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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) Lumped binding rates for each time-step, registered from one RyR during a single Random Walk simulation. In the simulation, one constantly open LCC channel was used and the RyR was positioned 10 nm from the center of the cleft. (B) Enlargement of panel A for t = [0, 0.01] ms. The mean binding rate fluctuates a lot for each time-step. (C) Filtered version of the binding rate. A Gaussian kernel with σ = 0.26 ms, corresponding to the scale of the registered IEIs, was chosen for the filtering. (A–C) Corresponding binding probabilities are given by the right y axis. For the ith time-step, this quantity is computed by .
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fig11: (A and B) Lumped binding rates for each time-step, registered from one RyR during a single Random Walk simulation. In the simulation, one constantly open LCC channel was used and the RyR was positioned 10 nm from the center of the cleft. (B) Enlargement of panel A for t = [0, 0.01] ms. The mean binding rate fluctuates a lot for each time-step. (C) Filtered version of the binding rate. A Gaussian kernel with σ = 0.26 ms, corresponding to the scale of the registered IEIs, was chosen for the filtering. (A–C) Corresponding binding probabilities are given by the right y axis. For the ith time-step, this quantity is computed by .

Mentions: During one simulation, we registered at each time-step. These values are plotted against time in Fig. 11 A. The right y axis gives the corresponding binding probabilities. The stochastic and discrete nature of the rates may be seen clearly in these chaotic data. The rate varies from time-step to time-step, as shown in the enlargement of the figure for t = [0, 0.01] ms, shown in Fig. 11 B. The mean rate registered for the whole run was ms−1. In 80% of the time-steps, the rate was smaller than this value, and in 11% of the time-steps, the rate equaled zero. In only 4.1% of the time-steps was the rate >10 ms−1 and the maximal registered rate for this run was 414 ms−1. These rates seem large but the resulting binding probabilities, were, as seen in the right y axis, all ≪1. We used the same size of time-step as earlier, Δt = 1.25 × 10−4 ms. The binding probability that corresponded with the mean rate for the whole run was 2.4 × 10−4. To be able to take the average of the binding rates over several time-steps, it has to make sense to take the sum of several binding rates. This measure is justified by the small binding probabilities that each receptor experiences every time-step (Fig. 11, A and B). The crucial issue was how the average binding rate fluctuates on a larger timescale, i.e., do the large variations in binding rates in each time-step average-out at a larger timescale and if so, how small can this timescale be?


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) Lumped binding rates for each time-step, registered from one RyR during a single Random Walk simulation. In the simulation, one constantly open LCC channel was used and the RyR was positioned 10 nm from the center of the cleft. (B) Enlargement of panel A for t = [0, 0.01] ms. The mean binding rate fluctuates a lot for each time-step. (C) Filtered version of the binding rate. A Gaussian kernel with σ = 0.26 ms, corresponding to the scale of the registered IEIs, was chosen for the filtering. (A–C) Corresponding binding probabilities are given by the right y axis. For the ith time-step, this quantity is computed by .
© Copyright Policy
Related In: Results  -  Collection

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

fig11: (A and B) Lumped binding rates for each time-step, registered from one RyR during a single Random Walk simulation. In the simulation, one constantly open LCC channel was used and the RyR was positioned 10 nm from the center of the cleft. (B) Enlargement of panel A for t = [0, 0.01] ms. The mean binding rate fluctuates a lot for each time-step. (C) Filtered version of the binding rate. A Gaussian kernel with σ = 0.26 ms, corresponding to the scale of the registered IEIs, was chosen for the filtering. (A–C) Corresponding binding probabilities are given by the right y axis. For the ith time-step, this quantity is computed by .
Mentions: During one simulation, we registered at each time-step. These values are plotted against time in Fig. 11 A. The right y axis gives the corresponding binding probabilities. The stochastic and discrete nature of the rates may be seen clearly in these chaotic data. The rate varies from time-step to time-step, as shown in the enlargement of the figure for t = [0, 0.01] ms, shown in Fig. 11 B. The mean rate registered for the whole run was ms−1. In 80% of the time-steps, the rate was smaller than this value, and in 11% of the time-steps, the rate equaled zero. In only 4.1% of the time-steps was the rate >10 ms−1 and the maximal registered rate for this run was 414 ms−1. These rates seem large but the resulting binding probabilities, were, as seen in the right y axis, all ≪1. We used the same size of time-step as earlier, Δt = 1.25 × 10−4 ms. The binding probability that corresponded with the mean rate for the whole run was 2.4 × 10−4. To be able to take the average of the binding rates over several time-steps, it has to make sense to take the sum of several binding rates. This measure is justified by the small binding probabilities that each receptor experiences every time-step (Fig. 11, A and B). The crucial issue was how the average binding rate fluctuates on a larger timescale, i.e., do the large variations in binding rates in each time-step average-out at a larger timescale and if so, how small can this timescale be?

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