Effects of Sarcolemmal Background Ca2+ Entry and Sarcoplasmic Ca2+ Leak Currents on Electrophysiology and Ca2+ Transients in Human Ventricular Cardiomyocytes: A Computational Comparison

The intricate regulation of the compartmental Ca²⁺ concentrations in cardiomyocytes is critical for electrophysiology, excitation-contraction coupling, and other signaling pathways. Research into the complex signaling pathways is motivated by cardiac pathologies including arrhythmia and maladaptive myocyte remodeling, which result from Ca²⁺ dysregulation. Of interest to this investigation are two types of Ca²⁺ currents in cardiomyocytes: 1) background Ca²⁺ entry, i.e., Ca²⁺ transport across the sarcolemma from the extracellular space into the cytosol, and 2) Ca²⁺ leak from the sarcoplasmic reticulum (SR) across the SR membrane into the cytosol. Candidates for the ion channels underlying background Ca²⁺ entry and SR Ca²⁺ leak channels include members of the mechano-modulated transient receptor potential (TRP) family. We used a mathematical model of a human ventricular myocyte to analyze the individual contributions of background Ca²⁺ entry and SR Ca²⁺ leak to the modulation of Ca²⁺ transients and SR Ca²⁺ load at rest and during action potentials. Background Ca²⁺ entry exhibited a positive relationship with both [Ca²⁺]_i and [Ca²⁺]_SR. Modulating SR Ca²⁺ leak had opposite effects of background Ca²⁺ entry. Effects of SR Ca²⁺ leak on Ca²⁺ were particularly pronounced at lower pacing frequency. In contrast to the pronounced effects of background and leak Ca²⁺ currents on Ca²⁺ concentrations, the effects on cellular electrophysiology were marginal. Our studies provide quantitative insights into the differential modulation of compartmental Ca²⁺ concentrations by the background and leak Ca²⁺ currents. Furthermore, our studies support the hypothesis that TRP channels play a role in strain-modulation of cardiac contractility. In summary, our investigations shed light on the physiological effects of the background and leak Ca²⁺ currents and their contribution to the development of disease caused by Ca²⁺ dysregulation.

Introduction

Ca²⁺ concentrations are dynamically controlled in cardiomyocytes by a complex regulatory system comprising ion channels, transporters, exchangers, regulatory proteins, and ion buffers. Intricately regulated levels of Ca²⁺ concentrations are critical for electrical activity, excitation-contraction coupling, and other signaling pathways. In many cardiac diseases, the delicate balance of Ca²⁺ cycling is perturbed. Ca²⁺ dysregulation underlies maladaptive cardiac remodeling. A complete understanding of all components of Ca²⁺ handling is essential for the development of therapeutic strategies to attenuate cardiac pathologies.

Many ion channels underlying the Ca²⁺ signaling in cardiomyocytes are well characterized. Ca²⁺ signaling related to excitation-contraction coupling primarily relies on sarcolemmal Ca²⁺ entry through voltage-gated L-type Ca²⁺ channels (LTCC) to trigger Ca²⁺-induced Ca²⁺ release from the sarcoplasmic reticulum (SR) through ryanodine receptors (RyR), and, subsequently, Ca²⁺ extrusion via the sodium-calcium exchanger (NCX) and reuptake into the SR through sarco/endoplasmic Ca²⁺-ATPase (SERCA). Other Ca²⁺ currents contribute to the modulation of Ca²⁺ concentrations in cardiomyocytes as well. Of interest to this study are background Ca²⁺ entry through the sarcolemma and SR Ca²⁺ leak. Understanding of the physiological role of these Ca²⁺ currents is still incomplete.

Beyond Ca²⁺ transport across the sarcolemma from the extracellular space into the cytosol through LTCC and NCX, sarcolemmal Ca²⁺ transport comprises background Ca²⁺ entry. In resting cardiomyocytes, [Ca²⁺]_i is of the order of 100 nM, which indicates the existence of a background Ca²⁺ entry pathway to balance the Ca²⁺ efflux of NCX (Eisner et al., 2020). Resting ventricular myocytes depleted of Ca²⁺ stores with caffeine are able to reload the SR by a mechanism that involves extracellular Ca²⁺, demonstrating further evidence of background Ca²⁺ entry (Terracciano and Macleod, 1996). Based on studies measuring background Ca²⁺ influx in rat ventricular myocytes of the order of 2–5 μmol/L per second (Choi et al., 2000) or 4 μmol/L per second (Sankaranarayanan et al., 2017), the background Ca²⁺ entry is approximately 10% of the influx through LTCC current (5–10 μmol/L each action potential) at normal heart rates (Eisner et al., 2020). The identity of background Ca²⁺ flux is still poorly defined, but several channels have been suggested as contributors.

One study identified a Ca²⁺ entry mechanism that is blocked by the nonspecific agent gadolinium (Gd³⁺) (Kupittayanant et al., 2006). Connexin hemichannels are candidates for background Ca²⁺ entry since they can be inhibited by Gd³⁺ (Stout et al., 2002). While connexin hemichannels primarily form pairs to allow ion fluxes between cells at intercalated discs, some are present as hemichannels in the surface membrane of a single cell (Wang et al., 2012; Leybaert et al., 2017) and may, therefore, provide a route for Ca²⁺ entry. However, primary candidates for background Ca²⁺ entry include members of the family of Transient Receptor Potential (TRP) channels, which are also sensitive to Gd³⁺. The mdx mouse model of muscular dystrophy exhibits [Ca²⁺]_i and [Na⁺]_i overload that can be blocked by Gd³⁺, and the increase in cation entry has been suggested to involve TRPC channels (Mijares et al., 2014). Myocytes from old mdx mice exhibit increased expression of a putative stretch-activated channel (SAC), TRPC1. Elevated [Ca²⁺]_i levels can also be reduced to [Ca²⁺]_i levels of WT myocytes when exposed to SAC blockers streptomycin or GsMTx-4 (Williams and Allen, 2007; Ward et al., 2008). Upregulated TRPC1 also contributes to increased [Ca²⁺]_i through SAC in hypertrophic myocardium of rats following isoproterenol injection (Chen et al., 2013). Background Ca²⁺ entry involved in maladaptive cardiac remodeling was more recently shown to critically depend on both TRPC1 and TRPC4 (Camacho Londono et al., 2015, Camacho Londoño et al., 2021). TRPC6 channels have also been shown to modulate cytosolic Ca²⁺ transients and SR Ca²⁺ load through sarcolemmal Ca²⁺ entry (Ahmad et al., 2020). Furthermore, TRPV4 can modulate Ca²⁺ transients and SR load, and participates in hypoosmotic stress-induced cardiomyocyte Ca²⁺ entry (Rubinstein et al., 2014; Jones et al., 2019).

Ca²⁺ transport across the SR membrane from the intracellular store into the cytosol is known as SR Ca²⁺ leak. During an action potential, activation of RyR clusters triggered by Ca²⁺ entry through LTCC results in a synchronized release of a large amount of Ca²⁺ from the SR that forms the Ca²⁺ transient and leads to cardiomyocyte contraction. Activation of RyR clusters causes Ca²⁺ sparks. These sparks are an important pathway for SR Ca²⁺ leak. RyR Ca²⁺ leak can occur also through a mechanism independent of sparks (Santiago et al., 2010). Further, total SR Ca²⁺ leak includes a component separate from RyRs (Zima et al., 2010). Many candidates have been suggested as components of this leak, yet it is still ill-defined. A candidate is the inositol 1,4,5-trisphosphate (IP₃) receptor (IP₃R), which is a Ca²⁺ release channel expressed at lower densities than RyRs in cardiomyocytes and found upregulated in heart failure (HF) (Go et al., 1995; Ai et al., 2005) and therefore may be a relevant contributor to SR Ca²⁺ leak (Zima et al., 2010). Members of the TRP family have also been suggested to contribute to SR Ca²⁺ leak. TRPC1 was found to operate as a SR Ca²⁺ leak channel in skeletal muscle (Berbey et al., 2009) and more recently in cardiomyocytes (Hu et al., 2020). Additional evidence suggests contribution of TRPC6, TRPM8, TRPP2, and TRPV1 to endoplasmic reticulum Ca²⁺ leak in various cell types, although characterization is still incomplete for cardiomyocytes (Lemos et al., 2021).

TRP channels constitute primary candidates for explaining background and leak Ca²⁺ currents through both the sarcolemma and the SR membrane. Interestingly, many members of the TRP family are known to be modulated by stretch (Inoue et al., 2009; Reed et al., 2014; Peyronnet et al., 2016). Cardiomyocyte contractility is known to respond to mechanical stretch in two phases: a rapid and a slow response (Calaghan and White, 1999). The rapid response is the cellular basis for the Frank-Starling Mechanism (FSM). It relies primarily on myofilament overlap and alteration of myofilament Ca²⁺ sensitivity, and does not involve changes in Ca²⁺ transients. The slow response has been termed slow force response (SFR) or stress-induced slow increase in contractility (SSC), describing the gradual increase in twitch force corresponding to an increase in [Ca²⁺]_i transients that develops over several minutes when stretch is sustained. Members of the TRP family are likely candidates for mechanotransduction of the SFR.

Though background and leak Ca²⁺ currents are still poorly defined, they may play important roles in regulating Ca²⁺ homeostasis and contractility and altered Ca²⁺ dynamics in cardiac disease. The lack of complete understanding of the identity and mechanics of these channels, in addition to their relatively small amplitude compared to the voltage-gated Ca²⁺ channels, make them difficult to study in vivo. A computational model provides the advantage to study the currents and their roles in isolation from other cellular mechanisms. In this study, we use a mathematical model of a human ventricular myocyte to analyze the individual contributions of background Ca²⁺ entry and SR Ca²⁺ leak to the modulation of Ca²⁺ transients and SR Ca²⁺ load. We also assess the effects of the Ca²⁺ currents on cellular electrophysiology.

Materials and Methods

Mathematical Model of Ventricular Myocyte

We applied a mathematical model of a human ventricular myocyte (Grandi et al., 2010). The model and subsequent analyses were executed in MATLAB (R2020b). The model includes subsarcolemmal and junctional compartments beyond the cytosol compartment. Total background Ca²⁺ current (I_Cabk) through the sarcolemma is defined as a summation of junctional (I_{Cabk, junc}) and subsarcolemmal (I_Cabk,sl) components:

$I_{Cab{k}_{junc}}=F_{junc}{G}_{bkg}\left(V_m-E_{C{a}_{junc}}\right),(1)$

$I_{Cab{k}_{sl}}=F_{sl}{G}_{bkg}\left(V_m-E_{C{a}_{sl}}\right),(2)$

$I_{Cabk}=I_{Cab{k}_{junc}}+I_{Cab{k}_{sl}},(3)$

where F_junc = 0.11 and F_sl = 0.89 are constants that determine the fraction of total background current corresponding to the junctional and subsarcolemmal spaces, respectively, G_bkg is the maximum conductance (5.513e-4 A/F) of the channels, V_m is the transmembrane voltage, and E_Ca,junc and E_Ca,sl are Nernst potentials corresponding to the junctional and subsarcolemmal spaces, respectively. SR Ca²⁺ leak (J_leak) describes the Ca²⁺ flux out of the SR:

${\mathit{J}}_{\mathrm{leak}}=K_{leak}\left({\left[C{a}^{2+}\right]}_{SR}-{\left[C{a}^{2+}\right]}_j\right),(4)$

where K_leak is the leak constant (5.348e-6/ms), and [Ca²⁺]_SR and [Ca²⁺]_j describe the Ca²⁺ concentrations in the SR and junctional space, respectively. In this study, we modulated G_bkg and K_leak independently and in conjunction to investigate the effects of altered sarcolemmal Ca²⁺ entry and altered SR Ca²⁺ leak, respectively.

A graphical summary of the Ca²⁺ currents in the cell model is shown in Figure 1. The total Ca²⁺ current into the junctional (I_Ca,tot,junc) and subsarcolemmal (I_Ca,tot,sl) compartments are determined by Eqs 5, 6, respectively:

$I_{C{a}_{totjunc}}=I_{C{a}_{junc}}+I_{Cab{k}_{junc}}+ I_{pC{a}_{junc}}-2\cdot I_{NC{X}_{junc}},(5)$

$I_{C{a}_{totsl}}=I_{C{a}_{sl}}+I_{Cab{k}_{sl}}+ I_{pC{a}_{sl}}-2\cdot I_{NC{X}_{sl}},(6)$

where I_Ca describes the L-type Ca²⁺ current, I_pCa describes the sarcolemmal Ca²⁺ pump current, and I_NCX describes the NCX current.

FIGURE 1

FIGURE 1. Diagram of Ca²⁺ handling components in human ventricular myocyte model (Grandi et al., 2010). In this mathematical model, the cytosol is divided into junction, sub-sarcolemmal and bulk compartments. Ca²⁺ currents through the sarcolemma enter both the junctional and sub-sarcolemmal compartments before diffusing to the bulk cytosol.

The rate of change of Ca²⁺ concentrations in the junctional compartment [Ca²⁺]_j, subsarcolemmal compartment [Ca²⁺]_sl, bulk cytosol [Ca²⁺]_i, and SR [Ca²⁺]_SR are given by:

$\frac{d{\left[C{a}^{2+}\right]}_j}{dt}=-I_{C{a}_{to{t}_{junc}}}\cdot \frac{C_{mem}}{V_{junc}\cdot 2\cdot F}+\frac{J_{C{a}_{juncsl}}}{V_{junc}}\cdot \left({\left[C{a}^{2+}\right]}_{sl}-{\left[C{a}^{2+}\right]}_j\right)-J_{C{a}_{B_{junction}}}+J_{RYR}\cdot \frac{V_{sr}}{V_{junc}}+J_{leak}\cdot \frac{V_{myo}}{V_{junc}},(7)$

$\frac{d{\left[C{a}^{2+}\right]}_{sl}}{dt}=-I_{C{a}_{to{t}_{sl}}}\cdot \frac{C_{mem}}{V_{sl}\cdot 2\cdot F}+\frac{J_{C{a}_{juncsl}}}{V_{sl}}\cdot \left({\left[C{a}^{2+}\right]}_j-{\left[C{a}^{2+}\right]}_{sl}\right)+\frac{J_{C{a}_{slmyo}}}{V_{sl}}\cdot \left({\left[C{a}^{2+}\right]}_i-{\left[C{a}^{2+}\right]}_{sl}\right)-J_{C{a}_{B_{sl}}},(8)$

$\frac{d{\left[C{a}^{2+}\right]}_i}{dt}=-J_{SERCA}\cdot \frac{V_{sr}}{V_{myo}}-J_{C{a}_{B_{cytosol}}}+\frac{J_{C{a}_{slmyo}}}{V_{myo}}\cdot \left({\left[C{a}^{2+}\right]}_{sl}-{\left[C{a}^{2+}\right]}_i\right),(9)$

$\frac{d{\left[C{a}^{2+}\right]}_{SR}}{dt}=J_{SERCA}-\left(J_{leak}\cdot \frac{V_{myo}}{V_{sr}}+J_{RyR}\right)-\frac{dCsq{n}_b}{dt},(10)$

where C_mem is the membrane capacitance, F is Faraday’s constant, V_junc is the volume of the junctional compartment, V_sl is the volume of the subsarcolemmal compartment, V_myo is the volume of the bulk cytosol, and V_sr is the volume of the SR. J_Ca,juncsl is the rate of Ca²⁺ flux from junctional to subsarcolemmal compartments and J_Ca,slmyo is the rate of Ca²⁺ flux from the subsarcolemmal compartment to bulk cytosol. J_RYR describes the SR Ca²⁺ release and J_SERCA describes the SR Ca²⁺ pump. Ca²⁺ buffering is described by J_{Ca,B,junction} in the junctional compartment, J_Ca,B,sl in the subsarcolemmal compartment, J_Ca,B,cytosol in the bulk cytosol, and dCsqn_b/dt in the SR. We refer to (Grandi et al., 2010) for more details and the equations describing the mathematical model.

Evaluating the Effects of Background and Leak Ca²⁺ Currents on Ca²⁺ Concentrations

To investigate the effects of altered background Ca²⁺ entry, we modulated G_bkg from 0 to 300% of its default value, 5.513e-4 A/F. To investigate the effects of altered SR Ca²⁺ leak, we modulated K_leak from 0 to 300% of its default value, 5.348e-6 ms⁻¹. Simulation durations were 1 min to establish steady state. We analyzed the last beat. We performed sensitivity analyses for the modulation of G_bkg and K_leak independently on measurements of resting V_m, maximum V_m, action potential duration to 90% repolarization (APD₉₀), diastolic [Ca²⁺]_i, systolic [Ca²⁺]_i, [Ca²⁺]_i amplitude, diastolic [Ca²⁺]_SR, systolic [Ca²⁺]_SR, and [Ca²⁺]_SR amplitude. Sensitivity analyses of each measured parameter were then normalized to the measurement of the default model values and fit to the quadratic equation:

$y\left(\mathrm{x}\right)=A{x}^2+Bx+C,(11)$

where x is the fraction (%) of default G_bkg or K_leak. We evaluated the factor of the quadratic term, A, the linear term, B, and the constant term, C, of this fit for relative comparisons of the effects of modulated background Ca²⁺ current or modulated SR Ca²⁺ leak. This analysis was repeated for the model ran at 1, 2, and 3 Hz electrical excitation.

We subsequently performed a dual-sensitivity analysis for the modulations of G_bkg and K_leak to understand how the two Ca²⁺ currents interact and influence Ca²⁺ handling in the cardiomyocyte together.

Results

Qualitative Changes in V_m, [Ca²⁺]_i, and [Ca²⁺]_SR When Background Ca²⁺ Entry is Modulated

We first examined the changes in V_m, [Ca²⁺]_i, and [Ca²⁺]_SR following modulation of the background Ca²⁺ entry current, I_Cabk, at pacing rates of 1, 2 and 3 Hz (Figure 2). Changes to features of the action potential were minimal (Figure 2A). Resting V_m exhibited a positive relationship with % G_bkg (Figure 2B), while maximum depolarization exhibited a negative relationship (Figure 2C) with increasing background Ca²⁺ current. APD₉₀ showed a positive relationship, where increased background Ca²⁺ entry lengthens the time for repolarization (Figure 2D). We measured diastolic and systolic values, as well as amplitude of [Ca²⁺]_i transients (Figure 2E). All measurements demonstrate positive relationships with increasing background Ca²⁺ entry; the increase in systolic [Ca²⁺]_i (Figure 2F) is greater than the increase in diastolic [Ca²⁺]_i (Figure 2G), especially at the slowest pacing rate, so the amplitude increases a considerable amount (Figure 2H). We measured the same parameters of [Ca²⁺]_SR transients, which also demonstrate an increase in amplitude caused by increased background Ca²⁺ entry (Figures 2I,L). The release was initiated from increased diastolic [Ca²⁺]_SR (Figure 2J) and ended in reduced systolic [Ca²⁺]_SR (Figure 2K). The relationships between background Ca²⁺ entry and features of the [Ca²⁺]_SR transient were also augmented at the slowest pacing rate.

FIGURE 2

FIGURE 2. Effects of modulating the background Ca²⁺ entry current, I_Cabk, on V_m, [Ca²⁺]_i, and [Ca²⁺]_SR. (A) Example action potentials for 50% and 200% G_bkg at 1 Hz pacing. Sensitivity analyses of modulating G_bkg on measurements of (B) resting V_m, (C) maximum V_m, and (D) time to 90% repolarization at 1, 2 and 3 Hz pacing. (E) Example [Ca²⁺]_i transients for 50% and 200% G_bkg at 1 Hz pacing. Sensitivity analyses of modulating G_bkg on measurements of (F) diastolic [Ca²⁺]_i, (G) systolic [Ca²⁺]_i, and (H) [Ca²⁺]_i amplitude at 1, 2 and 3 Hz pacing. (I) Example [Ca²⁺]_SR transients for 50% and 200% G_bkg at 1 Hz pacing. Sensitivity analyses of modulating G_bkg on measurements of (J) diastolic [Ca²⁺]_SR, (K) systolic [Ca²⁺]_SR, and (L) [Ca²⁺]_SR amplitude at 1, 2 and 3 Hz pacing.

Qualitative Changes in V_m, [Ca²⁺]_i, and [Ca²⁺]_SR When SR Ca²⁺ Leak is Modulated

Changes following independent modulation of the SR Ca²⁺ leak current, J_leak, were also examined (Figure 3). The effects of modulating K_leak on V_m were small (Figure 3A). Resting V_m exhibited a slight positive relationship (Figure 3B). Maximum V_m increased with increasing K_leak at 1 Hz pacing but decreased with K_leak at 2 and 3 Hz pacing (Figure 3C). Repolarization time measured as APD₉₀ was unaltered by modulations of K_leak (Figure 3D). Diastolic [Ca²⁺]_i was also relatively unaffected by modulations of K_leak (Figures 3E,F). However systolic [Ca²⁺]_i and thus also [Ca²⁺]_i amplitude decreased with increasing K_leak (Figures 3G,H). The effect was strongest at the slowest pacing frequency of 1 Hz. Alterations in [Ca²⁺]_SR were also strongest at 1 Hz pacing. Diastolic [Ca²⁺]_SR exhibited a negative relationship with K_leak (Figure 3J), while systolic [Ca²⁺]_SR exhibits a positive relationship with K_leak (Figure 3K), both contributing to reduced [Ca²⁺]_SR amplitude with increased K_leak (Figure 3L).

FIGURE 3

FIGURE 3. Effects of modulating the SR Ca²⁺ leak flux, J_leak, on V_m, [Ca²⁺]_i, and [Ca²⁺]_SR. (A) Example action potentials for 50% and 200% K_leak at 1 Hz pacing. Sensitivity analyses of modulating K_leak on measurements of (B) resting V_m, (C) maximum V_m, and (D) time to 90% repolarization at 1, 2 and 3 Hz pacing. (E) Example [Ca²⁺]_i transients for 50% and 200% K_leak at 1 Hz pacing. Sensitivity analyses of modulating K_leak on measurements of (F) diastolic [Ca²⁺]_i, (G) systolic [Ca²⁺]_i, and (H) [Ca²⁺]_i amplitude at 1, 2 and 3 Hz pacing. (I) Example [Ca²⁺]_SR transients for 50% and 200% K_leak at 1 Hz pacing. Sensitivity analyses of modulating K_leak on measurements of (J) diastolic [Ca²⁺]_SR, (K) systolic [Ca²⁺]_SR, and (L) [Ca²⁺]_SR amplitude at 1, 2 and 3 Hz pacing.

Summary and Comparison of Independent Modulation of Background and Leak Currents

For comparison of the effects of altered G_bkg or K_leak on features of the action potential, we normalized the measurements of resting V_m, maximum V_m, and APD₉₀ to the default measurements from the model for the given pacing frequency (Figures 4A–F). The relationships of measured vs. modified parameter, both represented as fraction (%) vs. default values, were fit to a 2nd order polynomial model for quantification of the effects (Figures 4G–I). The quadratic term of the fit is negligible for resting V_m and maximum V_m, demonstrating that these measurements exhibit primarily linear relationships with G_bkg and K_leak (Figure 4G). However, the polynomial fit of APD₉₀ to G_bkg has a strong quadratic term, demonstrating a non-linear relationship between APD₉₀ and G_bkg (Figure 4G). Since the relationships of resting V_m and maximum V_m are primarily linear, the linear term of the quadratic polynomial fit demonstrates the sensitivity of the measurements to changes in G_bkg or K_leak (Figure 4H). Both G_bkg and K_leak modulation have a positive relationship with resting V_m, but the changes are negligible for K_leak and minimal for G_bkg, never exceeding an increase greater than 2% of the default model’s resting V_m. Maximum V_m had a strong negative linear relationship with G_bkg for all pacing frequencies, especially at 1 Hz pacing. Maximum V_m had a positive linear relationship with K_leak at 1 Hz pacing but negatives linear relationship for 2 and 3 Hz. The constant of the quadratic polynomial model represents the measurements for the modulated currents set to 0 (Figure 4I). These results revealed that resting V_m was largely unchanged by pacing rate. Maximum V_m was slightly increased with no I_Cabk and slightly reduced with no J_leak at 1 Hz and slightly increased with no J_leak at 3 Hz. APD₉₀ decreased without I_Cabk but remained unchanged without K_leak.

FIGURE 4

FIGURE 4. Summary of effects of modulating G_bkg or K_leak independently on V_m. Sensitivity analyses of modulating G_bkg on measurements of (A) resting V_m, (B) maximum V_m, and (C) time to 90% repolarization normalized by the default model parameters. Sensitivity analyses of modulating K_leak on measurements of (D) resting V_m, (E) maximum V_m, and (F) time to 90% repolarization normalized by the default model parameters. Summary of the (G) quadratic term, (H) linear term and (I) constant term calculated by quadratic polynomial fits to the normalized sensitivity analyses in (A–H).

Figure 5 contains a summary of [Ca²⁺]_i normalized by measurements of the default model for the sensitivity analyses of G_bkg and K_leak modulation. The relationships of measured parameter vs. modified parameter, both represented as % default values, were fit to a 2nd order polynomial model for quantification of the effects. The quadratic term indicates nonlinearity of the relationship (Figure 5G). The quadratic term of the diastolic [Ca²⁺]_i fits were marginal, indicating a primarily linear relationship. For both G_bkg and K_leak modulations, we noticed a nonlinearity associated with systolic [Ca²⁺]_i and consequently the [Ca²⁺]_i amplitude, with the greatest degree of nonlinearity associated with the lowest pacing frequency. The linear term of the quadratic fit showed a strong linear sensitivity of the measured parameter to changes in G_bkg or K_leak (Figure 5H). The positive sign of this term for G_bkg modulations for each measurement, diastolic, systolic, and amplitude, demonstrates the positive relationship of these parameters with G_bkg, with greater slope of the relationship for systolic and amplitude measurements. The linear term for diastolic [Ca²⁺]_i sensitivity to K_leak was negative but very small, demonstrating that diastolic [Ca²⁺]_i exhibits negligible sensitivity to SR Ca²⁺ leak. The relationship of systolic [Ca²⁺]_i sensitivity to K_leak was negative, with the largest value for 1 Hz pacing. The same is true for [Ca²⁺]_i transient amplitude. The constant term of the quadratic polynomial fits corresponds to the % default if the modulated channel were 0 (Figure 5I). Elimination of I_Cabk resulted in a reduction of diastolic [Ca²⁺]_i, systolic [Ca²⁺]_i, and [Ca²⁺]_i amplitude, with the greatest reductions at 1 Hz pacing. Elimination of J_leak did not affect diastolic [Ca²⁺]_i, but results in increased systolic [Ca²⁺]_i and [Ca²⁺]_i amplitude, especially at 1 Hz pacing.

FIGURE 5

FIGURE 5. Summary of effects of modulating G_bkg or K_leak independently on [Ca²⁺]_i. Sensitivity analyses of modulating G_bkg on measurements of (A) diastolic [Ca²⁺]_i, (B) systolic [Ca²⁺]_i, and (C) [Ca²⁺]_i amplitude normalized by the default model parameters. Sensitivity analyses of modulating K_leak on measurements of (D) diastolic [Ca²⁺]_i, (E) systolic [Ca²⁺]_i, and (F) [Ca²⁺]_i amplitude normalized by the default model parameters. Summary of the (G) quadratic term, (H) linear term and (I) constant term calculated by quadratic polynomial fits to the normalized sensitivity analyses in (A–H).

Measurements of diastolic and systolic [Ca²⁺]_SR and [Ca²⁺]_SR transient amplitude, normalized by measurements of the default model, for the sensitivity analyses of G_bkg and K_leak modulations are summarized in Figure 6. Again, a 2nd order polynomial model for the normalized sensitivity analyses provided information about the relationships. The quadratic term of the fits demonstrates that systolic [Ca²⁺]_SR and [Ca²⁺]_SR transient amplitude both exhibited nonlinearity in the relationship to modulated G_bkg (Figure 6G). The linear term of the relationships demonstrated the best representation sensitivity of the measurements to G_bkg or K_leak modulation (Figure 6H). The sign of this term was opposite for each measured parameter for G_bkg vs. K_leak modulation but similar in amplitude. While the changes to [Ca²⁺]_SR transients all contributed to a positive correlation between G_bkg and [Ca²⁺]_SR transient amplitude, the opposite changes all contribute to a negative correlation between K_leak and [Ca²⁺]_SR transient amplitude. The constant term of the polynomial model, which corresponds to the % of default model measurements if the current is set to 0, showed that in the absence of I_Cabk, diastolic [Ca²⁺]_SR is lower, systolic [Ca²⁺]_SR is higher, and [Ca²⁺]_SR amplitude is reduced (Figure 6I). In the absence of J_leak, diastolic [Ca²⁺]_SR was slightly elevated, systolic [Ca²⁺]_SR was reduced, and the amplitude of [Ca²⁺]_SR was greater. The effects on all measurements of [Ca²⁺]_SR following modulation of G_bkg or K_leak were stronger at 1 Hz pacing than at faster pacing frequencies.

FIGURE 6

FIGURE 6. Summary of effects of modulating G_bkg or K_leak independently on [Ca²⁺]_SR. Sensitivity analyses of modulating G_bkg on measurements of (A) diastolic [Ca²⁺]_SR, (B) systolic [Ca²⁺]_SR, and (C) [Ca²⁺]_SR amplitude normalized by the default model parameters. Sensitivity analyses of modulating K_leak on measurements of (D) diastolic [Ca²⁺]_SR, (E) systolic [Ca²⁺]_SR, and (F) [Ca²⁺]_SR amplitude normalized by the default model parameters. Summary of the (G) quadratic term, (H) linear term and (I) constant term calculated by quadratic polynomial fits the normalized sensitivity analyses in (A–H).

Effects of Dual Modulation of Background Ca²⁺ Entry and SR Ca²⁺ Leak on Vm, [Ca²⁺]_i, and [Ca²⁺]_SR

With a thorough understanding of how G_bkg and K_leak modulations independently affect features of the action potential, [Ca²⁺]_i and [Ca²⁺]_SR transients, we subsequently performed a dual-parameter sensitivity analysis of G_bkg and K_leak together for 1 Hz pacing (Figure 7). Resting V_m is affected minimally by G_bkg, and negligibly by K_leak, apparent by the nearly horizontal contour lines (Figure 7A). Both increasing G_bkg and increasing K_leak resulted in increased V_m, but increasing G_bkg makes a larger contribution. Maximum V_m was modulated in opposite directions by G_bkg and K_leak (Figure 7B). Increasing G_bkg reduces maximum V_m while increasing K_leak increased maximum V_m, but G_bkg is the slightly more dominant effect. APD₉₀ experienced the greatest increase at large increases in G_bkg and small K_leak (Figure 7C). Diastolic [Ca²⁺]_i was primarily affected by a positive relationship to G_bkg, while K_leak appeared to make no contribution (Figure 7D). Interestingly, for all other [Ca²⁺]_i and [Ca²⁺]_SR measurements, the opposing effects of G_bkg and K_leak appeared to balance with a very similar linear relationship (Figures 7E–I). The contour lines of the dual-parameter sensitivity analyses for systolic [Ca²⁺]_i, [Ca²⁺]_i amplitude, diastolic and systolic [Ca²⁺]_SR and [Ca²⁺]_SR amplitude all exhibited strikingly similar slopes.

FIGURE 7

FIGURE 7. Dual-parameter sensitivity analysis of modulating G_bkg and K_leak at 1 Hz pacing. Surface plots with contour lines for measurements of (A) resting V_m, (B) maximum V_m, (C) APD₉₀, (D) diastolic [Ca²⁺]_i, (E) systolic [Ca²⁺]_i, (F) [Ca²⁺]_i amplitude, (G) diastolic [Ca²⁺]_SR, (H) systolic [Ca²⁺]_SR, and (I) [Ca²⁺]_SR amplitude.

If both background and leak currents were modulated along this relationship, measured parameters remained relatively unchanged and only diastolic [Ca²⁺]_i increased. An example of balanced background Ca²⁺ entry and SR Ca²⁺ leak demonstrates that a G_bkg value 200% of default and K_leak value of 270% default provided a balanced effect and canceled out the opposing modulations on systolic [Ca²⁺]_i and diastolic [Ca²⁺]_SR load, while diastolic [Ca²⁺]_i was elevated 7.9% (Figure 8).

FIGURE 8

FIGURE 8. Example Ca²⁺ transients of balanced background Ca²⁺ entry and SR Ca²⁺ leak. (A) 200% G_bkg and 270% K_leak balance their opposing modulations on systolic [Ca²⁺]_i while increasing diastolic [Ca²⁺]_i by 7.9%. (B) Zoomed-in region demonstrating 7.9% increase in diastolic [Ca²⁺]_i from default values (black line). (C) 200% G_bkg and 270% K_leak balance opposing effects on SR load.

Discussion

In this study, we evaluated the contributions of background Ca²⁺ entry and SR Ca²⁺ leak to V_m and Ca²⁺ concentrations in the SR and cytosol in cardiomyocytes in silico. Our investigations shed light on the differential effects of background and leak Ca²⁺ currents in physiology, and also provide insight into their contributions to disease development due to Ca²⁺ dysfunction. Below, we discussed background Ca²⁺ entry as a mechanism to positively modulate Ca²⁺ entry and SR Ca²⁺ leak as a critical balancing mechanism to maintain homeostasis.

Background Ca²⁺ Entry Positively Modulates Ca²⁺ Concentrations

Our results show that background Ca²⁺ entry has a positive relationship with diastolic [Ca²⁺]_i and [Ca²⁺]_SR, and the amplitude of their transients (Figure 2). The small increase in diastolic [Ca²⁺]_i is the direct result of an increase in background Ca²⁺ entry, and the increase in [Ca²⁺]_SR follows as SERCA responds to pump the extra Ca²⁺ into the SR. The small increase in diastolic [Ca²⁺]_i amplifies Ca²⁺-induced-Ca²⁺ release through RyRs, explaining the increased transient amplitudes. This supports the physiological role of background Ca²⁺ entry in increasing Ca²⁺ concentrations. These results replicate prior findings for TRP family members suggested as Ca²⁺ entry channels. For example, we provided evidence for TRPC6 as background Ca²⁺ entry in neonatal rat ventricular myocytes (Ahmad et al., 2020). Overexpression of TRPC6 contributes to both elevated [Ca²⁺]_i and [Ca²⁺]_SR. TRPC3 and TRPC6 have also both been implicated in the stress-induced slow increase in [Ca²⁺]_i and increased [Ca²⁺]_i transients contributing to the SFR (Seo et al., 2014; Yamaguchi et al., 2017; Yamaguchi et al., 2018). These studies highlight one potential role for Ca²⁺ entry in strained myocytes. Many of the candidates suggested as background Ca²⁺ entry channels are known to be modulated by stretch (Inoue et al., 2009; Reed et al., 2014; Peyronnet et al., 2016), so strained myocytes would exhibit an increase in background Ca²⁺ entry, leading to elevated diastolic levels and larger transients. This mechanism likely contributes to the SFR, increased contractile force following sustained stretch.

SR Ca²⁺ Leak Negatively Modulates Ca²⁺ Concentrations as a Balancing Mechanism

SR Ca²⁺ leak had only a marginal effect on diastolic [Ca²⁺]_i but reduced [Ca²⁺]_SR and the amplitude of Ca²⁺ transients (Figure 3). In general, modulating SR Ca²⁺ leak had the opposite effects of background Ca²⁺ entry, except for the weak effect on diastolic [Ca²⁺]_i. The leaked Ca²⁺ is removed from the cytosol effectively by NCX, which is why it does not affect free cytosolic levels but reduces SR Ca²⁺. On its own, this complicates the understanding of the physiological role of SR Ca²⁺ leak and the purpose of reduced SR load. When considering the dual-parameter sensitivity analysis, it became however evident that while background Ca²⁺ entry responds to increased needs with increased Ca²⁺, SR Ca²⁺ leak likely functions as a critical balancing component to regulate SR stores and maintain Ca²⁺ homeostasis. The combined effects of increasing both background Ca²⁺ entry and SR Ca²⁺ leak exhibit a linear relationship, represented by the contour lines of the dual sensitivity plot for systolic [Ca²⁺]_i, [Ca²⁺]_i amplitude, diastolic and systolic [Ca²⁺]_SR and [Ca²⁺]_SR amplitude (Figure 7). If both background and leak currents are modulated along this relationship, measured parameters remain relatively unchanged and only diastolic [Ca²⁺]_i increases. Examples of balanced background Ca²⁺ entry and SR Ca²⁺ leak in Figure 8 demonstrate that modulations in G_bkg and K_leak can be balanced in a way to cancel out the large opposing effects they have on Ca²⁺ transient amplitudes (Figure 8). This balancing mechanism of SR Ca²⁺ leak could be critical to prevent Ca²⁺ overload in the cell. Leak in the form of RyR sparks has been demonstrated as SR load regulator to prevent overload, with a steep dependency on [Ca²⁺]_SR (Shannon et al., 2002). However, large Ca²⁺ release events through RyR sparks increase sensitivity for arrhythmia (George, 2008). RyR sparks are most prevalent at high [Ca²⁺]_SR, but non-spark RyR leak and non-RyR leak do not appear to exhibit the same steep dependency on [Ca²⁺]_SR and therefore might function differently. Thus, a mechanism of non-RyR leak may be to regulate compartmental Ca²⁺ before the cells become overloaded, and RyR sparks increase as a more extreme measure.

Like channels involved in background Ca²⁺ entry in cardiomyocytes, candidates for SR Ca²⁺ leak also include members of the TRP family and were suggested to be mechano-modulated (Inoue et al., 2009; Reed et al., 2014; Peyronnet et al., 2016). This indicates that the background and leak Ca²⁺ currents could be modulated in conjunction. Non-RyR leak that can be modulated by stretch may provide a more moderate and steady regulation on a beat-by-beat basis in conjunction with background Ca²⁺ entry modulated by stretch. Recently we demonstrated that TRPC1 constitutes an SR Ca²⁺ leak channel, and its overexpression resulted in decreased SR Ca²⁺ load (Hu et al., 2020). TRPC1 channels are suggested to be modulated by stretch, indicating that the reduction in SR Ca²⁺ load could be a regulatory mechanism to match increased background Ca²⁺ entry through, e.g., TRPC6 channels. We speculate that background Ca²⁺ entry and SR Ca²⁺ leak fulfill a critical homeostatic function in the modulation of Ca²⁺ concentrations throughout the cardiomyocyte in response to strain.

Background and Leak Ca²⁺ Currents May Contribute to Hypertrophy and HF Under Chronic Pressure Overload

Both background Ca²⁺ entry and SR Ca²⁺ leak through TRP channels are likely to be modulated by cardiomyocyte strain (Inoue et al., 2009; Reed et al., 2014; Peyronnet et al., 2016). Under chronic pressure overload conditions, strain-modulation of TRPC channels could increase background Ca²⁺ entry and SR Ca²⁺ leak, and thus dysregulate Ca²⁺. Cardiac disease is perpetuated by Ca²⁺ dysregulation, and a stray from its homeostatic balance. Some of the suggested ion channels for these Ca²⁺ currents were found to be upregulated in models of cardiac disease, suggesting a role in pathogenesis (Ahmad et al., 2017; Hof et al., 2019). Diastolic [Ca²⁺]_i is elevated in HF causing diastolic dysfunction (Eisner et al., 2020). Elevated background Ca²⁺ entry could be a contributing factor. Two different models of HF with preserved ejection fraction (HFpEF) display increases in both diastolic and systolic [Ca²⁺]_i (Curl et al., 2018; Rouhana et al., 2019). A hypothesis is that a major difference in Ca²⁺ handling between HFpEF and HF with reduced ejection fraction (HFrEF) is preserved [Ca²⁺]_SR in HFpEF vs. reduced [Ca²⁺]_SR in HFrEF (Eisner et al., 2020). The decreased SR Ca²⁺ content contributes largely to the decrease in systolic [Ca²⁺]_i and contractile dysfunction (Bers, 2006). Based on the demonstration of a balancing mechanism between background Ca²⁺ entry and SR Ca²⁺ leak in this study, it is reasonable to speculate a difference between maintenance of this balance in HFpEF vs. a stray from this balanced relationship towards overcompensated leak in HFrEF. In addition to reducing SR Ca²⁺ available for release, causing systolic dysfunction, increased SR Ca²⁺ leak can be problematic, e.g., triggering arrhythmias and being energetically costly due to increased use of ATP to repump Ca²⁺ (Bers, 2014). Understanding the balance of background and leak Ca²⁺ currents in cardiomyocytes and how they affect Ca²⁺ homeostasis and remodeling in disease will be critical to develop effective drug therapies targeting Ca²⁺ channels.

Background and Leak Ca²⁺ Currents are More Effective at Modulating Ca²⁺ at Lower Frequency Pacing

In this study, we observed the well-established frequency dependency of Ca²⁺ transients. Increasing the rate of stimulation increases diastolic [Ca²⁺]_i in isolated myocytes (Frampton et al., 1991; Antoons et al., 2002; Dibb et al., 2007; Horváth et al., 2017; Sankaranarayanan et al., 2017). Background Ca²⁺ entry and SR Ca²⁺ leak also both exhibit a frequency effect. The parameters we measured are all more sensitive to modulations of the Ca²⁺ currents at slower pacing rates than at faster pacing rates (Figures 5, 6). The sensitivity of each measured [Ca²⁺]_i and [Ca²⁺]_SR parameter to G_bkg and K_leak is greatest in amplitude for 1 Hz pacing. An explanation is that at slower pacing rates, the background and leak currents have relatively more time to contribute to the total Ca²⁺ flux per beat vs. the voltage-gated ion channels that open during the action potential and are closed at rest.

Modulation of I_Cabk and J_leak has Marginal Effects on Action Potentials

Modulating K_leak had negligible effects on the action potential for any pacing frequency (Figure 4). For the values of K_leak tested, we found that the SR Ca²⁺ leak flux does not significantly contribute to sarcolemmal electrophysiology. Modulating G_bkg has marginal effects on features of action potentials (Figure 4). While G_bkg positively correlates with increased resting V_m, an increase to 300% G_bkg only resulted in <2% change from basal resting V_m. This minimal change in resting potential is unlikely to be functionally relevant. An increase to 300% G_bkg also reduces maximum depolarization by 7% for 1 Hz pacing. The largest effect is an increase in action potential duration (APD90) by around 10% for maximal G_bkg modulation. It has also been shown that APD prolongation leads to increased Ca²⁺ (Bouchard et al., 1995), suggesting a positive feedback loop for electrical and Ca²⁺ signaling. Another important note is that APD increase is known to be inotropic, e.g., in rat ventricular myocytes (Bouchard et al., 1995). This indicates another mechanism for the contribution of background Ca²⁺ entry to contractility. Conversely, prolonged APD can induce torsades de pointes tachycardia, leading to life-threatening ventricular fibrillation (Roden and Hoffman, 1985; Ravens and Cerbai, 2008).

Limitations

Mathematical modeling of cellular electrophysiology provides a valuable resource for studying how aspects of cellular physiology interact and affect one another. It provides a means to investigate questions that cannot be easily answered in vivo. However, there are also caveats of mathematical modeling that should be considered. It should be noted that the definitions of I_Cabk and J_leak used in this model are general simplifications and meant to reproduce poorly defined currents. The equations lack specific gating conditions of the currents. The current equations were not parameterized to match experimental data which is only incompletely characterized in human ventricular myocytes. Instead, the current equations are adjusted such that the model reproduces overall physiological action potentials and calcium transients. This is an important consideration, since the magnitudes of these currents could be largely different in living cells. Thus, interpreting the results of this study should focus on the qualitative trends. As the specific ion channels that contribute to Ca²⁺ entry and leak are identified and characterized, future work can aim to refine the current definitions and provide detailed current models to replace the general simplifications of I_Cabk and J_leak.

In a similar way, other ion currents in the model are not fully defined. For example, some K⁺ channels and isoforms of the Na⁺/K⁺-ATPase are modulated by localized Ca²⁺ concentrations (Tian and Xie, 2008; Weisbrod, 2020). However, the model does not include any Ca²⁺-dependent terms in the definitions of these currents. The inclusion of these interactions may alter the effects we see on V_m in this study. Future work could address this limitation.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author Contributions

MS and FS designed the study. MS implemented the modeling, analyzed simulation data and drafted the manuscript. All authors critically revised the manuscript and approved the version to be published.

Funding

This work was supported by Nora Eccles Treadwell Foundation.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphys.2022.916278/full#supplementary-material

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