A Calcium-Induced Calcium Release Mechanism Mediated by Calsequestrin

A calcium-induced calcium release mechanism mediated by calsequestrin

Young-Seon Lee ?† and James P. Keener † ∗ ? Department of Biomedical Sciences, Cornell University Ithaca, New York 14853, USA † Department of Mathematics, University of Utah Salt Lake City, Utah 84112, USA

∗Corresponding author. Address: Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, U.S.A., Tel.: (801)581-6089, Fax: (801)585- 1640, Email: [email protected]

1 Abstract

Calcium (Ca2+)-induced Ca2+ release (CICR) is widely accepted as the principal mechanism linking electrical excitation and mechanical contraction in cardiac cells. The CICR mechanism has been understood mainly based on binding of cytosolic Ca2+ with ryanodine receptors (RyRs) and inducing Ca2+ release from the sarcoplasmic reticulum (SR). However, recent experiments suggest that SR lumenal Ca2+ may also participate in regulating RyR gating through calsequestrin (CSQ), the SR lumenal Ca2+ buffer. We investigate how SR Ca2+ release via RyR is regulated by Ca2+ and calsequestrin (CSQ). First, a mathematical model of RyR kinetics is derived based on experimental evidence. We assume that the RyR has three binding sites, two cytosolic sites for Ca2+ activation and inactivation, and one SR lumenal site for CSQ binding. The open probability (P o) of the RyR is found by simulation under controlled cytosolic and SR lumenal Ca2+. Both peak and steady-state P o effectively increase as SR lumenal Ca2+ increases. Second, we incorporate the RyR model into a CICR model that has both a diadic space and the junctional SR (jSR). At low jSR Ca2+ loads, CSQs are more likely to bind with the RyR and act to inhibit jSR Ca2+ release, while at high SR loads CSQs are more likely to detach from the RyR, thereby increasing jSR Ca2+ release. Furthermore, this CICR model produces a nonlinear relationship between fractional jSR Ca2+ release and jSR load. These findings agree with experimental observations in lipid bilayers and cardiac myocytes.

Key words: CICR, calcium release unit, calsequestrin, ryanodine receptor

2 Introduction

In cardiac myocytes, Ca2+-induced Ca2+ release (CICR) is the process whereby Ca2+ influx through voltage-dependent Ca2+ channels induces a large Ca2+ release from the sarcoplasmic reticulum (SR) and consequential muscle contraction (Fabiato, 1983). It is generally accepted that a rapid local increase in Ca2+ concentration ([Ca2+]) in the diadic space between the sarcolemma and the SR membrane, called a “Ca2+ spark” (Cheng et al., 1993), is the basic unit of CICR in cardiac myocytes (Cannell et al., 1994). Since CICR is a primary link between the electrical activity of the membrane and the mechanical contraction of the muscle, i.e., excitation-contraction (EC) coupling, its abnormality may be a cause of cardiac arrhythmia (Boyden and ter Keurs, 2001; Chudin et al., 1999; Clusin, 2003). Because of the importance of CICR, the mechanism by which it is regulated is an area of active study. Fabiato suggested from experiments on skinned cardiac cells that CICR is regulated by time-and Ca2+ -dependent activation and inactivation through the SR Ca2+ channels or ryanodine receptors (RyRs) (Fabiato, 1985). Cytosolic Ca2+ binds to the activation site of the RyR and thus triggers SR Ca2+ release, providing a positive feedback. Although the termination mechanism of Ca2+ sparks remains controversial, Sham et al. (1998) suggested that the SR Ca2+ release is terminated by localized and Ca2+ -dependent inactivation in which Ca2+ binds to the inactivation site of the RyR, with a higher affinity at a lower rate constant than the activation site, leading to a negative feedback to terminate SR Ca2+ release. Indeed, considerable attention has been paid to the cytosolic aspects of Ca2+ release where cytosolic Ca2+ alone plays a role in regulating Ca2+ release through the RyR. However, some experiments from lipid bilayers suggest that SR lumenal Ca2+ can also regulate the RyR channel gating (Sitsapesan and Williams, 1997; Gyorke and Gyorke, 1998). These studies demonstrate that Ca2+ -sensing sites exist in the SR lumen (Ching et al., 2000), and high (low) SR Ca2+ load enhances (decreases) the open probability of the RyR. Therefore, SR Ca2+ load can be considered not only a Ca2+ store for release, but also influences the RyR channel gating. Furthermore, potentiation of SR Ca2+ release by increased SR Ca2+ load can produce a nonlinear relationship between the fractional SR Ca2+ release and the SR Ca2+ load (Bassani et al., 1995; Shannon et al., 2000b). So the question arises, how does the SR lumenal Ca2+ regulate (or modulate) the activity of the RyR channel? It has been suggested that the regulation occurs through either a direct

3 Ca2+ binding to the RyR inside the SR or via the SR lumenal auxiliary proteins calsequestrin (CSQ), triadin and junctin (Gyorke and Gyorke, 1998; Ching et al., 2000; Gyorke et al., 2004). In recent experiments with a lipid bilayer, Gyorke et al. (2004) proposed that CSQ, the SR lumenal Ca2+ buffer, inhibits (enhances) RyR activity at low (high) SR Ca2+ load through its association (dissociation) with the RyR. However, the precise molecular mechanisms by which CICR via the RyR is modulated by CSQ are still elusive. The local control theory of EC coupling (Stern, 1992) suggests that global Ca2+ release in the whole cell is primarily controlled by local Ca2+ in the junctional area between the T-tubule and the SR, where Ca2+ influx via L-type Ca2+ channels, or dihydropyridine receptors (DHPRs), triggers Ca2+ release through co-localized RyR channels. The junctional area is divided into two domains: diadic space and junctional SR (jSR) (see Fig. ??). The jSR is functionally separated from the network SR (NSR), which is a sparse and continuous subcompartment of the SR. Most SR Ca2+ pumps reside in the NSR for sequestering released Ca2+. CSQ is found mostly in the jSR, not in the network SR (Jorgensen and Campbell, 1984). Furthermore, experimental studies have suggested that CSQ actively participates in the regulation of RyR activity from the jSR (Beard et al., 2002; Beard et al., 2004; Beard et al., 2005; Dulhunty et al., 2006; Gyorke et al., 2004). Although several mathematical models of CICR incorporated the fact that the lumenal Ca2+ has a regulatory role in RyR gating directly (Laver, 2007; Sobie et al., 2002; Shannon et al., 2004) or indirectly mediated by CSQ (Snyder et al., 2000), none of those models take the interaction between RyR and CSQ (RyR-CSQ) in the regulation of CICR into account. Thus, our goal here is to develop a model of RYR gating that directly incorporates regulation by CSQ, and to incorporate this model into a CICR model with which to address the following questions: (a) What are the roles of CSQ in RyR channel gating and CICR? (b) How does the fractional SR Ca2+ release depend on the calcium load? (c) What is the effect of changing the concentration of total CSQ inside the jSR? Our CICR model shows that tight control of CICR can be obtained by a localized Ca2+ -dependent regulation, as well as by RyR-CSQ interaction. The description of RyR-CSQ in this model yields a good correlation with recent experimental observations, and this model produces a nonlinear relation between the fraction of jSR Ca2+ release and the jSR load that fits well with the experimental findings of Shannon et al. (2000b).

4 The Model

Overview We focus on CICR regulation by CSQ, which is primarily controlled by local Ca2+ in the junctional area between the T-tubule and the surface of SR membrane. To describe the behavior of Ca2+ release and transport, we construct a model of CICR, based on that of Rice et al. (1999), with modifications as follows. First, our kinetic model of RyR is a generalization of the four state model of Stern et al. (1999) that specifically incorporates cytosolic Ca2+ and CSQ binding. Second, we use master equations to determine the RYR open probability rather than use a fully stochastic model. Stochastic attrition of RyR channels may be an important factor for termination of the RyR activity (Stern, 1992), but local Ca2+ dependent inactivation, modeled in a deterministic way (via deterministic master equations), can also terminate Ca2+ release (Sham et al., 1998) and is used in our model. Third, a specified L-type Ca2+ current is used as the Ca2+ influx to stimulate CICR, allowing us to focus on Ca2+ release dynamics of RyR channels. Because of this third feature, our CICR model does not exhibit graded response; graded response appears to result primarily from the stochastic behavior of L-type calcium channels in response to voltage stimuli.

Experimental observations CSQ is a Ca2+ -binding protein with high capacity and moderate aﬃnity, and is localized in close proximity to the RyR channel in the lumen of jSR (Jorgensen et al., 1984). Each CSQ molecule has a capacity to bind about 40 to 50 Ca2+ ions (Maclennan and Wong, 1971). The equilibrium constant 2+ of CSQ binding to Ca (KD,CSQ = 0.63 mM) (Shannon and Bers, 1997) is known, but other kinetic constants of CSQ binding are not known yet (Gyorke et al., 2002). Donoso et al. (1995) reported that Ca2+ dissociation from or association to CSQ occurs on a much faster time scale than the Ca2+ release process. Contact between the RyR and CSQ is mediated by two junctional SR membrane proteins triadin and junctin (Guo et al., 1995; Jones et al., 1995). These four proteins, RyR, triadin, junctin, and CSQ, form a quarternary “RyR complex” as a unit of Ca2+ regulation in EC coupling (Zhang et al., 1997) (see Fig. ??). Triadin and junctin are anchoring proteins between CSQ

5 and RyR, but it has also been reported that CSQ can bind directly to RyR (Cho et al., 2007). Mitchell et al. (1988) observed that Ca2+ binding to CSQ causes a conformational change of CSQ, which prohibits CSQ interaction with the associated proteins in the RyR complex. Thus, [Ca2+] change in the jSR leads to a conformational change of CSQ, and this information is transmitted to the associated proteins and aﬀects RyR activity. When it associates with the RYR complex, which it does at low SR [Ca2+], CSQ inhibits the RyR channel opening (Ikemoto et al., 1989). Conversely, the open probability of the RyR channel is increased when CSQ dissociates from the RyR complex at high SR [Ca2+] (Gyorke et al., 2004). Finally, activation of the RyR induces Ca2+ dissociation from CSQ mediated by CSQ-RyR.

Kinetic model of the RyR channel Fabiato (1985) ﬁrst suggested that Ca2+ release through RyR channels is both time-and Ca2+-dependent, which means that the time constant of RyR activation is determined by how much Ca2+ is in the vicinity of the RyR. Fur- ther, since RyR is a tetrameric channel complex consisting of four monomers, it is reasonable to expect that RyR activation requires multiple Ca2+ ions to bind. In fact, data from lipid bilayers of single RyR channel suggest that RyR activation requires binding of 2 to 4 Ca2+ ions. (Sitsapesan and Williams, 1994; Zahradnikova et al., 1999). Here, we assume that two Ca2+ ions are required to bind for RyR activation, which is in accord with the idea of Fabiato (1985) and Stern et al. (1999). We further assume that inactivation results from binding of a single Ca2+ ion to the inactivation site of the RyR. Although the associated proteins triadin and junctin are suggested to play a role as mediators between the RyR and CSQ, their roles in Ca2+ release are not known. Hence, in this model, we do not include these mediators, but assume that the RyR has a direct binding site for CSQ molecule. Thus, we model of the RYR channel as having three binding sites, one for CSQ binding from the jSR lumen and two for Ca2+ binding from the diadic space, one for activation and one for inactivation of the receptor. This is analogous to the De Young and Keizer IP3R model which has one IP3 and two Ca2+ binding sites on the cytosolic side (De Young and Keizer, 1992). We let Sijk denote the binding states of the RyR, where i, j, k = 0 or 1, and 0 and 1 represent the unoccupied and occupied binding sites, respectively. The index i represents the CSQ binding site in the jSR lumen, the index

6 j denotes the cytosolic Ca2+ activation site, and the index k indicates the cytosolic Ca2+ inactivation site. The open (i.e., conducting) states of the RyR channel are those states S010 and S110 for which the activation site is bound by Ca2+ and Ca2+ is not bound to the inactivation site. As described below, the main differences between these two conducting states are the binding affinities for activating and inactivating Ca2+ . The rate constants of the RyR kinetics depend on the concentration of free CSQ (denoted as [CSQ] or q) in the jSR and free Ca2+ concentration ([Ca2+] or c) in the diadic space. The diagram of a general eight-state model of the RyR channel is shown in Fig. ??. We let xijk be the fraction of subunits in state Sijk. Using the law of mass action (Keener and Sneyd, 1998) we write a system of differential equations describing the dynamics of the fraction of channels in each state (equivalently, the probability that a single channel is in a particular state).

dx dx 100 = V − V − V , 000 = −V − V − V , (1) dt 1 11 4 dt 1 2 5 dx dx 010 = V − V − V , 110 = V + V − V , (2) dt 2 3 6 dt 3 11 7 dx dx 101 = −V + V + V , 001 = −V − V + V , (3) dt 12 8 4 dt 8 9 5 dx dx 011 = V − V + V , 111 = V + V + V , (4) dt 9 10 6 dt 10 12 7 where Vi are

2 V1 = k1qx000 − k−1x100, V2 = k2c x000 − k−2x010,

V3 = k3qx010 − k−3x110, V4 = k4cx100 − k−4x101,

V5 = k5cx000 − k−5x001, V6 = k6cx010 − k−6x011,

V7 = k7cx110 − k−7x111, V8 = k8qx001 − k−8x101, 2 V9 = k9c x001 − k−9x011, V10 = k10qx011 − k−10x111, 2 2 V11 = k11c x100 − k−11x110, V12 = k12c x101 − k−12x111.

For this model, there are 24 rate constants. To reduce this number, we have the following assumptions and restrictions:

1. Dissociation rates for the three binding sites are independent of Ca2+ or

7 CSQ binding:

k−2 = k−11 = k−9 = k−12, (5)

k−4 = k−5 = k−6 = k−7, (6)

k−1 = k−3 = k−8 = k−10. (7)

2. The rate of Ca2+ binding to the activation site is independent of whether the Ca2+ inactivation site is occupied or not, and the rate of Ca2+ binding to the Ca2+ inactivation site is independent of whether the Ca2+ activation site is occupied or not:

k2 = k9, (8)

k11 = k12, (9)

k4 = k7, (10)

k5 = k6. (11)

3. To satisfy detailed balance in each cycle of the eight-state model, the following conditions are required:

k11k−3k−2k1 = k−11k−1k2k3, (12)

k−1k5k8k−4 = k1k4k−8k−5, (13)

k−8k9k10k−12 = k8k12k−10k−9, (14)

k−6k3k7k−10 = k6k10k−7k−3, (15)

k−2k5k9k−6 = k2k6k−9k−5. (16) When combined with previous assumptions these reduce to the three independent constraints

k11k1 = k2k3, (17)

k5k8 = k1k4, (18)

k3k4 = k5k10. (19)

The most important assumptions relate to how CSQ binding aﬀects the opening and closing of the RYR. It is known that when it associates with the RYR complex, CSQ inhibits the RyR channel opening (Ikemoto et al., 1989). Conversely, the open probability of the RyR channel is enhanced when CSQ dissociates from the RyR complex (Gyorke et al., 2004). Thus, we assume that CSQ binding inhibits activation and promotes inactivation, as follows:

8 1. The rate of Ca2+ binding to the activation site with CSQ in the unbound state is larger than that with CSQ in the bound state (see Fig. ?? a in Discussion):

k2 = k9 > k11 = k12. (20)

2. The rate of Ca2+ binding to the inactivation site with CSQ in the bound state is larger than that with CSQ in the unbound state (see Fig. ?? b in Discussion): k4 = k7 > k5 = k6. (21)

It follows from (7), (20), and the requirements for detailed balance that the dissociation constant of CSQ binding is larger with Ca2+ bound to the activation site than with Ca2+ not bound to the same site, that is,

d k−3 d k−1 K3 = > K1 = (22) k3 k1 d k−10 d k−8 K10 = > K8 = . (23) k10 k8

d d d d Notice also that for detailed balance to hold, K10K1 = K8 K3 .

Reduction to a four-state model There is not suﬃcient experimental data to determine all of the binding rate constants for CSQ-RyR and Ca2+-RyR interactions. Most data obtained from lipid bilayer experiments have stationary rather than dynamic conditions with ﬁxed Ca2+ concentrations on the cytosolic and the SR lumenal sides. It is believed that Ca2+ dissociation from CSQ is much faster than other processes, and it may therefore also be that internal activities of CSQ- RyR and CSQ-Ca2+ occur on a faster time scale than the activation of the RyR, although this is by no means certain. However, in order to obtain further reductions of the model, we assume that the association and dissociation of CSQ with Ca2+ and the RyR are much faster than Ca2+ binding to the activation and inactivation sites in the RyR. This implies that on the time scale of the cytosolic Ca2+ binding to the RyR, CSQ binding to the RyR equilibrates to a quasi-steady state. Using this

9 approximation, we can reduce the eight-state model to a four-state model. We deﬁne the new state variables as

x00 = x000 + x100, x11 = x011 + x111, (24)

x10 = x010 + x110, x01 = x001 + x101. (25)

It follows from Eqs.(1)-(4) that

dx 00 = −V − V − V − V , (26) dt 11 4 2 5 dx 10 = V − V + V − V , (27) dt 2 6 11 7 dx 01 = −V + V − V + V , (28) dt 12 4 9 5 dx 11 = V + V + V + V , (29) dt 9 6 12 7 Now, the binding and unbinding reactions of CSQ to the RyR are taken to be in equilibrium (i.e., set (V1 = V3 = V8 = V10 = 0), which leads to the algebraic relations

d K1 k qx = k− x ⇒ x = x , (30) 1 000 1 100 000 q 100 d K3 k qx = k− x ⇒ x = x , (31) 3 010 3 110 010 q 110 d K8 k qx = k− x ⇒ x = x , (32) 8 001 8 101 001 q 101 d K10 k qx = k− x ⇒ x = x , (33) 10 011 10 111 011 q 111

d where Kj = k−j/kj for all j. Substituting these relations into Eqs. 24 and 25, we obtain

Kd Kd x = (1 + 1 )x , x = (1 + 10 )x , (34) 00 q 100 11 q 111 Kd Kd x = (1 + 3 )x , x = (1 + 8 )x . (35) 10 q 110 01 q 101

10 from which we determine that

d q K1 x100 = d x00, x000 = d x00, (36) q + K1 q + K1 d q K3 x110 = d x10, x010 = d x10, (37) q + K3 q + K3 d q K10 x111 = d x11, x011 = d x11, (38) q + K10 q + K10 d q K8 x101 = d x01, x001 = d x01. (39) q + K8 q + K8 Substituting Eqs. 36–39 into Eqs. 26–29, we obtain a reduced system of diﬀerential equations for the four-state model:

d d 2 dx00 k−2K3 + k−11 q (k2K1 + k11 q) c = d x10 − d x00 dt K3 + q K1 + q d d k−5K8 + k−4q (k5K1 + k4q) c + d x01 − d x00 , K8 + q K1 + q d 2 d dx10 (k2K1 + k11q) c k−2K3 + k−11 q = d x00 − d x10 dt K1 + q K3 + q d d k−6K10 + k−7 q (k6K3 + k7 q) c + d x11 − d x10 , K10 + q K3 + q d d dx11 (k6K3 + k7 q) c k−6K10 + k−7 q = d x10 − d x11 dt K3 + q K10 + q d 2 d (k9K8 + k12q) c k−9K10 + k−12 q + d x01 − d x11 , K8 + q K10 + q d d 2 dx01 k−9K10 + k−12 q (k9K8 + k12 q) c = d x11 − d x01 dt K10 + q K8 + q d d (k5K1 + k4q) c k−5K8 + k−4 q + d x00 − d x01 . K1 + q K8 + q Applying the conditions 5–11, we have a simpliﬁed four-state scheme of

11 the RyR (Fig. ??), which can be written as

dx00 2 = k− x − Act c x + (k− x − Inact c x ) , (40) dt 2 10 1 00 6 01 2 00 dx10 2 = Act c x − k− x + (k− x − Inact c x ) , (41) dt 1 00 2 10 6 11 1 10 dx11 2 = (Inact c x − k− x ) + Act c x − k− x , (42) dt 1 10 6 11 2 01 2 11 dx01 2 = k− x − Act c x + (Inact c x − k− x ) , (43) dt 2 11 2 01 2 00 6 01 where d k2K1 + k11q Act1 = d , (44) (K1 + q d k2K8 + k11q Act2 = d , (45) K8 + q d k6K3 + k7q Inact1 = d , (46) K3 + q d k6K1 + k7q Inact2 = d . (47) K1 + q

In this scheme, the state S10 represents the open state of the RyR, so that 2+ x10 is the open probability (Po) of the jSR Ca release ﬂux in the CICR model. For this model there are 9 undetermined parameters, the unbinding rate constants k−2 and k−6, the binding rate constants k2, k6, k7, and k11, and d d d the three equilibrium constants K1 , K3 , and K8 . However, detailed balance d d k6 d d k11 requires that K8 = K1 k7 and K3 = K1 k2 . The parameters k2, k−2, k4 = k7, k−4 = k−6 are taken from the four-state model of Stern et al. (1999). The remaining 3 free parameters k6, k11 and d K1 were chosen arbitrarily to give reasonable agreement between our CICR model and the data of Shannon et al. (2000b). The full list of parameter values is given in Table 1.

CICR model We now use this RyR kinetic model to construct a model of CICR. Fig. ?? gives the overview of Ca2+ ﬂuxes participating in CICR. [Ca2+] in the diadic

12 2+ space (ds) is determined by the L-type Ca influx (JDHPR), the jSR release flux (Jrel) and the diffusion flux from the diadic space to the myoplasm (Jxfer). 2+ 2+ The jSR [Ca ] changes due to jSR release flux (Jrel) and Ca diffusion 2+ flux from the NSR (Jtr). Thus, the Ca concentration changes in the two domains, diadic space (ds) and jSR, are described by

2+ d[Ca ]ds VjSR Vmyo = βds Jrel − Jxfer − JDHPR , (48) dt Vds Vds d[Ca2+] jSR = β (J − J ), (49) dt jSR tr rel where βds represents the percentage of total calcium flux that is not ab- 2+ sorbed by the buffer calmodulin (CMD), a cytosolic Ca buffer, and βjSR represents the percentage of total calcium flux that is not absorbed by the buffer CSQ, the jSR Ca2+ buffer. The formulas for these buffering terms are given below. The flux terms Jrel and Jxfer are scaled by the ratio of volumes to take into account that these fluxes are between compartments (jSR with volume VjSR and myoplasm with volume Vmyo to diadic space with volume Vds, respectively) with differing volumes.

Ca2+ fluxes For this discussion we are interested only in the behavior of the RyR release with a fixed Ca2+ input profile, thus the L-type Ca2+ channel (or dihydropyridine receptor, DHPR) flux is given by

ICa JDHPR = , (50) 2FVds 2+ where ICa is the L-type Ca current and F is Faraday’s constant, and where the L-type Ca2+ current is speciﬁed to be the piecewise exponential function

Ipeak(1 − exp(−t/τ1)) if 0.000 s ≤ t < 0.005 s ICa =  Ipeak(1 − exp(−0.005/τ1)) exp(−(t − 0.005)/τ2) if 0.005 s ≤ t < 0.200 s  0 s if 0.200 s ≤ t  with parameters Ipeak, peak amplitude, and time constants τ1 and τ2 (see Fig. ??). The jSR Ca2+ release ﬂux via the RyR channel is described as

2+ 2+ Jrel = v1Po([Ca ]jSR − [Ca ]ds), (51)

13 where v1 is the maximum permeability of the RyR channels, Po is the open 2+ 2+ probability of the RyR channel, and ([Ca ]jSR −[Ca ]ds) is the driving force from the jSR to the diadic space. The Ca2+ ﬂux from the diadic space to the myoplasm via diﬀusion is given by

2+ 2+ ([Ca ]ds − [Ca ]myo) Jxfer = , (52) τxfer where τxfer is the time constant for diffusion from the diadic space to the 2+ 2+ myoplasm and [Ca ]myo is set to 0.1 µM. The Ca diffusion flux from the NSR to the jSR is given by

2+ 2+ ([Ca ]NSR − [Ca ]jSR) Jtr = , (53) τtr

2+ where τtr is the time constant for diﬀusion from the NSR to the jSR. [Ca ]NSR 2+ is taken to be a constant value equal to the initial [Ca ]jSR.

Ca2+ -CSQ buﬀering CSQ is a dominant SR lumenal Ca2+ buﬀer (Bers, 2001). We suppose that each CSQ molecule has m Ca2+ binding sites, and that, because of coperativ- ity, all the binding sites are either simultaneously occupied or simultaneously empty. The reaction scheme of m Ca2+ ions binding to CSQ is

kon 2+ * 2+ mCa + CSQ ) mCa CSQ, (54) koff and Ca2+ bound to CSQ is described by the equation

d[mCa2+CSQ] = k [Ca2+]m [CSQ] − k [mCa2+CSQ], (55) dt on jSR off where kon and koff are the association and dissociation rate constants. We assume that the CSQ-Ca buﬀering process is faster than that of Ca2+ release (Donoso et al. 1995) , so that Eq. 55 can be taken to be in quasi-steady state:

2+ m 2+ kon[Ca ]jSR[CSQ] − koff [mCa CSQ] = 0. (56)

14 The total concentration of CSQ ([CSQ]tot) in the jSR is

2+ [CSQ]tot = [CSQ] + [mCa CSQ]. (57)

Eqs. 56 and 57 together imply that

m [CSQ]totkD,CSQ [CSQ] = m 2+ m , (58) kD,CSQ + [Ca ]jSR

1/m where kD,CSQ = (koff /kon) is the dissociation constant, found to be ap- proximately 400-650 µM (Shannon and Bers, 1997; Mitchell et al., 1988). 2+ According to Equation 58 [CSQ] varies inversely with [Ca ]jSR: at low 2+ [Ca ]jSR, [CSQ] is large, and vice versa (Fig. ??). We take m = 3. In general, the cooperativity of ligand (Ca2+) binding to protein (CSQ) is less than the number of binding sites (40 to 50) in the protein. The choice of m = 3 is suggested by the result of Pape et al. (2007), who determined experimentally a Hill coefficient for CSQ buffering of calcium of 2.95. It follows from the fast buffering approximation (Wagner and Keizer, 1994) that 1 βjSR = 2+ m−1 m m 2+ m 2 . (59) (1 + m[Ca ]jSR [CSQ]totkCSQ/(kCSQ + [Ca ]jSR) )

Ca2+ buﬀering in the diadic space Calmodulin (CMD) is a cytosolic Ca2+ binding protein, whose binding relation with Ca2+ can be derived similarly to that for Ca2+ -CSQ binding, with m = 1. We also assume that Ca2+ binds rapidly with CMD so that the rapid buﬀer approximation is applied, yielding 1 β = . (60) ds 2+ 2 (1 + [CMD]totkCMD/(kCMD + [Ca ]ds) )

There are other types of Ca2+ buffers such as sarcolemmal and SR membrane buffers, which are not included in this model. While the time course of Ca2+ transients may be changed somewhat by the addition of these buffers to the model, these will not change the qualitative behavior of our model in any substantial way.

15 Model simulation method We simulated Eqs. 40–43, 48, 49 numerically using MATLAB (The Math- Works, Natick, MA) v7.2.0 the solver ode23s, with the parameter values satisfying detailed balance given in Table 1 and 2. In order to focus on the effects of CSQ modulation in CICR, we examine a single release event of Ca2+ transient with a fixed L-type Ca2+ channel flux protocol (see Fig. ??).

Results

Open probability of RyR channel The properties of RyR channel are characterized by the peak and steady- state open probability (Po) in Fig. ??. Mimicking the conditions of the lipid bilayer experiments by Gyorke and Gyorke (1998), we simulated Eqs. 40-43 with Ca2+ concentrations held ﬁxed on two sides. Both peak and steady-state 2+ Po show eﬀective increases for increasing jSR lumenal [Ca ]. The steady- 2+ state Po with [Ca ]jSR = 20 µM, and 5 mM in Fig. ?? b showed 2.8-fold increase of the maximal steady-state Po between the two curves.

Behavior of the model transients To investigate the dynamical change of Ca2+ transients, we simulated Eqs. 40- 43, 48 and 49. Fig. ?? shows a Ca2+ transient with an initial jSR load 2+ 2+ ([Ca ]jSR) of 575 µM. The L-type Ca channel current is given as an ex- 2+ ponential function of Ipeak and τ1, τ2 (see Fig. ??) and induces Ca release through the RyR. Then [Ca2+] in the diadic space increases rapidly from the resting value of 0.101 µM to the peak ≈ 71.1 µM in 53 ms, and recovers in 200 ms (Fig. ?? a). In this model the Ca2+ transient in the diadic space is mostly determined by the open probability (Po) of the RyR. Thus the RyR 2+ Po displays a similar dynamical change to that of the Ca transient except for its scale and initial activation (Fig. ?? b). The peak Po, which is the 2+ maximum of x10, is ≈ 0.15 in (b). As shown in (c), [Ca ]jSR drops during 2+ 2+ the Ca release, and [Ca ]jSR has its minimum value at ≈ 211 µM as the Ca2+ release ﬂux decreases. The four states of the RyR kinetics are depicted in (b) and (d). Initially there is a large drop of x00, which corresponds to an increase in the other three fractions: x01, x10, x11. Except for its scale and delay, x10 changes in parallel with x11, while x00 changes in parallel with x01.

16 Fractional Ca2+ release The fractional jSR Ca2+ release is deﬁned as initial jSR load - minimum jSR load Fractional release (F %) = × 100. initial jSR load

The fractional Ca2+ release curves were obtained from the simulation with jSR [Ca2+] ranging from 300 µM to 700 µM for m = 1 and m = 3, as shown in Fig. ??. Both curves show a nonlinear and steep relationship between the initial jSR [Ca2+] and fractional Ca2+ release. Higher cooperativity of Ca2+ and CSQ binding (m = 3) produced a sigmoidal shaped curve with steeper slope than that for m = 1, with a slowly increasing plateau (m = 1) and a saturating plateau (m = 3). To explore the effect of the total CSQ content in the jSR, we simulated the model using different levels, specifically [CSQ]tot = 8 mM (A, dashed line), [CSQ]tot = 14 mM (B, solid line), and [CSQ]tot = 20 mM (C, dash- dot line) as three curves are shown in Fig. ??. This result shows that the fractional curve shifts up and to the left as total CSQ content is decreased. Three Ca2+ transients and jSR Ca2+ changes with three CSQ contents are simulated in Fig. ??. For the higher CSQ total content, the amplitudes of Ca2+ release were decreased, but activation time and duration of Ca2+ release were lengthened.

Discussion

Roles of CSQ in Ca2+ release Numerous experimental studies have suggested that regulation of SR Ca2+ release is determined not only by cytosolic Ca2+, but also by SR lumenal Ca2+. How- ever, the detailed mechanism by which SR lumenal Ca2+ regulates Ca2+ release is not well understood. A growing body of research suggests that CSQ is a modulator of RyR gating more than simply a Ca2+ buﬀer reducing Ca2+ concentration. In particular, a recent study in lipid bilayer, by Gy- orke et al. (2004), shed light on a possible mechanism of CSQ regulation. Based on this experimental evidence, we built a kinetic model of the RyR that includes the roles of CSQ binding. In contrast to other models, which are based only on cytosolic Ca2+ dependence of RyR gating, in our model,

17 CSQ binding to the RyR plays an important role in gating. The first role of CSQ in our model is as a Ca2+ sensor for jSR lumenal Ca2+. The rapid Ca2+ binding to CSQ leads to an equilibrium state, giving an algebraic relation between free CSQ concentration ([CSQ]) and jSR [Ca2+], described by Eq. 58 and depicted in Fig. ??. The second role of CSQ lies in the modulation of RyR gating. Since we have assumed that k2 > k11 (in fact, k2 = 10k11 in the model), the activation rates Act1 and Act2 in the kinetics of the RyR are decreasing functions of q (= [CSQ]) (see Fig. ?? a). On the other hand, the inactivation rates Inact1 and Inact2 in the kinetic model of the RyR are increasing functions of [CSQ] because k7 > k6 (see Fig. ?? b). Since [CSQ] is inversely related to jSR [Ca2+], the activation rates are increasing functions of jSR [Ca2+], while the inactivation rates are decreasing functions of jSR [Ca2+]. Thus, activation rates are increased at higher calcium jSR loads, while depletion of the jSR load increases the inactivation rates and enhances the termination of Ca2+ release. The relationships between jSR load and activation and inactivation rates are shown in Fig. ??. Note that our four-state model of the RyR channel is different from the “conventional” four-state model of Stern et al. (1999) in that it does not have a symmetric structure. This asymmetric structure results from our assumption that CSQ binding decreases activation (k2 > k11 ) and increases inactivation (k7 > k6), combined with the the requirement of detailed balance. In fact, the assumption that CSQ binding modifies activation (k2 =6 k11) forces a difference in the inactivation rates Inact1 and Inact2, because of the d d requirement of detailed balance that K1 =6 K3 . Similarly, the assumption that CSQ binding modifies inactivation (k6 =6 k7) forces a difference in the activation rates Act1 and Act2 because of the requirement of detailed balance d d that K8 =6 K1 .

Nonlinear fractional jSR Ca2+ release The experimental results of Bassani et al. (1995) and Shannon et al. (2000b) suggest that there is a nonlinear and steep relationship between fractional SR Ca2+ release and SR Ca2+ load. The nonlinear relationship is possible only if there is a modulatory eﬀect either from the cytosol or SR lumen. Shannon et al. (2004) assumed an “ad hoc” enzymatic role of SR lumenal Ca2+ in their RyR kinetic model, which produces a nonlinear fractional relationship. In contrast, our model relies on CSQ binding with the RyR to give a more mechanistic basis for the regulatory role of SR lumenal Ca2+ .

18 Simulation of our CICR model produced a nonlinear fractional release function of the jSR Ca2+ load, which is similar to that found in the experiment of Shannon et al. (2000b) (see Fig. ??). The maximal slope of the fractional Ca2+ release curve depends on the eﬀective number of Ca2+ ions that bind with CSQ, indicating that the Ca2+ binding capacity of CSQ can control Ca2+ release.

Termination mechanism of Ca2+ release The mechanism for termination of Ca2+ release is still unknown although several mechanisms have been suggested including stochastic attrition, total or partial SR depletion, and Ca2+ dependent inactivation (Sobie et al., 2002; Stern, 1992). Sham et al. (1998) suggest that local inactivation of RyR channels terminates Ca2+ release due to high and local [Ca2+]. Although one should not ignore modulatory effects on Ca2+ release by Mg+, FK-binding protein, calmodulin, etc., the focus here is on the role of localized, high [Ca2+] for the termination of Ca2+ release as Sham et al. suggest. Our model suggests that CSQ can play an important role in termination of Ca2+ release. Although the termination mechanism of RyR channel should be understood from time-dependent [Ca2+] changes of both the diadic space and jSR lumen, the steady-state behavior of RyR channel implies that RyR 2+ 2+ Po depends on [Ca ] in the diadic space and jSR lumenal Ca mediated by CSQ (Fig. ??). Increase of jSR Ca2+ from 20 µM to 5 mM (Fig. ?? b) enhances the sensitivity of RyR channel by a 2.8-fold increase in the steady- state Po which fits well to the 2.7-fold seen in the data of Gyorke and Gyorke (1998). Furthermore, Fig. ?? and Fig. ?? show that [CSQ] change resulting from the jSR Ca2+ change also has an effect on Ca2+ inactivation, so that inactivation is dependent on both diadic space Ca2+ and jSR lumenal Ca2+ .

Eﬀects of total CSQ levels on Ca2+ release Recent experimental data suggest that interaction between CSQ and RyR channel is an essential component to maintain stable cardiac function. A disruption or depression of CSQ binding to the RyR can cause abnormal Ca2+ release via RyR channels, resulting in cardiac arrhythmia. For example, it is suggested that reduced CSQ protein levels due to mutation of cardiac CSQ are associated with polymorphic ventricular tachycardia and sudden

19 cardiac death (Knollmann et al., 2006; Terentyev et al., 2003). On the other hand, overexpression of CSQ resulted in a significant decrease in cardiac contractility and amplitude of the Ca2+ transients usually found in cardiac hypertrophy and heart failure (Jones et al., 1998; Schmidt et al., 2000). To test the effect of total [CSQ] on Ca2+ release, we simulated the model for three different values of total [CSQ], [CSQ]tot = 8 mM (A), 14 mM (B), and 20 mM (C). The results are shown in Fig. ?? and Fig. ??. Time to peak Ca2+ release and the recovery of SR lumenal Ca2+ are markedly prolonged with increased CSQ level. Simultaneously, peak amplitudes of Ca2+ transients decrease with increased CSQ level. [CSQ]tot also affects the fractional release curve. Decreasing total [CSQ] shifts the fractional curve leftward and upward. Conversely, increased [CSQ] moves the fractional Ca2+ release curve to the right with decreased amplitude, in close correspondence to the experimental observations. (see Fig. 13). Terentyev et al. (2003) suggested that increased CSQ level slows lumenal Ca2+ -dependent closure of RyR channels. Similarly, it is found in our model that Ca2+ release is slowed by the higher CSQ content. However in this case the amplitude of the Ca2+ transient is smaller than that of the normal CSQ content. This result is in disagreement with the data of Teretyev et al. (2003) who found an increase in the amplitude of Ca2+ transient with CSQ overexpression. In contrast, several recent experiments proposed that CSQ overexpression increased SR Ca2+ content but reduced the amplitude of Ca2+ transients (Jones et al., 1998; Miller et al., 2005; Sato et al., 1998). They suggested that the inhibitory effect of CSQ may cause reduced SR Ca2+ release with a larger SR Ca2+ store. Our model shows that Ca2+ release is decreased when CSQ content is higher even though the SR Ca2+ content is larger than that with normal CSQ content. As this issue is still in debate, further theoretical and experimental studies are needed for its complete res- olution.

Comparison with other models The four-state RyR kinetic model presented here can be considered an ex- tension of the models of Fabiato (1992) or Stern et al. (1999). The formal shape and properties of our four-state RyR model are similar to the model of Shannon et al. (2004), although Shannon et al. formulated their model as- suming a Michaelis-type relation for SR lumenal [Ca2+] dependence without consideration of CSQ binding.

20 The rates of activation and inactivation in our four-state model of the RyR channel are asymmetric. This feature is a distinct difference from other models which are based only on cytosolic Ca2+. This property results from assumptions that Ca2+ bindings are not independent of CSQ binding, and similarly, that activation of the RyR is functionally coupled to CSQ-RyR interactions. It is noteworthy that this asymmetry also results partly from the requirement of detailed balance. Our CICR model focuses on local Ca2+ change in the diadic space and jSR through L-type Ca2+ influx, jSR Ca2+ release flux, efflux from the diadic space, and refilling flux to jSR from NSR (see Fig. ??). This conceptual structure is similar to the model of Rice et al. (1999), although we consider only the deterministic limit of the RyR channel gating. A major difference between our model and Rice et al. model is the inclusion here of a model for the interaction of CSQ binding with the RyR channel, leading to Ca2+ release mechanism regulated by jSR Ca2+. The Rice et al. RyR channel kinetics depend only on cytosolic Ca2+. Snyder et al. (2000) proposed bidirectional feedback between the RyR channel and SR lumenal Ca2+ through CSQ-Ca binding. Their model assumes that (a) Ca2+ binding to the RyR channel shifts the CSQ-Ca binding curve to trigger Ca2+ release from CSQ-Ca unbinding, and (b) after SR Ca2+ release, a reduction in [CSQ-Ca] decreases the dissociation constant of the cytosolic Ca2+ binding to the RyR at the inactivation site, thus shutting off the gate. However, in this model information of how CSQ-Ca binding is transmitted to the RyR gating is lacking. RyR channel model of the Sobie et al. (2002) has two states: open and closed state. The RyR closing rate is assumed to be independent of the diadic space [Ca2+], while our model assumes that inactivation rates depend on both the diadic space [Ca2+] and jSR lumenal [Ca2+]. In the model of Sobie et al., jSR lumenal dependence of RyR gating is modeled with a simple scaling factor for [Ca2+] in the diadic space. Thus, in their model, CSQ buffering does not directly play an active role in the gating of RyR channel, but indirectly mediates the change of jSR lumenal [Ca2+].

Limitations In our CICR model we used a ﬁxed Ca2+ current proﬁle to induce SR Ca2+ release. Because the Ca2+ current does not depend on voltage, our model cannot show graded response as a function of voltage (Stern et al. 1999). Graded response

21 is usually explained by the stochastic response of L-type channels to voltage stimuli. Our model is a deterministic model of triggered SR Ca2+ release, which makes the implicit assumption that the number of RyR channels in a diadic space is sufficiently large that stochastic effects are negligible. While there are certainly features of calcium release that cannot be replicated by deterministic models, such as graded response or spontaneous calcium release, many of the main features of RYR calcium release that relate to CSQ are appropriately modeled by a deterministic model. Indeed, we expect that 100- 200 RyRs in one diadic space is sufficient to expect deterministic triggered release behavior. An important issue is to understand how spontaneous calcium release is affected by the total CSQ content, and this remains a topic for future investigation. Our four state model was derived from an 8-state model using a quasi- steady state approximation for which we know no justification. Similarly, the derivation of our CICR model assumed fast buffering reactions, allowing significant simplifications. It may be that these assumptions are not justi- fied, however, we suspect that the qualitative features of the model are not changed significantly by these useful simplifications.

Conclusion

The model presented here incorporates the novel idea that CSQ plays a role not only as a jSR lumenal Ca2+ sensor, but also as a regulator of RyR channel activity by association with and dissociation from the RyR. CSQ senses the change of jSR Ca2+ and relays that information to the RyR through its interaction. RyR gating is controlled by the cytosolic Ca2+ activation and inactivation combined with CSQ regulation. The steady-state RyR Po shows about 2.8-fold increase in its maximal value with increased jSR Ca2+ , which agrees well with the data of lipid bilayer experiments. The model shows a nonlinear fractional jSR Ca2+ release curve, which shows a sharp transition as the jSR load increases. The cooperativity of Ca2+ with CSQ and total CSQ content can change the sensitivity of the jSR Ca2+ release, which might provide an explanation for increased spontaneous Ca2+ release with reduced CSQ. Also, increased CSQ content augments SR Ca2+ content, but prolongs Ca2+ release activation and duration with reduced amplitude. This is possible because of the inhibitory role of CSQ on RyR opening. Thus, this model

22 suggests a mechanism by which RyR channel activity is regulated tightly by both the diadic Ca2+ and jSR lumenal Ca2+ through CSQ binding.

Acknowledgement

The authors thank Robert F. Gilmour, Jr., Elizabeth Cherry and Flavio Fenton for reading and giving useful comments to this manuscript.

This research was supported in part by NSF Grant DMS-0211366 and NIH Grant HL 07515.

23 References

Bassani, J. W. M., Yuan, W., Bers, D. M., 1995. Fractional SR Ca release is regulated by trigger Ca and SR Ca content in cardiac myocytes. Am. J. Physiol. 268, C1313-C1329.

Beard, N. A., Sakowska, M. M., Dulhunty, A. F., Laver, D. R., 2002. Calse- questrin is an inhibitor of skeletal muscle ryanodine receptor calcium release channels. Biophys. J. 82, 310-320.

Beard, N. A., Laver, D.R., Dulhunty, A. F., 2004. Calsequestrin and the calcium release channel of skeletal and cardiac muscle. Prog. Biophys. Mol. Biol. 85, 33-69.

Beard, N. A., Casarotto, M. G., Wei, L., Varsanyi, M., Laver, D. R., Dul- hunty, A. F., 2005. Regulation of ryanodine receptor by calsequestrin: Eﬀect of high luminal Ca2+ and phosphorylation. Biophys. J. 88, 3444-3454.

Bers, D. M., 2001. Excitation-Contraction Coupling and Cardiac Contractile Force (2nd ed.). Dordrecht. The Netherlands: Kluwer Academic Press.

Boyden, P. A., ter Keurs, H. E., 2001. Reverse excitation-contraction coupling: Ca2+ ions as initiators of arrhythmias. J. Cardiovasc. Electrophysiol. 12, 382-385.

Cannell, M. B., Cheng, H., Lederer, W. J., 1994. Spatial non-uniformities in 2+ [Ca ]i during excitation-contraction coupling in cardiac myocytes. Biophys. J. 67, 1942-1956.

Cheng, H., Lederer, W. J., Cannell, M. B., 1993. Calcium sparks: Ele- mentary events underlying excitation-contraction coupling in heart muscle. Science. 262, 740-744.

Ching, L. L., Williams, A. J., Sitsapesan, R., 2000. Evidence for Ca2+ activation and inactivation sites on the luminal side of the cardiac ryanodine receptor complex. Circ. Res. 87, 201-206.

Cho, J. H., Ko, K. M., Gunasekaran, G., Lee, W., Kang, G. B., Rho, S.,

24 Park, B., Yu, J., Kagawa, H., Eom, S., Kim, D. H., Ahnn, J., 2007. Func- tional importance of polymerization and localization of calsequestrin in C. elegans. J. Cell. Sci. 120, 1551-1558.

Chudin, E., Goldhaber, J., Garinkel, A., Weiss, J., Kogan, B., 1999. Intra- cellular Ca2+ dynamics and the stability of ventricular tachycardia. Biophys. J. 77, 2930-2941.

Clusin, W. T., 2003. Calcium and cardiac arrhythmias: DADs, EADs, and alternans. Crit. Rev. Clin. Lab. Sci. 40, 337-375.

De Young, G. W., Keizer, J., 1992. A single pool IP3-receptor based model for agonist stimulated Ca2+ oscillations. Proc. Natl. Acad. Sci. USA. 89, 9895-9899.

Donoso, P., H. Prieto, Hidalgo, C., 1995. Luminal calcium regulates calcium release in triads isolated from frog and rabbit skeletal muscle. Biophys. J. 68, 507-515.

Dulhunty, A. F., Beard, N.A., Pouliquin, P., Kimura, T., 2006. Novel reg- ulators of RyR Ca2+ release channels: insight into molecular changes in genetically-linked myopathies. J. Muscle Res. Cell. Motil. 27, 351-365.

Fabiato, A., 1983. Calcium-induced release of calcium from the cardiac sarcoplasmic reticulum. Am. J. Physiol. 245, C1-C14.

Fabiato, A., 1985. Time and calcium dependence of activation and inactivation of calcium-induced release of calcium from the sarcoplasmic reticulum of a skinned canine cardiac purkinje cell. J. Gen. Physiol. 85, 247-289.

Fabiato, A. 1992, Two kinds of calcium-induced release of calcium from the sarcoplasmic reticulum of skinned cardiac cells. Adv. Exp. Med. Biol. 311, 245-262.

Guo, W., Campbell, K. P., 1995. Association of triadin with the ryanodine receptor and calsequestrin in the lumen of the sarcoplasmic reticulum. J. Biol. Chem. 270, 9027-9030.

25 Gyorke, I., Gyorke, S., 1998. Regulation of the cardiac ryanodine receptor channel by luminal Ca2+ involves luminal Ca2+ sensing sites. Biophys. J. 75, 2801-2810.

Gyorke, S., Gyorke, I., Lukyanenko, V., Terentyev, D., Viatchenko-Karpinski, S., Wiesner, T. F., 2002. Regulation of sarcoplasmic reticulum calcium release by luminal calcium in cardiac muscle. Front. Biosci. 7, d1454-1463

Gyorke, I., Hester, N., Jones, L. R., Gyorke, S., 2004. The role of calsequestrin, triadin, and junctin in conferring cardiac ryanodine receptor re- sponsiveness to luminal calcium. Biophys. J. 86, 2121-2128.

Ikemoto, N., Ronjat, M., Meszaros, L. G., Koshita, M., 1989. Postulated role of calsequestrin in the regulation of calcium release from sarcoplasmic reticulum. Biochemistry. 28, 6764-6771.

Ikemoto, N., Antoniu, B., Kang, J., Meszaros, L. G., Ronjat, M., 1991. In- travesicular calcium transient during calcium release from sarcoplasmic reticulum. Biochemistry. 30, 5230-5237.

Iyer, V., Mazhari, R., Winslow, R. L., 2004. A computational model of the human left-ventricular epicardial myocyte. Biophys. J. 87, 1507-1525.

Jones, L. R., Zhang, L., Sanborn, K., Jorgensen, A. O., Kelley, J., 1995. Puriﬁcation, primary structure, and immunological characterization of the 26-kDa calsequestrin binding protein (junctin) from cardiac junctional sarcoplasmic reticulum. J. Biol. Chem. 270, 30787-30787.

Jones, L. R., Suzuki, Y. J., Wang, W., Kobayashi, Y. M., Ramesh, V., Franzini-Armstrong, C., 1998. Regulation of Ca2+ signaling in transgenic mouse cardiac myocytes overexpressing calsequestrin. J. Clin. Invest. 101, 1385-1393

Jorgensen, A. O., Campbell, K. P., 1984. Evidence for the presence of calsequestrin in two structurally diﬀerent regions of myocardial sarcoplasmic reticulum. J. Cell Biol. 98, 1597-1602.

Keener, J. P., Sneyd, J., Mathematical Physiology. 1998. Springer, New

26 York.

Knollmann, B. C., Chopra, N., Hlaing, T., Akin, B., Yang, T., Ettensohn, K., Knollmann, B. E. C., Horton, K. D., Weissman, N. J., Holinstat, I., Zhang, W., Roden, D. M., Jones, L. R., Franzini-Armstrong, C., Pfeifer. K., 2006. Casq2 deletion causes sarcoplasmic reticulum volume increase, premature Ca2+ release, and catecholaminergic polymorphic ventricular tachycardia. J. Clin. Invest. 116, 2510-2520.

Laver, D. R., 2007. Ca2+ stores regulate ryanodine receptor Ca2+ release channels via luminal and cytosolic Ca2+ sites. Biophys. J. 92, 3541-3555.

Maclennan, D. H., Wong, P. T. S., 1971. Isolation of a calcium-sequestering protein from sarcoplasmic reticulum. Pro. Nat. Acad. Sci. 68, 1231-1235.

Miller, L. W., Currie, S., Loughrey, C. M., Kettlewell, S., Seidler, T., Reynolds, D. F., Hasenfuss, G., Smith, G. L., 2005. Eﬀects of calsequestrin overexpression on excitation-contraction coupling in isolated rabbit cardiomy- ocytes. Cardiovasc. Res. 67, 667-677.

Mitchell, R. D., Simmerman, H. K., Jones, L. R., 1988. Ca2+ binding eﬀects on protein conformation and protein interactions of canine cardiac calsequestrin. J. Biol. Chem. 263, 1376-1381.

Pape, P. C., Fenelon, K., Lamboley, C. H., Stachura, D., 2007. Role of calsequestrin evaluated from changes in free and total calcium concentrations in the sarcoplasmic reticulum of frog cut skeletal muscle ﬁbers. J. Physiol. 581, 319-367.

Rice, J. J., Jafri, M. S., Winslow, R. L., 1999. Modeling gain and gradedness of Ca2+ release in the functional unit of the cardiac diadic space. Biophys. J. 77, 1871-1884.

Sato, Y., Ferguson, D. G., Sako, H., Dorn, G. W., Kadambi, V. J., Yatani, A., Hoit, B. D., Walsh, R. A., Kranias, E. G., 1998. Cardiac-sepciﬁc overexpression of mouse cardiac calsequestrin is associated with depressed cardio- vascular function and hypertrophy in transgenic mice. J. Biol. Chem. 273, 28470-28477.

27 Schmidt, A. G., Kadambi, V. J., Ball, N., Sato, Y., Walsh, R. A., Kranias, E. G., Hoit, B. D., 2000. Cardiac-speciﬁc overexpression of calsequestrin results in left ventricular hypertrophy, depressed force-frequency relation and pulsus alternans in vivo. J. Mol. Cell. Cardiol. 32, 1735-1744

Sham, J. S., Song, L. S., Chen, Y., Deng, L. H., Stern, M. D., Lakatta, E. G., Cheng, H., 1998. Termination of Ca2+ release by a local inactivation of ryanodine receptors in cardiac myocytes. Proc. Natl. Acad. Sci. USA. 95, 15096-15101.

Shannon, T. R., Bers, D. M., 1997. Assessment of intra-SR free [Ca] and buﬀering in rat heart. Biophys. J. 73, 1524-1531.

Shannon, T. R., Ginsburg, K. S., Bers, D. M., 2000. Reverse mode of the sarcoplasmic reticulum calcium-pump and load-dependent cytosolic calcium decline in voltage-clamped cardiac ventricular myocytes. Biophys. J. 78, 322-333.

Shannon, T. R., Ginsburg, K. S., Bers, D. M., 2000. Potentiation of fractional sarcoplasmic reticulum calcium release by total and free intra-sarcoplasmic reticulum calcium concentration. Biophys. J. 78, 334-343.

Shannon, T. R., Wang, F., Puglisi, J., Weber, C., Bers, D. M., 2004. A mathematical treatment of integrated Ca dynamics within the ventricular myocyte. Biophys. J. 87, 3351-3371.

Sitsapesan, R., Williams, A. J., 1994. Gating of the native and puriﬁed cardiac SR Ca2+-release channel with monovalent cations as permeant species. Biophys. J. 67, 1484-1494.

Sitsapesan, R., Williams, A. J., 1997. Regulation of current ﬂow through ryanodine receptors by luminal Ca2+. J. Membr. Biol. 159, 179-185.

Snyder, S. M., Palmer, B. M., Moore, R. L., 2000. A mathematical models of cardiocyte Ca2+ dynamics with a novel representation of sarcoplasmic reticulum Ca2+ control. Biophys. J. 79, 94-115.

28 Sobie, E. A., Dilly, K. W., Dos Santos C. J., Lederer, W. J., Jafri, M. S., 2002. Termination of cardiac Ca2+ sparks: an investigative mathematical model of calcium-induced calcium release. Biophys. J. 83, 59-78.

Stern, M.D., Lakatta, E. G., Cheng, H., 1998. Termination of Ca2+ release by a local inactivation of ryanodine receptors in cardiac myocytes. Proc. Natl. Acad. Sci. USA. 95, 15096-15101.

Stern, M. D., 1992. Theory of excitation-contraction coupling in cardiac muscle. Biophys. J. 63, 497-517.

Stern, M. D., Song, L. S., Cheng, H., Sham, J. S., Yang, H. T., Boheler, K. R., Rios, E., 1999. Local control models of cardiac excitation-contraction coupling: A possible role for allosteric interactions between ryanodine receptors. J. Gen. Physiol. 113, 469-489.

Terentyev, D., Viatchenko-Karpinski, S., Gyorke, I., Volpe, P., Williams, S. C., Gyorke, S., 2003. Calsequestrin determines the functional size and stability of cardiac intracellular calcium stores: mechanism for hereditary arrhythmia. Proc. Natl. Acad. Sci. USA. 100, 11759-11764.

Wagner J., Keizer, J., 1994. Effects of rapid buffers on Ca2+ diffusion and Ca2+ oscillations. Biophys. J. 67, 447-456.

Zahradnikova, A., Zahradnik, I., Gyorke, I., Gyorke, S., 1999. Rapid activation of the cardiac ryanodine receptor by submillisecond calcium stimuli. J. Gen. Physiol. 114, 787-798.

Zhang, L., Kelly, J., Schmeisser, G., Kobayashi, Y. M., Jones, L. R., 1997. Complex formation between junctin, triadin, calsequestrin, and the ryanodine receptor. J. Biol. Chem. 272, 23389-23397.

29 Table 1: Parameters of the RyR channel

Parameters Deﬁnition Values Reference −2 −1 k2 S000 − S010 rate constant 0.045 µM s (Stern et al., 1999) −1 k−2 S010 − S000 rate constant 60 s (Stern et al., 1999) −1 −1 k4 = k7 S100 − S101 rate constant 0.47 µM s (Stern et al., 1999) −1 k−4 = k−6 S101 − S100 rate constant 5 s (Stern et al., 1999) −1 −1 k6 S000 − S001 rate constant 0.3 µM s Free −2 −1 k11 S100 − S110 rate constant 0.0045 µM s Free d K1 dissociation constant of S000 − S100 1000 µM Free d − d k11 K3 dissociation constant of S010 S110 10000 µM =K1 k2 d − d k6 K8 dissociation constant of S001 S101 638.3 µM =K1 k7

Table 2: Parameters of the CICR model

Parameters Definition Values Reference −12 Vmyo Myoplasmic volume 25.84 × 10 L (Iyer et al., 2004) −15 Vds diadic space volume 1.2 × 10 L (Iyer et al., 2004) −12 VjSR Junctional SR (jSR) volume 0.16 × 10 L (Iyer et al., 2004) F Faraday constant 0.0965 C/µmol (Rice et al., 1999) 2+ −1 v1 Maximal RyR channel Ca permeability 4000 s (Rice et al., 1999) [CSQ]tot Total jSR CSQ concentration 14 mM (Shannon et al., 1997) [CMD]tot Total diadic space CMD concentration 50 µM (Rice et al., 1999) 2+ kD,CSQ Ca half-saturation constant for CSQ 638 µM (Shannon et al., 2000a) 2+ kD,CMD Ca half-saturation constant for CMD 2.38 µM (Rice et al., 1999) 2+ −1 τ1 Time constant for L-type Ca channel 0.001 s 2+ −1 τ2 Time constant for L-type Ca channel 0.015 s −1 τtr Time constant for diffusion from NSR to jSR 0.01 s −1 τxfer Time constant for diffusion from 0.06 s diadic space to myoplasm Acap Capacitive membrane area 153.4 pF 2+ 2+ [Ca ]myo Concentration of myoplasmic Ca 0.1 µM 2+ 2+ [Ca ]ds(0) Initial concentration of diadic space Ca 0.101 µM x00(0) Initial value of state S00 0.998 x10(0) Initial value of state S10 0.000 x11(0) Initial value of state S11 0.000

30 Figure legends

Figure 1 2+ 2+ A diagram of local Ca transport at the junctional area. Ca inﬂux (JDHPR) enters the diadic space (indicated by the box with the dotted line) and in- 2+ 2+ duces SR Ca release (Jrel) from the junctional SR (jSR). The local Ca is 2+ removed from the diadic space to the cytosol (Jxfer). Ca is supplied to the jSR from the network SR (NSR) (Jtr).

Figure 2 Schematic description of RyR, CSQ, junctin and triadin. The RyR has two cytosolic binding sites (activation and inactivation) and one jSR lumenal CSQ binding site. CSQ can bind either with Ca2+ or the RyR mediated by the linkage of triadin and junction (T/J).

Figure 3 A diagram of general eight-state RyR kinetic model. c and q represent the 2+ cytosolic [Ca ] and free CSQ concentration ([CSQ]), respectively. Sijk denotes a kinetic state of the RyR. For example, S110 represents a state of CSQ bound, Ca2+ bound to the activation site and Ca2+ unbound to the inactivation site. The two states enclosed within each of the dotted-line boxes have the same cytosolic binding states. Sjk denotes the combined state of two states: S0jk and S1jk.

Figure 4 Diagram of four-state kinetic model of the RyR. Notations are used as d d d d Act1 = (k2K1 + k11q)/(K1 + q), Act2 = (k2K8 + k11q)/(K8 + q), Inact1 = d d d d (k6K3 + k7q)/(K3 + q), Inact2 = (k6K1 + k7q)/(K1 + q).

Figure 5 2+ Binding curve of CSQ to SR lumenal Ca at [CSQ]tot = 14 mM.

Figure 6 2+ ICa, L-type Ca current, with Ipeak = 721 pA, τ1 = 0.001, τ2 = 0.015.

Figure 7 Peak (a) and steady-state (b) open probability (Po) of the RyR as a function 2+ 2+ of [Ca ]ds for [Ca ]jSR = 20µM and 5 mM.

31 Figure 8 2+ 2+ Simulation results with an initial [Ca ]jSR = 575 µM. (a) Ca transient in the diadic space. (b) The open probability Po (x10 of the state S10) and state 2+ S11 (its fraction x11). (c) jSR [Ca ] . (d) Two states S00 (its fraction x00) and S01 (its fraction x01).

Figure 9 2+ 2+ Fractional jSR Ca release as a function of initial [Ca ]jSR for m = 1 and m = 3.

Figure 10 Rates constants Act1 (a) and Inact1 (b) in the RyR kinetics represented as functions of q (= [CSQ]). The asymptotes k2 = 0.045, k11 = 0.0045, k6 = 0.3, k7 = 0.47 represent upper and lower bounds for the curves.

Figure 11 Rates constants Act1 (a) and Inact1 (b) in the RyR kinetics shown as func- 2+ tions of [Ca ]jSR.

Figure 12 Simulation results of the diadic space Ca2+ transients and SR lumenal Ca2+ shown for diﬀerent values of [CSQ]tot = 8 mM (A), 14 mM (B) and 20 mM (C).

Figure 13 2+ 2+ Fractional jSR Ca release as a function of [Ca ]jSR shown for diﬀerent values of [CSQ]tot = 8 mM (A), 14 mM (B) and 20 mM (C).