Synergism of coupled subsarcolemmal Ca²⁺ clocks and sarcolemmal voltage clocks confers robust and flexible pacemaker function in a novel pacemaker cell model

Abstract

Recent experimental studies have demonstrated that sinoatrial node cells (SANC) generate spontaneous, rhythmic, local subsarcolemmal Ca²⁺ releases (Ca²⁺ clock), which occur during late diastolic depolarization (DD) and interact with the classic sarcolemmal voltage oscillator (membrane clock) by activating Na⁺-Ca²⁺ exchanger current (I_NCX). This and other interactions between clocks, however, are not captured by existing essentially membrane-delimited cardiac pacemaker cell numerical models. Using wide-scale parametric analysis of classic formulations of membrane clock and Ca²⁺ cycling, we have constructed and initially explored a prototype rabbit SANC model featuring both clocks. Our coupled oscillator system exhibits greater robustness and flexibility than membrane clock operating alone. Rhythmic spontaneous Ca²⁺ releases of sarcoplasmic reticulum (SR)-based Ca²⁺ clock ignite rhythmic action potentials via late DD I_NCX over much broader ranges of membrane clock parameters [e.g., L-type Ca²⁺ current (I_CaL) and/or hyperpolarization-activated (“funny”) current (I_f) conductances]. The system Ca²⁺ clock includes SR and sarcolemmal Ca²⁺ fluxes, which optimize cell Ca²⁺ balance to increase amplitudes of both SR Ca²⁺ release and late DD I_NCX as SR Ca²⁺ pumping rate increases, resulting in a broad pacemaker rate modulation (1.8–4.6 Hz). In contrast, the rate modulation range via membrane clock parameters is substantially smaller when Ca²⁺ clock is unchanged or lacking. When Ca²⁺ clock is disabled, the system parametric space for fail-safe SANC operation considerably shrinks: without rhythmic late DD I_NCX ignition signals membrane clock substantially slows, becomes dysrhythmic, or halts. In conclusion, the Ca²⁺ clock is a new critical dimension in SANC function. A synergism of the coupled function of Ca²⁺ and membrane clocks confers fail-safe SANC operation at greatly varying rates.

numerical models simulating dynamic interplay of multiple processes have been invaluable assets for the achievement of mechanistic insights into the complex function of excitable cells, including cardiac cells. Landmark quantitative models describing how membrane ion channels interact as a function of time and voltage underlie the quantitative theory of membrane excitation by Hodgkin and Huxley, which, in 1960, was modified to simulate spontaneous action potentials (APs) of cardiac pacemaker cells (46). Subsequently, a large cohort of essentially membrane-delimited models of sinoatrial nodal cells (SANC, the primary pacemakers of the heart) evolved with each newer model [12 models since 1980 (71)] specifying novel ion current properties commensurate with their discovery in voltage-clamp studies.

However, since seminal studies by Bozler (6) in 1943, substantial experimental evidence indicates that all cardiac cells, including pacemaker cells, possess another oscillator of subcellular origin that can work independently of sarcolemma (reviewed in Ref. 43). In 1979, summarizing results on oscillatory membrane currents under voltage clamp in cardiac Purkinje cells, Tsien et al. (63) suggested a new concept of cardiac pacemaker oscillations based on interactions of “a surface membrane oscillator” and “an internal oscillator” with “cycles of Ca²⁺ uptake and release by the sarcoplasmic reticulum” (SR). However, this concept was abandoned because recording the oscillatory Ca²⁺ signals by techniques of that time required conditions of strong Ca²⁺ signal amplification associated with Ca²⁺ overload and pathological function. Accordingly, the internal Ca²⁺ oscillator of some existing cardiac cell numerical models initiates abnormal cardiac impulses under the conditions of Na⁺ and Ca²⁺ overload (47).

The concept that interacting oscillators drive normal cardiac automaticity has recently emerged as further data, obtained with more precise experimental techniques, have demonstrated a substantial contribution of Ca²⁺ cycling by SR to the late phase of the diastolic depolarization (DD) of cardiac pacemaker cells under normal physiological conditions (reviewed in Ref. 41). Specifically, confocal imaging detects not only a global intracellular Ca²⁺ transient triggered by the AP, but also local, spontaneous Ca²⁺ releases (LCRs) from SR, which emerge beneath the cell membrane as multiple, relatively synchronous, locally propagating Ca²⁺ wavelets later in the cycle, i.e., during the late part of the DD (5, 21). LCRs activate the electrogenic operation of sarcolemmal Na⁺-Ca²⁺ exchanger (NCX), generating an inward current (I_NCX) that imparts a steep, exponentially rising phase to the DD (4, 5, 21, 42). Importantly, SR indeed behaves as a true intracellular Ca²⁺ oscillator (Fig. 1), generating spontaneous rhythmic LCRs observed in voltage-clamped and chemically skinned SANC (70). Under voltage clamp, the Ca²⁺ oscillator generates oscillatory membrane current spikes of 10-pA scale (70) that are sufficient to drive critical late DD acceleration [see details in recent review (32)].

Fig. 1.

Interacting Ca²⁺ clock and membrane clock in our model of rabbit sinoatrial node cells (SANC): schematic illustration of the cell compartments, major functional components, and their interactions approximated by our model. The physiological (full system) Ca²⁺ clock includes both the sarcoplasmic reticulum (SR) Ca²⁺ clock and balanced sarcolemmal Ca²⁺ fluxes of L-type Ca²⁺ current (I_CaL) and Na⁺-Ca²⁺ exchanger (NCX) (bold arrows); see text for details. Ca_i, Ca²⁺ in bulk cytosol; Ca_sub, Ca²⁺ in the subspace; SERCA, SR Ca²⁺ pump; Ca_nSR, Ca²⁺ in the network SR (nSR); Ca_jSR, junctional SR (jSR); j_SRCarel, SR Ca²⁺ release rate global variable; RyR, cardiac ryanodine receptor; AP, action potential; I_Kr, rapid delayed rectifier current; I_st, sustained current; I_NCX, NCX current; I_CaT, T-type Ca²⁺ current; I_f, hyperpolarization-activated or “funny” current.

Under normal conditions, Ca²⁺ oscillations in SANC are driven not by Ca²⁺ overload but by the enhanced functional status of the Ca²⁺ cycling proteins due to relatively high levels (compared with ventricular myocytes) of both cAMP-dependent PKA (67) and CaMK II (CaMKII; Ref. 69) signaling in the basal state of SANC. Variability of the SR oscillator is achieved via gradations in the PKA signaling (67) and represents a novel major mechanism contributing to the pacemaker rate regulation via both β-adrenergic receptors (β-ARs; rate increase; Refs. 4, 65, 67) and muscarinic acetylcholine receptors (m-AChRs; rate decrease; Ref. 38).

Whereas these recent experimental results clearly suggest a new fundamental paradigm of cardiac pacemaker function, the integration of the membrane voltage oscillator and SR Ca²⁺ oscillator in normal SANC function has not been tested within a dynamic numerical cell model. Existing theoretical SANC models [recently reviewed by Wilders (71)] lack terms for late diastolic LCRs in submembrane space and do not capture the significance of the coupling of LCRs with sarcolemmal ion channels in initiation and regulation of normal basal SANC automaticity. Here, we present a prototype SANC model of dynamically interacting physiological Ca²⁺ oscillator and the classic membrane voltage oscillator (Fig. 1), recently dubbed ”Ca²⁺ clock“ and ”membrane clock,“ respectively (recent reviews in Refs. 32, 41). The model predicts spontaneous diastolic SR Ca²⁺ release in submembrane space and NCX current resulting in late DD acceleration. The model simulations not only validate the plethora of recent experimental findings, but also have discovered that it is the coupled function of Ca²⁺ clock and membrane clock that confers robustness and flexibility to cardiac pacemaker function.

METHODS

Specific Aim of the Study and our Strategy to Develop and Explore the New Model

The specific aim of our study was to develop a new numerical SANC model featuring interactions of Ca²⁺ and membrane clocks to explore novel mechanistic insights of cardiac impulse initiation. Our new SANC model is based on only a single new assumption: the existence of SR-based Ca²⁺ clock that was proven experimentally. Thus a major new feature of our SANC model is a more advanced and tuned formulation of the SR function. In the present study, we applied a wide-scale parameter sensitivity analysis of the dynamic system behavior (28, 64). Since performing analysis for all or many model parameters is nonrealistic (at least in 1 study), we have kept many parameters at their original classic values but varied only those few that are 1) reportedly different in SANC vs. ventricular myocytes, and 2) likely enhanced by a relatively high level of basal of cAMP/PKA-dependent signaling as discovered experimentally (67).

Approximation of SR Ca²⁺ Release

Our model adopts the formulation of cardiac ryanodine receptor (RyR) function developed by Shannon et al. (Ref. 59; referred throughout the paper as Shannon model) based on numerous experimental data on SR Ca²⁺ cycling in rabbit cardiomyocytes and an RyR gating scheme previously suggested by Stern et al. (60). The new advanced RyR formulations feature 1) Markov chain processes describing RyR activation and restitution kinetics, and 2) luminal Ca²⁺ regulation. Since the cardiac type RyR2 is the common major Ca²⁺ release channel type in SANC and ventricular myocytes, Shannon model served as a reasonable starting point in our quest, which approximates SR Ca²⁺ release rate by one global variable, j_SRCarel. This rather simple approach was inspired by the experimental observations that individual multiple LCRs observed in SANC, in fact, become synchronized in SANC within 15–20% of the spontaneous cycle before subsequent membrane excitation (42, 70). Thus the integrated LCR signal mass provides a strong whole cell signal that can be described by a global variable, such as j_SRCarel. In other words, the signals of individual LCRs emerging during late DD in submembrane space are integrated by Ca²⁺ sensing sarcolemmal proteins (such as NCX) and thus can be lumped (by frequency, size, duration, and amplitude) as a single Ca²⁺ release signal of the SR. We have validated this approach numerically in our prior, rather naïve SANC model (42) by describing phenomenologically (approximated with a sinusoidal function) this diastolic j_SRCarel signal of the synchronized LCRs: grading the timing and amplitude of this Ca²⁺ release signal (according to experimental data) indeed modulated the AP firing rate within the entire physiological range for rabbit SANC.

Approximation of SR Ca²⁺ Pump, Cell Structure, and Ca²⁺ Buffering

We have kept formulations of Kurata et al. (27) model of primary rabbit SANC (referred throughout the paper as Kurata model) for the SR Ca²⁺ pump, which were originally developed by Luo and Rudy (36), and for ion buffering in all cell compartments, including Ca²⁺ buffering by calsequestrin in junctional SR (jSR). We have also retained the original cell structure of the Kurata model consisting of four compartments: submembrane space (subspace), cytosol, network SR (nSR), and jSR (Fig. 1).

Sarcolemmal Function

Our model of sarcolemmal function also adopts formulations of Kurata model. We chose this particular model because it has, in our opinion, the most advanced and updated numerical formulations of the membrane function specific to rabbit primary SANC. Our basal state firing model has ion current parameters (listed in the data supplement online at the AJP-Heart and Circulatory Physiology web site) similar to those of our previously published modification of Kurata model (42). We have used this modified membrane model in other previous studies (4, 42, 67, 70) and have demonstrated that it describes rabbit SANC function more accurately than the original Kurata model. Specifically, it better describes the absolute values of systolic and diastolic intracellular Ca²⁺ that we measured in rabbit SANC (42). Also, compared with the original Kurata model, our modified model is very sensitive to SR Ca²⁺ releases (42, 70) [including individual stochastic LCRs (4, 67)], recapitulating numerous recent experimental results that demonstrate the critical importance of LCRs in SANC function and regulation [see recent reviews (32, 41, 45)]. Specifically, the original Kurata model has a relatively high L-type Ca²⁺ current (I_CaL) conductance (g_CaL) of 0.58 nS/pF vs. 0.385 and 0.29 nS/pF in Demir et al. (10) and Zhang et al. (73) models, respectively. Thus one of our important model modifications of the Kurata model was to reduce g_CaL by 20% (to 0.464 nS/pF). The problem with experimentally reported I_CaL densities, however, is their large spread. Furthermore, I_CaL density varies substantially from cell to cell even within each study. For example, I_CaL density varied from 3.8 to 11.4 pA/pF (7.7 ± 2.3 pA/pF, mean ± SD) independently of cell size in a comprehensive study by Honjo et al. (17) of many rabbit SANC at 35°C and 0 mV (see Fig. 7B in Ref. 17). Our basal model simulates I_CaL of 6.9 pA/pF under these conditions, a value that is very close to their average. When Ca²⁺ clock is experimentally disabled by ryanodine (30 μM), the automaticity fails in majority of SANC [in 83% of cells (37)]. When we developed the model, the 20% decrease of g_CaL was required to reproduce this critical experimental result.

Fig. 7.

The mechanism of broad spontaneous AP rate modulation by P_up. Overlapped traces are simulations for Ca_nSR (A), Ca_jSR (B), j_SRCarel (C), I_NCX (D), and V_m (E) at various P_up (labels in B). Simulations at different P_up are shown by different colors. All traces were synchronized with the 1st AP crossing 0 mV at the same time (100 ms) and reasonably time-truncated to illustrate phase shifts. See text for details.

The importance of this and other changes to reproduce robust experimental results (e.g., with ryanodine) will be also clearly demonstrated in our wide-scale parametric system analysis (cf., results). We also compared AP characteristics and late DD ion currents of our basal firing model with those in prior SANC models and/or experimental data.

Model Equations, Parameters, Variables, and Initial Conditions

Our model is a system of 29 first-order differential equations. All model equations and parameter values are provided in the online data supplement. Table 1 lists all model variables (y₁–y₂₉) and their initial values. To avoid a lengthy transitional process in our parametric analysis, we set initial conditions for membrane clock gating variables (y₁₅–y₂₉) to ”ready-to-fire“ status, as if the sarcolemma was restituted at a low voltage (Table 1). Initial conditions for SR Ca²⁺ clock (y₁–y₁₄) were found as a set of instantaneous values of all variables for a mid-range Ca²⁺ oscillator (Fig. 3C), which will be described in results. The initial value for Ca²⁺ in the nSR (Ca_nSR, y₄) is 1.35 mM, which is within the physiological range of rabbit cardiac SR from 1 to 1.5 mM (58).

Table 1.

Model variables: description and initial values

	Variable	Initial Value	Description
SR Ca2+ Clock
y1	Cai	0.0001	Ca2+ in myoplasm, mM
y2	Casub	0.000223	Ca2+ in submembrane space, mM
y3	CajSR	0.029	Ca2+ in the junctional SR (jSR), mM
y4	CanSR	1.35	Ca2+ in the network SR (nSR), mM
y5	fTC	0.02	Fractional occupancy of the troponin Ca2+ site by Ca2+ in myoplasm
y6	fTMC	0.22	Fractional occupancy of the troponin Mg2+ site by Ca2 in myoplasm
y7	fTMM	0.69	Fractional occupancy of the troponin Mg2+ site by Mg2+ in myoplasm
y8	fCMi	0.042	Fractional occupancy of calmodulin by Ca2+ in myoplasm
y9	fCMs	0.089	Fractional occupancy of calmodulin by Ca2+ in submembrane space
y10	fCQ	0.032	Fractional occupancy of calsequestrin by Ca2+ in junctional SR
y11	R	0.7499955	RyR reactivated (closed) state
y12	O	3.4 × 10−6	RyR open state
y13	I	1.1 × 10−6	RyR inactivated state
y14	RI	0.25	RyR RI state (described in Refs. 59 and 60)
Membrane Clock
y15	Vm	−65	Membrane potential, mV
y16	dL	0	ICaL activation
y17	fL	1	ICaL voltage-dependent inactivation
y18	fCa	1	ICaL Ca2+-dependent inactivation
y19	paF	0	IKr fast activation
y20	paS	0	IKr slow activation
y21	pi	1	IKr inactivation
y22	n	0	IKs activation
y23	y	1	If activation
y24	dT	0	ICaT activation
y25	fT	1	ICaT inactivation
y26	q	1	Ito inactivation
y27	r	0	Ito and Isus activation
y28	qa	0	Ist activation
y29	qi	1	Ist inactivation

Our sinoatrial node cells (SANC) model is described by a system of 29 first-order differential equations given in the online data supplement (variables y₁ – y₂₉). RyR, ryanodine receptor; I_CaL, L-type Ca²⁺ current; I_Kr, rapid delayed rectifier current; I_st, sustained current; I_CaT, T-type Ca²⁺ current; I_f, hyperpolarization-activated or “funny” current; I_to, transient outward current; I_sus, sustained component of the 4-AP-sensitive current.

Fig. 3.

A major limitation (“the faster, the weaker”) of the isolated SR Ca²⁺ oscillator is its inability to sustain high-amplitude oscillations at high rates when k_s increases (A) or P_up decreases (B). C: model simulations at various k_s show that the mechanism of poor oscillator performance at high rates (i.e., decreasing amplitude) is linked to accumulation of Ca²⁺ in the submembrane space (Ca_sub, solid line). All membrane currents = 0.

Numerical Integration

Source code for the model was developed using Borland Delphi 7 software. The model integration was performed with a fixed time step of 0.01 ms using a Pentium 4 (2.8 GHz)-based PC. Decreasing the integration step to 0.001 ms did not change results of simulations but substantially increased computation time.

Finding Steady-State Solutions

Since our model considers dynamic variations of voltage and ion concentrations, it belongs to so-called ”second-generation“ models (71). Whereas models of this type describe complex and physiologically realistic interplay of electrophysiological and ionic mechanisms, some models have problems with 1) ”degeneracy,“ i.e., the existence of a continuum of equilibrium points, and 2) slow ion concentration drifts (Refs. 20, 24, 26 and reviewed in Ref. 71). For example, ion steady-state balance is achieved within hours of simulations in a model of atrial cell AP by Kneller et al. (24). A slow drift in Luo-Rudy dynamic model of ventricular cell was found to be due to the external stimulation current (20). Since our model does not apply external currents, a practical solution to the problems of drift and degeneracy was previously suggested to set the intracellular Na⁺ and K⁺ to constant values and treat them as independent model parameters (13, 71). Thus, as previously suggested by Kurata et al. (28), we fixed intracellular Na⁺ and K⁺ to 10 and 140 mM, respectively. Our parametric analysis of the system behavior was performed by a computer algorithm (online with simulations) that distinguished three types of steady-states: 1) steady rhythmic oscillations; 2) no oscillations or damped oscillations; and 3) chaotic oscillations. Steady oscillations were identified by the algorithm suggested previously by Kurata et al. (28): numerical integration was continued until the relative differences in both amplitude and period between the newly calculated cycle and the preceding cycle became <0.0005 [2-dimensional (2-D) analysis] or <0.0001 (1-dimensional analysis).

Comparisons with Prior Models

We compared some of our model predictions to those of previously published SANC models. For this comparison, we used traces from either a recently published review by Wilders (71) or from our own simulations using Cellular Open Resource (COR) developed and maintained by Alan Garny at Oxford University (http://cor.physiol.ox.ac.uk). COR can run numerous cell models, including classic SANC models, in CellML format available at http://www.cellml.org.

Terminology

Spontaneous Ca²⁺ release.

Since our rather simple model of integrated Ca²⁺ dynamics does not describe stochastic, locally propagating Ca²⁺ releases within the subsarcolemmal space, we use the term ”spontaneous release“ here to describe the model SR Ca²⁺ release (variable j_SRCarel), which is not induced by Ca²⁺ influx via I_CaL during APs, but rather initiated by SR itself (e.g., during DD or when all membrane currents are set to 0).

System robustness.

We determine robustness as the ability of the pacemaker cell system to perform, i.e., to generate normal, high-amplitude APs, when one or more key parameters of the system change. The larger the parameter variation without compromising the system performance, the more robust is the system.

RESULTS

SANC SR is a Release-Pumping-Delay Oscillator

It has been experimentally documented in chemically skinned and voltage-clamped SANC, in which effects of voltage-activated sarcolemmal ion currents are excluded, that the isolated SR is capable to spontaneously and rhythmically release Ca²⁺ (67, 70). Thus our numerical SR subsystem (for variables y₁–y₁₄, ”SR Ca²⁺ Clock“ in Table 1) of the entire SANC model was first explored in isolation of membrane clock (all ion currents = 0) and tuned to operate as a self-sustained intracellular Ca²⁺ oscillator in line with experimental results.

SR system behavior was numerically analyzed by grading two key parameters: SR Ca²⁺ uptake rate (P_up) and SR Ca²⁺ release rate constant (k_s). We chose these parameters because: 1) they are expected to change from their original values of Luo-Rudy model and Shannon model of ventricular myocytes; and 2) prior experimental studies suggested that they likely mediate the regulation of SR Ca²⁺ clock ticking speed (67). PKA has a relatively high activity in SANC (compared with myocytes) in the basal state, resulting in phosphorylation of phospholamban and RyRs (67), which enhance SR Ca²⁺ pumping and affect SR Ca²⁺ release, respectively.

We found that the isolated SR can indeed operate as a self-sustained Ca²⁺ oscillator (53). An example of such an oscillator is shown in Fig. 2A, and its operation can be described by a simple “release-pumping-delay” mechanism: a small spontaneous Ca²⁺ release from jSR to the subspace occurs as the primary or initiating event. When Ca²⁺ in the subspace (Ca_sub) increases to a sufficient level, it amplifies the Ca²⁺ release via Ca²⁺-induced Ca²⁺ release (CICR); this relatively strong, secondary Ca²⁺ release simultaneously depletes (i.e., resets) jSR. The released Ca²⁺ is pumped into the nSR. The delay between releases is determined by the Ca²⁺ pumping rate and Ca²⁺ diffusion from subspace to cytosol and also from nSR to jSR. As Ca²⁺ in jSR (Ca_jSR) slowly increases, RyRs are restituted, and the next release is ultimately initiated, etc.

Fig. 2.

Emergence and dynamic characteristics of the isolated Ca²⁺ SR oscillator (osc.) with a simple “release-pumping-delay” mechanism in our new model of rabbit SANC (all membrane currents = 0). A: model simulations illustrating that SR can operate as a self-sustained Ca²⁺ oscillator, P_up = 20 mM/s. Note: for this and other simulations, we provide only those model parameters, which are different from our basal firing model, which is fully described in the online data supplement. B and C: results of parametric analysis for the isolated SR Ca²⁺ oscillator with simultaneously varying SR Ca²⁺ release rate and SR Ca²⁺ pumping rate are shown in terms of oscillation frequency (B) or amplitude (C). Each image shows the result of 161 × 161 = 25,921 simulations (each pixel = 5 μs⁻¹ × 0.5 mM/s) with the resultant frequency or amplitude coded by the color map shown on each image top; the parametric space yielding no sustained Ca²⁺ oscillations is shown in blue. Small green box represents the Ca²⁺ oscillator shown in A. k_s, SR Ca²⁺ release rate constant.

Dynamic Characteristics and Major Limitations of the Isolated SR Ca²⁺ Oscillator

Our broad 2-D-parametric analysis of simultaneous k_s and P_up changes revealed that the SR Ca²⁺ oscillations persist over a wide range of amplitudes and frequencies from 1.3 to 6.1 Hz (graded by red-white color in Fig. 2, B and C) that surpasses the entire range of heart physiological rates. This extended range of rates of the Ca²⁺ oscillator is limited by bifurcation border of the cone in Fig. 2B. The highest rates are reached when the oscillator approaches to dynamic equilibrium (steady release) at the bottom border of the cone. The lowest rates are reached when the oscillator approaches to static equilibrium (no release) at the upper border of the cone.

Static equilibrium occurs when either the release rate is too small or Ca²⁺ pumping rate is too fast, i.e., jSR becomes highly loaded, but Ca_sub level is low and insufficient to induce secondary Ca²⁺ releases (Fig. 3C, left). A small steady background RyR release is well-balanced by Ca²⁺ pumping. The system behaves as a trigger in this instance: a short Ca²⁺ injection into subspace produces a strong Ca²⁺ transient via CICR (data not shown), similar to CICR in ventricular myocytes during an AP.

The oscillator behavior is simple–the higher its frequency, the less is its amplitude (Fig. 2, B vs. C)–and can be easily explained as Ca²⁺ release-pumping interplay in our simulations. Increasing k_s at a fixed P_up (Fig. 3, A and C) or decreasing P_up at a fixed k_s (Fig. 3B) boosts the oscillation frequency but simultaneously causes Ca²⁺ accumulation in the subspace (Ca_sub in Fig. 3C, solid traces) that facilitates subsequent CICR. In turn, the oscillation amplitude decreases as the maximum loading of jSR drops (Fig. 3C, dotted traces) because of less time available for the SR Ca²⁺ reloading. As k_s increases (or P_up decreases) further, Ca²⁺ is not removed sufficiently quickly from the subspace: Ca²⁺ oscillations become damped at high levels of Ca_sub with high, persistent Ca²⁺ release (Fig. 3C, right, ”damped oscillator“). Thus, to sustain oscillations, a higher release rate must be matched with the faster Ca²⁺ removal from the subspace by faster pumping (Fig. 2, B and C, “Persistent oscillations”).

The above-described paradigm, “the higher the rate, the lower the amplitude,” represents the major limitation of the isolated Ca²⁺ oscillator: it is unable to generate high-amplitude oscillations at higher rates. Our further analysis will show that this problem is solved within the full (i.e., physiological) system of Ca²⁺ and membrane clocks.

Operation of Surface Membrane Clock within the System of Two Clocks

After we tuned parameters of the SR toward its experimentally observed Ca²⁺ oscillatory regime, we next explored the function of the system comprised of the membrane clock and the SR Ca²⁺ clock. Before our detailed parametric analysis, we first tested conceptually whether and how the membrane clock operates within such a system using an example of SR Ca²⁺ clock shown in Fig. 2A. Our simulations showed that membrane clock (Fig. 4, top traces) is indeed functional and apparently behaves in its well-known oscillatory pattern, as previously described (see review in Ref. 71). In short, I_CaL generates the AP upstroke, and rapid delayed rectifier current (I_Kr; in rabbit) repolarizes the cell membrane. Inactivation of I_Kr produces an early DD phase, followed by contributions from activation of hyperpolarization-activated or “funny” current (I_f), T-type Ca²⁺ current (I_CaT), sustained current (I_st), I_NCX, and sub-AP-threshold I_CaL during late DD. The cumulative action of these multiple DD mechanisms brings the membrane potential to the AP threshold, etc. Whereas our functional paradigm of SANC membrane clock includes well-known ion current types, their contributions, however, fundamentally differ from those in prior SANC models because of timely and strong interactions of SR Ca²⁺ clock with membrane clock in our SANC model system (vertical arrows in Fig. 4).

Fig. 4.

Model simulations showing major dynamic interactions (vertical lines) of membrane clock and SR Ca²⁺ clock during spontaneous AP firing in rabbit SANC. Shown are simulated traces of voltage (V_m), major ion currents, Ca²⁺ concentrations, and the Ca²⁺ release flux (j_SRCarel) during 2 pacemaker cycles. Dashed lines show 0 levels; P_up = 20 mM/s. DD, diastolic depolarization.

Major Interactions of SR Ca²⁺ Clock and Membrane Clock

SR Ca²⁺ and membrane clocks intimately interact biophysically and biochemically via coupling factors, such as Ca_sub, cAMP/PKA, and CaMKII, which influence function of both clocks (Fig. 5A). More specifically, I_CaL of membrane clock is modulated by high basal activities of CaMKII (69) and cAMP-dependent PKA (52, 67); I_CaL inactivation is accelerated by Ca_sub; I_f is modulated by cAMP by shifting its voltage-dependent activation (12); and I_NCX is greatly increased by spontaneous intracellular Ca²⁺ releases (5). As to SR Ca²⁺ clock, its phospholamban, SR Ca²⁺ pump (SERCA), and RyRs are also modulated by Ca²⁺, PKA, and CaMKII (Fig. 5A). The interactions of clocks via Ca_sub and cAMP/PKA result in robust generation of rhythmic APs in the model, as described below. Coupling via CaMKII was not specifically simulated here because of limited data in SANC (but see Refs. 33, 69).

Fig. 5.

Enhanced flexibility of the rate modulation in our model is based on mutual functional dependence of Ca²⁺ and membrane clocks. A illustrates factors that couple the 2 clocks comprising the novel pacemaker mechanism. B: damped SR Ca²⁺ release oscillations and I_NCX oscillations when membrane clock is instantly disabled by a voltage clamp to −65 mV; P_up = 20 mM/s. C: different patterns of Ca²⁺ release dynamics at different P_up of 20 and 40 mM/s in the isolated Ca²⁺ SR oscillator (all membrane currents = 0), simulated with the initial cell Ca²⁺ load of steady-state AP firing such as shown Fig. 4. The steady oscillations at 40 mM/s are shown after 40 s of simulated SR activity. D: simulations of V_m and I_NCX dynamics when Ca_sub was fixed at the maximum diastolic potential (MDP) (solid curves) or when I_CaL was completely blocked at the MDP (dotted curves), resulting in afterhyperpolarization or delayed afterdepolarization (DAD), respectively; P_up = 20 mM/s.

We identified the essential interactions of SR Ca²⁺ clock and membrane clock by analyzing dynamics of the key components of the system (Fig. 4). We also compared Ca²⁺ dynamics in the full system with those in the isolated SR Ca²⁺ clock (Fig. 4 vs. Fig. 2A, with SR having the same parameters).

1. AP ignition by I_NCX during late DD.

Spontaneous primary releases during mid-DD, and then secondary releases during late DD occur in the full system as in isolated SR Ca²⁺ clock, but the secondary releases activate I_NCX (Fig. 4, vertical red arrows). This critical interaction, predicted by the model, is in good agreement with experimental results in rabbit SANC demonstrating that the late diastolic I_NCX is strongly activated by the occurrence of spontaneous local SR Ca²⁺ releases into the submembrane space (5). It has also been experimentally demonstrated that these spontaneous, local releases are, in fact, Ca²⁺ wavelets (5), which are driven by propagating CICR and described by secondary release in the model. I_NCX activation occurs at the critical phase of the pacemaker cycle, the late DD, when I_Kr and I_f DD mechanisms have faded out, but I_CaL is not yet appreciatively activated. Whereas I_NCX, I_CaL, I_CaT, and I_st contribute to the late DD, I_NCX contribution is the greatest in our model (Fig. 4) as demonstrated in experimental studies (4, 5, 74). DD acceleration via I_NCX activation brings the membrane to its excitation threshold and thereby basically ignites rhythmic spontaneous APs (Fig. 4, “DD acceleration”). Acceleration of late DD by I_NCX was also demonstrated experimentally in rabbit SANC (4, 5).

2. SR depletion/resetting.

Compared with Ca²⁺ dynamics of isolated SR Ca²⁺ clock (Fig. 2A), the full system generates additional Ca²⁺ release that occurs during AP via the conventional I_CaL-related CICR (Fig. 4, “Third release”), resulting in conventional Ca²⁺ transient and strong jSR depletion, resetting SR Ca²⁺ cycling. In contrast to a relatively slow SR “self-resetting” (Fig. 2A), I_CaL resets SR quicker and more efficiently as it almost instantly boosts Ca_sub (Fig. 4, Ca_sub blue trace).

3. SR refueling.

Isolated Ca²⁺ clock (Fig. 2A) has a fixed total amount of Ca²⁺ that is defined by the model operator, which chooses initial conditions for simulations (mainly by the initial Ca_nSR level). However, when the SANC model is spontaneously firing rhythmic APs, the amount of Ca²⁺ in the system (the “fuel”) is mainly determined by the balance of Ca²⁺ efflux by NCX and influx by I_CaL. In other words, the system of interacting SR and sarcolemma tunes and equilibrates a specific appropriate amount of cell Ca²⁺ at a given rate. For example, the full system in Fig. 4 tunes its Ca_nSR to ∼2.2 mM, that is ∼63% higher than it was set initially in the system (1.35 mM; see Table 1). Refueling/rewinding the Ca²⁺ cycling is achieved during the AP, as Ca²⁺ influx of I_CaL is made available for pumping into nSR (Fig. 4, SR “refueling”).

4. Subspace depletion/resetting.

NCX assists SR Ca²⁺-ATPase in Ca²⁺ removal from the subspace, including during the DD, so that Ca_sub declines after AP to the levels that are below than the Ca²⁺ in bulk cytosol (Ca_i) (Fig. 4, bottom, right blue arrow). In contrast, when SR Ca²⁺ clock operates without the membrane clock, Ca_sub remains always higher than Ca_i (Fig. 2A). This NCX-assisted “cleanup” of Ca²⁺ in submembrane space during early and mid-DD delays the Ca_sub increase caused by primary Ca²⁺ release and thereby prevents premature secondary Ca²⁺ release in our model. This, in turn, gives sufficient time to reload jSR to higher levels, culminating in strong synchronized spontaneous release during later DD and larger Ca²⁺ oscillation amplitude (e.g., 472 vs. 255 μM for Ca_jSR oscillation amplitude in Fig. 4 vs. Fig. 2A).

The Concept of System Ca²⁺ Clock

Thus a major result of our analysis so far is that a reductionist approach separating functional paradigm of either clock makes no sense for the physiological system of strongly coupled oscillators during spontaneous AP firing. The two oscillators tune their operation and phases toward mutually entrained operation so that they operate in antiphase (spontaneous release → AP → spontaneous release, etc.). In this instance, the Ca²⁺ clock paradigm is not limited to the SR Ca²⁺ cycling but also includes SR-sarcolemma interactions. Thus the system Ca²⁺ clock (or physiological Ca²⁺ clock; “System Ca²⁺ clock” in Fig. 1) in addition to its SR Ca²⁺ loop (“SR Ca²⁺ clock” in Fig. 1) also includes the sarcolemmal Ca²⁺ loop (bold arrows in Fig. 1) formed by function of Ca²⁺ channels and NCX.

The importance of this additional loop for Ca²⁺ cycling is easily tested in our model by an instant voltage clamp to −65 mV. These conditions halt activation of voltage-dependent ion currents and their balanced sarcolemmal Ca²⁺ fluxes. If those fluxes were minor for the system Ca²⁺ clock function, Ca²⁺ oscillations would persist similarly to that shown in Fig. 2A. However, our model predicts that the Ca²⁺ clock with instantly disrupted physiological links damps after a few oscillations rather than persists (Fig. 5B). This model prediction recapitulates prior experimental findings in rabbit SANC of the damped oscillatory submembrane Ca²⁺ signals under voltage clamp at −65 mV (70).

Since cell Ca²⁺ load is critically important for function of Ca²⁺ clock, another method to test the importance of the membrane Ca²⁺ loop is to compare function of SR Ca²⁺ clock and the system Ca²⁺ clock under the same Ca²⁺ load that is achieved during steady-state AP firing shown in Fig. 4. We calculated steady-state averages for Ca²⁺ in all four cell compartments: cytosol (Ca_i = 219 nM), submembrane space (Ca_sub = 456 nM), jSR (Ca_jSR = 0.217 mM), and nSR (Ca_nSR = 2.2 mM), represented by variables y₁–y₄ in Table 1. A given steady-state was insured by our standard criterion (see methods) of <0.01% change (cycle to cycle) in AP amplitude and cycle length. All four steady-state averages were then used as initial conditions for respective variables in new simulations of the isolated SR Ca²⁺ dynamics (all sarcolemmal currents were set to 0). As a result, we got damping of Ca²⁺ oscillations (Fig. 5C, top), a response basically similar to that obtained with instant voltage clamp to −65 mV (Fig. 5B). We also found that steady Ca²⁺ oscillations of isolated SR under these cell Ca²⁺ loading conditions are still possible but require a substantial P_up increase from 20 to >29 mM/s (Fig. 5C, bottom). Thus these simulations provided additional evidence for our concept of the full system Ca²⁺ clock: although isolated SR can persistently oscillate any given Ca²⁺ load with a respective increase in P_up, the synergism of SR and sarcolemma within entire SANC pacemaker system allows robust Ca²⁺ oscillations at much lower (i.e., physiologically reasonable) P_up values.

Ca²⁺ Clock Confers System Flexibility: the AP Firing Rate is Broadly Modulated by the Ca²⁺ Clock Ticking Speed

Although the system Ca²⁺ clock clearly depends on sarcolemmal Ca²⁺ fluxes, the critical late DD Ca²⁺ signals are still generated by spontaneous SR Ca²⁺ releases. Simulations under voltage clamp at −65 mV (Fig. 5B) revealed the true spontaneous nature of these releases and their I_NCX counterparts/spikes at the time of late DD (“would-be” DD). Next, we evaluated the “power” of those NCX spikes, i.e., how much depolarization they can produce in our model. We compare membrane depolarizations and I_NCX in two simulations: 1) when Ca_sub was instantly fixed at its maximum diastolic potential (MDP) level of 116 nM (no Ca²⁺ release); and 2) when I_CaL was inhibited also at MDP (Fig. 5D). In its essence, the first simulation (solid trace in Fig. 5D) shows a smooth afterhyperpolarization without ignition, and the second simulation (dotted trace in Fig. 5D) shows an oscillatory delayed afterdepolarization produced by the oscillatory spontaneous release on the top of this afterhyperpolarization. The I_NCX difference at its first peak was 18.4 pA, and depolarization difference was 14.7 mV. Thus Ca²⁺ release-induced I_NCX spikes are sufficient to timely bring the membrane potential to −38.8 mV, i.e., well above I_CaL activation threshold, from the level of afterhyperpolarization (−53 mV) reached by the early and mid-DD mechanisms (mainly I_Kr inactivation and activation of I_f and I_CaT). Prior experimental studies showed that an ion current of only a few picoamperes scale can indeed drive the critical DD phase (11). Thus the spontaneous Ca²⁺ releases can be interpreted as Ca²⁺ clock ticks and their NCX spikes as ignition currents. Our model simulations thus support ideas about importance of late DD and I_NCX for cardiac cell pacemaker function in general (55, 74) and for function of rabbit SANC specifically (4, 5).

We then numerically tested an hypothesis that grading Ca²⁺ clock ticking speed strongly modulates the AP firing rate in our model. Grading P_up from 1 to 40 mM/s, indeed, greatly (2.3-fold) increased the AP firing rate from 1.8 to 4.2 Hz (Fig. 6A). Our additional simulations (Fig. 7) revealed the mechanism of the broad rate modulation by P_up, which includes coordinated changes of phase and amplitude for the diastolic Ca²⁺ release (i.e., Ca²⁺ clock ticks; Fig. 7C). At higher pumping rates, both the amplitude and the frequency of Ca²⁺ oscillations increase due to a substantial shift in the net cell Ca²⁺ balance: I_CaL is activated more frequently, resulting in a larger net Ca²⁺ influx. This larger Ca²⁺ influx combined with faster pumping increases the average Ca_nSR level (Fig. 7A) and rate of jSR refilling (Fig. 7B). Higher jSR Ca²⁺ loads are achieved in shorter times (Fig. 7B), thereby insuring larger amplitude of diastolic Ca²⁺ release (Figs. 6B and 7C) and larger NCX currents (i.e., ignition currents; Fig. 7D) to quickly bring membrane potential to AP excitation threshold, i.e., to ignite APs (Fig. 7E). Larger NCX currents, in turn, simultaneously balance increased Ca²⁺ influx via I_CaL.

Fig. 6.

A: the AP firing rate is broadly and smoothly modulated by P_up, yielding a unique parametric solution for the 3-Hz basal AP rate (black circle). B: interactions of Ca²⁺ and membrane clocks result in higher amplitude Ca²⁺ oscillations at higher P_up values. Since higher P_up values result in higher AP rates (A), the system clock operates following a Bowditch-like pattern: “the faster rate, the stronger release” (inset) thus overcoming a major functional limitation of the isolated SR Ca²⁺ clock (“the faster, the weaker” in Fig. 3).

The interactions of SR and sarcolemma described above overcome the major limitation of the isolated SR Ca²⁺ clock, i.e., its inability to sustain high-amplitude oscillations at high rates (i.e., “the faster, the weaker”; Fig. 3). In contrast, the oscillation amplitude of the full system Ca²⁺ clock raises as the oscillation rate increases (i.e., “the faster, the stronger”; Fig. 6B, inset).

Finding the System Solution for the Basal State AP Firing by Interacting Clocks

Since P_up modulates AP firing rate so smoothly and broadly (Fig. 6A), the P_up-AP rate modulation curve yields a unique parametric solution basically for any given reasonable value of the basal AP firing rate. A conventional, mid-range AP basal rate of 3 Hz measured in rabbit SANC (Table 2) is reached in our model somewhere in the middle of the most effective, almost linear range (1–25 mM/s) of rate modulation by P_up. The respective unique parametric solution of P_up = 12 mM/s for this basal rate (3 Hz) is shown in Fig. 6A by black circle. This solution is further explored as basal state firing SANC model.

Table 2.

Basic AP characteristics of our basal firing model (in bold) are similar to those of some previous SANC models and within experimental data

Characteristics	Noble and Noble (49) Central SANC Model	Zhang et al. (73) Central SANC Model	Demir et al. (10) SANC Model	Kurata et al. (27) Central SANC Model	Present Model of Central SANC	The Range of Experimental Values
MDP, mV	−61	−58	−61	−59	−62.7	−66.2 to −51.9
APA, mV	84	79	96	75	75.9	66–78
CL, ms (AP rate, Hz)	263 (3.8)	327 (3.06)	263 (3.8)	307.5 (3.26)	333 (3)	247–389 (2.6–4.1 Hz)
APD50, ms	70	139	86	107	101	72.7–111
dV/dtmax, V/s	4.7	2.7	9.6	6.4	4.8	4.8–27

All data for prior models were taken from Table 3 in Ref. 71. The ranges of experimental values were evaluated based on data in Table 1 in Ref. 27 (compiled from 12 studies that used physiological temperatures). MDP, maximum diastolic potential; APA, action potential (AP) amplitude; APD₅₀, AP duration at its half amplitude; dV/dt_max, the maximum rate of rise of membrane potential during AP upstroke; CL, cycle length.

Major Electrophysiological Characteristics of Basal Firing SANC Model

Basic AP characteristics (Table 2) of our basal AP firing model are within the range of experimental data and similar to those of previous SANC models. Predictions for major ion current densities during spontaneous SANC firing for different models are listed in Table 3. Our model has typical densities for I_f and I_CaL. Its predictions for I_CaT and I_st densities are smaller than some but not all prior models; our values for these currents are reasonable, given the experimental facts: 1) I_CaT was detected only in 20% of SANC (in 5 of 25 cells tested; Ref. 44), and its experimental blockade has small effect on the spontaneous beating rate (see review in Ref. 71); and 2) molecular origin and specific blockers for I_st have not yet been identified; I_st, in fact, exhibits many properties of I_CaL and I_NCX and, therefore, may not be a discrete entity. Furthermore, I_st is not incorporated in many classic models (71), including the Kurata model (Ref. 27; its version with I_st = 0). A larger density of the late diastolic I_NCX in our model is related to NCX activation by spontaneous diastolic Ca²⁺ release (Ca²⁺ clock ticks, discussed above). This larger diastolic I_NCX density of 0.458 pA/pF is within experimental estimates of the ryanodine-sensitive current during DD from 0.3 pA/pF (68) to 1.6 pA/pF (5). Whereas the Kurata model also predicts substantial diastolic I_NCX (0.328 pA/pF; Table 3), its time course is fundamentally different from that in our model (Fig. 8). The I_NCX dynamic in Kurata model (Fig. 8, dotted lines) exhibits a decrease followed by a plateau throughout almost the entire DD; the subsequent I_NCX surge during the AP upstroke is driven by I_CaL activation and a conventional AP-induced Ca²⁺ transient. In contrast, the I_NCX dynamic in our model exhibits substantial increase throughout the mid- to late DD, i.e., clearly before I_CaL activation (Fig. 8, solid lines). This I_NCX increase is driven by spontaneous diastolic Ca²⁺ release from SR (Fig. 7C) that is also observed in experimental studies of rabbit SANC (5, 68) but missing in the original Kurata model. It is important to note that it is ion current dynamics (i.e., the change in ion current balance), rather than just ion current instant values, that drive the DD. For example, although K⁺ delayed rectifier current has an outward direction, it greatly contributes to the early DD when its conductance inactivates (46). Thus the strongly increasing I_NCX during DD is a major unique feature of our basal firing model that makes it superior in predicting numerous, highly reproducible experimental findings related to Ca²⁺ cycling and its interaction with the sarcolemma in SANC.

Table 3.

Amplitudes of ion currents [contributing to mid- to late diastolic depolarization (DD)] predicted by our basal firing model (in bold) compared with some previous SANC models

Characteristics	Noble and Noble (49) Central SANC Model	Zhang et al. (73) Central SANC Model	Demir et al. (10) SANC Model	Kurata et al. (27) Central SANC Model	Present Model of Central SANC
Maximum diastolic If, pA/pF	0.061	0.1	0.073	0.109	0.068
Maximum ICaT, pA/pF	No ICaT	0.353	0.188	0.227	0.09
Maximum peak ICaL, pA/pF	4.95	3.46	11.00	6.87	5.35
Maximum peak Ist, pA/pF	No Ist	No Ist	No Ist	0.40	0.1
Late diastolic INCX, pA/pF	0.06	0.03	0.15	0.328	0.458

Late diastolic Na⁺-Ca²⁺ exchanger current (I_NCX; bottom row) was measured at 85% of cycle length from the upstroke at 0 mV of the prior AP (e.g., Fig. 8, I_NCX). The current densities of previous models were estimated from simulations presented in review by Wilders (71).

Fig. 8.

In contrast to Kurata model (dotted curves), our basal firing model (“Present model,” solid curves) exhibits strong increase of inward I_NCX during diastolic depolarization before activation of I_CaL. Simulated dynamics of membrane potential (V_m), I_NCX, and I_CaL at a steady-state AP firing are shown for 2 pacemaker cycles beginning at V_m = 0. The cycle lengths were 307.5 and 333 ms for Kurata model and our basal model, respectively. The vertical dash-dot line of 75% of cycle length illustrates approximate timing for I_CaL activation, i.e., when I_NCX was measured in Table 3.

Basal Firing Model Predicts Effects of Experimental Inhibition of Ca²⁺ Clock, NCX, and I_f

Effect of SR Ca²⁺ pump inhibition.

Two recent pharmacological studies (56, 66) provided new, direct evidence of the importance of SR Ca²⁺ cycling kinetics for basal rate in SANC. In both studies, inhibition of SR Ca²⁺ pump by cyclopiazonic acid (CPA) resulted in substantial decrease of SANC beating rate by 30% (guinea pig) and 26% (rabbit), respectively. Inhibition of Ca²⁺-ATPase by CPA in rabbit SANC can be easily interpreted and tested in our model in terms of a reduction in the SR Ca²⁺ pumping rate, P_up. The P_up-AP rate modulation curve predicted by our model extends to very low rates, down to 1.8 Hz, i.e., 40% decrease (Fig. 9A) from its basal rate value of 3 Hz. Thus our parametric (P_up) analysis yields a unique solution for the CPA (3 μM) experimental result, with P_up decreasing to 4 mM/s, i.e., to approximately one-third of its basal value of 12 mM/s (Fig. 9A, “Present model”). An example of model simulation for the CPA effect is shown in Fig. 9B. It is important to note that this prediction is an example of the unique property of our new model, i.e., the model exhibits a distinguishably wide range of rate modulation by Ca²⁺ cycling. All prior SANC models have extremely narrow (if any) rate modulation by SR Ca²⁺ pumping and therefore have no solution for the experimental CPA effect. For example, as shown in Fig. 9A, Zhang et al. (73) model has no equations for Ca²⁺ cycling; the Kurata model predicts almost negligible (only 3.5%) rate decrease on complete SR Ca²⁺ pump inhibition; and, paradoxically, the Kyoto model of guinea pig SANC (57) predicts a slight AP rate increase rather than decrease as SR Ca²⁺ pump is inhibited.

Fig. 9.

Predictions of chronotropic perturbations by our basal AP firing model. A: finding the system solution (Present model, black box) for the rate reduction produced experimentally by a specific SR Ca²⁺ pump inhibition with cyclopiazonic acid (CPA) (66). Also shown: predictions of small (if any) rate modulation produced by changes in the Ca²⁺ SR pumping rate in Kyoto model (Ref. 57; “Kyoto”), Zhang et al. (73) model (“Zhang”), and Kurata model. B–E: predictions of our basal model for voltage (V_m) dynamics, with instant model parameter changes (black bars) simulating various chronotropic interventions indicated by respective labels in the panels. k_NCX, scaling factor for I_NCX.

Effect of short-term inhibition of NCX by lithium.

Our basal state model reproduces the effect of the “skipped” AP on a short-term experimental inhibition of NCX by a rapid and brief substitution of Li⁺ for Na⁺ (lithium spritz; Ref. 5; Fig. 9C).

Effect of large concentrations of ryanodine.

It is widely accepted that ryanodine, which interferes with normal SR Ca²⁺ release, slows basal AP firing rate of rabbit SANC. Depending on its concentration and time after administration, the range of the effect varies from moderate AP rate slowing to complete halt (reviewed in Ref. 30). The most recent study by Lyashkov et al. (37) showed that spontaneous AP firing fails in 83% of cells after 15 min at a high ryanodine concentration of 30 μM. In line with this experimental result, our basal state model fails to generate rhythmic APs when SR Ca²⁺ cycling is completely inhibited (Fig. 9D). This is another example illustrating that Ca²⁺ and membrane clocks are strongly coupled during basal AP firing: once rhythmic ignition currents initiated by Ca²⁺ clock are halted, membrane clock undergoes a lengthy transitional process including subthreshold oscillations, which eventually generate dysrhythmic excitations.

Modest effect of I_f blockade by Cs⁺.

In our basal model, blockade of I_f (Fig. 9E) results in the rate reduction by 5.0%, i.e., close to 5.2% measured experimentally in single rabbit SANC when I_f was blocked by Cs⁺ (65).

Ca²⁺ Clock Confers the System Robustness

Next, we tested a hypothesis that clock interactions enhance robustness of the SANC pacemaker system. For this purpose, we examined the system behavior on simultaneous variations of key system parameters of P_up and g_CaL. This analysis (Fig. 10A) identified P_up-g_CaL areas of rhythmic APs, chaotic APs, and no firing of the entire system. The areas of both chaotic firing and no firing greatly narrow toward lower g_CaL values as P_up rises, and, vice versa, gradually increasing g_CaL extends the P_up range of rhythmic firing. As defined in methods, the larger the parameter variations, in the absence of the system performance compromise, define the more robust system. Thus our analysis shows that although the membrane-delimited clock (at P_up = 0, i.e., along the abscissa) can indeed generate rhythmic APs within a limited g_CaL range, the system of Ca²⁺ and membrane clocks is more robust because the combined system greatly performs in the range of low g_CaL values, where the classic membrane-delimited clock fails. For example, at P_up = 12 mM/s, the system is able to generate rhythmic APs at extremely low I_CaL conductances, down to 0.295 nS/pF (green dash line); but when the Ca²⁺ clock is disabled (P_up = 0), the cell membrane needs a substantial (58.5%) increase of its I_CaL conductance to 0.4675 nS/pF for spontaneous APs just to emerge (on the right of the red dash line in Fig. 10A). Also, whereas the full system generates high-quality frequent APs at various g_CaL, the membrane clock operating alone halts, generates chaotic APs, or greatly slows (Fig. 10B).

Fig. 10.

Ca²⁺ clock confers the system robustness. A: our parametric system analysis with simultaneous variations in I_CaL conductance and SR Ca²⁺ pumping rate. The image shows the result of 97 (x) × 61 (y) = 5,917 simulations (each pixel = 0.0025 nS/pF × 0.2 mM/s). The resultant AP firing rate of the system is illustrated by graded color coded by the color map on the right; areas of no firing and chaotic firing are also marked and separated by yellow line. B: simulation traces for specific points of interest in A, illustrating more robust system operation when Ca²⁺ clock is functional. C: simulations of the effect of moderate PKA inhibition by 1.7 μM PKI (represented by vector bd in A). M clock, membrane clock.

Next, we compared the results of g_CaL-P_up parametric analysis at different I_f conductances (g_If). This comparison revealed that although I_f provides only small contribution to basal firing (Fig. 9E), it greatly increases the area of rhythmic firing, especially at low rates (along the entire border line of rhythmic firing in Fig. 11) and thus also greatly enhances the system robustness. Thus our overall parametric analysis for simultaneous changes of g_If, g_CaL, and P_up (Figs. 10 and 11) reveals that the existence of strong interactions of respective multiple pacemaker mechanisms (i.e., I_CaL, I_f, and I_NCX activated by Ca²⁺ clock) greatly enhances the robustness of the coupled SANC pacemaker system. Specifically, any enhancement of either mechanism broadens the fail-safe ranges for the other two mechanisms (Figs. 10 and 11): 1) increasing P_up broadens fail-safe changes for g_CaL and g_If; 2) increasing g_CaL broadens fail-safe changes for g_If and P_up; and 3) increasing I_f broadens fail-safe changes for g_CaL and P_up.

Fig. 11.

I_f increases the system robustness at low AP rates. Our parametric system analysis with simultaneous variations in I_CaL conductance and SR Ca²⁺ pumping rate was performed at 2 extreme settings: no I_f (A) and large I_f (B). The system featuring I_f has smaller areas of no firing and chaotic AP firing. The results are illustrated by images, with the size (5,917 simulations) and the pixel size (0.0025 nS/pF × 0.2 mM/s) being identical to those in Fig. 10A. Simulations for some specific points of interest within the explored parametric space are depicted on the left. g_If, I_f conductance.

High I_CaL and I_f Conductances Limit Pacemaker Flexibility

The results and logic of the above section lead, however, to an apparent paradox. The natural evolution of the pacemaker system has somehow culminated in a less robust membrane clock, whereas its higher robustness could be very easily achieved by a simple increase in g_CaL and/or g_If, e.g., an additional Ca²⁺ channel and/or f-channel expression. To solve the paradox, we asked the question, “what might be an additional functional advantage of our modeled coupled clock system with a moderate g_CaL and/or g_If vs. a system with very high g_CaL and/or g_If?” Our model simulations show that the range of rate regulation by P_up substantially shrinks as g_CaL or g_If increases (Fig. 12). We interpret this result as follows: although higher g_CaL or g_If enhances the system robustness, this substantially limits the flexibility of the pacemaker function (at least via P_up modulation). Thus the experimental data (e.g., ryanodine effect) show, and our new model validates the experimental data, that the real SANC membrane clock operating alone is more fragile compared with that portrayed by previous essentially membrane-delimited SANC models (all those models had no or little dependence on Ca²⁺ cycling); but, at the same time, the coupled clock system is very robust and, importantly and advantageously, also very flexible. In other words, the membrane clock seems to forgo some its high robustness to allow the smooth and wide range of rate regulation via interactions of Ca²⁺ clock and membrane clock. “In return” the Ca²⁺ clock greatly enhances (i.e., recovers) the coupled system robustness (Fig. 10A) so that the coupled system likely achieves the highest possible balance of robustness and flexibility that is critical for effective pacemaker function (and thus for survival).

Fig. 12.

The range of regulation of spontaneous (Spont.) AP firing rate by P_up substantially shrinks as g_CaL (A) or g_If (B) increases. The curves illustrate the results of our model sensitivity analysis (basal model) for spontaneous AP rate change, with P_up varying from its basal value of 12 mM/s. The respective conductance values are shown at the curves, with 2× and 3× representing their multiples.

Prediction of Rate Reduction by PKA Inhibition

Recent experimental studies discovered that emergence of spontaneous Ca²⁺ releases (i.e., Ca²⁺ clock ticking) in rabbit SANC is driven by a relatively high PKA activity in the basal state (67). More specifically, critical experimental evidence for this basal clock mechanism and its importance for both basal AP firing and rate regulation was obtained by using a PKA inhibitor PKI: grading basal PKA activity results in graded phospholamban dephosphorylation paralleled with graded rate decrease down to complete halt (67). In those experiments, a physiological-scale rate reduction by 22.6% was achieved by a moderate PKA inhibition with 1.7 μM PKI, and we tested whether our model can reproduce this important result.

The function of the SR Ca²⁺ pump and its regulatory protein phospholamban are mimicked by P_up variations in our model. Accordingly, PKI effect can be interpreted and tested in terms of P_up decrease. Since P_up broadly modulates the AP firing rate (Fig. 9A), a P_up reduction from 12 to 5.1 mM/s (vector bf in Fig. 10A) is sufficient to explain the 22.6% rate reduction. However, this P_up reduction by 57.5% is more than one might expect from the half-maximum effect of phospholamban dephosphorylation found somewhere between 1.7 and 5 μM PKI (67), pointing to a possible contribution of other mechanism(s).

Could it be sarcolemmal ion channels? In fact, I_f is modulated directly by cAMP (12) but not by PKA. I_Kr is not modulated by PKA but rather PKC (14); but slow component of the delayed rectifier K⁺ current (I_Ks), which is modulated by PKA (62), has a negligible basal density in rabbit SANC. The most plausible candidate is I_CaL, but experimental results on I_CaL modulation by PKA inhibition and/or by acetylcholine in SANC greatly vary from extremely strong inhibition by 85% (52) to no effect in at least two studies (19, 50). The difference in results can be due at least in part to technical difficulties to measure and to interpret I_CaL inhibition when the current undergoes substantial rundown in whole cell patch clamp (e.g., >50% during 5 min in Ref. 52).

Thus we tested the effect of a rather small inhibition of basal g_CaL such as 15.5% [half-maximum I_CaL inhibition reported by Zaza et al. (72)]. In practice, such small current changes are difficult to detect by patch-clamp technique because of registration noise and aforementioned I_CaL rundown. Surprisingly, with this small g_CaL change, the 22.6% rate reduction can be achieved in the model with a much smaller, 41.6% P_up decrease (Fig. 10A, vector bd vs. bf; simulation traces in Fig. 10C). Since the effect of g_CaL reduction per se is very small (only 2.8%, vector ba in Fig. 10A), g_CaL and P_up changes exhibit a noticeable synergism in their rate slowing. This result of the smaller P_up reduction combined with the small g_CaL reduction (rather than of the large P_up reduction alone) is in a better agreement with experimental result in rabbit SANC that the half-maximum effect of phospholamban dephosphorylation is reached between 1.7 and 5 μM PKI (67).

Extreme Rate Acceleration by PKA Activation with cAMP (Phosphodiesterase Inhibition)

We also tested whether our new model predicts important recent experimental results of extreme rate acceleration by phosphodiesterase inhibition, specifically a ∼47% rate increase by phosphodiesterase 3 inhibitor milrinone in rabbit SANC (68). It was shown that this effect is linked to dramatic (9-fold) increase of basal cAMP resulting in strong PKA activation. We introduced changes for ion currents that were reported under these conditions: g_CaL increased by 45%; I_Kr increased by 12%, and its activation was shifted by 5 mV toward negative potentials (68); and a saturation shift of I_f activation curve by cAMP to its half-maximum activation voltage (V_If,1/2) of −58.06 mV, predicted by Eq. 7 in Demir et al. (9) model, was also included.

Although the uptake rate P_up could not be directly measured experimentally in SANC, both cAMP and phospholamban phosphorylation were found to be dramatically increased by milrinone (68). Therefore, in addition to ion current changes noted above, we also gradually increased P_up (“Ca & M clocks” curve in Fig. 13A) in attempt to find the system parametric solution for this extreme case. Indeed, the system yielded a unique solution for the 47% rate increase when P_up was increased ∼4-fold (from 12 to 46 mM/s; black square in Fig. 13A). The system solution was also found at a slightly higher P_up of 55 mM/s, when membrane clock was not changed (“Ca clock only,” black circle in Fig. 13A).

Fig. 13.

Our model predicts extreme rate acceleration by PKA activation with cAMP increase (upon phosphodiesterase 3 inhibition with milrinone). A: finding system solution for the reported milrinone effect (47% basal rate increase) when membrane clock (I_CaL, I_f, and I_Kr) was changed according previously reported experimental data (68) and P_up was graded (“Ca & M clocks,” black box) or when only P_up was graded (“Ca clock only,” black circle) from basal state firing (“Basal”). Model predicts 12.8% rate increase when only membrane clock was changed (“M clock only”). B: simulations illustrating the mechanism of the milrinone effect. Compared are simulated traces of V_m, I_f, I_NCX, j_SRCarel, and Ca_i kinetics for basal state AP firing (“Basal,” dotted lines) vs. respective traces for the milrinone effect (“Milrinone,” solid lines), which is depicted by black box in the plot in A.

In contrast, when Ca²⁺ clock was not changed, membrane clock has a limited range of rate modulation: the aforementioned changes of membrane clock alone increased the AP firing rate by only 12.8% (black diamond in Fig. 13A), in line with experimental results that demonstrate the major role of Ca²⁺ clock in the SANC rate acceleration by phosphodiesterase inhibition (68). In contrast to precisely timed AP ignitions by Ca²⁺ clock, ion channels modulate AP rate via a complex interplay of their effects on both DD and AP durations. This leads to a paradoxical behavior of some classic models lacking Ca²⁺ clock: these models decrease their AP firing rate when I_CaL increases, i.e., opposite to experimental results in cardiac pacemaker cells (55). For example, aforementioned g_CaL increase by 45% produces a 3% rate decrease in the Kurata model, as the AP duration increase prevails (data not shown). Despite a prominent increase in the action potential duration at 50% repolarization (APD₅₀; from 101 to 113.4 ms), our basal firing model predicts a 3% rate increase in this instance. Thus our model overcomes the paradox of some classic models because in addition to its direct electrogenic contribution, increased I_CaL shortens DD by acting as a part of the system Ca²⁺ clock (Fig. 1). As to I_f, it does not affect AP duration, but its activation shift alone (to V_If,1/2 = −58.06 mV) produces only a minor rate increase in our basal model or Kurata model: 2.9 or 4.1%, i.e., when Ca²⁺ clock is unchanged or lacking, respectively.

We performed additional model simulations to further explore the mechanism of the milrinone effect (Fig. 13B). Interestingly, our model predicts notable decrease in diastolic I_f despite its activation shift and a much faster rate of rise. This diastolic I_f amplitude decrease occurs because of insufficient time for I_f to activate at this extremely high AP firing rate and short DD. In contrast, diastolic j_SRCarel amplitude approximately tripled, and the respective diastolic I_NCX more than doubled (17.7 → 47.3 pA or 0.55 → 1.48 pA/pF). This model prediction is very close to the experimental estimate of the I_NCX increase from 0.7 to 1.5 pA/pF measured in rabbit SANC under similar conditions (68). We also compared simulations of cytosolic Ca²⁺ (Ca_i) transient kinetics in control and with strong cAMP increase produced by milrinone (Fig. 13B, bottom). Milrinone shortened duration of the simulated AP-induced Ca²⁺ transients (from AP upstroke to 90% transient decay) by ∼20%, i.e., close to ∼25% measured experimentally in rabbit SANC under these conditions (68). Thus our model and experimental data validate each other to indicate that P_up likely substantially increases in SANC when intracellular cAMP surges.

It is important to note, again, that the model predictions shown in Fig. 13 are the unique property of the present model. Previous SANC models have very small sensitivity (if any) to variations in SR pumping rate (Fig. 9A) and, therefore, do not provide solutions either for rate slowing with CPA or for the extreme rate acceleration by phosphodiesterase inhibition.

DISCUSSION

We developed a novel unique numerical model of normal cardiac cell pacemaker function, which is based on classic formulations of SR and membrane function. The model requires a single, straight-forward assumption: the existence of submembrane Ca²⁺ clock in SANC, an assumption based on solid experimental evidence of the emergence and critical functional importance of spontaneous rhythmic subsarcolemmal releases in rabbit SANC under normal physiological conditions [original studies (4, 5, 37, 38, 42, 65, 67, 68, 70), editorials (2, 7, 34, 40), and recent reviews (31, 32, 41, 43, 45)]. By varying rates of SR Ca²⁺ pumping and release, we determined the parametric space of the oscillatory SR behavior and then explored how this modeled Ca²⁺ oscillator integrates with classic membrane voltage oscillator of SANC to insure robust and flexible spontaneous AP firing.

The concept of interacting oscillators has been also previously proposed as a potential mechanism of entrainment of individual SANC within sinoatrial node (22) and to pathological rhythms under Ca²⁺ overload conditions (63) or formed by more than single pacemaker area (“ectopic” pacemakers; Ref. 1) or by uncoupling of Ca²⁺ cycling and excitation waves within myocardium (54).

The Coupled Oscillators Model Validates Recent Experimental Results

Surprisingly, our numerical system of coupled oscillators reproduced basically all recently discovered new behavioral details of cardiac cell pacemaker function (listed below), none of which can be predicted by existing numerical SANC models [including Zhang et al. (73) model, Kurata model, and Kyoto model; Fig. 9A] that do not incorporate the Ca²⁺ clock.

1. AP ignition involves cooperative action of both Ca²⁺ and membrane clock.

The model predicts the occurrence of spontaneous submembrane releases (Ca²⁺ clock ticks) resulting in I_NCX activation (ignition currents) and late DD acceleration (Fig. 4, red arrows) as observed experimentally in rabbit SANC (4, 5, 42). A recent morphological study of SANC showed that RyRs are indeed strongly expressed and colocalized with NCX (37). Thus, in both our model formulation and spontaneously firing SANC, RyR-NCX cross talk insures effective coupling of the rhythmic spontaneous Ca²⁺ signals of the synchronous RyR openings to the late DD acceleration that is sufficient to reach the AP threshold from the levels of the afterhyperpolarizations (Figs. 5D and 7E) produced by early and mid-DD mechanisms such as I_Kr inactivation and I_f activation. It is important to note that prediction of the timing of spontaneous Ca²⁺ release occurrence is a specific characteristic of the novel model; this timing could not be predicted by our prior model of Ca²⁺ diastolic release driven by a simple sinusoidal function (42).

2. High level of PKA activity and phospholamban phosphorylation in the basal state in rabbit SANC (67).

In our model, P_up variations mimic the function of the SR Ca²⁺ pump and its regulatory protein phospholamban. Accordingly, the high level phospholamban phosphorylation is described well by our model because P_up solution for the 3-Hz basal rate was found in the middle of the P_up-rate modulation curve (Fig. 9A).

3. The enhanced system flexibility via Ca²⁺ clock modulation confers the novel pacemaker rate regulation mechanism.

The high level of PKA activity in the basal state insures basal Ca²⁺ clock ticking. Therefore, grading PKA activity up and down (e.g., via β-ARs and m-AChRs) varies the Ca²⁺ clock ticking speed up and down, resulting in broad rate regulation in rabbit SANC (Refs. 67 and 38, respectively). Although the present study does not specifically address the complex interplay of Ca²⁺ and membrane clocks in autonomic regulation of cardiac pacemaker function, our novel model strongly supports this new PKA-dependent (phospholamban phosphorylation) rate regulation mechanism: decelerating or accelerating Ca²⁺ clock ticking by grading P_up modulates the pacemaker rate within an extremely broad range from −40 to +53% from a basal rate of 3 Hz, respectively (Figs. 9A and 13A).

It is important to note, however, that the full-scale rate increase of +53% cannot be probably explained by the P_up modulation solely by the phospholamban phosphorylation mechanism. Indeed, such rate boost requires a P_up increase of up to ∼4-fold (Fig. 13A), whereas the P_up modulation via phospholamban phosphorylation is limited only to 2–3 times [depending on cytosolic Ca²⁺ (8)]. This indicates that, to reach the extreme rate acceleration, phospholamban/PKA signaling is likely further amplified and/or complemented by some additional factors, such as cell Ca²⁺ per se and CaMKII signaling (Fig. 5A). On the other hand, a physiological rate regulation range of ±20% of basal beating rate can be easily achieved in the model with physiologically reasonable (i.e., moderate) P_up changes via phospholamban/PKA signaling (either alone or better together with respective moderate ion channel changes). Indeed, a 22.6% rate reduction is achieved with a 41.6% of P_up decrease (vector bd in Fig. 10A); but +20% is achieved with a ∼33% or ∼75% P_up increase (Fig. 13A, “Ca & M clocks” or “Ca clock only,” respectively). We also want to note that the hypothetical physiological rate regulation by P_up is very effective in our model within the above mentioned moderate range but become saturated (i.e., ineffective) at higher P_up values (Fig. 13A).

4. Major rate and rhythm disturbances on Ca²⁺ clock obstruction.

These included effects of SR Ca²⁺ pump inhibition with CPA (Fig. 9B), NCX inhibition by acute extracellular Na⁺ replacement by Li⁺ (Fig. 9C), ineffective RyR function with ryanodine (Fig. 9D), and inhibition of PKA-dependent phosphorylation with PKI (Fig. 10C, vector bd in Fig. 10A).

Additional Novel Model Insights

Our new model not only validates recent experimental findings, but also provides novel mechanistic insights into cardiac pacemaker function, which can be neither logically deduced from the existing experimental data nor predicted by previous numerical modeling of SANC:

1. The first model of a physiological Ca²⁺ clock.

As mentioned in the Introduction, a contemplative mechanism of cardiac Ca²⁺ clock as cycles of Ca²⁺ uptake and release by the SR had been suggested as early as in 1979 by Tsien et al. (63). However, spontaneous Ca²⁺ oscillations in cardiac cells are traditionally associated with pathological conditions (61), i.e., specifically with high intracellular Na⁺ leading to Ca²⁺ overload (47). The novel and original finding of our study is that the physiological Ca²⁺ clock is not limited to the SR Ca²⁺ clock but also includes balanced sarcolemmal Ca²⁺ fluxes of I_CaL and NCX (Fig. 1, bold arrows) that substantially enhance the Ca²⁺ clock performance. The coupled physiological Ca²⁺ clock exhibits the robust functional paradigm “the faster, the stronger” (Fig. 6B, inset), whereas the isolated SR Ca²⁺ clock cannot sustain high-amplitude oscillations at higher rates, i.e., “the faster, the weaker” (Fig. 3). Our coupled Ca²⁺ clock operates at a physiological intracellular Na⁺ (10 mM) and physiological cell Ca²⁺ loads (reported by Ca_nSR levels). A reasonable range of basal AP firing rates from 2.56 to 3 Hz is achieved in our model with physiological Ca_nSR levels from 1 to 1.5 mM (Fig. 7), which were reported for rabbit cardiac cells (58). At higher rates, the Ca²⁺ load in our model increases (Fig. 7A), also in a good agreement with experimental results of larger caffeine-induced transients at higher AP firing rates, e.g., on β-AR stimulation (70) or phosphodiesterase inhibition (68).

In contrast to Shannon model of one compartment SR (i.e., Ca_nSR = Ca_jSR) of ventricular myocytes (59), an important characteristic of two-compartment SR function in our model is that the amplitude of Ca_nSR oscillations remains small compared with Ca_nSR average level (Fig. 7A). This allows maintenance of a substantial Ca²⁺ reserve in the system (in other words, the system keeps some fuel reserve in its “fuel tank”); recapitulating an experimental observation of strong caffeine-induced Ca²⁺ transients at any oscillation cycle phase in rabbit SANC (70). Our model simulations indicate that this might be important for robust pacemaker function: when Ca²⁺ pump is blocked (Fig. 9D), this Ca²⁺ reserve insures generation of additional/reserved AP ignitions before SR gets empty and system fails. Future biophysical measurements in SANC are required to clarify importance of this cell Ca²⁺ reserve as well as specific contributions of inter-SR diffusion, P_up, RyR restitution, Ca²⁺ channels, and NCX to the Ca²⁺ clock amplitude and ticking speed.

2. The fine structure of the SR Ca²⁺ release in SANC has three distinct phases (Fig. 4, red trace in bottom).

Our recent pilot study using a high-speed camera in rabbit SANC (51) has demonstrated Ca²⁺ sparks as initial form of Ca²⁺ release during DD, followed by propagating release before the AP-initiated Ca²⁺ transient.

3. Mechanisms of arrhythmic AP firing in the system are associated with uncoupling of Ca²⁺ and membrane clocks or uncoupling of components of membrane clock (e.g., I_CaL and I_f; Figs. 10 and 11, areas of “Chaotic firing”).

Our model predictions added I_f importance for the system performance when its firing rate decreases (Fig. 11), in line with recent surprising findings in HCN4 (I_f channel) knockout mice (15). It also supports an earlier concept first proposed by Noble et al. (48), i.e., I_f has more time to activate within longer cycle lengths and thus acts as a “stabilizer” of the pacemaker rate. As intrinsic cycle length is substantially longer in atrioventricular node vs. that in sinoatrial node, this concept also explains a substantially stronger effect of I_f blockade in the atrioventricular node (35).

4. Ca²⁺ clock confers robustness to the SANC pacemaker system.

For the first time, we described Ca²⁺ clock as a new critical dimension in robust initiation of cardiac impulse. Our broad parametric analysis demonstrates that any enhancement of either mechanism (P_up, I_CaL, or I_f) broadens the fail-safe ranges for the other two mechanisms (Figs. 10 and 11). The robust operation of the SANC pacemaker system is insured by tight cooperation of Ca²⁺ and membrane clocks as they commit their moderate combined resources for SANC basal firing (Fig. 10A, “Basal state”), preserving substantial reserve to effectively handle extreme perturbations. In our example of simulation of an extreme rate increase produced by phosphodiesterase inhibition (Fig. 13), the preserved reserve of Ca²⁺ clock (i.e., reserve in P_up rate modulation) is required and sufficient to effect a remarkable, 53% AP rate increase. Under these conditions, intracellular cAMP raises by ninefold (68) and supposedly saturates an I_f activation voltage shift and thereby increases I_f rate of rise; however, the diastolic I_f amplitude still decreases because of insufficient time for activation during shorter DD duration (Fig. 13B, I_f). Thus reserve recourses of membrane clock alone (when Ca²⁺ clock parameters are fixed in our model) insure rate increases of only up to 12.8% (black diamond in Fig. 13A), i.e., insufficient to reach such higher (+53%) rates.

About Balance of Robustness and Flexibility in the Present and Prior SANC Models

We interpreted the results of our simulations in the section High I_CaL and I_f Conductances Limit Pacemaker Flexibility (Figs. 10–12) to indicate that the system of coupled clocks with moderate (i.e., physiologically reasonable) values of g_CaL and g_If likely achieves the highest possible balance of robustness and flexibility of SANC function. However, a high balance of robustness and flexibility has been also achieved in some previous, essentially membrane-delimited models. For example, Demir et al. (9) model also exhibits robust AP firing and simultaneously reproduces full-scale rate regulation via m-AChRs and β-ARs. However, the Demir et al. model was based on its earlier (1994) basal AP firing model version (10) that incorporated the knowledge about SANC electrophysiology at that time. That older model included a relatively large voltage-gated sodium (I_Na) conductance and I_CaT conductance, which, according to more recent data, are absent or much less in primary rabbit SANC, respectively (44, 71). Both I_Na and I_CaT have low voltage activation thresholds and, therefore, strongly amplified the initial cAMP-sensitive signaling of I_f-induced early depolarization in the Demir et al. (9) model. Another, more recent pacemaker regulation mechanism is via a sustained, nonselective, inward current I_st in the Kyoto model of guinea pig SANC (16, 57). However, as we noted earlier, molecular origin and specific blockers for I_st have not yet been identified; I_st exhibits many properties of I_CaL and I_NCX and, therefore, might not even be a discrete entity. Thus the difference in the models is in their late DD mechanisms, which are: “LCR-induced I_NCX + I_CaL” in the present model; “I_Na + I_CaT + I_CaL” in the Demir et al. (9) model; and “I_st + I_CaL” in the Kyoto model (16, 57). If we exclude I_Na, I_CaT, and I_st as major regulation players in primary SANC, then the regulation only via I_f and/or I_CaL seems to be insufficient to explain the high balance of robustness and flexibility of SANC function (40) so that our LCR-induced I_NCX mechanism may be suggested as the missing link in the mystery of cardiac automaticity (see recent review in Ref. 32). A minor role of I_f in pacemaker rate regulation has been also demonstrated by recent functional genetics studies with I_f knockout (15).

Model Limitations

Our presentation is limited by the scope of a research paper as well as by the specific aim to develop and to initially explore the first prototype SANC model of dynamically interacting Ca²⁺ and membrane clocks and having minimum changes in classic formulations. Thus, although it is extremely tempting for us to further evaluate the importance of various parameters for behavior of the new modeled system, we limited our analysis to only a few key parameters. Since many model parameters have ascended from prior models of ventricular and atrial myocytes, our SANC-specific formulations for the coupled Ca²⁺ and membrane clocks will require further development and tuning as new experimental data in SANC emerge.

An important future model development is related to SR geometry under sarcolemma that ultimately determines the fine spatiotemporal structure of the Ca²⁺ release in SANC that could not be approximated here. In fact, the released Ca²⁺ in our model is instantly diluted in a relatively large subspace volume that was originally defined in Kurata model as a 20-nm space below the entire sarcolemma. This explains, in part, why SR Ca²⁺ oscillations require a substantial increase in the release rate constant k_s compared with that in Shannon model of ventricular cell SR operating as an ensemble of Ca²⁺ release units (CRUs).

Actually, RyRs seem to be arranged in clusters under sarcolemma in SANC (Fig. 2 in Ref. 37) and thus probably form subsarcolemmal CRUs (SANC lack T-tubules). In this instance, RyRs release Ca²⁺ into a relatively small volume of cleft space where individual jSRs of CRUs approach sarcolemma. Our pilot experimental study of 2-D Ca²⁺ signals showed that wavelet-like LCRs in SANC are indeed formed by the stochastic Ca²⁺ sparks propagating in a “fire-diffuse-fire” manner under sarcolemma (51). One recent numerical realization of such advanced Ca²⁺ release mechanism in SANC includes local Ca²⁺ diffusion and propagating CICR within dynamic corridors of refractoriness for the CRUs under sarcolemma (39). Thus future modeling of SANC function will probably include a stochastic mechanism of LCR generation by CRUs. It merits emphasis, however, that results of such detailed modeling will not necessarily contradict to the results of the present study. Indeed, experimental studies showed that it is the lumped signal of synchronized LCRs, well-approximated in the present model by variable j_SRCarel, that drives both basal AP firing and the AP rate modulation of SANC (38, 42, 65, 67).

Another potentially important model limitation is that we fixed intracellular Na⁺ concentration; modeling of Na⁺ dynamics was not a specific aim of our study and requires further special consideration.

Summary

The pacemaker of the heart generates billions of uninterrupted beats at greatly varying rates during the life span and therefore must be an extremely robust and simultaneously flexible system. A rather simple numerical release-pumping-delay SR Ca²⁺ clock described here (Figs. 1–3) possesses an extreme flexibility (1.3–6.1 Hz), but its robustness is limited, as it cannot sustain high-amplitude oscillations at high rates in the absence of interactions with membrane clock. The SANC pacemaker system of the two coupled oscillators keeps the advantage of the superior flexibility of the SR Ca²⁺ clock but simultaneously rebalances cell Ca²⁺ (via sarcolemmal Ca²⁺ fluxes of I_CaL and NCX; Figs. 1 and 7) to insure higher Ca²⁺ oscillation amplitudes at higher rates (Fig. 6B, inset, and Fig. 7). Operating in antiphase fashion, Ca²⁺ clock, in turn, evokes rhythmic APs by critically timed (short but sufficiently strong) NCX ignition currents (Fig. 5D) during late DD (Fig. 4, red arrow) when the prior AP is completing its recovery (i.e., afterhyperpolarization; Fig. 5D).

Robustness is a fundamental property of evolvable complex biological systems (23); a simple mechanism cannot handle extreme perturbations because it does not possess safety reserves. The heterogeneous redundancy of the oscillators (chemical Ca²⁺ clocks of the SR and electrical clocks of the surface membrane ion channels) allows basal state AP firing with moderate recourses of each clock in our numerical model of SANC pacemaker system, embracing smaller P_up, I_CaL, and/or I_f, including those at which membrane clock, operating alone, fails (Figs. 10 and 11). This provides broad safety margins and confers extremely high robustness to the AP firing, including instances of extreme chronotropic perturbations (Fig. 13).

Thus our model offers a new integrative framework for conceptualizing SANC function and regulation as a robust and flexible system of the dynamically coupled Ca²⁺ and membrane oscillators. This new vision resolves existing apparent controversy of two extreme pacemaker concepts of cardiac automaticity driven by either oscillator alone (Refs. 25, 34; see also Ref. 18 vs. 30).

The present study also provides a theoretical basis for a novel general theory of regulation of heart contraction and rhythm (29, 43): Ca²⁺ cycling into and from SR by proteins common to both ventricular myocytes and pacemaker cells regulate the duty cycle of both cell types. Our simulations show that the Bowditch phenomenon known for ventricular myocytes as the faster, the stronger is also operational and critically important for pacemaker cells (29): whereas the stronger Ca²⁺ release at higher pacing rates results in stronger contractions of myocytes, the stronger diastolic Ca²⁺ release results in stronger accelerating DD in spontaneously firing SANC (Fig. 7). Thus Ca²⁺ releases directly regulate primary functions of these specialized cardiac cells resulting in stronger and more frequent heart beats, thereby comprising the effective regulation of blood pumping by the heart.

Future Perspectives

Another level of complexity and robustness of cardiac pacemaker involves cell interactions within sinoatrial node. The late DD acceleration, in fact, is the earliest event within the central sinoatrial node pacemaker region (3) thus indicating that the LCR/NCX signals ignite spontaneous APs not only in isolated SANC, but also in the initiation site of the cardiac pacemaker tissue. Since SANC have copies of RyRs, NCX, and SERCA2 throughout the sinoatrial node, including those small-sized central cells (37), our model may be useful with respect to the development of new sinoatrial node tissue model to probe the role of the coupled Ca²⁺ cycling and membrane excitation to heart rate regulation (via PKA and CaMKII signaling) in silico, vis-à-vis multiple ambitious projects of physiome, or in vivo by biological pacemakers. Targeting cellular dynamics instead of specific molecules offers a new opportunity to understanding and design of novel therapies to treat pacemaker abnormalities and arrhythmias (61).

GRANTS

This research was supported by the Intramural Research Program of the National Institutes of Health, National Institute on Aging.