Modeling the mechanism of Ca²⁺ release in skeletal muscle by DHPRs easing inhibition at RyR I1-sites

The activity of the skeletal muscle of vertebrates is regulated in the main by voltage sensors/dihydropyridine receptors (DHPRs) in the membrane of the transverse (t-) tubular system (Schneider and Chandler, 1973; Ríos and Brum, 1987; Ríos and Pizarro, 1991; Melzer et al., 1995) and the intracellular [Mg²⁺], which exerts a powerful inhibitory action on the ryanodine receptors (RyRs)/sarcoplasmic reticulum (SR) Ca²⁺-release channels in terminal cisternae of the SR (Lamb and Stephenson, 1991, 1994; Lamb, 2002a, 2002b; Dulhunty, 2006; Stephenson, 2006). DHPRs physically interact with RyRs in the narrow junctional space (JS) between t-system and SR membranes at triads, where t-tubules are flanked in close apposition by two terminal cisternae of the SR. The size of the gap at triads between t-tubule and SR membranes is critical for enabling functional crosstalk between DHPRs and RyRs and is primarily determined by junctophilin-1 proteins, JPH-1s, that directly interact at their tubular membrane end (N-terminal region) with DHPRs and at their SR membrane end (C-terminal region) with the respective DHPR-coupled RyR1 (Landstrom et al., 2014; Perni, 2022; Lehnart and Wehrens, 2022). The RyRs in the JS are organized in checkered arrays (Block et al., 1988; Franzini-Armstrong and Nunzi, 1983; Franzini-Armstrong et al., 1998) with every second RyR being physically coupled to a group of four DHPRs forming a tetrad, such that each DHPR in the tetrad is in physical contact with one of the four subunits (protomers) of a RyR. There is broad agreement about a direct line of communication that ensures fast signal transmission from DHPRs activated by t-system membrane depolarization to DHPR-coupled RyRs, but the exact nature of the molecular interactions between skeletal DHPRs and RyRs in the JS remains unclear.

All RyRs immediately adjacent to each other, together with the associated proteins form a couplon (Stern et al., 1997; Franzini-Armstrong et al., 1999), such that triads effectively have two couplons, one on each side of the t-tubule that function independently of each other. In addition to couplons localized in the JS of all vertebrates, non-mammalian skeletal muscles also have variable groups of RyRs on the SR terminal cisternae located outside the JS in parajunctional arrays that do not face t-tubules and do not have contact with DHPRs or RyRs in couplons (Felder and Franzini-Armstrong, 2002).

Skeletal muscles of adult mammals contain almost exclusively the RyR1 isoform, which is one of the three RyR isoforms (RyR1–3) expressed in mammalian tissues. In contrast, skeletal muscles from non-mammalian vertebrates, like frogs—which served for many years as the animal model of choice for investigating the mechanism of excitation–contraction coupling (ECC) in skeletal muscle—express in similar amounts two RyR isoforms: α-RyR and β-RyR, which are homologs of the mammalian isoforms RyR1 and RyR3, respectively (Sutko and Airey, 1996; Murayama et al., 2000; Lanner et al., 2010; Murayama and Kurebayashi, 2011).

There is strong evidence that all couplons in skeletal muscles of mammals and non-mammalian vertebrates consist of RyR1s/α-RyRs and that the parajunctional groups of RyRs found predominantly in non-mammalian muscles consist of RyR3s/β-RyRs (Felder and Franzini-Armstrong, 2002; Perni et al., 2015). The abundant parajunctional arrays of β-RyRs in non-mammalian vertebrate skeletal muscle appear to support and possibly amplify the DHPR-activation-induced Ca²⁺ release that occurs at the level of couplons, considering, for example, that there are two triads (i.e., four RyR1 couplons) per sarcomere in mammalian skeletal muscle, but only one triad (i.e., two α-RyRs couplons and several β-RyRs parajunctional arrays) per sarcomere of similar length in frog skeletal muscle.

The relative ease of purification of the RyR1 isoform from skeletal muscles of mammals together with its relevance to human conditions contributed to the fact that the RyR1 isoform is the most extensively studied RyR isoform and its ionic control mechanisms are best understood. Manifestly, skeletal muscle fibers of mammals are uniquely suited for investigating how voltage-dependent DHPR activation interacts with ionic control mechanisms of RyR1 activation.

Cully et al. (2018) (see also Barclay and Launikonis, 2022) have shown that the average [Ca²⁺] in the JS ([Ca²⁺]_js) of human skeletal muscle fibers at rest is considerably greater than that in the bulk of the cytosol ([Ca²⁺]_c) and that [Ca²⁺]_js approaches [Ca²⁺]_c only when the RyR1s are fully blocked. Similarly, Despa et al. (2014) showed a standing diastolic [Ca²⁺] difference in cardiac myocytes between the cleft (which is equivalent to the JS in skeletal muscle fibers) and the cytosol at large, produced primarily by the diastolic SR Ca²⁺ leak through the cardiac RyR2s. Furthermore, Sanchez et al. (2021) used a Ca²⁺-sensitive fluorescent probe targeted to the JS in mouse skeletal muscle fibers and obtained results that are consistent with [Ca²⁺]_js, being several times greater than [Ca²⁺]_c at rest. These observations suggest that a diffusional barrier exists between the JS and most of the cytosol (JS/cytosol barrier), causing JS to function as an intermediate compartment between the SR cisternae and the cytosol at large.

JS, as part of the cytosol, plays a central role in the regulation of SR Ca²⁺ release because all cytoplasmic regulatory sites on the RyR1s/α-RyRs in a couplon are exposed to the JS rather than to the bulk of the cytosol. For each of the four protomers of one RyR1, there is a Ca²⁺ activation site (Ca²⁺ A-site), an ATP-binding site, a caffeine-binding site, and a low-affinity Ca²⁺/Mg²⁺ inhibitory I1 site. The location of the Ca²⁺ A-sites, the ATP sites, and the caffeine-binding sites on the cytoplasmic shell of the RyR1 has been specifically determined in cryo-EM reconstructions by Des Georges et al. (2016). These sites are located close to each other and are part of a “control hub” that regulates the opening of the RyR1 gate (Des Georges et al., 2016). After ATP is bound to its specific sites on the RyR1s in muscle, the control hub is primed to open the RyR1 gate when Ca²⁺ binds to the A-sites and the inhibitory low-affinity Ca²⁺/Mg²⁺ I1-sites are unoccupied (Nayak and Samsó, 2022). The low-affinity Ca²⁺/Mg²⁺ I1-sites are situated in the four E-F domains located outside the control hub, at the base of the huge mushroom-like cytosolic shell of the RyR1 (Laver et al., 1997a; Nayak et al., 2024), which rotates outward and downward toward its basis when the control hub is activated to open the RyR1 pore (Des Georges et al., 2016).

Employing a mathematical model based on the role of ionic control mechanisms of RyR1 function developed in the Appendix, I explore here the hypothesis that a sudden rise in Ca²⁺ leak in the JS through DHPR-coupled RyR1s in couplons, caused by reduction of inhibition at the RyR1 Ca²⁺/Mg²⁺ inhibitory I1-sites when the associated DHPRs are activated, is sufficient to enable synchronized responses that trigger a regenerative rise of Ca²⁺ release that remains under voltage control.

Data S1, S2, S3, S4, S5, and S6 consist of simulations of SR Ca²⁺ release in skeletal muscle using the mathematical model described in the article under the various conditions mentioned in the text, performed with Wolfram Mathematica online. The program scripts and simulation outputs refer to model predictions for Ca²⁺ release upon stimulation (1) under physiological conditions when the SR is either normally loaded, or severely depleted of Ca²⁺, without SR Ca²⁺ uptake (Data S1), (2) under physiological conditions with strong SR Ca²⁺ uptake (Data S2), (3) in the presence of 10 mM BAPTA when the SR is either normally loaded, or severely depleted of Ca²⁺ (Data S3), (4) in the presence of 10 mM EGTA when the SR is either normally loaded or severely depleted of Ca²⁺ (Data S4), (5) under physiological conditions at submaximal levels of DHPR-activation (Data S5), and (6) in the presence of 20 mM BAPTA at increased ionic strength with SR severely depleted of Ca²⁺ (Data S6).

Data S1, S2, S3, S4, S5, and S6 provide Wolfram Mathematica online program scripts.

Since the cytosolic regulatory sites on RyR1s/α-RyRs in situ are exposed to the JS rather than to the cytosol at large, it follows that the open probability of both DHPR-noncoupled and DHPR-coupled RyRs are sensitive to the ionic concentrations in the JS environment. Eqs. 6 and 7 describe the dependency of the open probability of DHPR-noncoupled RyR1s, ^ncRyR1s, (^ncP₀), and DHPR-coupled RyR1s, ^cRyR1s (^cP₀), respectively, on [Ca²⁺]_js, [Mg²]_js, total [Ca²⁺+ Mg²⁺]_js= [CaMg]_js, monovalent metallic cation concentration ([M_m⁺]_js), and SR luminal Ca²⁺ ([Ca²⁺]_SR). The equations are derived for steady-state conditions in Appendix 1 as Eqs. A14 and A15 within the framework of the dual-inhibition model by Mg²⁺ of RyR1 activity at the Ca²⁺ A-sites and low-affinity inhibitory Ca²⁺/Mg²⁺ I1 sites in the presence of millimolar ATP, considering structural features of the RyR1 molecules. As described in Appendix 1, Eq. 6/Eq. A14 quantitatively explains the RyR1 open probability measurements made on isolated RyR1s incorporated in lipid bilayers (Laver et al., 1997b; Laver, 2018) and Ca²⁺ fluxes from Ca²⁺-loaded heavy SR vesicles (Meissner, 1984, 1986), while Eq. 7/Eq. A15 considers evidence that DHPRs can exert an additional inhibitory action on (DHPR-coupled) RyR1s in skeletal muscle fibers at rest (see Kirsch et al., 2001; Zhou et al., 2006; and references therein).

The first term in Eqs. 6 and 7, P_A0, describes the probability of activation at the Ca²⁺ A-sites of either ^ncRyR1s or ^cRyR1s when the four protomeric A-sites of the RyR1s are activated. The values of the protomeric parameters ^1pP_i, ^1pP_max, ^1pK_Mm², ^1pK_CaA, ^1pK_Mg, and ^1pK_L are shown in Table 1 and were derived from experimental observations as described in Appendix 1. ^1pP_i characterizes the probability of an individual A-site activation in the absence of Ca²⁺ and Mg²⁺ in the JS (or cytosol) when the A-site is free or has two M_m⁺ bound to it, while ^1pP_max describes the probability of an individual Ca²⁺ A-site activation when the A-site is occupied by Ca²⁺. ^1pK_Mm² is the dissociation constant of two M_m⁺ from the Ca²⁺ A-site, considering that a divalent cationic site is likely to bind two M_m⁺, ^1pK_CaA is the Ca²⁺ dissociation constant from the Ca²⁺ A-site and ^1pK_Mg is the Mg²⁺ dissociation constant from the Ca²⁺ A-site when the SR is depleted of Ca²⁺ ([Ca²⁺]_SR= 0), and ^1pK_L is the Ca²⁺ dissociation constant from the luminal SR Ca²⁺ site, whose occupancy by Ca²⁺ alters the Mg²⁺ dissociation constant of the corresponding A-site as discussed in Appendix 1.

The second terms, ^ncP_I0, in Eq. 6 and ^cP_I0 in Eq. 7, describe the probabilities that the pores of ^ncRyR1s or ^cRyR1s channels are not blocked from opening by inhibition at their respective Ca²⁺/Mg²⁺ I1-sites. As explained in the Appendix, the Hill Equation with a Hill coefficient of 2 that was used to fit experimental results on Ca²⁺/Mg²⁺ inhibition at the I1-sites of isolated RyR1s incorporated in lipid bilayers was deemed to also describe inhibition at the ^ncRyR1s in muscle. In contrast, the enhanced inhibitory action exerted by DHPRs on ^cRyR1s was considered to result from an increased level of cooperativity between the four protomers in the ^cRyR1 that produces quasi-simultaneous binding and dissociation of four Ca²⁺/Mg²⁺ ions by a two-state allosteric mechanism (see e.g., Viappiani et al., 2014) at the Ca²⁺/Mg²⁺ inhibitory I1-sites, such that ^cRyR1 channels are blocked from opening at the Ca²⁺/Mg²⁺ inhibitory I1-sites when all four I1-inhibitory sites are occupied, and the channels are allowed to open when all four I1-inhibitory sites are free.

Barclay and Launikonis (2022) estimated the Ca²⁺ inflow into the JS from the SR through the RyR1s (k_SRJS([Ca²⁺]_SR–[Ca²⁺]_js) at 3.8 µM ms⁻¹ in skeletal human muscle fibers at rest when [Ca²⁺]_SR= 400 µM (Ziman et al., 2010) and [Ca²⁺]_c was buffered at 67 nM with 50 mM EGTA, which has its main molecular forms: H₂EGTA²⁻ and Ca-EGTA²⁻ negatively charged. Under these conditions, k_diff, which characterizes the diffusiveness of Ca²⁺ across the JS/cytosol barrier, was evaluated at 28 ms⁻¹. Using these values in Eq. 2, [Ca²⁺]_js = 203 nM (3.8 µM ms⁻¹/28 ms⁻¹ + 67 nM), and from Eq. 4, k_SRJS= 9.525 s⁻¹ (28,000 s⁻¹ × 0.136 µM/399.8 µM) at rest.

Note that the ability of a steady Ca²⁺ leak Lk = 3.8 µM ms⁻¹ into the JS to generate a difference between [Ca²⁺]_js and [Ca²⁺]_c in the order of 10–130 nM in the presence of 50 mM total EGTA ([EGTA]_T) when [Ca²⁺]_c = 67 nM (Cully et al., 2018; Barclay and Launikonis, 2022) is only possible if the diffusiveness of H₂EGTA²⁻ and Ca-EGTA²⁻ across the JS/cytosol barrier (k_diffEGTA) is several orders of magnitude lower than that for Ca²⁺ (k_diff = 28 ms⁻¹). This is because the difference between [Ca²⁺]_js and [Ca²⁺]_c generated by a steady Ca²⁺ leak into the JS is given by the ratio $Lkkdiff+kdiffEGTAKDEGTATKD+Ca2+jsKD+Ca2+c$⁠, where K_D is the equilibrium dissociation constant of Ca²⁺ from CaEGTA²⁻ (∼185 nM). Accordingly, a Ca²⁺ leak of 3.8 µM ms⁻¹ when [Ca²⁺]_c = 67 nM would generate a [Ca²⁺]_js–[Ca²⁺]_c difference of 10 or 130 nM when k_diffEGTA= 2.5 s⁻¹ (= 8.97 × 10⁻⁵k_diff) or 0.0127 s⁻¹ (= 4.56 × 10⁻⁷k_diff), respectively. Notice that a diffusiveness of H₂EGTA²⁻ and Ca-EGTA²⁻ across the barrier in the order of 0.0127 s⁻¹ necessitates several minutes for [EGTA]_T to equilibrate between JS and the rest of the cytosol and indicates that little H₂EGTA²⁻ and Ca-EGTA²⁻ is exchanged across the barrier over <100 ms. In principle, a situation where the diffusiveness of H₂EGTA²⁻ and Ca-EGTA²⁻ can be orders of magnitude smaller than that for Ca²⁺ would occur if the JS/cytosol barrier in muscle fibers at rest was negatively charged with relatively small pores.

Regarding the magnitude of k_diff, the value of 28 ms⁻¹ estimated by Barclay and Launikonis (2022) in human skeletal muscle fibers under resting conditions can only support a maximum rate of Ca²⁺ release from the JS into the cytosol at large of 2.3 µM ms⁻¹ (⁠$kdiff×maximum[Ca2+]js×R$⁠, where R is the ratio between the JS volume and the volume of cytosolic water) since [Ca²⁺]_js cannot be greater than endogenous [Ca²⁺]_SR ≈ 400 µM and R ≈ 1/4,930 according to Table 3 in Barclay and Launikonis (2022). In comparison, the maximal SR Ca²⁺ release rate into the cytosol of mouse muscle fibers ranges between 55 and 200 µM ms⁻¹ (see e.g., Sztretye et al., 2011; Baylor and Hollingworth, 2012), indicating that k_diff needs to rise strongly above its resting level during stimulation to support much higher SR Ca²⁺ release rates.

Experiments with mechanically skinned muscle fibers showed that the ability of DHPRs, when activated, to open RyR1s/α-RyRs in the presence of millimolar ATP, is facilitated when [Mg²⁺]_c is decreased below 1 mM, is curtailed when [Mg²⁺]_c is raised above 1 mM, and is effectively blocked at 10 mM [Mg²⁺]_c (Lamb and Stephenson, 1991, 1992, 1994). The fact that maximal activation of the DHPRs cannot open the RyR1s/α-RyRs pores in the presence of 10 mM [Mg²⁺]_c strongly suggests that the inhibition exerted by Mg²⁺ on the RyR1s/α-RyRs must be removed (Lamb and Stephenson, 1991, 1994) before pores can open. Based on these experiments Lamb and Stephenson proposed that activation of DHPRs by depolarization of the t-system membrane induces a 10–20-fold decrease in the affinity of Mg²⁺ binding to sites on RyR1s/α-RyRs that inhibit the opening of the RyR1s/α-RyRs (Lamb and Stephenson, 1991, 1994, Stephenson 1996, 2006; Stephenson et al., 1998; Lamb, 2000, 2002a, 2002b; Laver 2018; and references therein).

Support for voltage- and implicitly DHPR-activation-dependent change in [Mg²⁺]-induced inhibition of SR–Ca²⁺ release observed in mechanically skinned muscle fibers has been further obtained from experiments on isolated triads (Ritucci and Corbett, 1995) and voltage-clamped cut fibers of frog (Jacquemond and Schneider, 1992) and rat (Jóna et al., 2001). For example, Jóna et al. (2001) reported a 10-fold decrease in the apparent affinity of Mg²⁺ binding to sites on rat RyR1s when the membrane potential of the muscle fiber changed from normal polarization to full depolarization.

Thus, one can envisage that a sudden reduction of inhibition at the low-affinity Ca²⁺/Mg²⁺ I1 sites of ^cRyR1s in a couplon caused by activation of the DHPRs, could initiate a positive feedback loop that increases Ca²⁺ leak into the JS, raising [Ca²⁺]_js in the micromolar range that subsequently activates more ^cRyR1s at their A-sites to open their pores, which in turn, further increases the Ca²⁺ leak/flux into the JS.

For proof of principle, it is assumed, in the first instance that (1) couplons are initially equilibrated under resting conditions [[Ca²⁺]_js = 203 nM, [Ca²⁺]_c= 67 nM, [Ca²⁺]_SR = 400 µM, [Mg²⁺]_js = 1 mM, [M_m⁺]_js= 150 mM], (2) [Ca²⁺]_c, [Mg²⁺]_js, and [M_m⁺]_js do not change following maximal activation of the DHPRs, (3) Ca²⁺ binding to calsequestrin is in rapid equilibrium with [Ca²⁺]_SR and does not affect RyR1 function, (4) there is no putative activation of the ^ncRyR1s by their neighboring ^cRyR1s when the latter become activated by DHPR activation, and (5) the SR–Ca²⁺ pump is blocked. For modeling, the kinetics of activation of the ^cRyR1s it was considered that the functional state of the RyR1s is determined by the level of ion occupancy at the Ca²⁺ activation A-sites and the low-affinity Ca²⁺/Mg²⁺ inhibition I1 sites on the four protomers of a RyR1 at time t, and that the binding rate constants of Ca²⁺, Mg²⁺, and M_m⁺ to RyR1 sites (k_on) are diffusion-limited, with k_on = 4 × 10⁸ M⁻¹ s⁻¹, like those for BAPTA (4.5 × 10⁸ M⁻¹ s⁻¹ for Ca²⁺; Naraghi, 1997) and ATP (4 × 10⁸ M⁻¹s⁻¹ for Mg²⁺; Pecoraro et al., 1984). The dissociation rate constants (k_off) of Ca²⁺, Mg²⁺, and M_m⁺ from specific RyR1 sites were then obtained from the equilibrium dissociation constants (K_D) listed in Table 1 (k_off = k_onK_D), which were derived from experimental results as described in Appendix 1.

Note that as indicated in Table 1, ^4pK_in = 0.05 mM when DHPRs are not activated and 1.5 mM when DHPRs are maximally activated.

Time-dependent changes of specific properties associated with the functional state of ^cRyR1s and ^ncRyR1s in a couplon following perturbations induced by sudden 30-fold rise in apparent dissociation constant at the inhibitory Ca²⁺/Mg²⁺ I1 sites of ^cRyR1s were obtained by numerically solving the set of four differential Eqs. 12, 13, 14, and 15 in [Ca²⁺(t)]_js, [Ca^1pA(t)]/[^1pA_Total], [Mg^1pA(t)]/[^1pA_Total], and [Ca²⁺(t)]_SR with Wolfram Mathematica online (program scripts provided in Data S1, S2, S3, S4, S5, and S6).

Fig. 1 A shows the predicted timecourse of [Ca²⁺(t)]_js after the sudden reduction of inhibition at the low-affinity Ca²⁺/Mg²⁺ I1 sites of ^cRyR1s in a couplon caused by maximal activation of the DHPRs in a mammalian muscle fiber. At time zero, ^4pK_in suddenly rises from 0.05 to 1.5 mM in the ^cRyR1s of a couplon induced by the simultaneous maximal activation of the associated DHPRs (as considered in Eq. 12). Mg²⁺ dissociates very quickly from the Ca²⁺/Mg²⁺ I1 sites of ^cRyR1s with a rate constant k_off in the order of 6 × 10⁵ s⁻¹ (4 × 10⁸ M⁻¹ s⁻¹ × 1.5 mM), ensuring the swift reduction of inhibition at the I1 sites of ^cRyR1s within few µs that, in turn, induces a 581-fold rise of [Ca²⁺]_js from 0.203 nM to 70 µM within 0.5 ms (see inset in Fig. 1 A). The very large rise in [Ca²⁺]_js facilitates the rapid displacement of Mg²⁺ from the A-sites on all RyR1 protomers in the couplon so as to increase the Ca²⁺ occupancy at the A-sites from 14.6% at rest to 97% within 2 ms following the DHPR-induced rise in ^4pK_in (Fig. 1 B). With the SR Ca²⁺ pump blocked, there is a rapid SR Ca²⁺ depletion such that [Ca²⁺]_SR decreases 89% of the initial value after 20 ms and 99.2% after 500 ms of maximal DHPR activation (Fig. 1 C). Meanwhile, the open probability of the DHPR-coupled RyR1s (^cP_o(t)) shoots up from 1.2 × 10⁻⁸ to 0.62 within 4 ms (while ^ncP_o(t) only rises from 4.8 × 10⁻⁶ to 1.7 × 10⁻³) causing the SR Ca²⁺ permeability to increase from 0.0095 to 1,200 ms⁻¹ in 3.5 ms (inset in Fig. 1 C). The large increase in ^cP_o(t) supports an SR Ca²⁺ flux into the cytosol at large with a peak of 61.0 µM ms⁻¹ (expressed as Ca²⁺ released per volume of cytoplasmic water) after 1.8 ms following excitation (Fig. 1 D). Note that the maximal rate of SR Ca²⁺ release into the cytosol can exceed 100 µM ms⁻¹ if all ^cRyR1s in the couplon become activated and induce the simultaneous activation of all adjacent ^ncRyR1s as suggested by experiments where two or more RyR1s that are physically connected when incorporated in lipid bilayers can open and close simultaneously (Marx et al., 1998; Laver et al., 2004; Porta et al., 2012).

The inset in Fig. 1 D shows the time course of the SR Ca²⁺ flux into the cytosol when the SR Ca²⁺ pump is very active, returning 200 μM Ca²⁺ (relative to SR volume) ms⁻¹ (1 − [Ca²⁺t)]_SR/400 µM) that would fill the empty SR in a resting fiber in a matter of 200 ms. [Ca²⁺]_SR decreases by 82.5% of its initial value after 20 ms of maximal stimulation and then equilibrates at 59.5 µM. The reduced Ca²⁺ gradient across the SR membrane causes the SR Ca²⁺ release flux to drop to a lower level without the pump having any significant effect on the peak of the SR Ca²⁺ release flux into the cytosol (inset Fig. 1 D). Thus, as the simulation shows, the SR Ca²⁺ flux will decrease from an early peak to a lower level at constant stimulation when the SR-Ca²⁺ release flux exceeds the Ca²⁺ return flux by the SR-Ca²⁺ pump. The reduction in the Ca²⁺ flux entering the JS further reduces [Ca²⁺]_js(t), causing ^cP₀ and SR membrane permeability to decrease, the outcome resembling a Ca²⁺ inactivation process of RyR1s.

According to simulations shown in Fig. 2, [Ca²⁺]_js(t) and SR Ca²⁺ release flux are determined by the fraction of DHPR-activated ^cRyR1s. Consequently, the charge movement associated with DHPR activation of ^cRyR1s is a measure of DHPR-activated ^cRyR1s. The close association between SR Ca²⁺ release and fraction of DHPR-activated ^cRyR1s shown in Fig. 2 will therefore result in voltage control of Ca²⁺-release through voltage-dependent DHPR-activation.

Let us assume that 10 mM of a fast Ca²⁺ buffer such as BAPTA is present in the cytosol at large in a resting mammalian muscle fiber at the time of stimulation. The Ca²⁺-bound form of BAPTA (CaBAPTA²⁻) carries two negative charges suggesting that the diffusiveness of CaBAPTA²⁻ across a negatively charged JS/cytosol barrier is orders-of-magnitude smaller compared with that of Ca²⁺ as it is the case for CaEGTA²⁻, discussed above. The very large difference between the diffusiveness of Ca²⁺ and CaEGTA²⁻ or CaBAPTA²⁻ ensures that a steady Ca²⁺ leak into the JS generates a [Ca²⁺]_js–[Ca²⁺]_c difference that is about the same at steady-state in the absence as in the presence of mM BAPTA or EGTA.

The time course of [Ca²⁺(t)]_js and that of the other parameters associated with the functional state of ^cRyR1s and ^ncRyR1s in couplons following the sudden DHPR-activation–dependent decreased inhibition at the I1 sites of ^cRyR1s in the presence of 10 mM BAPTA was predicted by numerically solving the system of Eqs. 13, 14, 15, 16, and 17 with Wolfram Mathematica online for two sets of conditions when the “fiber” was at rest, before being stimulated. In one case, the values of the parameters in the fiber at rest were same as those in the absence of 10 mM BAPTA ([Ca²⁺]_SR = 400 µM, [Ca²⁺]_js = 203 nM, [Ca²⁺]_c= 67 nM, [Mg²⁺]_js = 1 mM and [M_m⁺]_js= 150 mM). In the other case, the SR was severely depleted of Ca²⁺ with [Ca²⁺]_SR = 100 µM, but [Ca²⁺]_js, [Mg²⁺]_js and [M_m⁺]_js were kept the same as in the first case to permit more direct examination of the effect of SR Ca²⁺ depletion on the activation of SR Ca²⁺ release. The marked reduction in the Ca²⁺ leak into the JS at rest caused by the lower luminal SR Ca²⁺ can only sustain a difference of ∼5 nM between [Ca²⁺]_js and [Ca²⁺]_c. Therefore, [Ca²⁺]_c at rest was deemed to be 198 nM (= 203–5 nM) in this case. All other assumptions detailed in the paragraph prior to the description of Eq. 11 were also adopted for computing the time course of Ca²⁺ release in the presence of 10 mM BAPTA.

When the SR is endogenously loaded with Ca²⁺ at the time of DHPR activation, the [Ca²⁺(t)]_js response is slightly delayed in 10 mM BAPTA (Fig. 3 A and inset) than in the absence of BAPTA (Fig. 1 A and inset), but it reaches the same peak (70 µM) after 3 ms with BAPTA and after 0.6 ms without BAPTA. The SR Ca²⁺-release flux into the cytosol (inset in Fig. 3 B) reaches a peak that is just a fraction higher in the presence than in the absence of BAPTA (61.5 versus 61.0 µM per volume of cytosol ms⁻¹, respectively) due in part to the slightly delayed decrease in [Ca²⁺(t)]_SR (Fig. 3 B).

In contrast, when the SR is severely depleted of Ca²⁺ at time of maximum DHPR activation ([Ca²⁺(0)]_SR = 100 µM) in 10 mM BAPTA, there is a relatively long delay after stimulation (∼35 ms) for [Ca²⁺(t)]_js to reach the threshold for producing a spike (Fig. 3 C trace a). The inset in Fig. 3 C (trace b) displays the time course of [Ca²⁺(t)]_js when [Ca²⁺]_SR(0) = 100 µM in the absence of BAPTA, showing that the presence of BAPTA is responsible for the delay, but does not affect [Ca²⁺(t)]_js peak height. Similarly, in the presence of 10 mM BAPTA, there is a delay before [Ca²⁺]_SR starts to decrease more rapidly (Fig. 3 D) and before SR Ca²⁺-release flux into the cytosol reaches its peak (inset in Fig. 3 D). When [Ca²⁺]_SR(0) = 100 µM in the absence of BAPTA, [Ca²⁺]_SR decreases rapidly without delay after stimulation, while the SR Ca²⁺-release flux reaches the same peak height as in the presence of BAPTA 9.1 ms after stimulation (Fig. 3 E). Significantly, if 10 mM BAPTA is replaced with 10 mM EGTA, which binds and releases Ca²⁺ slower than BAPTA (^EGTAk_on = 0.00167 µM⁻¹ ms⁻¹, ^EGTAk_off= 0.00031 ms⁻¹) and the SR is severely depleted of Ca²⁺ at time of DHPR activation, the spike in [Ca²⁺(t)]_js occurs without delay and the SR Ca²⁺-release flux starts rising within 2 ms (Data S4).

There is an apparent discrepancy between the ∼35-ms delay predicted by the model for the onset of [Ca²⁺]_SR decrease after stimulation in the presence of 10 mM BAPTA when [Ca²⁺]_SR(0) = 100 µM (Fig. 3 D) and the absence of such delay in experiments of Olivera and Pizarro (2018), who measured the time course of [Ca²⁺]_SR in frog muscle fibers after stimulation following SR Ca²⁺ depletion to [Ca²⁺]_SR = 100 µM in the presence of 20 mM BAPTA (see Fig. 3 B in their paper). The depletion of [Ca²⁺]_SR in these experiments occurred after the addition of 20 mM BAPTA (2.04 mM CaBAPTA²⁻ and 17.96 mM BAPTA⁴⁻ at 20 nM [Ca²⁺] with Cs⁺ as counterion) to an internal Cs-based solution of ∼150 mM ionic strength. The addition of 20 mM BAPTA would have increased the overall ionic strength of the internal solution by about 186–336 mM, which, in turn, would have raised K_in to ≥2.5 mM according to measurements of Laver et al. (2004) using Cs-based solutions of similar ionic strength (see their Fig. 1 and Table 1). A greater K_in would reduce inhibition at the Ca²⁺/Mg²⁺ I1 sites of non-DHPR-coupled RyRs at constant cytosolic [Mg²⁺] (∼1 mM). Consequently, P_I0 in the muscle fibers at rest would have risen considerably in the experiments of Olivera and Pizarro (2018) after the addition of 20 mM BAPTA and would have caused the SR Ca²⁺ leak into the JS and the rest of the cytosol to rise in spite of a gradual decrease of [Ca²⁺]_SR from 400 to 100 µM, as was indeed reported by Olivera and Pizarro (2018). As a result, [Ca²⁺]_js at time of stimulation would have been sizably greater in the experiments of Olivera and Pizarro (2018) than 203 nM considered in the model simulation shown in Fig. 3 D. In the model simulation shown in Fig. 3 E, a rise in K_in (and ^4pK_in) from 0.05 to 0.25 mM (rather than to 2.5 mM) caused [Ca²⁺]_js to rise to 1.6 µM when [Ca²⁺]_SR= 100 μM at the time of stimulation and produce a prompt decline of [Ca²⁺]_SR upon stimulation, as reported by Olivera and Pizarro (2018) in their experiments (Data S6). Thus, apparent controversies between experimental observations and model-based predictions could be simply due to differences between perceived values of parameters at the time of stimulation in experimental settings and actual values used in model-based simulations.

It is also important to mention that the proposed mechanism of Ca²⁺ release predicts that maximal DHPR activation induces a regenerative Ca²⁺ release spike even when [Ca²⁺]_SR is reduced to 10 µM in the presence of 10 mM BAPTA in the JS, and [Ca²⁺]_js ≈ [Ca²⁺]_c = 0.10 µM. In this case, according to simulation, [Ca²⁺(t)]_js reaches 1.3 µM after 12.9 s of continuous stimulation if the DHPRs do not become inactivated and the diffusiveness of BAPTA across the JS/cytosol barrier under resting conditions is negligible over this period of 12.9 s (Data S3).

Taken together, these results highlight the robustness of the proposed hypothesis.

Central to the proposed hypothesis is the presence of a diffusional JS/cytosol barrier that supports an elevated [Ca²⁺]_js relative to [Ca²⁺]_c in the presence of a small Ca²⁺ leak into the JS. Sudden activation of DHPRs on DHPR-coupled RyR1s/α-RyRs induces a sudden, localized disturbance at the low-affinity Ca²⁺/Mg²⁺ I1-inhibitory sites of the coupled RyR1s/α-RyRs by allosteric regulation. The signal between the sites of physical interaction of the four DHPRs with the four protomers of the ^cRyR1 and the base of the RyR1, where the I1-sites are located is possibly transmitted via long levers that provide long-range allosteric pathway within the cytosolic shell of the RyR1s as recently proposed by Nayak et al. (2024). The disturbance at the I1 sites raises the Ca²⁺/Mg²⁺ equilibrium dissociation constant causing a small fraction of DHPR-activated ^cRyR1s to open their pores at the prevalent [Ca²⁺]_js when inhibition at their I1 sites is eased. Consequently, the Ca²⁺ flux/leak into the JS rapidly rises and the presence of the JS/cytosol diffusional barrier facilitates the rise of [Ca²⁺]_js that starts a regenerative Ca²⁺ activation process confined to the fraction of voltage-activated coupled RyR1s/α-RyRs. The rising [Ca²⁺]_js causes Ca²⁺ to displace Mg²⁺ at the A-sites of RyR1s/α-RyRs leading to full activation of the DHPR-activated fraction of RyR1s/α-RyRs. As shown in Fig. 2, [Ca²⁺]_js(t) and SR Ca²⁺ release flux are governed by the fraction of DHPR-activated ^cRyR1s, which ensures voltage control of Ca²⁺-release through voltage-dependent DHPR-activation. Increased Ca²⁺ diffusiveness across the JS/cytosol barrier by a mechanism analogous to that that opens floodgates in a reservoir when the level of water in the reservoir exceeds a certain level greatly enhances the Ca²⁺ flux from the JS into the bulk cytosol. Deactivation or inactivation of DHPRs rapidly reinstates inhibition at the Ca²⁺/Mg²⁺ I1 sites of the DHPR-coupled RyR1s/α-RyRs in the presence of 1 mM [Mg²⁺]_js. Consequently, the vast majority of DHPR-coupled RyR1s/α-RyRs close, Ca²⁺ leak into the JS and [Ca²⁺]_js declines, and Mg²⁺ binds to the A-sites of all RyRs in the couplons, further reducing the size of the Ca²⁺ leak into the JS and [Ca²⁺]_js.

Essential to the proposed model for the mechanism of Ca²⁺ release in skeletal muscle by DHPRs is the existence of a diffusional barrier between the narrow space, where cytosolic Ca²⁺ and Mg²⁺ regulatory sites are located on RyRs, and the rest of the cytosol. There is evidence that this barrier displays low diffusiveness for Ca²⁺ at low [Ca²⁺]_js (Barclay and Launikonis, 2022) and even a much lower diffusiveness (by orders of magnitude) for divalent anions such as CaEGTA²⁻ and EGTA²⁻ (as detailed in this paper), features that are typical of cation exchange membranes (Tekinalp et al., 2023). Such membranes carry fixed negative charges that severely reduce the diffusiveness of divalent anions relative to cations (Tekinalp et al., 2023). The extremely low diffusiveness of anions ensures that little exchange takes place between anions in the JS and those in the rest of the cytosol for relatively short periods of time after stimulation. As shown by simulations of the proposed model, the orders-of-magnitude lower diffusiveness of the barrier for divalent anions than for Ca²⁺ allows Ca²⁺ induced Ca²⁺-release to produce a spike following stimulation even in the presence of large concentrations of negatively charged Ca²⁺ buffers like CaEGTA²⁻/EGTA²⁻ or CaBAPTA²⁻/BAPTA⁴⁻. The precise nature of the barrier is not known, although it is likely to incorporate proteins present in the JS including DHPRs, RyRs, JPHs, and possibly metabolic enzymes that are densely packed around myofibrils at triads. For the model to produce Ca²⁺ fluxes with peak values reported in the literature, the diffusiveness of Ca²⁺ across the barrier needs to increase by orders-of-magnitude as [Ca²⁺]_js rises. Movement in the cytosolic shells of RyRs when control hubs associated with A-sites become activated may alter interactions between RyRs with activated control hubs, activated DHPRs, and JPHs to increase the junctional gap between the SR and tubular membranes and change the properties of the barrier such as to allow Ca²⁺ to diffuse more freely from the JS into the rest of the cytosol. As suggested below, new experiments are needed to determine the nature of the barrier and characterize more fully its properties.

Two types of experiments are suggested to test the foundations on which this hypothesis is based.

The diffusional JS/cytosol barrier in mammalian skeletal muscle should be further characterized preferably using Ca²⁺ probes targeted to the JS (Despa et al, 2014; Luo and Hill, 2014; Sanchez et al, 2021) under conditions where the magnitude of the steady RyR1 Ca²⁺ leak is altered by changing the SR Ca²⁺-loading, altering [Mg²⁺]_c, [Ca²⁺]_c, or introducing modulators of RyR1 activity such as caffeine and low-level voltage-dependent activation of DHPRs. Alternatively, impermeant, low-affinity, negatively charged Ca²⁺-dyes that diffuse very slowly across JS/cytosol barrier could be used with immobilized, freshly mechanically skinned muscle fibers with intact ECC (Lamb and Stephenson, 2018). The dye is first loaded for ∼30–60 min into the JS from a cytosolic solution containing the dye. The dye is then thoroughly washed out from the cytosol at large with several rinses. Changes in the dye signal predominantly from the JS upon rapid alteration of the Ca²⁺ leak could provide new information in support or against the presence of a selective diffusional barrier between the JS and the cytosol at large on which the hypothesis is based.

According to predictions based on the proposed hypothesis, there is a delay before a regenerative Ca²⁺ release spike occurs in the presence of 10 mM BAPTA (but not in the presence of 10 mM EGTA) after stimulation that causes DHPR activation when [Ca²⁺]_SR≤ 0.1 mM. Measurements of SR Ca²⁺ release in mammalian muscle fibers, similar to those performed by Olivera and Pizarro (2018), under conditions that [Ca²⁺]_SR≤ 0.1 mM at the time of stimulation in the presence of [Ca²⁺]_c < 10 nM and critically that the ionic strength in the presence of 10 mM BAPTA is kept within the physiological range with K⁺ as the main cation in solution can provide evidence in support or against the proposed hypothesis.

The dual-inhibition model of RyR1 activity by Mg²⁺ in the presence of millimolar ATP specifies that RyR1s are inhibited when either Mg²⁺ is bound to cytosolic Ca²⁺ A-sites of the RyR1s or when cytosolic Ca²⁺/Mg²⁺ I1-sites on the RyR1s are occupied by Ca²⁺ or Mg²⁺ (see reviews by Laver 2018; Meissner 2017; and references therein). Conversely, the RyR1s become maximally activated when the overall probability is highest for cytosolic Ca²⁺ A-sites to be occupied by Ca²⁺ and the cytosolic Ca²⁺/Mg²⁺ I1 sites are free of Ca²⁺ or Mg²⁺.

The activation term P_A0 expresses the probability that the control hub of the RyR1 molecule containing the Ca²⁺ A-sites causes the RyR1 pore to open when Ca²⁺/Mg²⁺ I1 sites on the RyR1 are not occupied by either Ca²⁺ or Mg²⁺. On the other hand, the inhibition term P_I0 expresses the probability that the pore of the RyR1 channel is not inhibited at the low-affinity Ca²⁺/Mg²⁺ I1 sites.

Importantly, millimolar ATP activates RyR1s in the absence of Ca²⁺ ([Ca²⁺] = 1 nM) and Mg²⁺ (Smith et al., 1986; Meissner et al., 1986; Laver et al., 2004), and Mg²⁺ also potently inhibits RyR1 activation induced by ATP. These observations were quantitatively explained by reasoning that the pore of the RyR1 channel can open in the presence of millimolar ATP not only when Ca²⁺ is bound to the A-sites (as is the case in the absence of ATP) but also when the A-sites are free, or when two M_m⁺ are bound to them (and inhibition at the Ca²⁺/Mg²⁺ I1 sites is not complete) (Laver et al., 2004; see caption to Table II in Laver et al., 2004). P_i and P_max express the level of RyR1 activation (0 to 1) at the A-sites when the A-sites are free or have two M_m⁺ bound to them, and when the A-sites are saturated with Ca²⁺, respectively.

All parameters associated with individual/protomeric ion binding sites on the RyR1 molecule are preceded by the “1p” superscript to distinguish them from corresponding parameters associated with activation of isolated RyR1 molecules that have four integrated protomers. The latter parameters are not preceded by a superscript.

Based on the experimental results of Laver et al. (2004) on the open probability of isolated RyR1s incorporated in lipid bilayers in the presence of 2 mM ATP and Eqs. A2, A3, and A4, one can derive all parameter values necessary to quantitatively describe activation at individual A-sites under different ionic conditions with respect to luminal Ca²⁺ and cytosolic Ca²⁺, Mg²⁺, and M_m⁺ concentrations, and quantitatively verify how well predictions based on Eqs. A2 and A3 can explain experimental observations made on isolated RyR1s.

The value of ^1pP_max can be derived from the maximum RyR1 open probability, P_Max, when [Ca²⁺]_c is raised in the absence of Mg²⁺ using Eq. A2. According to Table I of Laver et al. (2004), P_Max = 0.86 at both 0.01 and 1 mM [Ca²⁺]_L. Substituting P_A0 with P_Max and ^1pP_P0 with ^1pP_max in Eq. A4, it follows that ^1pP_max = 0.963 and is independent of [Ca²⁺]_L in the range 0.01–1 mM.

^1pP_i refers to the level of individual A-site activation in the absence of cytosolic Ca²⁺ and Mg²⁺. According to Eq. A2^1pP_A0=^1pP_i in the absence of cytosolic Ca²⁺([Ca²⁺]_c≤ 1 nM) and Mg²⁺. Similarly, P_i refers to the level of individual A-site activation in the absence of cytosolic Ca²⁺ and Mg²⁺, hence P_A0= P_i. Moreover, since all Ca²⁺/Mg²⁺ I1 sites on the RyR1 molecule are free in the absence of cytosolic Ca²⁺ and Mg²⁺, and therefore, P_I0= 1, it follows from Eq. A1 that P₀= P_A0(= P_i). Furthermore, since ^1pP_A0= (P_A0)^0.25 in Eq. A4, it means that ^1pP_i=^1pP_A0= (P_A0)^0.25 = (P₀)^0.25 in the absence of cytosolic Ca²⁺ and Mg²⁺. Laver et al. (2004) reported P₀ values of 0.21 ± 0.16 and 0.25 ± 0.07 for RyR1s in the presence of 50 and 250 mM cytosolic [Cs⁺], respectively, when [Ca²⁺]_L = 0.1 mM, [Ca²⁺]_c= 1 nM, and [Mg²⁺]_c= 0 (Table III in Laver et al. (2004), Control column). Since the mean P₀ values are close to each other and not statistically significantly different from each other, it indicates that P₀ is not sensitive to [M_m⁺]_c in the range 50–250 mM in the absence of cytosolic Ca²⁺ and Mg²⁺. Using the mean average value for P₀= 0.23 in the absence of cytosolic Ca²⁺ and Mg²⁺ when [Ca²⁺]_L = 0.1 mM, it follows that ^1pP_i ≈ (0.23)^0.25 = 0.6925. Similarly, the values reported by Laver et al. (2004) in Table III for P₀ at [Ca²⁺]_L = 1 mM in 50 mM (0.45 ± 0.20) and 250 mM (0.58 ± 0.06) cytosolic [Cs⁺] in the absence of cytosolic Ca²⁺ and Mg²⁺ are relatively close and not statistically significantly different from each other, supporting the view that P_i is not sensitive to [M_m⁺]_c in the range 50–250 mM in the absence of cytosolic Ca²⁺ and Mg²⁺. Taking the mean average value of 0.515 for P₀ in the absence of cytosolic Ca²⁺ and Mg²⁺ when [Ca²⁺]_L = 1 mM, it follows that ^1pP_i ≈ (0.515)^0.25 = 0.8471.

When [Ca²⁺]_L was raised to 3 mM in the absence of cytosolic Ca²⁺ ([Ca²⁺]_c= 1 nM) and Mg²⁺ in 250 mM cytosolic [Cs⁺], Laver et al. (2004) reported in Table III a P₀ value of 0.63 ± 0.07. Using the same approach, the derived value for ^1pP_i (= (P₀)^0.25) is 0.891 when [Ca²⁺]_L= 3 mM. Laver et al. (2004) also reported in Table I that P_i = 0.1 ± 0.07 in 250 mM [Cs⁺] when [Ca²⁺]_L was 0.01 mM. The derived value for ^1pP_i (= (P_i)^0.25) based on Eq. A4 is 0.562. Thus, P_i is sensitive to [Ca²⁺]_L but not to the cytosolic [M_m⁺] in the range 50–250 mM.

According to Table III in Laver et al. (2004), the P₀ values measured at [Ca²⁺]_L = 0.1 mM in the absence of Ca²⁺ ([Ca²⁺]_c = 1 nM) and Mg²⁺ were reduced 50% by 8 µM cytosolic [Mg²⁺]_c in the presence of [Cs⁺]_c = 50 mM and by 20 µM [Mg²⁺]_c in the presence of [Cs⁺]_c = 250 mM. Since inhibition at the Ca²⁺/Mg²⁺-I1 sites is negligible (<0.7%) at these low [Mg²⁺]_c in the absence of cytosolic Ca²⁺, it follows that P₀ = P_A0 in the absence of [Ca²⁺]_c and [Mg²⁺]_c and P₀≈ P_A0 when [Mg²⁺]_c ≤ 8 μM at 50 mM and ≤20 μM at 250 mM [M_m⁺]_c. Consequently, ^1pP_A0= 0.6925 when P₀ = P_A0= 0.23 and is not sensitive to [M_m⁺]_c in the range 50–250 mM in the absence of [Ca²⁺]_c and [Mg²⁺]_c. The 50% reduction of P₀ from 0.23 to 0.115 corresponds to the reduction of ^1pP_A0 from 0.695 to ≈ 0.582 (= (0.115)^0.25) based on Eq. A4. ^1pP_i is also insensitive to the [M_m⁺]_c in the range 50–250 mM at constant [Ca²⁺]_L since by definition ^1pP_i = ^1pP_A0 in the absence of Ca²⁺ and Mg²⁺. The value of the left term in Eq. A6(^1pP_i/^1pP_A0− 1) (^1pK_Mg(1 + [Ca²⁺]_L/^1pK_L)) is therefore the same at 50 and 250 mM [M_m⁺]_c for conditions where ^1pP_A0 is reduced to 0.582 at 0.1 mM [Ca²⁺]_L. The rightside term in Eq. A6 ([Mg²⁺]_c/(1 + [M_m⁺]_c²/^1pK_Mm²) must have the same value when P_A0 (0.23) decreases by the same fraction (50%) in 50 and 250 mM [Cs⁺]_c, i.e., 8 µM/(1 + 2,500 mM²/^1pK_Mm²) = 20 µM/(1 + 62,500 mM²/^1pK_Mm²). Consequently, ^1pK_Mm²= 37,500 mM², ^1pK_Mm= 193.6 mM, and the value of the right side term of Eq. A6 is 7.5 µM (8 µM/1.06(6) = 20 µM/(2.6(6)), meaning that (^1pP_i/^1pP_A0− 1) (^1pK_Mg(1 + [Ca²⁺]_L/^1pK_L)) = 7.5 µM. Since (^1pP_i/^1pP_A0− 1) = (0.6925/0.582 − 1) = 0.18986 and [Ca²⁺]_L = 0.1 mM, it follows that ^1pK_Mg(1 + 0.1 mM/^1pK_L) = 39.5 µM.

^1pK_Mm² can also be evaluated from the P₀ measurements of Laver et al. (2004) in Table III for 1 mM [Ca²⁺]_L at 50 and 250 mM [Cs⁺]_c. The values of P₀ measurements made at 1 mM [Ca²⁺]_L for 50 and 250 mM [Cs⁺]_c in the absence of Ca²⁺ and Mg²⁺ were similar and not statistically significantly different from each other, indicating that P₀ is not sensitive to [M_m⁺]_c in the range 50–250 mM. Consequently, the mean value of the two sets of measurements (P₀ = 0.515) was taken to describe the open RyR1 probability in the absence of Ca²⁺ and Mg²⁺ at 1 mM [Ca²⁺]_L. Since in this case there is no inhibition at the I1 sites, P_A0 = P₀= 0.515 and ^1pP_i = ^1pP_A0= (0.515)^0.25 = 0.8471. The value of P₀ was reduced by 50% in the presence of 20 µM [Mg²⁺]_c in the 50 mM [Cs⁺]_c solution and 72 µM [Mg²⁺]_c in the 250 mM [Cs⁺]_c solution in the absence of cytosolic Ca²⁺ ([Ca²⁺]_c≤1 nM) when [Ca²⁺]_L was kept at 1 mM. Note that inhibition at the I1 sites was estimated to be ∼4% in the presence of 20 µM [Mg²⁺]_c at 50 mM [Cs⁺]_c and negligible (<0.1%) in the presence of 72 µM [Mg²⁺]_c at 250 mM [Cs⁺]_c. In view of this, P₀ = P_A0 = (0.515/2) and ^1pP_A0 = (0.515/2)^0.25 = 0.712 for 250 mM [Cs⁺]_c, and P_A0 = 0.2678 (= 1.04 × 0.515/2) and ^1pP_A0 = (0.2678)^0.25 = 0.719. Dividing the left side term of Eq. A6 for 50 mM [Cs⁺]_c by the left side term of Eq. A6 for 250 mM [Cs⁺]_c and the right side term of Eq. A6 for 50 mM [Cs⁺]_c by the right side term of Eq. A6 for 250 mM [Cs⁺]_c, one obtains the following expression: (0.8471/0.719 − 1)/(0.8471/0.712 − 1) = (20 µM/72 µM)(1 + 62,500/^1pK_Mm²)/(1 + 2,500 mM²/^1pK_Mm²) or 3.384 (1 + 2,500 mM²/^1pK_Mm²) = (1 + 62,500 mM²/^1pK_Mm²) and ^1pK_Mm²= 22,668 mM² with ^1pK_Mm = 150.6 mM. Furthermore, from Eq. A6, it follows that ^1pK_Mg(1 + 1 mM/^1pK_L) = ([Mg²⁺]_c/(1 + [M_m⁺]_c²/^1pK_Mm²)) (^1pP_A0/(^1pP_i–^1pP_A0)) = 119.1 µM.

The ^1pK_L and [Mg²⁺]_c values were obtained by solving the system of equations: ^1pK_Mg(1 + 0.1 mM/^1pK_L) = 39.5 µM and ^1pK_Mg(1 + 1 mM/^1pK_L) = 119.1 µM, which yielded ^1pK_L= 346.6 µM and ^1pK_Mg= 30.65 µM.

The values for ^1pK_Mm obtained from the two sets of independent measurements were 193.6 and 150.6 mM. The mean value for ^1pK_Mm obtained from the two sets of measurements (172.1 mM) was used for quantitative predictions of RyR1 activation.