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Stochastic binding of Ca2+ ions in the dyadic cleft; continuous versus random walk description of diffusion.

Hake J, Lines GT - Biophys. J. (2008)

Bottom Line: With these results we demonstrate that the stochasticity and discreteness of the Ca(2+) signaling in the dyadic cleft, determined by single binding events, can be described using a deterministic model of Ca(2+) diffusion together with a stochastic model of the binding events, for a specific range of physiological relevant parameters.Time-consuming RW simulations can thus be avoided.We also present a new analytical model of bimolecular binding probabilities, which we use in the RW simulations and the statistical analysis.

View Article: PubMed Central - PubMed

Affiliation: Simula Research Laboratory, Lysaker, Norway. hake@simula.no

ABSTRACT
Ca(2+) signaling in the dyadic cleft in ventricular myocytes is fundamentally discrete and stochastic. We study the stochastic binding of single Ca(2+) ions to receptors in the cleft using two different models of diffusion: a stochastic and discrete Random Walk (RW) model, and a deterministic continuous model. We investigate whether the latter model, together with a stochastic receptor model, can reproduce binding events registered in fully stochastic RW simulations. By evaluating the continuous model goodness-of-fit for a large range of parameters, we present evidence that it can. Further, we show that the large fluctuations in binding rate observed at the level of single time-steps are integrated and smoothed at the larger timescale of binding events, which explains the continuous model goodness-of-fit. With these results we demonstrate that the stochasticity and discreteness of the Ca(2+) signaling in the dyadic cleft, determined by single binding events, can be described using a deterministic model of Ca(2+) diffusion together with a stochastic model of the binding events, for a specific range of physiological relevant parameters. Time-consuming RW simulations can thus be avoided. We also present a new analytical model of bimolecular binding probabilities, which we use in the RW simulations and the statistical analysis.

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(A and B) Number of registered binding events from 100 runs each, where we altered different parameters. The data were collected from a receptor 30 nm from the center and are represented by the box-plots together with a 95% confidence interval for the true means (red horizontal lines). The blue solid circles represent the expected number of binding events that are predicted by the continuous model. (A) We scaled the number of Ca2+ ions that enter the cleft, i.e.,  together with the diffusion constant D, with a factor represented by the x axis. The spatial resolution was constant for these simulations, σ = 5 nm. The blue asterisk denotes a statistical difference between the continuous model and the RW model for scale = 0.1. (B) We kept the scale constant at 0.1, but altered the spatial resolution (see the x axis). Here, the difference between the RW model and the continuous model increased as the mean value of the collected binding events declined with the spatial resolution. (C) We ran the simulation 100 times. We collected the mean binding rates for each run that the receptor were exposed to. The data from each set of 100 runs are presented as 95% confidence intervals for the true means. The blue horizontal lines represent the binding rates collected from runs in which we registered binding events, as in panel B. The red horizontal lines represent binding rates collected from runs in which we did not register binding events, only the rate. In these runs we could not differentiate statistically between the registered binding rates and the rates predicted from the continuous model.
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fig12: (A and B) Number of registered binding events from 100 runs each, where we altered different parameters. The data were collected from a receptor 30 nm from the center and are represented by the box-plots together with a 95% confidence interval for the true means (red horizontal lines). The blue solid circles represent the expected number of binding events that are predicted by the continuous model. (A) We scaled the number of Ca2+ ions that enter the cleft, i.e., together with the diffusion constant D, with a factor represented by the x axis. The spatial resolution was constant for these simulations, σ = 5 nm. The blue asterisk denotes a statistical difference between the continuous model and the RW model for scale = 0.1. (B) We kept the scale constant at 0.1, but altered the spatial resolution (see the x axis). Here, the difference between the RW model and the continuous model increased as the mean value of the collected binding events declined with the spatial resolution. (C) We ran the simulation 100 times. We collected the mean binding rates for each run that the receptor were exposed to. The data from each set of 100 runs are presented as 95% confidence intervals for the true means. The blue horizontal lines represent the binding rates collected from runs in which we registered binding events, as in panel B. The red horizontal lines represent binding rates collected from runs in which we did not register binding events, only the rate. In these runs we could not differentiate statistically between the registered binding rates and the rates predicted from the continuous model.

Mentions: For a long time, continuous and deterministic models have been used to study Ca2+ dynamics in the dyadic cleft (21,28–31), and its role in the release of Ca2+. Two recent studies of Ca2+ dynamics use a discrete RW model to describe the Ca2+ diffusion in the cleft (12,32). Koh et al. (12) uses MCell and argues that few Ca2+ ions in a small volume cannot properly be simulated with a continuous model of diffusion. However, they do not present any results that support this claim. Tanskanen et al. (32) present an impressive study that includes physiological details on a microscale level, such as the electrostatic force from the sarcolemmal and the geometrical structures of the large membrane proteins in the cleft, while integrating the Ca2+ release from many clefts, and thus obtaining a measure of the Ca2+ release from the whole cell. In contrast to Koh et al. (12), they explicitly address the difference between their model and an equivalent model that uses a deterministic description of Ca2+ diffusion. They do this by measuring the effect on the excitation-contraction coupling (ECC) gain when they vary the diffusion constant of Ca2+, together with the parameters that determine the influx of Ca2+ ions to the cleft. They show that the ECC gain varies with the parameters (see Fig. 12 in (32)). This result points to a “subtle but potentially significant difference in predicted macroscopic behavior arising from the underlying stochastic simulation of Ca2+ motion in the dyad” (32). The rationale for this statement is that if they had changed the same parameters in an equivalent model using a deterministic description of Ca2+ diffusion, they would not have registered any differences in ECC gain because the receptors situated in the cleft would have experienced the same level of Ca2+ concentration. In our study we examine the discrete events in the cleft that are actually modeled differently in a continuous versus a RW model of diffusion in the dyadic cleft; namely, the binding of single Ca2+ ions to single receptors. By doing this, we strip the model of Ca2+ dynamics in the dyadic cleft of many important physiological details that affect the generation and termination of a spark (12,21,32), but the comparison between the actual differences between the two diffusion models become clearer.


Stochastic binding of Ca2+ ions in the dyadic cleft; continuous versus random walk description of diffusion.

Hake J, Lines GT - Biophys. J. (2008)

(A and B) Number of registered binding events from 100 runs each, where we altered different parameters. The data were collected from a receptor 30 nm from the center and are represented by the box-plots together with a 95% confidence interval for the true means (red horizontal lines). The blue solid circles represent the expected number of binding events that are predicted by the continuous model. (A) We scaled the number of Ca2+ ions that enter the cleft, i.e.,  together with the diffusion constant D, with a factor represented by the x axis. The spatial resolution was constant for these simulations, σ = 5 nm. The blue asterisk denotes a statistical difference between the continuous model and the RW model for scale = 0.1. (B) We kept the scale constant at 0.1, but altered the spatial resolution (see the x axis). Here, the difference between the RW model and the continuous model increased as the mean value of the collected binding events declined with the spatial resolution. (C) We ran the simulation 100 times. We collected the mean binding rates for each run that the receptor were exposed to. The data from each set of 100 runs are presented as 95% confidence intervals for the true means. The blue horizontal lines represent the binding rates collected from runs in which we registered binding events, as in panel B. The red horizontal lines represent binding rates collected from runs in which we did not register binding events, only the rate. In these runs we could not differentiate statistically between the registered binding rates and the rates predicted from the continuous model.
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC2480677&req=5

fig12: (A and B) Number of registered binding events from 100 runs each, where we altered different parameters. The data were collected from a receptor 30 nm from the center and are represented by the box-plots together with a 95% confidence interval for the true means (red horizontal lines). The blue solid circles represent the expected number of binding events that are predicted by the continuous model. (A) We scaled the number of Ca2+ ions that enter the cleft, i.e., together with the diffusion constant D, with a factor represented by the x axis. The spatial resolution was constant for these simulations, σ = 5 nm. The blue asterisk denotes a statistical difference between the continuous model and the RW model for scale = 0.1. (B) We kept the scale constant at 0.1, but altered the spatial resolution (see the x axis). Here, the difference between the RW model and the continuous model increased as the mean value of the collected binding events declined with the spatial resolution. (C) We ran the simulation 100 times. We collected the mean binding rates for each run that the receptor were exposed to. The data from each set of 100 runs are presented as 95% confidence intervals for the true means. The blue horizontal lines represent the binding rates collected from runs in which we registered binding events, as in panel B. The red horizontal lines represent binding rates collected from runs in which we did not register binding events, only the rate. In these runs we could not differentiate statistically between the registered binding rates and the rates predicted from the continuous model.
Mentions: For a long time, continuous and deterministic models have been used to study Ca2+ dynamics in the dyadic cleft (21,28–31), and its role in the release of Ca2+. Two recent studies of Ca2+ dynamics use a discrete RW model to describe the Ca2+ diffusion in the cleft (12,32). Koh et al. (12) uses MCell and argues that few Ca2+ ions in a small volume cannot properly be simulated with a continuous model of diffusion. However, they do not present any results that support this claim. Tanskanen et al. (32) present an impressive study that includes physiological details on a microscale level, such as the electrostatic force from the sarcolemmal and the geometrical structures of the large membrane proteins in the cleft, while integrating the Ca2+ release from many clefts, and thus obtaining a measure of the Ca2+ release from the whole cell. In contrast to Koh et al. (12), they explicitly address the difference between their model and an equivalent model that uses a deterministic description of Ca2+ diffusion. They do this by measuring the effect on the excitation-contraction coupling (ECC) gain when they vary the diffusion constant of Ca2+, together with the parameters that determine the influx of Ca2+ ions to the cleft. They show that the ECC gain varies with the parameters (see Fig. 12 in (32)). This result points to a “subtle but potentially significant difference in predicted macroscopic behavior arising from the underlying stochastic simulation of Ca2+ motion in the dyad” (32). The rationale for this statement is that if they had changed the same parameters in an equivalent model using a deterministic description of Ca2+ diffusion, they would not have registered any differences in ECC gain because the receptors situated in the cleft would have experienced the same level of Ca2+ concentration. In our study we examine the discrete events in the cleft that are actually modeled differently in a continuous versus a RW model of diffusion in the dyadic cleft; namely, the binding of single Ca2+ ions to single receptors. By doing this, we strip the model of Ca2+ dynamics in the dyadic cleft of many important physiological details that affect the generation and termination of a spark (12,21,32), but the comparison between the actual differences between the two diffusion models become clearer.

Bottom Line: With these results we demonstrate that the stochasticity and discreteness of the Ca(2+) signaling in the dyadic cleft, determined by single binding events, can be described using a deterministic model of Ca(2+) diffusion together with a stochastic model of the binding events, for a specific range of physiological relevant parameters.Time-consuming RW simulations can thus be avoided.We also present a new analytical model of bimolecular binding probabilities, which we use in the RW simulations and the statistical analysis.

View Article: PubMed Central - PubMed

Affiliation: Simula Research Laboratory, Lysaker, Norway. hake@simula.no

ABSTRACT
Ca(2+) signaling in the dyadic cleft in ventricular myocytes is fundamentally discrete and stochastic. We study the stochastic binding of single Ca(2+) ions to receptors in the cleft using two different models of diffusion: a stochastic and discrete Random Walk (RW) model, and a deterministic continuous model. We investigate whether the latter model, together with a stochastic receptor model, can reproduce binding events registered in fully stochastic RW simulations. By evaluating the continuous model goodness-of-fit for a large range of parameters, we present evidence that it can. Further, we show that the large fluctuations in binding rate observed at the level of single time-steps are integrated and smoothed at the larger timescale of binding events, which explains the continuous model goodness-of-fit. With these results we demonstrate that the stochasticity and discreteness of the Ca(2+) signaling in the dyadic cleft, determined by single binding events, can be described using a deterministic model of Ca(2+) diffusion together with a stochastic model of the binding events, for a specific range of physiological relevant parameters. Time-consuming RW simulations can thus be avoided. We also present a new analytical model of bimolecular binding probabilities, which we use in the RW simulations and the statistical analysis.

Show MeSH
Related in: MedlinePlus