Cardiac myocyte calcium signaling is often modeled using deterministic ordinary differential equations (ODEs) and mass-action kinetics. However, spatially restricted “domains” associated with calcium influx are small enough (e.g., 10^{−17} liters) that local signaling may involve 1–100 calcium ions. Is it appropriate to model the dynamics of subspace calcium using deterministic ODEs or, alternatively, do we require stochastic descriptions that account for the fundamentally discrete nature of these local calcium signals? To address this question, we constructed a minimal Markov model of a calcium-regulated calcium channel and associated subspace. We compared the expected value of fluctuating subspace calcium concentration (a result that accounts for the small subspace volume) with the corresponding deterministic model (an approximation that assumes large system size). When subspace calcium did not regulate calcium influx, the deterministic and stochastic descriptions agreed. However, when calcium binding altered channel activity in the model, the continuous deterministic description often deviated significantly from the discrete stochastic model, unless the subspace volume is unrealistically large and/or the kinetics of the calcium binding are sufficiently fast. This principle was also demonstrated using a physiologically realistic model of calmodulin regulation of L-type calcium channels introduced by Yue and coworkers.

Concentration changes of physiological ions and other chemical species (such as kinases, phosphatases, and various modulators of cellular activity) influence and regulate cellular responses [^{−17} liters, with approximately 20,000 diadic subspaces per cell [

Previous studies have compared discrete-state (stochastic) and continuous-state (deterministic) models in the analysis of biological and chemical systems, including models of biochemical networks, enzyme kinetics, and population dynamics [

Because of recent interest in the physiological relevance of spatially localized control of voltage- and calcium-regulated calcium influx and sarcoplasmic reticulum calcium release in cardiac myocytes [

To answer this question, we constructed and analyzed a minimal Markov model of a calcium-regulated calcium channel and associated subspace. We compared the expected steady-state subspace calcium concentration in this stochastic model (a result that accounts for the small subspace volume) with the result obtained using the corresponding deterministic ODE model (an approximation that assumes large system size). Section

We begin with the case of a single calcium channel that is associated with a spatially restricted subspace but not regulated by subspace calcium (Figure ^{−1} to the constant bulk concentration of

Diagram of the components and fluxes in a minimal subspace model. Calcium influx ^{−1}). Bulk calcium at the concentration

In the corresponding stochastic description of calcium influx into a diadic subspace, the state variable is the number of calcium ions in the subspace (a discrete quantity that we will denote by

If we write

To find the steady-state probability distribution of

To see how the subspace calcium concentration fluctuations predicted by this minimal model depend on the parameters

Figure

Steady-state probability distribution of the number of calcium ions (^{−1},

Diagram of the components and fluxes in a subspace model that includes calcium-regulated calcium influx. A single calcium channel (with two calcium binding sites) is associated with a subspace of volume

Most importantly, the deterministic and stochastic descriptions of this minimal subspace model agree in the following sense: the expected value of the fluctuating calcium concentration in the stochastic model

The numerical results presented above can be obtained analytically by considering the dynamics of the moments of the number of calcium ions in the subspace, defined as

This section augments the subspace model presented above to include calcium regulation of a calcium channel (see Figure ^{−1}, ^{−2} ms^{−1}, and the dissociation constant for calcium binding, denoted by

Let us denote the states of the stochastic system by

where

Let us write

By differentiating (

Note that as the volume increases (

The moment analysis in the previous section suggests that the expected calcium concentration in the subspace given by

Figures

Subspace volume-dependence of calcium fluctuations and open probability of a calcium-activated channel. Steady-state probability distribution for ^{−2} ms^{−1}) [^{−2} ms^{−1}) with

Figure

Both the open probability and expected calcium concentration asymptotically approach values in a range that are easily precalculated. For example,

In order to further characterize the effect of subspace volume on the calcium-regulated channel and subspace dynamics, we defined the small system deviation

Percentage small system deviation (^{−2} ms^{−1}) and calcium-inactivated channel (^{−2} ms^{−1}).

For comparison, Figure

The previous section analyzed the effect of subspace volume when the influx pathway involves calcium regulation of a single channel. In this section, we assume that the total number of channels increases with subspace volume (see Figure

Illustration of two possible volume scalings. For the single channel volume scaling, calcium influx

Assuming as before that two free calcium ions

Bifurcation diagram showing the steady-state calcium concentration

Following the notation developed in the previous section, we write

Figure

Subspace volume-dependence of concentration fluctuation and channel open probability for multiple calcium-activated channels. (A) Steady-state probability distribution for ^{−2} ms^{−1}). For each panel, the dashed black line denotes the conditional expected concentration (^{−2} ms^{−1}). The fast/large and slow system limits are shown in red and blue, respectively. (b) Steady-state ^{−2} ms^{−1}. In the bistable system, the larger of the two stable equilibrium (large system limit) is shown in red. The smaller equilibrium is approximately equal to the slow system limit (shown in blue).

Figure

Figure

Subspace volume-dependence of concentration fluctuation and channel open probability for multiple calcium-inactivated channels. (A) Steady-state probability distribution for ^{−2} ms^{−1}). (B) Steady-state ^{−2} ms^{−1}).

Figure

Percentage small system deviation (^{−2} ms^{−1}) and (b) calcium-inactivated channels (^{−2} ms^{−1}).

In the previous sections, we demonstrated that the expected steady-state subspace concentration determined using a minimal model of a calcium-activated or -inactivated channel was volume-dependent and could greatly differ from the steady-state concentration computed from deterministic ODEs. In this section, we show similar results for a state-of-the-art model of calmodulin-mediated calcium regulation.

Both the N-lobe and C-lobe of calmodulin have two binding sites for calcium. Depending on the calcium channel type (L, N, or P/Q), calcium binding to the C-lobe has been shown to be responsible for either activation or inactivation of the channel, while N-lobe binding appears to be primarily responsible for channel inactivation [

Calmodulin regulation of the calcium channel at steady-state. (A) State diagram of calmodulin regulation of a calcium channel (modified from [

Using this published model as a starting point, we formulated the corresponding discrete Markov model. The elementary reactions for calmodulin-mediated regulation of the channel are

Figure

Figure

Small system deviation for a single calmodulin-activated and -inactivated channel using “slow CaM” and “SQS” parameters (see text).

The parameter space for the calmodulin-inactivated channel differed somewhat from our simplified model (Figure

We also calculated the small system deviation

Small system deviation for multiple calmodulin-activated and -inactivated channels, using “slow CaM” and “SQS” parameters.

We developed a minimal model of a calcium-regulated channel in a small subspace and formulated a Markov model in which each possible discrete state is represented. For small subspace volumes, we found that the value predicted by a continuous-state, deterministic ODE model often deviated from the expected steady-state calcium concentration in the discrete-state, stochastic model. We analyzed how this deviation depends on channel binding kinetics, subspace volume, and calcium influx rate. We demonstrated that the deterministic description also deviated from the stochastic model in a physiologically realistic model of calmodulin-mediated calcium channel regulation.

Many studies have modeled the influence of signaling proteins on intracellular and transmembrane ion channel/receptor kinetics, such as calcium/calmodulin-dependent kinase II phosphorylation [

In addition to demonstrating that a discrete/stochastic model of calcium-regulated calcium influx often deviates from a continuous/deterministic description, we analyzed how subspace volume and concentration fluctuations influence channel dynamics. Because calmodulin effectively colocalizes with the L-type calcium channel [

Prior work by our lab has investigated calcium channel regulation through a host of various mechanisms. Groff and Smith investigated the influence of inactivation on calcium spark dynamics in a channel regulated by both calcium-activation and -inactivation [

Only a few previous studies have utilized a discrete representation of calcium ions in the context of cardiac myocyte subspace dynamics. Winslow and colleagues simulated the spatial location of discrete diffusing calcium ions, as well as the spatial structure and geometry of the L-type calcium channel and ryanodine receptor in the cardiac dyad [

Previous studies have modeled biochemical reaction networks using master equations and compared results with deterministic ODE models. McQuarrie demonstrated in 1963 that for first-order reactions, the expected steady-state concentrations derived from the chemical master equation and deterministic ODEs agree [

Darvey et al. demonstrated for several generic second-order reactions, the expected concentration computed from the chemical master equation may deviate from the corresponding ODE model [

We found that the small system size deviation was particularly complex in cases where the deterministic ODE, that is, the model appropriate for the large system size limit, is bistable (Figure

The two-state kinetic models of the calcium channel introduced in Section

The most significant limitation in the model formulation is our neglect of spatial dynamics of calcium diffusion within the dyadic subspace and the details of the spatial arrangement of the ryanodine receptors [

Another limitation of the present work is that we focus on

Our findings demonstrate the physiological relevance of concentration fluctuations in both minimal and realistic models of a calcium-regulated channels associated with subspaces of small volume. The take home message is:

The authors do not have any conflict of interests.

The work was supported in part by National Science Foundation Grant DMS-1121606 and the Biomathematics Initiative at The College of William & Mary. The authors acknowledge stimulating discussions with graduate students Xiao Wang and Mary Mohr. This study was inspired by comments made at the 2012 UC Davis Cardiovascular Symposium.