Moment Closure for Local Control Models of Calcium-Induced Calcium Release in Cardiac Myocytes

Abstract

In prior work, we introduced a probability density approach to modeling local control of Ca²⁺-induced Ca²⁺ release in cardiac myocytes, where we derived coupled advection-reaction equations for the time-dependent bivariate probability density of subsarcolemmal subspace and junctional sarcoplasmic reticulum (SR) [Ca²⁺] conditioned on Ca²⁺ release unit (CaRU) state. When coupled to ordinary differential equations (ODEs) for the bulk myoplasmic and network SR [Ca²⁺], a realistic but minimal model of cardiac excitation-contraction coupling was produced that avoids the computationally demanding task of resolving spatial aspects of global Ca²⁺ signaling, while accurately representing heterogeneous local Ca²⁺ signals in a population of diadic subspaces and junctional SR depletion domains. Here we introduce a computationally efficient method for simulating such whole cell models when the dynamics of subspace [Ca²⁺] are much faster than those of junctional SR [Ca²⁺]. The method begins with the derivation of a system of ODEs describing the time-evolution of the moments of the univariate probability density functions for junctional SR [Ca²⁺] jointly distributed with CaRU state. This open system of ODEs is then closed using an algebraic relationship that expresses the third moment of junctional SR [Ca²⁺] in terms of the first and second moments. In simulated voltage-clamp protocols using 12-state CaRUs that respond to the dynamics of both subspace and junctional SR [Ca²⁺], this moment-closure approach to simulating local control of excitation-contraction coupling produces high-gain Ca²⁺ release that is graded with changes in membrane potential, a phenomenon not exhibited by common pool models. Benchmark simulations indicate that the moment-closure approach is nearly 10,000-times more computationally efficient than corresponding Monte Carlo simulations while leading to nearly identical results. We conclude by applying the moment-closure approach to study the restitution of Ca²⁺-induced Ca²⁺ release during simulated two-pulse voltage-clamp protocols.

INTRODUCTION

The key step linking electrical excitation to contraction in cardiac myocytes is Ca²⁺-induced Ca²⁺ release (CICR), in which Ca²⁺ current flowing across the cell membrane triggers the release of additional Ca²⁺ from the sarcoplasmic reticulum (SR). In ventricular cells, CICR occurs as a set of discrete microscopic events known as Ca²⁺ sparks (1), with each spark triggered by local, rather than cell-wide, increases in myoplasmic [Ca²⁺]. As a consequence of this local-control mechanism of CICR, the cellular SR Ca²⁺ release flux is not a function of a single quantity, such as spatially averaged intracellular [Ca²⁺], but instead depends on thousands of different local Ca²⁺ concentrations, each of which can fluctuate with stochastic openings and closings of nearby Ca²⁺ channels in the sarcolemmal and SR membranes. The picture is further complicated by the fact that dynamic changes in local SR [Ca²⁺], which are also spatially heterogeneous, are thought to influence the gating of SR Ca²⁺ release channels known as ryanodine receptors (RyRs).

Computational models have been developed in which SR Ca²⁺ release depends directly on the average myoplasmic [Ca²⁺] (2–4). These so-called common-pool models (5) display SR Ca²⁺ release that occurs in an all-or-none fashion, contrary to experiments showing that release is smoothly graded with changes in Ca²⁺ influx (6–8). On the other hand, several published models achieve graded Ca²⁺ release using nonmechanistic formulations, such as having SR Ca²⁺ release depend explicitly on Ca²⁺ currents rather than on local [Ca²⁺] (9–13).

Models of EC coupling are able to reproduce graded Ca²⁺ release mechanistically by simulating the stochastic gating of channels in Ca²⁺ release sites using Monte Carlo methods. In these approaches, one or more L-type Ca²⁺ channels interact with a cluster of RyRs through changes in [Ca²⁺] in a small diadic subspace between the sarcolemmal and SR membranes. These models also generally consider local changes in junctional SR [Ca²⁺], because these changes are thought to be important for Ca²⁺ spark termination and refractoriness (14–16). Realistic cellular SR Ca²⁺ release can be simulated by computing the stochastic triggering of sparks from hundreds to thousands of such Ca²⁺ release units (CaRUs) (5,15–17). However, Monte Carlo simulations of local control of EC coupling can be computationally demanding, making it difficult to augment these models with representations of the ionic currents responsible for action potentials, and impractical to use this approach for simulations of phenomena occurring over the course of many heartbeats.

We recently demonstrated that an alternative probability-density approach can be used to simulate graded, locally controlled SR Ca²⁺ release mechanistically (18). In this prior work, coupled advection-reaction equations were derived relating the time-dependent probability density of subsarcolemmal subspace and junctional SR [Ca²⁺] conditioned on CaRU state. By numerically solving these equations using a high-resolution finite difference scheme and coupling the resulting probability densities to ordinary differential equations (ODEs) for the bulk myoplasmic and sarcoplasmic reticulum [Ca²⁺], a realistic but minimal model of cardiac excitation-contraction coupling was produced. This new approach to modeling local control of EC coupling is often computationally more efficient than Monte Carlo simulation, particularly if the dynamics of subspace [Ca²⁺] are much faster that those of junctional SR [Ca²⁺], allowing the bivariate probability density functions for subspace and junctional SR [Ca²⁺] to be replaced with univariate densities for junctional SR [Ca²⁺]. However, the probability density approach can lose its computational advantage when the number of states in the CaRU model is large or the dynamics of local [Ca²⁺] are such that numerical stability requires a refined mesh for solving the advection-reaction equations.

We therefore aimed to develop methods for improving upon the probability-density approach, and in this study, we describe a moment-closure technique that leads to significant computational advantages. After briefly reviewing the Monte Carlo and probability density approaches to modeling local control of EC coupling in cardiac myocytes, the new methodology begins with a derivation of a system of ODEs describing the time-evolution of the moments of the univariate probability density functions for junctional SR [Ca²⁺] jointly distributed with CaRU state. This open system of ODEs is then closed using an algebraic relationship that expresses the third moment of junctional SR [Ca²⁺] in terms of the first and second moments. In this manner, the partial differential equations describing the univariate probability densities of junctional SR [Ca²⁺] jointly distributed with CaRU state are replaced with ODEs describing the time-evolution of the moments of these distributions. In simulated voltage-clamp protocols using 12-state CaRUs that respond to the dynamics of both subspace and junctional SR [Ca²⁺], this moment-closure approach to simulating local control of EC coupling produces high-gain Ca²⁺ release that is graded with changes in membrane potential, a phenomenon not exhibited by common pool models. Benchmark simulations indicate that this moment-closure technique for local control models of CICR in cardiac myocytes is nearly 10,000-times more computationally efficient than corresponding Monte Carlo simulations, while leading to nearly identical results. We conclude by applying the moment-closure approach to study the restitution of Ca²⁺-induced Ca²⁺ release during simulated two-pulse voltage-clamp protocols.

MODEL FORMULATION

The focus of this article is a moment-closure technique to modeling local control of CICR in cardiac myocytes. The whole cell model of EC coupling that will be used to demonstrate the method closely follows our prior work in which we presented traditional Monte Carlo simulations of graded, locally controlled SR Ca²⁺ release to validate a novel probability density approach that represents the distribution of diadic subspace and junctional SR Ca²⁺ concentrations with a system of partial differential equations (18). Below we briefly review the Monte Carlo and probability density formulations, emphasizing minor adjustments that were required to implement the moment-closure technique. The Results section begins with the derivation of the moment-closure equations and follows with the validation and benchmarking of the moment-closure technique for local control models of CICR in cardiac myocytes by comparison to Monte Carlo simulation.

Monte Carlo formulation

The Monte Carlo model of local control of CICR in cardiac myocytes describes the dynamics of bulk myoplasmic [Ca²⁺], network SR [Ca²⁺], N diadic subspace Ca²⁺ concentrations, and N junctional SR domain Ca²⁺ concentrations through a system of ODEs. These are coupled to N Markov chains representing the stochastic gating of each CaRU that consists of one L-type Ca²⁺ channel (DHPR) and one RyR megachannel coupled through the local diadic subspace (c_ds) [Ca²⁺]. While a complete description of CICR would include stochastic gating of roughly N = 10,000 CaRUs, each containing multiple L-type Ca²⁺ channels (1–10) (19) and RyRs (30–300) (20), Monte Carlo simulations of EC coupling focusing on local control have often used Markov models of reduced complexity (5,16,21). This level of resolution will suffice to introduce the moment-closure technique.

Concentration balance equations

The Monte Carlo model consists of N+2 ODEs representing the time-evolution of [Ca²⁺] in the bulk myoplasm (c_myo), network SR (c_nsr), and N junctional SRs compartments. Consistent with Fig. 1, the concentration balance equations for these compartments are

(1)

(2)

(3)

where 1 ≤ n ≤ N and λ_nsr and λ_jsr are volume fractions (see the Appendix). The flux through the RyR megachannel associated with the n^th CaRU is given by

(4)

where is a stochastic variable that takes the value 1 or 0 depending on whether the n^th RyR megachannel is open or closed, and is the associated diadic subspace concentration defined below (Eq. 9). Similarly, diffusion from the network SR to each junctional SR compartment is given by

(5)

FIGURE 1.

Diagram of model components and fluxes. Each Ca²⁺ release unit consists of two restricted compartments (the diadic subspace and junctional SR with [Ca²⁺] denoted by c_ds and c_jsr, respectively), a two-state L-type Ca²⁺ channel (DHPR), and a six-state Ca²⁺ release site. The t-tubular [Ca²⁺] is denoted by c_ext and the fluxes J_in, J_ncx, J_serca, and J_leak are described in the text and the Appendix.

The total refill flux occurring in Eq. 2 includes the contribution from each CaRU and is given by

(6)

while the total flux out of the N diadic subspaces is given by

(7)

The remaining four fluxes that appear in Eqs. 1–3 and Fig. 1 include (influx into the diadic subspaces via L-type Ca²⁺ channels which are functions of the random variable ), J_in (background Ca²⁺ influx), J_ncx (Na⁺-Ca²⁺ exchange), J_serca (SR Ca²⁺-ATPases), and J_leak (the network SR leak). The functional form of these four fluxes can be found in the Appendix.

Diadic subspace Ca²⁺ concentration

Note that a concentration balance equation is not included for diadic subspace [Ca²⁺], because in our previous study we observed that model parameters lead to rapid equilibrium of the diadic subspace [Ca²⁺] with the [Ca²⁺] in the junctional SR and bulk myoplasm (18). Thus, in each diadic subspace we assume a [Ca²⁺] () that balances the fluxes in and out of that compartment,

(8)

that is,

(9)

where 1 ≤ n ≤ N and

(10)

(11)

In these expressions, the quantities and indicate whether the channel is open or closed, and v_ryr = and and are functions of plasma membrane voltage defined by

(12)

where the L-type Ca²⁺ channel flux, is given by Eq. 56 (see the Appendix).

Twelve-state CaRU model

The RyR model used here is similar to the two-state minimal model of an RyR megachannel used in prior work (18). Consistent with several studies indicating that the gating of the RyR cluster associated with each CaRU is essentially all-or-none (5,15,16), the two-state RyR megachannel model used in Williams et al. (18) included transition rates that were nonlinear functions of diadic subspace (c_ds) and junctional SR (c_jsr) [Ca²⁺], thereby allowing for Ca²⁺-dependent activation of RyR gating as well as spark termination facilitated by localized depletion of junctional SR [Ca²⁺]. Because the moment-closure approach is most easily presented when all Ca²⁺-mediated transitions in the CaRU model are bimolecular association reactions, the six-state RyR megachannel model used here employs sequential binding of diadic subspace Ca²⁺ ions to achieve highly cooperative Ca²⁺-dependent opening of the RyR megachannel. Similarly, an explicit junctional SR Ca²⁺-dependent transition is included so that depletion of luminal Ca²⁺ decreases the open probability of the megachannel,

(13)

Parameters were chosen (see Table 3) so that the behavior of this minimal six-state RyR megachannel model approximated the above-mentioned two-state model.

TABLE 3.

Ca²⁺ release unit parameters (L-type Ca²⁺ channel and RyR cluster)

Parameter	Definition	Value
	Total RyR cluster release rate	0.9 s−1
	Total DHPR permeability	3.5 × 10−5 cm s−1
	DHPR activation threshold	−10 mV
σdhpr	DHPR activation parameter	6.24 mV
	Maximum rate of DHPR opening	556 s−1
	Closing rate of DHPR opening	5000 s−1
	Rate of DHPR closing	5000 s−1
	Rate of RyR activation	2000 μM −1s−1
	Rate of RyR deactivation	1600 s−1
	Rate of RyR opening	40 μM −1s−1
	Rate of RyR closing	500 s−1
α	Cooperativity factor	2
β	Cooperativity factor	2

As in prior work (18), we use a two-state model of the L-type Ca²⁺ channel (DHPR),

(14)

where and represent closed and open states, is the voltage-dependent activation rate (10) given by

(15)

and is the constant deactivation rate that sets the mean open time (0.2 ms) and maximum open probability (0.1) of the channel. Although this two-state DHPR model ignores voltage- and Ca²⁺-dependent inactivation of L-type Ca²⁺ channels, these processes do not significantly influence the triggering of CICR during the whole-cell voltage clamp protocols that are used in this article to validate the moment-closure technique.

Combining the six-state RyR megachannel model with the two-state L-type channel model yields a 12-state CaRU model that takes the form

(16)

where horizontal and vertical transitions are governed by Eqs. 13 and 14, respectively, and the first character ( or ) indicates the state of the DHPR while the second character ( or ) refers to the state of the RyR megachannel.

Note that the 12×12 infinitesimal generator matrix (sometimes called the Q-matrix) that collects the rate constants of the CaRU model (Eq. 16) can be written compactly in the form

(17)

where the elements of K_φ(V) are the Ca²⁺-independent transitions (both voltage-dependent and voltage-independent with units of time⁻¹), and the elements of K_ds and K_jsr are the association rate constants for the transitions mediated by diadic subspace (c_ds) and junctional SR (c_jsr) [Ca²⁺], respectively (with units of concentration⁻¹ time⁻¹). Although noncooperative binding of Ca²⁺ is not a formal requirement for the application of the moment-closure technique, for simplicity we will assume the CaRU model is written in the form of Eq. 17.

Univariate probability density model

The moment-closure technique begins with the equations for a univariate probability density model of local control of Ca²⁺-induced Ca²⁺ release in cardiac myocytes (18). We write ρⁱ(c_jsr, t) to denote probability density functions for the distribution of [Ca²⁺] in a large number of junctional SR compartments jointly distributed with CaRU state, that is,

(18)

where i is an index over CaRU state, and the tilde in and indicate random quantities. For these densities to be consistent with the dynamics of the Monte Carlo model of cardiac EC coupling as N → ∞, they must satisfy a system of advection-reaction equations of the form (18,22,23)

(19)

where 1 ≤ i ≤ M, M = 12 is the number of states in the CaRU model, Q is the M×M generator matrix (Eq. 17), the row-vector ρ(c_jsr, t) = (ρ¹, ρ², ···, ρ^M) collects the time-dependent probability densities for the junctional SR [Ca²⁺] jointly distributed with CaRU state (Eq. 18), and [ρQ]ⁱ is the i^th element of the vector-matrix product ρQ.

Note that the factor in Eq. 19 describes the deterministic aspect of the time-evolution of c_jsr when the CaRU is in state i. That is, consistent with Eq. 3 we have

(20)

where 1 ≤ i ≤ M and is a function of CaRU state, the local junctional SR [Ca²⁺], and the bulk myoplasmic [Ca²⁺] analogous to Eqs. 9–11,

(21)

where

(22)

(23)

In these expressions, the quantities and take values of 0 or 1 depending on whether the respective component of the CaRU model is closed or open, and and are functions of plasma membrane voltage defined by

(24)

where is the total flux through the L-type Ca²⁺ channels (Eq. 58).

Conversely, the reaction terms ([ρQ]ⁱ) on the right-hand side of Eq. 19 correspond to the stochastic aspect of the CaRU dynamics (i.e., changes in probability due to the stochastic gating of the RyR megachannel and DHPRs). This term involves processes that may depend on the junctional SR [Ca²⁺] directly (as in the transition ) or indirectly (as in the transition ), as well as terms dependent on the membrane voltage (such as the transition ). Using the decomposition of Q given by Eq. 17, one can see that [ρQ]ⁱ is a function of V and c_jsr given by

(25)

where K_φ(V) provides the voltage-dependence, the superscripts of K_φ^{j, i}, and indicate row and column indices of these matrices, ρ^j(c_jsr, t) is the probability density for state j, and and are given by Eqs. 21–23.

The concentration balance equations governing the bulk myoplasmic (c_myo) and network SR (c_nsr) [Ca²⁺] in the probability density formulation are identical to those used in the Monte Carlo approach (Eqs. 1–2), except that the fluxes and are dependent on the densities that is,

(26)

(27)

where is a function of c_jsr (Eq. 21).

RESULTS

Moments of junctional SR [Ca²⁺]

The application of the moment-closure technique to the local control model of Ca²⁺-induced Ca²⁺ release (CICR) in cardiac myocytes presented above begins by writing the q^th moment of the univariate probability density function, ρⁱ(c_jsr, t), as

(28)

where the nonnegative integer q indicates the moment degree in and is an exponent in (c_jsr)^q. As defined in Eq. 18, ρⁱ(c_jsr, t) is the distribution of [Ca²⁺] in a large number of junctional SR compartments jointly distributed with CaRU state. Thus, the zero^th moment corresponds to the probability—denoted as πⁱ(t) in Williams et al. (18)—that a randomly sampled CaRU is in state i; that is,

where conservation of probability implies Because the joint probability densities do not individually integrate to unity, the first moment,

is related to the expected value of the junctional SR [Ca²⁺] conditioned on CaRU state through

(29)

while the conditional variance of the junctional SR [Ca²⁺] is

(30)

Expressing fluxes in terms of moments

Considering Eqs. 1 and 2 and Eqs. 26–27, one sees that the fluxes and mediate the influence of the distribution of diadic subspace and junctional SR [Ca²⁺] on the dynamics of the bulk myoplasmic [Ca²⁺] (c_myo) and the network SR [Ca²⁺] (c_nsr). Using the definition of the moments of junctional SR [Ca²⁺] (Eq. 28), these fluxes become functions of the zero^th and first moments,

(31)

Similarly, the total flux through all the L-type Ca²⁺ channels ( Eq. 24) and the RyR Ca²⁺ channels become

(32)

and

(33)

Note that the average diadic subspace and junctional SR Ca²⁺ concentrations can also be written in terms of the moments,

(34)

(35)

and and when expressed in terms using these quantities.

Derivation of moment equations

Differentiating Eq. 28 with respect to time and using the equations of the univariate probability density approach (Eqs. 19–25), we obtain a system of ODEs that describe the time-evolution of these moments defined in Eq. 28,

(36)

where M = 12, 1 ≤ i ≤ M, q = 0, 1, 2, …, and and are given by Eqs. 22 and 23. In this expression the CaRU model is specified by the M × M matrices K_φ, K_ds, and K_jsr defined in Eq. 17, and the superscripts in K_φ^j,i, and indicate the transition rate or bimolecular rate constant in the j^th row and i^th column of these matrices. Note that, in the M equations for the zero^th moments the first two terms evaluate to zero because q = 0. When q ≥ 1, the first term depends on both the network SR [Ca²⁺] (c_nsr) and the bulk myoplasmic [Ca²⁺] (c_myo) through The terms in the first summation have a similar dependence on c_myo and this can affect transitions mediated by diadic subspace Ca²⁺ and the magnitude of these terms depends also on voltage through Perhaps most importantly, the presence of diadic subspace and junction SR Ca²⁺-mediated transitions in the CaRU model implies that is a function of whenever or is nonzero. That is, Eq. 36 is an open system of the form

(37)

(38)

where we write as a shorthand for Consequently, Eq. 36 is unusable in its current form, because to determine the time-evolution of the q^th moments one needs to know the value of the (q+1)^th moments.

Moment closure

To utilize Eq. 36, we truncate the open system at the second moment (q = 2) and close the system of ODEs by assuming that the third moment can be expressed as an algebraic function φ of the lower moments that is,

(39)

(40)

(41)

The remainder of this section derives the required expression of the form (Eqs. 48–53). This is accomplished by specifying the function φ in a manner that would be strictly correct if the probability density functions were scaled β-distributions. Note that choosing this form of φ to perform the moment closure given by Eqs. 39–41 is not equivalent to assuming that the probability density functions are well approximated by β-distributions. What we are assuming is that the relationship between and the lower moments is similar to the relationship observed in the β-distribution. This assumption is validated a posteriori by evaluating the accuracy of results obtained using this approach (see Figs. 2–6).

FIGURE 2.

The response of the whole cell model during a 20-ms step depolarization from a holding potential of −80 mV to −10 mV (bar) with the Monte Carlo and moment-closure results indicated as a shaded line and solid line, respectively. (From top to bottom) Average diadic subspace [Ca²⁺] total Ca²⁺ flux via the DHPR Ca²⁺ channels total Ca²⁺-induced Ca²⁺ release flux and average junctional SR [Ca²⁺] The Monte Carlo simulation used N = 1000 Ca²⁺ release units and parameters as in Tables 2–4.

FIGURE 3.

Solid lines show the dynamics of bulk myoplasmic (c_myo) and network SR (c_nsr) [Ca²⁺] in the whole-cell voltage-clamp protocol of Fig. 2 with step potential of −10 mV (note longer timescale). The dashed and solid lines are the Monte Carlo and moment-closure results, respectively.

FIGURE 4.

Comparison between results obtained from Monte Carlo (shaded line) simulations and moment-closure approach (solid line) for the probability (πⁱ), the conditional expectation of c_jsr and the conditional variance of c_jsr for three selected CaRU states, (left column), (middle column), and (right column). The Monte Carlo simulation used N = 2000 Ca²⁺ release units.

FIGURE 5.

Histograms of junctional SR [Ca²⁺] conditioned on CaRU state obtained by Monte Carlo simulation (t = 30 ms in Fig. 2). Solid diamonds show β-distributions with same mean and variance. Each panel corresponds to one of four agglomerated states of the CaRU: DHPR and RyR megachannel both closed; DHPR open and RyR megachannel closed; DHPR closed and RyR megachannel open; and DHPR and RyR megachannel both open.

FIGURE 6.

Summary of whole-cell voltage-clamp simulations such as those presented in Figs. 2–4 normalized to emphasize gradedness of Ca²⁺ release with respect to membrane potential and Ca²⁺ influx. Moment-closure results (solid and broken lines) agree with Monte Carlo calculations (open symbols) for a range of test potentials. Integrated Ca²⁺ influx via L-type channels () is shown as open circles (Monte Carlo) and dotted line (moment closure). Integrated RyR flux () is shown as open squares (Monte Carlo) and dashed line (moment-closure). EC coupling gain ( right axis) is shown as open diamonds (Monte Carlo) and solid line (moment-closure).

The derivation begins by considering a random variable that is functionally dependent on through

(42)

In this expression, the minimum and maximum values of junctional SR [Ca²⁺] are given by and where are the steady-state values of c_jsr found by setting in Eq. 20,

where and are given by Eqs. 22 and 23. In this way, the maximum and minimum junctional SR Ca²⁺ concentrations are determined to be

(43)

(44)

where If the probability density for conditioned on CaRU state i were β-distributed, then

(45)

where the β-function B(αⁱ, βⁱ) appears as a normalization constant and αⁱ(t), and βⁱ(t) are all functions of time. Under this assumption, the first several conditional moments of would be

(46)

(47)

and inverting these expressions gives

(48)

(49)

Note that Eq. 42 implies the following relationship between the conditional moments of and

(50)

(51)

where we have used for q = 0, 1, and 2; consequently, αⁱ and βⁱ can be found as a function of and These parameters allow us to approximate the third conditional moment of

(52)

which, in turn, allows us to approximate the third conditional moment of junctional SR [Ca²⁺] given by where

After some simplification one obtains

(53)

which is an expression that takes the form as required by Eq. 41, because is a function of and given by Eqs. 48–52.

Note that the expression derived above is one of several possibilities that we tested, but the only one that could be validated. For example, when φ was chosen in a manner that would be strictly correct if the probability densities were scaled normal or log-normal distributions, the resulting moment closure did not perform well (not shown). Using the β-distribution to derive φ makes sense because it is a continuous distribution defined on a finite interval. In addition, for particular values of αⁱ and βⁱ, the β-distribution (while remaining integrable) diverges at the boundaries ( or 1). Similarly, prior work has established that the densities ρⁱ(c_jsr, t) can accumulate probability at the minimum and maximum junctional SR Ca²⁺ concentrations (Eqs. 43 and 44) and diverge as or (18,22). As mentioned above, the use of the β-distribution to derive φ is ultimately validated by evaluating the accuracy of results obtained using this approach (see Figs. 2–6).

Representative Monte Carlo and moment-closure results

Fig. 2 shows representative results from the minimal whole cell model of EC coupling described above. In this simulated voltage-clamp protocol, the holding potential of −80 mV is followed by a 20-ms duration test potential to −10 mV. The Monte Carlo result (shaded line) which involves a large but finite number of Ca²⁺ release units (N = 1000) can be easily spotted by the fluctuations due to the stochastic gating of the CaRUs. The moment-closure result (solid line) that assumes N → ∞ lacks these fluctuations. The top and bottom panels of Fig. 2 show the average diadic subspace and junctional SR Ca²⁺ concentrations in the Monte Carlo calculation (shaded lines) as well as the corresponding quantities from the moment-closure calculation (solid lines, Eqs. 34 and 35). The middle two panels of Fig. 2 show the total Ca²⁺ influx through L-type Ca²⁺ channels and the total Ca²⁺ release from the RyR Ca²⁺ channels for the Monte Carlo calculation (shaded lines) as well as the corresponding quantities for the moment-closure result (solid lines, Eqs. 32 and 33). In both the Monte Carlo and moment-closure calculations, the test potential of −10 mV leads to 16× gain, here defined as the ratio where the overbar indicates an average over the duration of the pulse.

Fig. 3 shows [Ca²⁺] in the bulk myoplasm (c_myo) and network SR (c_nsr) before, during, and after the −10 mV voltage pulse (note change in timescale). In both cases the moment-closure result is shown as a solid line while the Monte Carlo is displayed as a dashed line (note agreement). While junctional SR depletion develops rapidly after the initiation of the voltage pulse (not shown), refilling the junctional SR compartments via diffusion of Ca²⁺ from the network SR ( in Eq. 2) depletes this compartment (c_nsr), which does not fully recover until ∼300 ms after the termination of the voltage pulse.

Taken together, Figs. 2 and 3 validate our implementation of both the Monte Carlo and moment-closure approaches. Also note that the similarity of these results to Figs. 2 and 3 in Williams et al. (18) indicates that the six-state RyR megachannel model (Eq. 13)—used here because it takes the form of Eq. 17—has behavior similar to the two-state model of Williams et al. (18).

Dynamics of the moments of junctional SR [Ca²⁺]

The top row of Fig. 4 shows the time evolution of the probability of three selected CaRU states during the simulated voltage-clamp protocol of Figs. 2 and 3, as calculated using both the Monte Carlo (shaded lines) and moment-closure (solid lines) methods. Before the voltage pulse, the probability of state (DHPR in state and RyR in state see Eqs. 13–16) is ∼1, but during the voltage pulse to −10 mV this probability drops to ∼0.78 (20–40 ms). Conversely, the probability of CaRU state (DHPR open and RyR open) and (DHPR closed and RyR open) both increase during the voltage pulse. The dynamics of voltage-dependent activation of DHPRs and subsequent triggering of the opening of RyR megachannels is similar in both the Monte Carlo (shaded lines) and moment-closure (solid lines) calculations.

The second row of Fig. 4 shows the mean junctional SR [Ca²⁺] conditioned on CaRU state for the Monte Carlo (shaded line) and the moment-closure (solid line) techniques. In the Monte Carlo calculation this conditional mean is given by

(54)

where Nⁱ(t) is the number of CaRUs in state i at time t and so that the sum includes only those CaRUs in state i. The corresponding quantity in the moment-closure technique is the conditional expectation (Eq. 29). Note that before the voltage pulse the expectation of SR [Ca²⁺] is ∼1000 μM when conditioning on CaRU state 851 μM when conditioning on CaRU state and 306 μM when conditioning on CaRU state That is, at the holding potential of −80 mV, the stochastic gating of CaRUs leads to depletion of junctional SR [Ca²⁺] associated with release sites with open RyR megachannels (more pronounced in than because the former state is longer-lived). However, the probability of CaRU states and is very low at −80 mV and, consequently, the expectation of junctional SR [Ca²⁺] irrespective of CaRU state given by the weighted average

in the Monte Carlo model and

in the moment-closure calculation is ∼1000 μM, consistent with Fig. 2. Also note that during the voltage pulse the conditional expectation of junctional SR [Ca²⁺] decreases for CaRU states and but first increases and then decreases for CaRU state presumably because the increasing probability of state during the pulse is due to CaRU transitions into this state from others (such as ) that have higher resting junctional SR [Ca²⁺].

The third row of Fig. 4 shows the variance of the junctional SR [Ca²⁺] conditioned upon the CaRU state for the Monte Carlo (shaded line) and the moment-closure (solid line) techniques. For the Monte Carlo calculation

where Nⁱ and nⁱ(t) are defined as in Eq. 54, while the corresponding conditional variance of the junctional SR [Ca²⁺] in the moment-closure calculation is (Eq. 30). Note that during the voltage pulse the conditional variance of c_jsr increases, as the dynamics of EC coupling lead to increased heterogeneity of junctional SR [Ca²⁺], and that the moment-closure technique accurately accounts for this heterogeneity (compare shaded and solid lines).

The distribution of junctional SR [Ca²⁺] conditioned on CaRU state

Fig. 5 shows a snapshot of the distribution of junctional SR [Ca²⁺] (c_jsr) conditioned upon the state of the Ca²⁺ release unit at t = 30 ms, midway through the voltage pulse protocol of Figs. 2–4. For clarity, the five closed states of the RyR megachannel ( in Eq. 13) have been lumped resulting in a contracted presentation with four CaRU states: and where and (Eq. 16). Thus, the two histograms on the bottom of Fig. 5 indicate the distribution of JSR [Ca²⁺] when the DHPR is open while the two histograms on the right of Fig. 5 indicate the distribution of JSR [Ca²⁺] when the RyR megachannel is open

Fig. 5 shows a broad range of junctional SR [Ca²⁺] regardless of CaRU state, consistent with the high variances at t = 30 ms in Fig. 4. For example, when the RyR megachannel is closed ( and left panels), a randomly sampled junctional SR is likely to be replete, as indicated by the large vertical bar at c_jsr ≈ 1000 μM. However, one can also find depleted junctional SR [Ca²⁺] associated with closed RyR megachannels, where RyRs have recently opened and the junctional SR has not had time to refill. Conversely, when the RyR is open ( and right panels), the probability mass has shifted to lower junctional SR [Ca²⁺].

The diamonds of Fig. 5 show β-distributions with the same mean and variance as the histograms obtained from Monte Carlo simulation. While the agreement is noteworthy, this correspondence is not required for the moment-closure technique to work well. What is required is that the relationship between the third and lower moments in the histograms is similar to that observed in the β-distribution. For example, the histogram junctional SR [Ca²⁺] for CaRU state at t = 30 ms has moments of μM, and When moments 0–2 are used to estimate the third moment using Eq. 53 with and (Eqs. 43 and 44), one obtains for a relative error of only 0.1%. It is this low error that is responsible for the excellent agreement between the moment-closure result and the Monte Carlo calculation observed in Figs. 2–4.

The model displays gain and gradedness

To further validate the moment-closure approach by comparison to Monte Carlo simulation, Fig. 6 summarizes a large number of simulated whole-cell voltage-clamp protocols such as those presented in Figs. 2–4. The open circles of Fig. 6 show the trigger Ca²⁺ influx via L-type Ca²⁺ channels integrated over the 20-ms voltage step to test potentials in the range −40 to 40 mV using 1000 CaRUs (the plot is normalized to maximum value of The dotted line of Fig. 6 shows that the trigger Ca²⁺ influx in the moment-closure calculation agrees with the Monte Carlo simulations. Similarly, the open squares of Fig. 6 show the voltage-dependence of the Ca²⁺ release flux (normalized to maximum value of while the dashed lines of Fig. 6 show that the Ca²⁺ release flux observed in the moment-closure calculation agrees with the Monte Carlo simulations. Note that the Monte Carlo and moment-closure calculations exhibit graded Ca²⁺ release. Furthermore, the EC coupling gain () is a decreasing function of voltage, in the range of 32–15× for test potentials between −40 and 0 mV. Most importantly, the Monte Carlo and moment-closure calculations are nearly identical (compare open diamonds and solid line).

Computational efficiency of the moment-closure approach

While the previous sections have shown that the moment-closure and Monte Carlo calculations are essentially equivalent in terms of the dynamic cellular responses they predict, it is important to note that the moment-closure approach is significantly faster than Monte Carlo simulation. The Monte Carlo simulations presented above are performed using Δt = 0.01 μs, a value chosen so the probability of transition occurring in each CaRU is <5% per time step. Table 1 shows that the run time for these 60-ms simulations increases approximately linearly with the number of CaRU units; for example, an N = 10,000 simulation takes ∼11 times longer than a N = 1000 simulation. When our current implementation of the moment-closure method is employed using a nonadaptive time step of Δt = 0.01 μs, the run time is 95 min, which is ∼100 times faster than Monte Carlo simulations with a physiologically realistic number of CaRUs (e.g., N = 10,000). However, a time step of 0.01 μs is much smaller than required for integrating the moment-closure ODEs. When this artificial constraint is removed and the moment-closure approach is benchmarked using a nonadaptive time step as large as numerical stability will allow, the calculations are 8755:0.9 = 9728 times faster than Monte Carlo simulations containing N = 10,000 CaRUs. That is, the computational efficiency of the moment-closure approach is nearly four orders-of-magnitude superior to physiologically realistic Monte Carlo simulations, while leading to nearly identical results (see Figs. 2–4, and 6). Furthermore, integration methods that utilize adaptive time-stepping are likely to further enhance the computational advantage of the moment-closure approach to modeling local control of EC coupling.

TABLE 1.

Run times required for a 60-ms simulation such as that presented in Fig. 2 using both Monte Carlo and moment-closure approaches

	Δt (μs)	N	Time (min)
Monte Carlo	0.01	100	50
	0.01	1000	794
	0.01	10,000	8755
Moment closure	0.01	—	95
	1	—	0.9

TABLE 2.

Model parameters: volume fractions, Ca²⁺ buffering, and exchange between restricted domains and the bulk, physical constants, and fixed ion concentrations

Parameter	Definition	Value
N	Number of diadic subspaces	50–20,000
Vnsr	Network SR volume	3.15 × 10−7μL
Vmyo	Myoplasmic volume	2.15 × 10−5μL
	Total diadic subspace volume	2 × 10−8μL
	Total junctional SR volume	2.45 × 10−8μL
Cm	Capacitive membrane area	1.534 × 10−4μF
βds	Subspace buffering factor	0.5
βjsr	Junctional SR buffering factor	0.065
βnsr	Network SR buffering factor	1.0
βmyo	Myoplasmic buffering factor	0.05
	Junctional SR refilling rate	0.018 s−1
	Diadic subspace efflux rate	5.2 s−1
F	Faraday's constant	96,480 Coul mol−1
R	Gas constant	8314 mJ mol−1 K−1
T	Absolute temperature	310 K
cext	Extracellular Ca2+ concentration	1.8 mM
[Na+]ext	Extracellular Na+ concentration	140 mM
[Na+]myo	Intracellular Na+ concentration	10.2 mM

TABLE 4.

Model parameters: Na⁺-Ca²⁺ exchange current, SERCA pumps, and background Ca²⁺ influx

Parameter	Definition	Value
Kfs	Forward half-saturation constant for SERCA pump	0.17 μM
Krs	Reverse half-saturation constant	1702 μM
ηfs	Forward cooperativity constant	0.75
ηrs	Reverse cooperativity constant	0.75
vserca	Maximum SERCA pump rate	8.6 μM s−1
	Magnitude of Na+-Ca2+ exchange current	150 μA μF−1
Kncx, n	Na+ half-saturation constant	87.5 × 103μM
Kncx, c	Ca2+ half-saturation constant	1.38 × 103μM
	Saturation factor	0.1
ηncx	Voltage dependence of Na+-Ca2+ exchange	0.35
vleak	SR Ca2+ leak rate constant	2.4 × 10−6 s−1
gin	Maximum conductance of background Ca2+ influx	9.6 × 10−5 mS μF−1

Restitution of CICR studied using moment-closure approach

To show how the computational efficiency of the moment-closure approach facilitates studies that can provide biophysical insight, we present a study of the restitution of Ca²⁺-induced Ca²⁺ release during simulated two-pulse voltage-clamp protocols (see (24)). As diagrammed in the inset, Fig. 7 A plots the ratio of the integrated release during the two pulses () as a function of time between the end of the first pulse and beginning of the second (denoted by τ). Using the standard value for the maximum reuptake flux ( μM⁻¹ s⁻¹), the time constant for recovery of CICR is ∼93 ms. Increasing or decreasing by 20% (dashed and dotted lines) leads to a time constant for CICR recovery of 80 ms and 120 ms, respectively. This result is qualitatively consistent with the results of Szentesi et al. (24), and the hypothesis that restitution of calcium release depends primarily on refilling of local SR calcium stores (24–26). As in Fig. 2, the moment-closure approach was validated by comparison to Monte Carlo simulation using these alternate values of and an interpulse interval of τ = 20 ms (not shown).

FIGURE 7.

CICR restitution study using a simulated two-pulse voltage-clamp protocol and different values of the maximum reuptake flux (A) Ratio of the integrated release during the two pulses () as a function of time between the end of the second pulse and beginning of the first (τ). Parameters: (dotted line), 8.60 (solid line), and 10.32 μM⁻¹ s⁻¹ (dashed line), and as in Tables 2–4. (Inset) Timing of voltage pulses from

–80 to −10 mV. (B) Distributions of junctional SR [Ca²⁺] consistent with the conditional expectations and variances observed in the moment-closure model at the beginning of the second pulse when τ = 0.02, 0.06, 0.1, and 0.2 s. Dotted, solid, and dashed lines indicate value of as in panel A. Solid symbols indicate the expected value of junctional SR [Ca²⁺] given by and Eq. 29. The solid line and open triangles correspond to the initial (and fully recovered) distribution.

The solid symbols in the four panels of Fig. 7 B show that in each of these three cases the expected value of the junctional SR [Ca²⁺] at the beginning of the second pulse is an increasing function of the interpulse interval τ. Also shown are the distributions of junctional SR [Ca²⁺] consistent with the conditional expectations and variances observed in the moment-closure model at the time of the second pulse begins when τ = 0.02, 0.06, 0.1, and 0.2 s. Note that the rightmost extent of these distributions indicates the network SR [Ca²⁺] in the corresponding simulation (c_jsr ≤ c_nsr), and the fully recovered distribution (dotted lines) has an expectation of ∼1000 μM (open triangle). Note that the variance of the junctional SR [Ca²⁺] decreases as a function of the interpulse interval τ (compare widths of distributions).

Fig. 8 shows the recovery of the network SR [Ca²⁺] (dotted line), the junctional SR [Ca²⁺] (solid line), and the average concentration when the two compartments are aggregated according to their effective volumes (dashed line). This last measure represents the total SR content as would be assessed experimentally via the rapid application of caffeine. Importantly, the restitution of CICR as probed by the ratio of the integrated release ( solid circles) is consistent with the recovery of the junctional SR [Ca²⁺], but not consistent with recovery of the network SR [Ca²⁺] or the aggregate concentration.

FIGURE 8.

Recovery of the network SR [Ca²⁺] (c_jsr, dotted line), the junctional SR [Ca²⁺] (c_nsr, solid line), and the average concentration when the two compartments are aggregated according to their effective volumes (c_nsr&jsr, dashed line). (Inset) Timing of voltage pulses from −80 to −10 mV and representative c_jsr trace. Solid circles show CICR restitution observed in Fig. 7 A. Standard value of μM⁻¹ s⁻¹ used.

Fig. 9 A is similar to Fig. 7 A except that, in this case, the rate of calcium diffusion from network SR to junctional SR is modified from the standard value of μM⁻¹ s⁻¹. Despite the fact that the restitution of CICR follows the recovery of junctional SR [Ca²⁺] (see Fig. 8), the time constant of CICR restitution is less sensitive to the junctional SR refill rate () than the maximum SERCA pump rate For example, increasing or decreasing by a factor of 2 (dashed and dotted lines) leads to a time constant for CICR recovery of 91 and 105 ms (similar to the standard value of 93 ms). Conversely, the extent of junctional SR depletion at the end of the first pulse ranges from 51–65% in Fig. 9 A and 58–59% in Fig. 7, and thus appears to be more sensitive to the value of than (a range proportional to the parameter variation in Fig. 9 A in the former range would span 2.5 rather than 14%).

FIGURE 9.

Summary of CICR restitution study using a simulated two-pulse voltage-clamp protocol and different values of the junctional SR refill rate given by (dotted line), 0.018 (solid line), and 0.036 μM⁻¹ s⁻¹ (dashed line). Standard value of μM⁻¹ s⁻¹ used. See legend to Fig. 7.

Consistent with these observations, Fig. 9 B shows that the expected value of junctional SR [Ca²⁺] increases with increasing interpulse interval τ and that decreased values of lead to increased depletion (compare solid triangles). Comparison of the reconstructed distributions indicates that decreased slows the recovery of junctional SR [Ca²⁺] and leads to increased heterogeneity, i.e., higher variance in junctional SR [Ca²⁺] (compare dotted and solid lines).

DISCUSSION

In previous work (18) we showed that the probability density approach to modeling local control of Ca²⁺ release in cardiac myocytes can be 30–650 times faster than traditional Monte Carlo simulations when the probability densities are univariate (i.e., functions of the junctional SR [Ca²⁺] but not explicitly functions of diadic subspace [Ca²⁺]). The derivation of the moment-closure technique presented in this article begins with a univariate probability density formulation, but the resulting simulations are nearly 10,000 times faster than Monte Carlo (see Table 1). For the whole-cell model that is the focus of this article, the moment-closure technique is thus significantly more efficient that our previously presented univariate probability density method (18).

Although the computational efficiency of the moment-closure technique in this local control context is exciting, it is important to note that the relative merits of Monte Carlo, probability density, and moment-closure methods are in general model-dependent. For example, the run time required for the Monte Carlo simulations such as Fig. 2 is, at least ultimately, an almost linear function of the number of CaRUs (see Table 1). Similarly, we have observed that the computational efficiency of the univariate probability density calculation presented in Williams et al. (18) scales linearly with the number of Ca²⁺ release unit states (M) and the number of mesh points used to discretize the junctional SR [Ca²⁺]. Because the moment-closure approach results in 2 + 3M ODEs (bulk myoplasmic [Ca²⁺], network SR [Ca²⁺], and and for each CaRU state), the computational demand of the moment-closure approach is expected to scale linearly with M. That is, increasing the number of CaRU states could reduce the computational advantage of the moment-closure approach relative to Monte Carlo.

While the CaRU model used here to introduce and validate the moment-closure approach includes a two-state DHPR model and a six-state RyR megachannel (Eq. 16), the success of the moment equations (Eq. 36 and Eqs. 39–41) and moment-closure using given by Eqs. 48–53 does not depend on the CaRU model; rather, any CaRU that takes the form of Eq. 17 could be employed. For example, a more realistic DHPR model that includes voltage and Ca²⁺-dependent inactivation would allow integration of the moment-closure approach to modeling local control of CICR and action potentials modeled using Hodgkin-Huxley-style membrane currents. Similarly, a more realistic CaRU model could be constructed as the composition of multiple RyR single channel models. This approach will therefore allow for the development of mechanistic, local control models that can examine phenomena such as stochastic SR calcium leak and bidirectional interactions between calcium transients and action potential morphology. However, to maintain the computational advantage of the moment-closure approach relative to Monte Carlo, the state-space explosion that inevitably occurs in compositional models is an important practical consideration. For example, one 12-state L-type channel and 12 four-state RyRs leads to a CaRU model with M = 5460 distinguishable states and thus >16,000 ODEs, a value approaching the 20,000 ODEs required in a 10,000 CaRU Monte Carlo simulation.

The moment-closure approach presented here begins with a univariate probability density approach to modeling heterogeneous junctional SR [Ca²⁺]. This was motivated by our previous work in which we observed that model parameters lead to rapid equilibrium of the diadic subspace [Ca²⁺] with the [Ca²⁺] in the junctional SR and bulk myoplasm (18). When junctional SR [Ca²⁺] was also assumed to be rapidly equilibrated with the bulk myoplasmic and network SR Ca²⁺ concentrations so that these local Ca²⁺ concentrations could be expressed as algebraic functions of c_myo, c_nsr, and CaRU state (21,27), the resulting model did not exhibit high gain Ca²⁺ release that is graded with membrane potential (not shown). That is, the assumption that both diadic subspace [Ca²⁺] and junctional SR [Ca²⁺] are in quasi-static equilibrium with bulk myoplasmic and network SR Ca²⁺ leads to unacceptable errors and cannot be employed to accelerate this particular whole-cell model. This approximation has, however, been successfully employed in previous studies of cardiac CICR (21,27). It therefore seems likely that the model simplifications that can be employed depend on the details of both RyR gating and local concentration changes, issues that are currently being extensively studied.

In situations where rapid equilibration of diadic subspace [Ca²⁺] does not occur, the appropriate starting point for the moment-closure approach is a bivariate probability density model (18). While it is straightforward to derive the open system of ODEs analogous to Eq. 36 for the time-evolution of the moments of bivariate probability densities ρⁱ(c_ds, c_jsr, t) defined by

we have yet to find a moment-closure method that works well in the bivariate case. This would be an important further development of the moment-closure approach as a computationally efficient alternative to Monte Carlo simulation of the local control of EC coupling in cardiac myocytes.

APPENDIX: WHOLE CELL MODEL OF EC COUPLING—FLUXES AND VOLUME RATIOS

The whole cell model of EC coupling that is the focus of this article includes several fluxes that directly influence the dynamics of the bulk myoplasmic and network SR [Ca²⁺]. For example, the Na⁺-Ca²⁺ exchanger current that appears in Eqs. 1 and 2 identical in the Monte Carlo, probability density, and moment-closure formulations and takes the form J_ncx = −A_mI_ncx/F where (2,10,28)

A_m = C_mβ_myo/V_myo, c_ext is the extracellular Ca²⁺ concentration, and [Na⁺]_myo and [Na⁺]_ext are the intracellular and extracellular sodium concentrations, respectively (for parameters see (18)). The SERCA-type Ca-ATPase flux that appears in Eqs. 1 and 2 includes both forward and reverse modes (29) and is given by

with parameters as in Williams et al. (18). In addition, Eqs. 1 and 2 include a leakage Ca²⁺ flux given by

Following Rice et al. (28), Eq. 1 includes a constant background Ca²⁺ influx that takes the form J_in = −A_mI_in/zF, where I_in = g_in(V – E_Ca) and E_Ca = (RT/2F) ln(c_ext/c_myo).

The effective volume ratios λ_nsr and λ_jsr that appear in Eqs. 2 and 3 are defined with respect to the physical volume (V_myo) and include a constant-fraction Ca²⁺ buffer capacity for the myoplasm (β_myo). For example, the effective volume ratio associated with the network SR is

with effective volumes defined by and Because each individual junctional SR compartment is assumed to have the same physical volume (V_jsr) and buffering capacity (β_jsr), the effective volume ratio that occurs in Eq. 3 is

(55)

where the second expression defines λ_jsr in terms of the total physical volume of all the junctional SR compartments in aggregate Similar assumptions and equations apply for the diadic subspaces so that the definition of λ_ds follows Eq. 55. However, when rapid equilibration of diadic subspace [Ca²⁺] is assumed, the volume ratio λ_ds no longer influences the steady state (see Eqs. 8–11 and Eqs. 21–23).

In the Monte Carlo model the trigger Ca²⁺ flux into each of the N diadic spaces through DHPR channels ( in Eq. 8) is given by

(56)

where A_m = C_mβ_myo/V_myo. The inward Ca²⁺ current is given by

(57)

where V_θ = RT/zF, is the total (whole cell) permeability of the L-type Ca²⁺ channels, and is a random variable that is 0 when the L-type Ca²⁺ channel associated with the n^th CaRU is closed and 1 when this channel is open. Thus, the quantities and required to evaluate (Eq. 10) and (Eq. 11) are defined through

consistent with Eq. 12. In the univariate probability density approach and moment-closure method the total flux through L-type Ca²⁺ channels is given by

(58)

and the quantities and are used to evaluate (Eq. 22) and (Eq. 23).