been observed in mice lacking functional ankyrin-B [31,32] and could be a key contributing factor to the emerging irregularities in SAN function. We hypothesized that impairment in both transporters could act synergistically, whereby smaller concomitant reductions in vNKA and vNCX are required to disrupt SAM firing activity compared to those needed when NKA (≥60%) and NCX (≥75%) are altered individually. To test our hypothesis, we simulated various combinations of NKA and NCX block, ranging from normal function to complete block, and assessed the impact on average FR and FR variability ( Figure 7). As hypothesized, our simulations predicted developing irregular pacemaking activity, including bursting and permanent loss of automaticity, with a combined ≥50% reduction of vNKA and vNCX. We previously described the key role of ICaL in the generation of SAM APs, and the deleterious consequences of increasing CDI (Figures 5 and 6). Notably, ICaL was found reduced in dysfunctional SAMs from both NCX knockout and ankyrin-B-deficient mice [29,32], potentially due to a compensatory mechanism to reduce Ca 2+ load [48]. To assess how changes in ICaL influence SAM response in the face of combined NKA and NCX block, we expanded the parametric analysis in Figure 7 by simulating the effects of variations in

Summary of the Results
Intracellular Na + homeostasis is critical in regulating cardiac electrophysiology and contraction in atrial and ventricular myocytes [14,15]. Elevated cardiomyocyte [Na + ]i has been reported in diseases such as HF and diabetes [16], contributing to arrhythmogenesis

Summary of the Results
Intracellular Na + homeostasis is critical in regulating cardiac electrophysiology and contraction in atrial and ventricular myocytes [14,15]. Elevated cardiomyocyte [Na + ] i has been reported in diseases such as HF and diabetes [16], contributing to arrhythmogenesis and impaired relaxation. SAN dysfunction is often associated with these diseases, but the role of [Na + ] i in the regulation of cardiac pacemaking is largely unexplored in normal physiology, let alone in disease. Here, we sought to investigate how Na + homeostasis affects SAM automaticity and to test whether [Na + ] i dysregulation may contribute to SAN dysfunction. We simulated mouse SAM APs using the Kharche et al. model [34], fitted to our experimental dataset recently collected in mice at a physiologic temperature [39] ( Figure 1). We showed that disrupting Na + homeostasis by reducing NKA or NCX function leads to an array of phenotypes, including enhanced FR, bursting behavior, and complete loss of automaticity (Figure 3). Our model-based analysis revealed that these behaviors are due to the coupling of Na + homeostasis to Ca 2+ handling (via NCX) and to membrane potential dynamics (via I CaL CDI) in SAMs (Figures 5 and 6). We also found that NKA and NCX block displays a synergistic pro-arrhythmic effect (Figure 7), and disruption of SAM automaticity is also facilitated by downregulation of I CaL (Figure 8).

Feedback of Na + and Ca 2+ Signals and Membrane Clock Dynamically Modulates SAM Automaticity
Our results reveal complex interactions of intracellular Na + homeostasis, Ca 2+ handling, and AP dynamics in SAMs ( Figure 9A). At baseline, modest increases in Ca 2+ and Na + loading enhance FR. Mild inhibition of NKA leads to enhanced automaticity via a direct effect on membrane dynamics (due to a reduced outward current in diastole) and a Na + -mediated increase in Ca 2+ load and consequent NCX increase that accelerates the Ca 2+ clock. Modest NCX inhibition leads to opposite and smaller changes in [Na + ] i ( Figure S2), but the impaired Ca 2+ extrusion favors Ca 2+ accumulation and results in a comparable chronotropic effect. Strong NKA and NCX function reductions lead to excessive Ca 2+ accumulation, which increases CDI and prevents I CaL and AP firing. For intermediate degrees of NKA and NCX block, SAMs display intermittent firing activity, in which slow changes in [Na + ] i and diastolic [Ca 2+ ] i (which increase during firing and decrease during the pause) drive the switch between predominantly positive or negative coupling between increased Ca 2+ loading and coupled membrane and Ca 2+ clock. In the case of the 60% NKA block, the sudden decrease in NKA causes slow Na + accumulation. This increased [Na + ] i in turn outwardly shifts the NKA current and causes FR to increase but also increases Ca 2+ loading (due to reduced NCX) and favors CDI. Thus, negative feedback exists between the slow Na + dynamics and the fast [Ca 2+ ] i -FR subsystem. In phase plots of FR and [Na + ] i , the fast [Ca 2+ ] i -FR subsystem shows bistable FR (FR nullcline, orange lines in Figure 9B, central panel). The slow [Na + ] i variable increases monotonically with FR ([Na + ] i nullcline, black dots). The intersection of these two nullclines, which identifies the system's fixed point, does not occur in either stable branches of the FR nullcline but crosses the unstable region ( Figure 9B, central panel). Thus, the system oscillates, and the FR-Na + phase plot forms a hysteresis loop, corresponding to the quasi-periodic FR fluctuations (i.e., bursting) observed at the steady-state ( Figure 5A). When NKA block is modest (i.e., 40%) and our simulations predict regular steady-state firing activity, Na + and FR nullclines cross within the FR-Na + phase plot, and the fixed point of the system anchors on the stable branch corresponding to regular AP firing ( Figure 9B, left panel). Conversely, with strong NKA reduction (i.e., 70%), the firing is stably suppressed ( Figure 9B, right panel). Similar dynamics were observed in ventricular myocytes, where feedback of [Ca 2+ ] i and [Na + ] i that influence membrane voltage could explain the intermittency of early after-depolarizations [27]. supraventricular arrhythmias [50]. Although cardiac glycosides decrease heart rate due to vagomimetic and anti-adrenergic effects [50], studies in isolated SAN multicellular preparations, devoid of neurohormonal control, reported an increase in FR (a phenomenon called "digitalis-induced sinus tachycardia"), development or irregular activity, and arrest of pacemaking function [46,50,51]. While our simulations reproduce these phenotypes, future experimental work should investigate whether reduction of Na + load attenuates the impact of cardiac glycosides on SAM function. Our simulations also recapitulated data from mouse models of atrial-selective NCX knockout or ankyrin-B syndrome [30,31], thus suggesting that interventions aimed at restoring Na + homeostasis and/or its consequences for Ca 2+ -dependent processes can reduce the susceptibility for pacemaking irregularities. Indeed, our simulation of complete NCX block predicted loss of automaticity, as observed in isolated SAMs from NCX knockout mice [45]. Loss of function of ankyrin-B impairs targeting and stabilization of NKA, NCX, and inositol trisphosphate (InsP3) receptors at the transverse-tubule/SR sites in cardiomyocytes [31], leading to a broad set of cardiac dysfunctions, including impaired pacemaking [31][32][33]. Since the analysis of SAMs isolated from ankyrin-B-deficient mice suggested concomitant downregulation of NKA and NCX [32], SAN irregularities in this animal model

Disruption of Na + Homeostatic Processes Contributes to SAN Dysfunction in Animal Models and Patients
Our model analysis suggests that pharmacological NKA inhibition can have opposite consequences in SAN myocytes, depending on the degree of induced Na + (and Ca 2+ ) rise, similar to experimental observations in ventricular myocytes upon administration of cardiac glycosides [49]. Cardiac glycosides are used in treating congestive HF to promote inotropy. Moderate Na + accumulation has a positive inotropic effect in ventricular myocytes and a positive chronotropic effect in our SAN cell simulations. Excessive [Na + ] i displays pro-arrhythmic side effects, including an increased propensity for spontaneous SR Ca 2+ release and delayed after-depolarizations, and decreased lusitropy in ventricles [49], and disrupts simulated SAN pacemaker activity, which may contribute to arrhythmia due to ectopic pacemakers. Notably, patients intoxicated by cardiac glycosides also develop supraventricular arrhythmias [50]. Although cardiac glycosides decrease heart rate due to vagomimetic and anti-adrenergic effects [50], studies in isolated SAN multicellular preparations, devoid of neurohormonal control, reported an increase in FR (a phenomenon called "digitalis-induced sinus tachycardia"), development or irregular activity, and arrest of pacemaking function [46,50,51]. While our simulations reproduce these phenotypes, future experimental work should investigate whether reduction of Na + load attenuates the impact of cardiac glycosides on SAM function.
Our simulations also recapitulated data from mouse models of atrial-selective NCX knockout or ankyrin-B syndrome [30,31], thus suggesting that interventions aimed at restoring Na + homeostasis and/or its consequences for Ca 2+ -dependent processes can reduce the susceptibility for pacemaking irregularities. Indeed, our simulation of complete NCX block predicted loss of automaticity, as observed in isolated SAMs from NCX knockout mice [45]. Loss of function of ankyrin-B impairs targeting and stabilization of NKA, NCX, and inositol trisphosphate (InsP 3 ) receptors at the transverse-tubule/SR sites in cardiomyocytes [31], leading to a broad set of cardiac dysfunctions, including impaired pacemaking [31][32][33]. Since the analysis of SAMs isolated from ankyrin-B-deficient mice suggested concomitant downregulation of NKA and NCX [32], SAN irregularities in this animal model could be facilitated by the synergistic effect described for the combined NKA and NCX block (Figure 7).
While we simulated acute changes induced by NCX and NKA inhibition, long-term chronic changes (e.g., due to transcriptional and post-translational regulation) are likely to occur in these extremely high Na + and Ca 2+ levels. Indeed, experiments in both NCX knockout and ankyrin-B-deficient mice also revealed a~50% decrease in I CaL [29,32]. While reduced I CaL is likely a mechanism to limit cellular Ca 2+ overload [48], our analysis predicts that this maladaptive alteration further impairs SAN pacemaking by reducing the NKA and NCX block ranges that remain compatible with stable SAM function (Figure 8). Future experimental investigations could test whether increasing I CaL can attenuate the pathologic phenotype observed in these mice. Notably, level of serum (or extracellular) Ca 2+ concentration ([Ca 2+ ] o ) influences Ca 2+ influx via I CaL and the function of other Ca 2+ handling processes in SAMs. Development of hypocalcemia and hypercalcemia have been documented in several pathologic states, and both conditions have been associated with increased pro-arrhythmic risk [52]. Our simulations show that increasing [Ca 2+ ] o enhances SAM automaticity (Figure 10), in agreement with recent computational work that also has suggested that this effect is even more pronounced in humans vs. small mammals [53,54]. Our results further demonstrate that increasing [Ca 2+ ] o increased SAM susceptibility to irregularities induced by combined NKA and NCX block (Figure 10), confirming recent experimental observations reporting that hypercalcemia not only increases FR but can also increase the propensity of SAN dysfunction in mice [55].
to occur in these extremely high Na and Ca levels. Indeed, experiments in both NCX knockout and ankyrin-B-deficient mice also revealed a ~50% decrease in ICaL [29,32]. While reduced ICaL is likely a mechanism to limit cellular Ca 2+ overload [48], our analysis predicts that this maladaptive alteration further impairs SAN pacemaking by reducing the NKA and NCX block ranges that remain compatible with stable SAM function (Figure 8). Future experimental investigations could test whether increasing ICaL can attenuate the pathologic phenotype observed in these mice. Notably, level of serum (or extracellular) Ca 2+ concentration ([Ca 2+ ]o) influences Ca 2+ influx via ICaL and the function of other Ca 2+ handling processes in SAMs. Development of hypocalcemia and hypercalcemia have been documented in several pathologic states, and both conditions have been associated with increased pro-arrhythmic risk [52]. Our simulations show that increasing [Ca 2+ ]o enhances SAM automaticity (Figure 10), in agreement with recent computational work that also has suggested that this effect is even more pronounced in humans vs. small mammals [53,54]. Our results further demonstrate that increasing [Ca 2+ ]o increased SAM susceptibility to irregularities induced by combined NKA and NCX block (Figure 10), confirming recent experimental observations reporting that hypercalcemia not only increases FR but can also increase the propensity of SAN dysfunction in mice [55].
We assessed the translatability of our findings in mouse to human physiology by simulating the Loewe et al. model of the human SAM [53]. Human simulations confirmed our observations in murine SAMs that inhibition of NCX or NKA can disrupt SAM automaticity via Ca 2+ overload and consequent ICaL inhibition ( Figure S3). As observed with our mouse model ( Figure 5), preventing the accumulation of Na + due to NKA block restores regular automaticity in human SAMs (see [Na + ]i-clamp simulation in Figure S3C). Clamping CDI of ICaL to the values predicted before block restores fast (but irregular) firing activity after disruption of human SAM function induced by NCX or NKA block (see CDIclamp simulations in Figure S3C,D). However, the human model did not predict the chronotropic effect induced by mild NKA block in murine SAMs (Figure 4), and neither NCX nor NKA block (or their combined inhibition) led to developing the bursting activity. These results suggest that intracellular Na + levels affect pacemaking function in both human and murine SAMs. However, interspecies differences likely exist in the strengths and relative roles of the processes linking Na + , Ca 2+ , and membrane potential homeostasis depicted in Figure 9 that warrant further investigation.  We assessed the translatability of our findings in mouse to human physiology by simulating the Loewe et al. model of the human SAM [53]. Human simulations confirmed our observations in murine SAMs that inhibition of NCX or NKA can disrupt SAM automaticity via Ca 2+ overload and consequent I CaL inhibition ( Figure S3). As observed with our mouse model ( Figure 5), preventing the accumulation of Na + due to NKA block restores regular automaticity in human SAMs (see [Na + ] i -clamp simulation in Figure S3C). Clamping CDI of I CaL to the values predicted before block restores fast (but irregular) firing activity after disruption of human SAM function induced by NCX or NKA block (see CDI-clamp simulations in Figure S3C,D). However, the human model did not predict the chronotropic effect induced by mild NKA block in murine SAMs (Figure 4), and neither NCX nor NKA block (or their combined inhibition) led to developing the bursting activity. These results suggest that intracellular Na + levels affect pacemaking function in both human and murine SAMs. However, interspecies differences likely exist in the strengths and relative roles of the processes linking Na + , Ca 2+ , and membrane potential homeostasis depicted in Figure 9 that warrant further investigation.

Limitations and Future Directions
Despite the modifications made to closely recapitulate our experimental dataset, our model maintains the same structure of the original Kharche et al. version and inherits its main limitations previously discussed in [34]. Notably, this framework does not include several components proposed to actively regulate SAM electrophysiology in control and, more so, pathologic conditions. Those include ion channels (e.g., small-conductance Ca 2+ -activated K + channels [56]), transporters (e.g., InsP 3 receptors [57], and Na + /H + exchanger [58,59]), and intracellular signaling pathways (e.g., CaMKII- [60] and PKAdependent cascades [61]).
Neurohumoral regulation is a major determinant of pacemaking function in vivo [62], and β-adrenergic signaling critically regulates the coupling between the membrane and Ca 2+ clocks in human SAMs [63]. Thus, future work should investigate the role of Na + homeostasis in mediating SAM response to the β-adrenergic challenge when increases in PKA-dependent phosphorylation of phospholemman enhances NKA function [64,65]. The consequent Na + depletion expected in this case could, for example, be involved in the isoproterenol-induced restore of automaticity observed in dormant SAMs [63,66].
Future investigations should also extend our analysis to incorporate the structural modifications that have been associated with SAN dysfunction, including remodeling in subcellular ultrastructure [30] and functional microdomains [67] and tissue-level fibrosis [68]. Experiments in intact SAN tissue isolated from NCX knockout mice revealed bursting activity [29]. Since heterogeneous Na + loading has been associated with repolarization abnormalities in ventricular tissue simulations [27], future work should explore whether intercellular [69] and inter-regional variability [44,70] affect SAN function at the tissue level.

Conclusions
In this study, we show that [Na + ] i dynamically modulates SAM automaticity during regular and irregular firing regimes, and reveal new mechanisms by which aberrant Na + signaling plays an important role in generating cardiac disorders [71]. Experimental characterization of the mechanisms controlling SAM Na + homeostasis, including testing our model predictions, may advance our understanding of cardiac pacemaking function and dysfunction and aid in identifying new therapeutic targets [72].

Experimental Data
Experimental data previously collected by the Proenza laboratory [35][36][37]39] were used to constrain model reparameterization. Briefly, SAMs were isolated from 2-3 months old male C58BL/6 J mice [73,74], and membrane currents and spontaneous APs were recorded at 35 • C in whole-cell and amphotericin perforated patch configurations, respectively [35,36,39]. AP characteristics (defined in Table A2) were determined for each cell from average waveforms from 5 s recording windows using the software ParamAP [40].

Model Development
The Kharche et al. model of murine SAMs [34], implemented in MATLAB (The Math-Works Inc., Natick, MA, USA), provided the basis for our simulations. We first modified the formulation of the following currents to match our experimental data ( Figure 1A):  [35]. This was obtained by negatively shifting (−7 mV) the voltage-dependence of activation and inactivation and decreasing the maximal conductance by one-third. (ii) I to and I sus . Maximal conductances of both components of the outward K + currents were increased by 2.5-fold [36]. (iii) I f . The original I f the formulation of the Kharche et al. model was replaced with our recently updated version [37]. Briefly, we modified and extended the Hodgkin-Huxley type I f model originally present in the Kharche et al. framework based on our novel data in murine SAMs describing voltage-dependence of I f availability and activation/deactivation kinetics. Notably, in our novel formulation, activation and deactivation of I f exhibit both fast and slow kinetics [37].
Next, to reproduce the average AP properties observed in our experiments [39], we applied an established population-based optimization method [38]. We created a population of 1000 model variants by perturbing selected parameters (listed in Table A1) with random scaling factors chosen from a log-normal distribution with a median value of 1 and a standard deviation of 0.1 [75]. We assessed AP characteristics in each model in the population and then performed reverse multivariable regression [38] to identify the parameters scaling factors required to best reproduce the AP biomarkers experimentally observed ( Figure 1B). We used the following set of target values: upstroke velocity of 10 mV/ms; repolarization rate of −3 mV/ms; maximum diastolic potential of −65 mV; AP amplitude of 75 mV; AP duration at 90% repolarization of 60 ms; AP duration at 50% repolarization 25 ms; cycle length of 150 ms; diastolic duration of 75 ms; diastolic depolarization rate of 100 mV/s.

Simulation Protocols and Analysis
Using our newly developed dataset-specific version of the Kharche et al. model [34], we simulated various degrees of reduction of NKA and NCX function by modulating their respective maximal transport rates v NKA and v NCX , individually or in combination. In a separate set of simulations, we superimposed changes in I CaL properties (i.e., ±50% in G CaL and ±4 mV shift in voltage-dependence) or extracellular [Ca 2+ ]. We simulated the effect of parameter perturbations for 120 s and quantified the impact on FR (average and variability) over the last 60 s. Specifically, in each simulation, we assessed the FR at 7 time points (every 10 s, from 60 to 120 s after block), analyzing the voltage signal over a time interval of 1.8 s, and then calculated average and standard deviation from the 7 samples.
To identify the subcellular mechanisms involved in regulating SAM automaticity upon NKA and NCX block, we also performed simulations in which [Na + ] i or CDI of I CaL were clamped to desired values. In "CDI-clamp" simulations, we used the subsarcolemmal [Ca 2+ ] signal observed in the absence of any block as input for the calculation of the changes in the state variable Fca, which represents the CDI-related gate in the Hodgkin-Huxley type I CaL model implemented in Kharche et al. [34]. Note that Fca, which varies between 0 and 1, decreases as CDI increases and vice versa. Therefore, CDI values were estimated in our simulations as CDI = 1 − Fca. Finally, voltage-clamp simulations were performed to assess how changes in SAM automaticity influence [Na + ] i ("FR-clamp"). Specifically, the AP obtained with the updated baseline model was used as voltage command in AP-clamp simulations and stretched/compressed in time to simulate different FRs within the range of 300-600 bpm. Na + levels predicted in the absence of firing activity were determined by clamping the transmembrane potential at −40 mV.
Consequences of NKA and NCX block on human SAM electrophysiology were investigated simulating the model recently developed by Loewe et al. [53]. Table A2. List and definition of action potential characteristics. The values of these biomarkers were determined as described in [40].