Despite its importance and abundance of experimental data, the molecular mechanism of RyR2 activation by calcium is poorly understood. Recent experimental studies involving coexpression of wild-type (WT) RyR2 together with a RyR2 mutant deficient in calcium-dependent activation (Li, P., and S.R. Chen. 2001. J. Gen. Physiol. 118:33–44) revealed large variations of calcium sensitivity of the RyR tetramers with their monomer composition. Together with previous results on kinetics of Ca activation (Zahradníková, A., I. Zahradník, I. Györke, and S. Györke. 1999. J. Gen. Physiol. 114:787–798), these data represent benchmarks for construction and testing of RyR models that would reproduce RyR behavior and be structurally realistic as well. Here we present a theoretical study of the effects of RyR monomer substitution by a calcium-insensitive mutant on the calcium dependence of RyR activation. Three published models of tetrameric RyR channels were used either directly or after adaptation to provide allosteric regulation. Additionally, two alternative RyR models with Ca binding sites created jointly by the monomers were developed. The models were modified for description of channels composed of WT and mutant monomers. The parameters of the models were optimized to provide the best approximation of published experimental data. For reproducing the observed calcium dependence of RyR tetramers containing mutant monomers (a) single, independent Ca binding sites on each monomer were preferable to shared binding sites; (b) allosteric models were preferable to linear models; (c) in the WT channel, probability of opening to states containing a Ca2+-free monomer had to be extremely low; and (d) models with fully Ca-bound closed states, additional to those of an Monod-Wyman-Changeaux model, were preferable to models without such states. These results provide support for the concept that RyR activation is possible (albeit vanishingly small in WT channels) in the absence of Ca2+ binding. They also suggest further avenues toward understanding RyR gating.

## Introduction

The SR calcium release channel (also known as the ryanodine receptor, RyR) is composed of four identical subunits (Lai et al., 1989). It forms characteristic foot structures (Inui et al., 1987) that span the gap between t-tubules and junctional SR in skeletal and cardiac muscle (Block et al., 1988; Sun et al., 1995). The massive cytoplasmic domain of RyR has fourfold rotational symmetry as revealed by electron cryomicroscopy and image reconstruction (for review see Wagenknecht and Samsó, 2002). This domain contains binding sites for numerous cytoplasmic regulators of RyR activity, including calcium and ATP (for review see Meissner, 2002, 2004).

In the physiological context, the most important regulator of the cardiac RyR isoform (RyR2) is cytoplasmic calcium. RyR2 activation by calcium forms the basis of calcium-induced calcium release, a mechanism responsible for excitation–contraction coupling in cardiac muscle (Fabiato, 1985). The activation of the RyR by calcium ions has been the subject of extensive experimental work (for review see Coronado et al., 1994; Meissner, 2004). When studied in planar lipid bilayers, RyR2s exhibit a steady-state calcium dependence with EC50 of ∼1 μM (in the absence of Mg and ATP; Coronado et al., 1994; Meissner, 2004). Kinetic studies with application of fast photolytically generated Ca2+ spikes showed a steep calcium dependence of activation, implying that binding of several Ca2+ ions (n ∼ 4) must take place for transition of the resting channel to an open state (Zahradníková et al., 1999). These and other studies led to construction of numerous gating models of the channel (Ashley and Williams, 1990; Sitsapesan and Williams, 1994; Tang and Othmer, 1994; Sachs et al., 1995; Schiefer et al., 1995; Zahradníková and Zahradník, 1996; Keizer and Levine, 1996; Villalba-Galea et al., 1998; Stern et al., 1999; Fill et al., 2000; Saftenku et al., 2001; Sobie et al., 2002; Villalba-Galea et al., 2002; Rosales et al., 2004). These models, while adequate in describing phenomenological aspects of RyR gating, lack clear-cut means of attributing particular gating transitions to calcium binding at the respective calcium binding sites on the RyR, or, in other words, lack realistic structural context. To our knowledge, there are only three published models of RyR that take into account the tetrameric structure of RyR when describing its calcium activation: the adaptation gating model (AG; Cheng et al., 1995), our extended minimal gating model (EMG; Zahradníková et al., 1999), and a subset of EMG containing only the fast-access states of the H-mode (HM; Dura et al., 2003). In contrast, models of other channel types often possess direct correlates between channel structure and function (Marks and Jones, 1992; Bezprozvanny and Ehrlich, 1993; Goulding et al., 1994; Edelstein et al., 1996; Galzi et al., 1996; Varnum and Zagotta, 1996; Cox et al., 1997; Tibbs et al., 1997; Mak et al., 2003).

The homotetrameric structure of RyR (a), its fourfold rotational symmetry (b), and clear experimental evidence that binding of several calcium ions is required for RyR2 activation (c) all suggest that RyR2 activation by calcium is an allosteric process. Therefore, it may be treated according to the allosteric concept of Monod et al. (1965) in analogy to the elegant treatment of RyR1 activation by the DHPR voltage sensors (Rios et al., 1993). Models describing activation of RyR2 by calcium might carry a structurally distinct Ca2+ binding site on each monomer. Alternatively, each monomer might contribute partially to the formation of calcium sensor(s) shared by several monomers (as suggested by Li and Chen, 2001).

Recently Li and Chen (2001) have probed the molecular basis of RyR2 calcium-dependent activation by coexpressing wild-type (WT) RyR2 together with an RyR2 mutant deficient in calcium-dependent activation (E3987A). Importantly, five different channel variants were observed, each with a distinct calcium sensitivity that differed by up to several orders of magnitude. The five variants supposedly represented different combinations of WT and mutant monomers in the channel tetramer. The new dimension of these studies, the variation of calcium sensitivity with monomer composition of the RyR tetramer, provides an opportunity to test and compare RyR models with different mechanisms of activation by Ca2+.

In the present study, we analyzed the effect of monomer substitution by its calcium-insensitive mutant counterpart on the calcium dependence of activation of the resulting RyR tetramers using the three published tetrameric models of RyR, and several new models with various implementations of the channel calcium sensor(s). We show, using mathematical modeling, that for description of RyR gating in mixtures of WT and mutant monomers, models in which four independent calcium binding sites are responsible for the calcium-dependent activation of the channel are preferable to models in which two or four subunits create a joint calcium binding site. In the optimal model, calcium binding and channel opening were allosterically coupled. Opening of the WT channel to states with partially occupied binding sites had very low probability. The model also required inclusion of further, fully calcium-bound closed states, in addition to the states representing the open and closed conformations of the channel at all configurations of calcium binding site occupation. These results provide new insights into the molecular mechanisms of RyR activation by calcium and offer important clues regarding RyR regulation in health and disease.

## Materials And Methods

### Optimization of Models

Calculations of the theoretical calcium dependence of channel open probability for individual RyR variants corresponding to different combinations of WT and mutant (M) monomers in individual gating schemes were performed in Mathematica (Ver. 5.0, Wolfram Research) as described previously (Zahradníková and Zahradník, 1996; Zahradníková et al., 1999), by solving the system of equations describing the equilibria between channel states via their dissociation constants. The resulting formulae, expressing channel open probability as a function of monomer composition and calcium concentration, are presented in the electronic supplement . They were compiled, and their variable parameters were fitted in OriginPro (Ver. 7.0, OriginLab) to the data taken from Fig. 7 D of Li and Chen (2001). Only the three most sensitive RyR variants were taken into account because of the experimentally observed overlap between calcium activation and calcium inactivation range of the two least sensitive variants (Li and Chen, 2001) that prevented reliable parameter estimation.

### Statistical Analysis

The values of χ2 were determined from the sum of squares of differences between experimental data and model prediction, and from the experimental variance. The quality of data description by the models was characterized by the χ2 test (Press et al., 1992) as described previously (Zahradníková et al., 1999). The significance of differences between the models was assessed by means of the F-test (Press et al., 1992).

Dose–response parameters of the data and the models were approximated by the Hill equation. The 95% confidence interval for the parameters was assumed to be equal to ±2.5 times their standard error of the fit.

### Notation

Channel gating schemes are shown in a simplified notation (Fig. 1). Closed states (C) are depicted as squares, and open states (O) are depicted as circles. All channel transitions are reversible and obey the principle of detailed balance. The indexes of individual states, denoting the number of bound calcium ions or a particular mode of activity, are shown as numbers/letters inside the squares/circles. When two types of calcium binding sites are present, the first index corresponds to the binding site that undergoes mutation. Calcium-dependent channel transitions are depicted as thick lines, which are solid if the transition is affected by mutation, and dotted, if the transition is unaffected by mutation. Other transitions between states are depicted as thin lines. The arrows characterizing the direction of the transitions are omitted. All equilibria between channel states were described using dissociation constants (i.e., the reciprocal values of the respective equilibrium constants).

### Online Supplemental Material

The gating schemes of tested channel models with 0, 1, and 2 mutant monomers are presented in the online supplemental material together with the relationships between parameter values and steady-state probabilities of individual channel states . The fitting functions, describing the open probability of channel models as a function of parameter values and calcium concentration are also presented.

## Results

In the present study we analyzed three previously published tetrameric RyR models, two new simple models with shared calcium sensor(s), and where possible, their allosteric counterparts. Calculations of the calcium dependence of RyR2 activation for tetramers containing zero to two M monomers were based on modification of these gating models as detailed below.

### Selection of Models for the Study

The relationships between individual models are shown in Fig. 1 A. The models are described in the order of increasing complexity. A detailed description of model parameters and their relationships to probability of channel states is presented in the online supplement . The simplest model, the HM model (Dura et al., 2003), contains four independent Ca2+ binding sites, one on each monomer. When all binding sites are occupied by Ca2+, the channel can make the transition into the open state. These transitions result in the gating scheme shown in Fig. 1 B (top). The gating of the channel can be fully described by two parameters: KCa, the microscopic Ca2+ dissociation constant of individual binding sites; and KO4, the dissociation constant of the open state (O4, depicted as 4 inside a circle) relative to the last closed state (C4, depicted as 4 inside a square) and thus determines the maximum open probability of the channel (see the online supplemental material). The conceptual scheme of calcium binding sites within the tetrameric structure of the channel in relation to monomer composition is depicted in the first row of Fig. 2 (top). The effect of mutation on channel gating was implemented as a decrease in the affinity of the M Ca2+ binding site (yellow circles) with no change in the remaining equilibria (see supplemental data for detailed descriptions of all gating schemes of channels containing mutant monomers).

The Monod-Wyman-Changeux (MWC; Monod et al., 1965) allosteric principle describes the effect of conformational changes on the monomers (in our case, calcium binding) on the concerted conformational change of the whole protein (in our case, channel opening), which are considered separate but allosterically coupled steps. Extension of the HM model by application of the MWC principle resulted in the allosteric HM model (aHM; “a” standing for allosteric will be used as a prefix for the sake of consistency in all allosteric models). The model has four independent Ca2+ binding sites, one on each monomer, with Ca2+ dissociation constants KCa. In contrast to the HM model, the channel can open independently of the occupation of Ca2+ binding sites, resulting in four additional open states with respect to the HM model (Fig. 1 B, second row; O0–O3, denoted as numbers inside circles). The tendency for opening at different levels of Ca2+ binding is determined by two thermodynamic constants. KO0 is the dissociation constant of the calcium-free open state (O0) relative to the calcium-free closed state (C0), and the allosteric parameter f is equal to the ratio of calcium dissociation constants of the open states relative to those of closed states (the smaller the value of f, the lower the microscopic calcium dissociation constant of open states relative to closed states). Maximum open probability is unequivocally determined by the value of KO4 = f4 KO0 (see the online supplemental material). Therefore at a constant value of maximum open probability in the limiting case of f→0, the aHM model becomes equivalent to the HM model (KO0→∞). Again, the effect of mutation (Fig. 2, top) was expressed as a decrease in the affinity of the corresponding Ca2+ binding site (yellow circles) without any change in the remaining equilibria.

The shared binding sites (SBS) model was created from the HM model by postulating that the four Ca2+ binding sites of the channel are each created by sharing residues of two neighboring monomers. The gating scheme of the wild-type channel has two parameters and is identical to that of the HM model (Fig. 1 B, top). Upon combination of WT and M monomers (Fig. 2, second row), semi mutant (SM) binding sites (red circles) form in addition to WT (black circles) and M binding sites (yellow circles). The effect of mutation was implemented as a progressive decrease in the affinity of the corresponding SM and M Ca2+ binding sites (red and yellow circles) without any change in the remaining equilibria.

The allosteric SBS (aSBS) model was created from the SBS model by allowing the channel to open independently of the occupation of Ca2+ binding sites (i.e., the relationship between the SBS and aSBS models is the same as the relationship between the HM and aHM models). The gating scheme of the wild-type channel has three parameters and is identical to that of the aHM model (Fig. 1 B, second row). Upon combination of WT and M monomers (Fig. 2, second row), SM binding sites (red circles) form in addition to WT (black circles) and M binding sites (yellow circles). At a constant value of maximum open probability, the aSBS model becomes equivalent to the SBS model in the limiting case of f→0. The effect of mutation was implemented as a progressive decrease in the affinity of the corresponding SM and M Ca2+ binding sites (red and yellow circles) without any change in the remaining equilibria.

The EMG model, developed for description of single-channel kinetics and modal behavior of the RyR (Zahradníková et al., 1999), is an extension of the HM model containing additional, fully calcium-occupied open and closed states that constitute the two additional activity modes, the L and I modes (Zahradníková and Zahradník, 1996; Zahradníková et al., 1999; Fig. 1 B, third row). The calcium dependence of RyR activation is described by five parameters. As in the HM model, the microscopic Ca2+ dissociation constant of individual binding sites on the monomers is determined by the parameter KCa, and the parameter KO4 defines the dissociation constant of the open state relative to the last closed state. Three additional dissociation constants KOL, KCL, and KCI (see the online supplemental material) define the equilibria involving states of the L and I modes of activity, OL, CL, and CI (corresponding to states O2, C5, and I in the notation of Zahradníková et al., 1999). For the purpose of the present study, the most important effect of these additional states is a decrease in the maximal open probability of the channel. The EMG model becomes equivalent to the HM model as the probabilities of the L and I modes approach 0 (i.e., as the dissociation constants KOL, KCL, and KCI approach infinity). In this model, the effect of mutation was implemented as a decrease in the affinity of the corresponding Ca2+ binding site (yellow circles in Fig. 2, first row) without any change in the remaining equilibria.

The allosteric EMG (aEMG) model was created from the EMG model by allowing the channel to open independently of the occupation of Ca2+ binding sites (i.e., the relationship between the EMG and aEMG models is the same as the relationship between the HM and aHM models; Fig. 1 B, fourth row). The calcium dependence of RyR activation in the aEMG model is described by six parameters. As in the aHM model, the microscopic calcium dissociation constant of the calcium binding sites on the monomers is defined by the parameter KCa, KO0 is the dissociation constant of the calcium-free open state (O0) relative to the calcium-free closed state (C0), and the allosteric parameter f is equal to the ratio of calcium dissociation constants of the open states relative to those of closed states. The additional dissociation constants KOL, KCL, and KCI (see online supplemental material) are analogous to those of the EMG model. At a constant value of the maximum open probability, the aEMG model becomes equivalent to the EMG model in the limiting case of f→0. It becomes equivalent to the aHM model as the probabilities of the L and I mode approach 0. The effect of mutation was implemented as a decrease in the affinity of the M calcium binding site (yellow circles in Fig. 2, first row) without any change in the remaining equilibria.

The joint binding site (JBS) model (Fig. 1 B, fifth row) was derived from the HM model by adding a fifth Ca2+ binding site, created jointly by all four monomers. The model is fully described by four parameters. Calcium binding to the channel is characterized by the parameters KCa, the dissociation constant of the joint Ca2+-binding site, and KCaI, the microscopic dissociation constant of the independent Ca2+-binding sites on individual monomers. The parameters KO0 and KO1 are dissociation constants of the open states when the joint calcium binding site is empty and occupied, respectively.

The four independent Ca2+ binding sites, corresponding to those of the HM model, are not affected by the mutation (squares in Fig. 2, third row). The replacement of each WT monomer by an M monomer was assumed to progressively decrease the calcium affinity of the joint binding site (blue, red, green, and yellow circle, respectively) without affecting any other equilibria. The common feature of the JBS and aHM models is that the channel can open in the presence as well as in the absence of Ca2+ binding to the binding site that is affected by the mutation.

The AG model of Cheng et al. (1995) is not directly related to the other models described above. It has been conceived to describe the mechanism of RyR adaptation. It has two types of Ca2+ binding sites (Fig. 1 B, bottom). Ca2+ binding to the O sites (first index), characterized by the microscopic dissociation constant KCa, leads to channel activation. Ca2+ binding to the A sites (second index), characterized by the microscopic dissociation constant KA, leads to channel adaptation, i.e., to channel closure. Individual Ca2+ binding sites are independent. The channel is open when at least one of the monomers has Ca2+ bound to its O site, and when the number of occupied A sites is less than or equal to the number of occupied O sites. The published parameter values are KCa = 0.5 μM and KA = 0.1 μM (Cheng et al., 1995). The effect of mutations in individual monomers on the calcium binding sites is shown in Fig. 2 (bottom). The A sites are denoted as solid squares. The effect of mutation was implemented as a decrease in the affinity of the O sites (yellow circles) without any change in the remaining equilibria. As the maximum open probability of this model is by definition equal to 1, for the purpose of this study it was assumed that there is a separate, independent process that limits POmax to the experimentally observed value of 0.85 (in analogy to Mak et al., 2003).

### Derivation of Models Containing Mutant Subunits

The calcium-binding equilibria (in the case of allosteric models, the calcium-binding equilibria of the resting state of the channel) were described using microscopic dissociation constants of individual calcium binding sites. Models of channels containing M monomers were derived by postulating that the calcium affinity of the mutant binding site(s) is decreased, while calcium binding to the wild-type binding sites remains intact. In the case of the JBS, SBS, and aSBS models, the calcium affinities of quarter mutant (QM), SM, three-quarter mutant (TM), and M binding sites were considered independent.

In the case of HM, aHM, EMG, aEMG, and AG models, the binding sites are located on individual monomers. Therefore, the equilibria between channel states differing in the number of bound calcium ions (Eq. 1),

$\mathrm{S}_{\mathrm{i,j}}+\mathrm{Ca}^{2\mathrm{+}}{\iff}\mathrm{S}_{\mathrm{i}+1,\mathrm{j}}\mathrm{,}$
(1)

where Si,j denotes the state with i and j Ca2+ ions bound to the WT and M monomers, respectively, were calculated using probability factors for association and dissociation of calcium (Eq. 2):

$\frac{\left[\mathrm{S}_{\mathrm{i,j}}\right]{\cdot}\left[\mathrm{Ca}^{2\mathrm{+}}\right]}{\left[\mathrm{S}_{\mathrm{i}+1,\mathrm{j}}\right]}=\mathrm{K}_{\mathrm{Ca}}^{\mathrm{WT}}{\cdot}\frac{\mathrm{n}{-}\mathrm{i}}{\mathrm{i}+1}\mathrm{,}$
(2)

where n and m stand for the number of WT and M monomers, respectively (m + n = 4).

In the case of the SBS and aSBS models, the binding sites are jointly formed by two neighboring monomers. Therefore, upon combination of WT and M monomers, three types of binding sites, WT, SM, and M, are formed. The equilibria between channel states differing in the number of bound calcium ions (Eq. 3),

$\mathrm{S}_{\mathrm{i,j,k}}+\mathrm{Ca}^{2\mathrm{+}}{\iff}\mathrm{S}_{\mathrm{i}+1,\mathrm{j,k}}\mathrm{,}$
(3)

where Si,j,k denotes the state with i, j, and k Ca2+ ions bound to the WT, SM, and M binding sites, respectively, were therefore calculated using probability factors for association and dissociation of calcium (Eq. 4):

$\frac{\left[\mathrm{S}_{\mathrm{i,j,k}}\right]{\cdot}\left[\mathrm{Ca}^{2\mathrm{+}}\right]}{\left[\mathrm{S}_{\mathrm{i}+1,\mathrm{j,k}}\right]}=\mathrm{K}_{\mathrm{Ca}}^{\mathrm{WT}}{\cdot}\frac{\mathrm{n}{-}\mathrm{i}}{\mathrm{i}+1}\mathrm{,}$
(4a)

$\frac{\left[\mathrm{S}_{\mathrm{i,j,k}}\right]{\cdot}\left[\mathrm{Ca}^{2\mathrm{+}}\right]}{\left[\mathrm{S}_{\mathrm{i,j}+1,\mathrm{k}}\right]}=\mathrm{K}_{\mathrm{Ca}}^{\mathrm{SM}}{\cdot}\frac{\mathrm{m}{-}\mathrm{j}}{\mathrm{j}+1}\mathrm{,}$
(4b)

$\frac{\left[\mathrm{S}_{\mathrm{i,j,k}}\right]{\cdot}\left[\mathrm{Ca}^{2\mathrm{+}}\right]}{\left[\mathrm{S}_{\mathrm{i,j,k}+1}\right]}=\mathrm{K}_{\mathrm{Ca}}^{\mathrm{M}}{\cdot}\frac{\mathrm{l}{-}\mathrm{k}}{\mathrm{k}+1}\mathrm{,}$
(4c)

where n, m, and l stand for the number of WT, SM, and M binding sites, respectively (l + m + n = 4).

In the case of the JBS model, the binding site is jointly formed by all four monomers. Therefore, upon combination of WT and M monomers, five types of binding sites, WT, QM, SM, TM, and M, are formed. The equilibria between channel states with empty and occupied joint calcium binding site (Eq. 5),

$\mathrm{S}_{\mathrm{0,j}}^{\mathrm{XX}}+\mathrm{Ca}^{2\mathrm{+}}{\iff}\mathrm{S}_{1,\mathrm{j}}^{\mathrm{XX}}\mathrm{,}$
(5)

where Si,jXX denotes the state of the RyR tetramer with the binding site XX and with i Ca2+ ions bound to the joint binding site and j Ca2+ ions bound to the binding sites corresponding to those of the HM model, was calculated as

$\frac{\left[\mathrm{S}_{0,\mathrm{j}}^{\mathrm{XX}}\right]{\cdot}\left[\mathrm{Ca}^{2\mathrm{+}}\right]}{\left[\mathrm{S}_{1,\mathrm{j}}^{\mathrm{XX}}\right]}=\mathrm{K}_{\mathrm{Ca}}^{\mathrm{XX}}\mathrm{,}$
(6)

### Approximation of Experimental Data by the Models

#### The HM Model

The gating of individual channel variants was described by optimizing the parameters KCaWT and KCaM, the microscopic Ca2+ dissociation constants of individual WT and M binding sites. The parameter KO4 = 0.176 was determined from the preset value of POmax= 0.85 (Li and Chen, 2001). Parameters of the optimal model are given in Table I. The calcium sensitivity of the M monomer was decreased ∼400× relative to the WT monomer.

The effect of consecutive substitutions of the WT monomers by M monomers on the calcium dependence of RyR open probability is depicted in Fig. 3 (top left) overlaid on the data points replotted from Fig. 7 D of Li and Chen (2001). It can be seen that while the calcium dependence of variants with nM = 0 and nM = 2 is well described by the model, the model exhibits significant deviations from the experimentally observed behavior for the variant with nM = 1.

#### The aHM Model

The gating of individual channel variants was described by optimizing the parameters KCaWT and KCaM, the microscopic Ca2+ dissociation constants of individual WT and M binding sites, and the allosteric parameter f, characterizing the calcium affinity of open states relative to that of closed states. The parameter KO0 = 0.176/f4 was determined from the preset value of POmax = 0.85 (Li and Chen, 2001). Parameters of the optimal model are given in Table I. The calcium sensitivity of the M monomer was decreased ∼500× relative to the WT monomer.

The effect of consecutive substitutions of the WT monomers by M monomers on the calcium dependence of RyR open probability is depicted in Fig. 3 (top right). It can be seen that the calcium dependence of all variants is quite satisfactorily described by the model. For the WT channel, the open state with four Ca2+ ions bound was predominant at Ca2+ concentrations above the apparent calcium dissociation constant, but at lower values of Ca2+ ∼50% of open probability was due to open states with three or less Ca2+ ions bound (black hatched area). The open probability in the absence of Ca2+ was POmin = 0.0011. Open states with less than four bound Ca2+ ions became quite prominent in the variant with nM = 1 (red hatched area), while their abundance in the variant with nM = 2 (green hatched area) was similar to that of the WT channels.

It was not possible to describe the observed changes in calcium sensitivity of mutant channels if KCaM was set equal to KCaWT and the mutation was supposed to produce only changes in the allosteric parameter f. If both KCa and f were allowed to vary between WT and M monomers, the quality of fit was not improved (unpublished data).

#### The SBS Model

The gating of individual channel variants was described by optimizing the parameters KCaWT, KCaSM, and KCaM, the microscopic Ca2+ dissociation constants of individual WT, SM, and M binding sites. The parameter KO4 = 0.176 was determined from the preset value of POmax = 0.85 (Li and Chen, 2001). Six different RyR variants (Fig. 2, second row) formed upon combination of WT and M monomers. A satisfactory fit of the calcium-dependent open probabilities of RyR variants could be obtained only for KCaM > 15 mM, and under these conditions the calcium sensitivity of WT2M2trans (Fig. 2, second row, third panel), containing four semimutant binding sites, was significantly higher than that of WT2M2cis (fourth panel), which contained one fully mutant and three wild-type binding sites. In the examined calcium range of 0.01–2 mM, neither WT1M3 nor M4 activated significantly, so that this model produced five levels of apparent calcium sensitivity similar to those observed by Li and Chen (2001)(unpublished data). Parameters of the optimal model are given in Table I. The calcium sensitivity of the SM binding sites was decreased ∼130× relative to the WT binding sites.

The effect of consecutive substitutions of the WT monomers by M monomers (nM ≤ 2) on the calcium dependence of RyR open probability is depicted in Fig. 3 (second row left). It can be seen that while the calcium dependence of variants with nM = 0 and nM = 2 is well described by the model, the model exhibits significant deviations from the experimentally observed behavior for the variant with nM = 1.

#### The aSBS Model.

The gating of individual channel variants was described by optimizing the parameters KCaWT, KCaSM, and KCaM, the microscopic Ca2+ dissociation constants of individual WT, SM, and M binding sites, and the allosteric parameter f, characterizing the calcium dissociation constants of open states relative to that of closed states. The parameter KO0 = 0.176/f4 was determined from the preset value of POmax = 0.85 (Li and Chen, 2001). As with the SBS model, six different RyR variants (Fig. 2, second row) formed upon combination of WT and M monomers which, however, produced only five levels of calcium sensitivity in the examined range of calcium concentrations. Parameters of the optimal model are given in Table I. The calcium sensitivity of the SM binding sites was decreased ∼130× relative to the WT binding sites.

The effect of consecutive substitutions of the WT monomers by M monomers on the calcium dependence of RyR open probability is depicted in Fig. 3 (second row right). The optimal model was significantly activated in the absence of calcium (POmin = 0.029). It can be seen that at low Ca2+ all variants exhibit significant deviations from the experimentally observed behavior due to this basal activation, while the activity of channels is relatively well described by the model at higher calcium concentrations. For the WT channel, open probability at Ca2+ concentrations below the apparent calcium dissociation constant was mostly due to open states with three or less Ca2+ ions bound (black hatched area), and the open state with four Ca2+ ions bound started to predominate only at [Ca2+] > 1 μM. Open states with less than four bound Ca2+ ions were still more prominent in the variant with nM = 1 (red hatched area), and their abundance in the variant with nM = 2 (green hatched area) was similar to that of the WT channels.

#### The EMG Model

The gating of individual channel variants was described by optimizing the parameters KCaWT and KCaM, the microscopic Ca2+ dissociation constants of individual WT, SM, and M binding sites. The parameters KOL, KCL, and KCI were kept identical to the published ones (Zahradníková et al., 1999). The parameter KO4 = 0.00219 was determined from KOL, KCL, and KCI and from the preset value of POmax = 0.85 (Li and Chen, 2001). Parameters of the optimal model are given in Table I. Calcium sensitivity of the M monomer was decreased ∼1000× relative to the WT monomer.

The effect of consecutive substitutions of the WT monomers by M monomers on the calcium dependence of RyR open probability for selected values of POmax is depicted in Fig. 3 (third row left). It can be seen that while the calcium dependence of variants with nM = 0 and nM = 2 is well described by the model, the model exhibits significant deviations from the experimentally observed behavior for the variant with nM = 1 at calcium concentrations below the apparent KCa.

#### The aEMG Model

In addition to parameters of the aHM model, the aEMG model contains parameters KOL, KCL, and KCI. The value of KCI had only minor effects on the calcium dependence of open probability, while optimizing either both KOL and KCL, or either one of these parameters in isolation produced similar results (unpublished data). Therefore, for description of activity of individual RyR variants, we optimized the parameters KCaWT, KCaM, f, and KCL, while the parameters KOL and KCI were kept identical to the published values (Zahradníková et al., 1999). The value of KO0 = 0.225KCL/(f4[3.4 + 0.825KCL]) was calculated from the values of the remaining parameters. Parameters of the optimal model are given in Table I. Calcium sensitivity of the M monomer was decreased ∼800× relative to the WT monomer.

The effect of consecutive substitutions of the WT monomers by M monomers on the calcium dependence of RyR open probability for selected values of POmax is shown in Fig. 3 (third row right). It can be seen that the calcium dependence of all variants is quite satisfactorily described by the model. For the WT channel, the open state with four Ca2+ ions bound (O40) was predominant at Ca2+ > 0.2 μM, i.e., at all calcium concentrations that evoked significant activation of the channel. Open states with less than four Ca2+ ions bound did not contribute substantially to open probability in the WT channel (black hatched area in Fig. 3), while they became quite prominent in the variant with nM = 1 (red hatched area in Fig. 3, contributing >50% of total PO at Ca2+ < 5 μM) and (to a lesser extent) in the variant with nM = 2 (green hatched area in Fig. 3, contributing >50% of total PO at <30 μM). The open probability in the absence of Ca2+ was POmin = 0.0001.

As with the aHM model, it was not possible to describe the observed changes in calcium sensitivity of mutant channels if KCaM was set equal to KCaWT and the mutation was supposed to produce only changes in the allosteric parameter f. If both KCa and f were allowed to vary between WT and M monomers, the quality of fit was not improved (unpublished data).

#### The Joint Binding Site (JBS) Model

For description of activity of individual RyR variants, we optimized the KCaJ parameters characterizing the dissociation constant of the wild-type, quarter mutant, and semi mutant joint Ca2+-binding site (KCaWT, KCaQM, and KCaSM, respectively), at different combinations of parameter values for KCaI and KO0. The parameter KO1 = 0.176 was determined from the preset value of POmax = 0.85 (Li and Chen, 2001). Parameters of the optimal model are given in Table I. Only models with KO0 ≥ 20, in which the maximum open probability in the absence of Ca2+ binding to the joint binding site was <0.05, provided a satisfactory fit to the data. As a result, the calcium dependence of open probability of the channel was only weakly dependent on the parameter KCaI. At KCaI = 0.1 μM and KO0 = 100, the calcium dissociation constant of channel variants with one or two mutant subunits was decreased ∼15 and ∼500 times, respectively, relative to that of the WT channel.

The effect of consecutive substitutions of the WT monomers by M monomers on the calcium dependence of RyR open probability for selected values of POmax is shown in Fig. 3 (bottom left). It can be seen that the calcium dependence of all variants is quite satisfactorily described by the model.

#### The AG Model

The gating of individual channel variants was described by optimizing the calcium dissociation constants of wild-type and mutant monomers, KCaWT and KCaM, and the parameter KA. Parameters of the optimal model are given in Table I.

The effect of consecutive substitutions of the WT monomers by M monomers on the calcium dependence of RyR open probability is shown in Fig. 3 (bottom right). Model channels containing M monomers displayed nonmonotonic calcium dependence, with calcium-dependent inactivation above 1 μM, caused by Ca2+ binding to the A sites of the channel, followed by a second increase in PO above 10 μM due to Ca2+ binding to the O sites of the M monomer(s). Such a bi-modal calcium dependence was not observed experimentally (Li and Chen, 2001), and is especially at odds with the data of the channel variant with nM = 2. In this model, Ca2+ binding to the O sites of the channel leads to channel opening if the corresponding A sites are not occupied by Ca2+, and therefore open probability at a wide range of Ca2+ concentrations was due to open states with three or less Ca2+ ions bound for all channel variants (black, red, and green hatched areas for nM = 0, 1, and 2, respectively).

#### Dose–Response Parameters of RyR Variants in the Tested Models

In seven out of eight models, substitution of WT monomers in the RyR tetramer resulted in a progressive decrease in calcium sensitivity. Individual models differed in the relative effect of single and double substitution on the resulting calcium sensitivity. They also differed in the effect of monomer substitution on the steepness of the calcium dependence of open probability. In this section, we compared the predicted apparent calcium dissociation constants and Hill slopes of the three tested WT/M variants in individual models. The AG model displayed nonmonotonic calcium dependence of PO in channels containing mutant monomers and therefore could not be analyzed by this method.

Fig. 4 summarizes the predicted dose–response parameters of models compared with those of the data (Li and Chen, 2001). Parameters significantly different from those of the data are marked by asterisks. The quality of description of the data by the models falls into three categories. In the aHM and aEMG model, one out of six parameters was significantly different from those of the data. In the HM, SBS, aSBS, EMG, and JBS models, two out of six parameters were significantly different from those of the data. Based on this analysis, the aHM and aEMG models are preferential to all other models.

We have examined for each of the models how apparent calcium affinities and apparent Hill slopes vary with the number of mutant monomers in the studied RyR variants (black, red, and green bars for nM = 0–2). It is to be noted that the available experimental data (Li and Chen, 2001; Fig. 4 A, first set of bars) suggest that substitution of one monomer in the WT RyR by an M monomer results in only an ∼10-fold decrease of calcium sensitivity, while substitution of the second monomer by an M monomer results in a further 20-fold decrease of Ca2+ sensitivity. The aHM, aSBS, EMG, aEMG, and JBS models (Fig. 4 A) were able to simulate the change of calcium sensitivity upon increase of the number of mutant monomers in the RyR tetramer, i.e., the logarithmic decrease in sensitivity of tetramer variants with increasing number of mutant monomers.

Furthermore, the experimental data (Li and Chen, 2001; Fig. 4 B, first set of bars) suggest that substitution of one monomer in the WT RyR by an M monomer results in a prominent decrease of the steepness of calcium dependence of activation, while substitution of the second monomer by an M monomer results in increase of the steepness. Differences between models were more prominent in the relationship between nM and the apparent Hill slope for calcium dependence of channel activation (Fig. 4 B). The aEMG model, and to some extent also the models aHM, aSBS, and EMG, showed the experimentally observed nonmonotonous dependence of the apparent Hill slope on nM. On the other hand, in the JBS model the value of the apparent Hill slope decreased monotonously with nM, and in the HM and SBS models, it changed only slightly with nM.

#### Evaluation of Models by Statistical Criteria

We have judged the goodness of fit of the above models by the χ2 test. The following ranking of models was obtained (in the order of decreasing quality of fit): aEMG > aHM > JBS > aSBS > EMG > HM > SBS. The residuals of models aEMG, aHM, and JBS were normally distributed, and these models were not significantly different based on the F-test at P = 0.05. However, only the aEMG model could be accepted by the χ2 test at P = 0.05 (see Table I).

## Discussion

This theoretical study was undertaken to explain the alterations of RyR channel Ca sensitivity observed upon substitution of individual monomers with a Ca-insensitive mutant within the tetrameric structure of the channel protein (Li and Chen, 2001). By analyzing the ability of several different RyR models to reproduce the available experimental data on Ca dependency of various WT/mutant hybrids, we provide evidence that in RyRs, calcium binding and channel opening are allosterically coupled. We introduce an RyR channel model, the aEMG model, that not only accounts for the calcium dependence of RyR activation but, for the first time, reflects the homotetrameric nature of the RyR channel.

The major finding of this work is that only models based on the MWC allosteric concept provided a satisfactory description of the calcium dependence for RyR channels that are composed of wild-type monomers and monomers deficient in calcium-dependent activation. There is a major conceptual difference between allosteric models (see Cox et al., 1997 for the Ca2+-activated BK channel) and linear, Hodgkin-Huxley type fully cooperative models (Sitsapesan and Williams, 1994; Zahradníková et al., 1999; Dura et al., 2003). In linear models, the channel opens only when all the activation sites are occupied by Ca, that is, when the last calcium-dependent gate is unlocked. In allosteric models the channel may open with any number of calcium ions bound.

In the WT channel, the presence or absence of MWC states did not influence the calcium dependence of open probability. Indeed, as visually apparent from the black traces in Fig. 3, all models except aSBS and AG provided satisfactory fit to the wild-type data (χ2 test, unpublished data). Only the presence of the mutant monomer exposed the existence of the MWC states, and therefore, for the whole dataset, all allosteric models provided a significantly better fit than their linear counterparts (see Table I).

Of the three tested allosteric models, only the aEMG model passed all tests. This model, in addition to allosteric coupling between calcium binding and channel opening, incorporates fully calcium-bound channel closed states of the L- and I-mode (Zahradníková and Zahradník, 1996) that limit its maximal open probability. In the absence of the L- and I-mode states, allosteric coupling was not sufficient to describe the experimental observations (the aHM model, Table I). Similarly, in the absence of allosteric coupling, L- and I-mode states were not sufficient to describe the experimental observations (the EMG model, Table I). While in the aEMG model the MWC states are necessary for satisfactory description of mutant channel gating, occupancy of these states in the wild-type RyR was negligible (Fig. 3, third row right), much lower than in the aHM and aSBS models. This explains why previous kinetic experiments revealed that in the wild-type channel, binding of four calcium ions is required before RyR opening (Zahradníková et al., 1999).

It is noteworthy that several previously suggested RyR models share some characteristics with the aEMG model. Out of the three best-ranking gating schemes of the extensive analysis of RyR gating by Rosales et al. (2004), two of the models contained fully Ca-bound closed states following the open states, and the third model (McManus and Magleby, 1991) contained interconnected open states with not fully Ca-liganded channel. A gating scheme sharing both features was proposed for explaining kinetic properties of RyR activation by Ca2+ (Fill et al., 2000). Neither of these models, however, attributed particular Ca2+-dependent gating transitions to individual channel monomers and therefore could not be tested against the data of Li and Chen (2001).

Allosteric gating models have not been previously proposed for activation of the ryanodine receptor by Ca2+. Nevertheless, analogous models were useful for description of RyR regulation by the DHPR voltage sensors (Rios et al., 1993) and for regulation of the inositol trisphosphate receptor (IP3R) by ATP (Bezprozvanny and Ehrlich, 1993) as well as by IP3 and calcium (Mak et al., 2003). In this work, we have applied a similar approach and extended it by (a) considering the effect of nonequivalent, i.e., wild-type and mutant monomers, (b) examining the tested models in the whole parameter space by nonlinear fitting, and (c) applying statistical methods to analyze the quality of models.

### The Model and the Identity of the Binding Site

The parameters of the aEMG model relate to four functional elements of the channel: the calcium binding site on the RyR monomer (KCa), the resting-active equilibrium (KO0), coupling between calcium binding and channel opening (f), and L- and I-mode equilibria (KOL, KCL, and KCI). In the optimized model, the mutation of E3987 selectively decreased the calcium affinity of the binding site by three orders of magnitude, without affecting the remaining equilibria.

Niu and Magleby (2002) showed that a binding site created by a single amino acid would result in mutant subunits having no calcium affinity, which would lead to a progressive decrease of the Hill coefficient with increasing number of mutant monomers, a feature not observed experimentally (Li and Chen, 2001). Therefore, it seems safe to conclude that either the Ca2+ binding pocket of the RyR monomer is created by several amino acids, one of which is the glutamate E3987, or that this amino acid allosterically affects the calcium dissociation constant of the binding site while affecting neither the transitions between the closed and open conformations of the channel, nor coupling between calcium binding and channel opening. In this context it is noteworthy that the E2100 residue of the IP3R1 receptor, homologous with E3987 of the RyR2 channel, was proposed to play a direct role in Ca2+ binding (Tu et al., 2003).

### Implications of the Allosteric Gating Principle

The allosteric nature of RyR regulation by Ca2+ has profound implications for our understanding of RyR2 gating at resting calcium concentrations. In contrast to previously proposed mechanisms, where the RyR channel could open only upon Ca2+ binding, the aEMG model predicts nonzero (POmin = 1 × 10−4) open probability in the absence of Ca2+. This might explain the discrepancy between the substantial resting calcium spark frequency (100 pl−1 s−1 at 10 nM Ca2+; Li et al., 2002) and the low calcium sensitivity (high apparent KCa) of the RyR2 under physiological conditions (∼10 μM; Cannell et al., 1994; Lukyanenko and Györke, 1999). The proposed concept of RyR activation provides a unifying picture that can help to understand several phenomena, such as nonzero open probabilities observed in planar lipid bilayers at very low Ca2+ concentrations in the skeletal RyR (Lee et al., 2002), activation of RyRs by agonists such as sulmazole (Williams and Holmberg, 1990), caffeine (Sitsapesan and Williams, 1990), and ATP (Kermode et al., 1998) in the absence of Ca2+, or the observed increase in resting open probability caused by an RyR2 mutation implicated in hereditary arrhythmia (Jiang et al., 2002).

### Implications of Monomer-dependent RyR Regulation

Allosteric regulation of the RyR, discussed in the previous paragraph, operates at the level of the whole tetramer, i.e., at the concerted closed-to-open transition of the RyR (parameters KO0 and f). On the other hand, agents that affect the calcium affinity of individual monomers would be expected to modulate channel activity by acting on each monomer independently, i.e., in a manner analogous to that of the E3987A mutation. Similarly, RyR monomers differing in their phosphorylation status (for reviews see Meissner 2004; Wehrens et al., 2005), FKBP binding (Kaftan et al., 1996, Marx and Marks, 2002), and/or calmodulin binding (for reviews see Meissner 2002, 2004) might be independently modulated as well. These and similar effects might contribute to the often-observed heterogeneity in the behavior of single RyR channels (Ma, 1995; Zahradníková and Zahradník, 1995; Zahradníková et al., 1995; Copello et al., 1997).

### Further Directions

Several important questions pertaining to steady-state activity of the RyR are not addressed by this study. Most importantly, the inactivation mechanisms, that is, the molecular mechanisms that are responsible for the additional gating states of the L- and I-mode (Zahradníková and Zahradník, 1995) and for the nature of channel inhibition by Ca2+ and Mg2+ (Laver et al., 1997; Zahradníková et al., 2003) still remain unresolved. It is also not clear whether the calcium affinity decrease in the E3987A mutation is caused by differences in the calcium off-rate, on-rate, or both. In other channel types (Zhou et al., 1996; Reimann et al., 2001; Powe et al., 2002), as well as in the E-F hand calcium binding protein calmodulin (Black et al., 2000; Tikunova et al., 2001), variations in the on- as well in the off-rate of binding were observed upon mutation of binding sites. The exact mechanism of the affinity change in RyR2 mutants could be determined by kinetic studies using caged calcium (Zahradníková et al., 1999, 2003, 2004) or caged calcium buffers (Velez et al., 1997).

### Conclusions

We have found that the calcium dependence of activation in RyR channels containing subunits deficient in calcium binding can be best explained by the existence of allosteric coupling between calcium binding and channel opening. These results open new insights into the molecular mechanisms of RyR activation by calcium and provide important clues for understanding RyR regulation in health and disease. Taken together, the modeling approach described in this work may prove to be valuable for analysis and description of RyR modulation based on monomer-dependent phenomena.

## Acknowledgments

The research of Alexandra Zahradníková was supported in part by a Howard Hughes Medical Institute International Scholar's Award. This work was supported by grants VEGA 2/4150/04 (to A. Zahradníková), NIH-FIRCA R03 TW05543 (to S. Györke), and APVT-51-31104 (to I. Zahradník).

David C. Gadsby served as editor.

Abbreviations used in this paper: a, allosteric; AG, adaptation gating; EMG, extended minimal gating; HM, H-mode; JBS, joint binding site; M, mutant; MWC, Monod-Wyman-Changeux; QM, quarter mutant; SBS, shared binding sites; SM, semi mutant; TM, three-quarter mutant; WT, wild-type.

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