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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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Simulation results from the continuous model (black lines) and from the Random Walk (RW) model (colored lines). The results represent the average concentrations from the whole cleft. The same simulation setup was used for the two models, including buffers from Table 1. One LCC is open from the start. After 2 ms, one more opens. Then, after 4 ms, both close. The black lines are the results from one simulation of the continuous model. Each line, solid, dashed, and dash-dotted, represents the concentration of Ca2+, mobile buffer, and stationary buffer, respectively. The right y axis shows the scale for the stationary buffer. The colored lines are the mean concentrations from 40 runs of the RW model (red lines), and the concentrations from a single RW simulation (green lines).
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fig7: Simulation results from the continuous model (black lines) and from the Random Walk (RW) model (colored lines). The results represent the average concentrations from the whole cleft. The same simulation setup was used for the two models, including buffers from Table 1. One LCC is open from the start. After 2 ms, one more opens. Then, after 4 ms, both close. The black lines are the results from one simulation of the continuous model. Each line, solid, dashed, and dash-dotted, represents the concentration of Ca2+, mobile buffer, and stationary buffer, respectively. The right y axis shows the scale for the stationary buffer. The colored lines are the mean concentrations from 40 runs of the RW model (red lines), and the concentrations from a single RW simulation (green lines).

Mentions: To confirm that the solution from the continuous model coincided with the mean concentrations from the RW model, we did one run with the continuous model and 40 runs of the RW model, using the same parameters (Fig. 7 for result). The black lines are the concentration in the cleft given by the continuous model of, respectively, Ca2+ (solid line), mobile buffer (dashed line), and stationary buffer (dash-dotted line). The colored lines, partly covered by the black lines, are 1), the concentration results from a single RW simulation (green lines); and 2), the average results from 40 RW runs (red lines). Note that the scale for the stationary buffer traces is given in the right y axis. One LCC was opened at t = 0, to act as a Ca2+ source in the cleft. After ∼1 ms, the steady state, in which most of the stationary buffers were bound to Ca2+, was achieved. After 2 ms, a second LCC was opened. This time, the steady state occurred more quickly, due to the fact that less stationary buffer was available. We see that the [Ca2+] in the single RW run fluctuates a great deal in the steady-state period, but the mean concentration does not. After 4 ms, both LCCs were turned off and the Ca2+ left the cleft quickly. Some Ca2+ remained, due to the unbinding of Ca2+ from the stationary buffer.


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

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

Simulation results from the continuous model (black lines) and from the Random Walk (RW) model (colored lines). The results represent the average concentrations from the whole cleft. The same simulation setup was used for the two models, including buffers from Table 1. One LCC is open from the start. After 2 ms, one more opens. Then, after 4 ms, both close. The black lines are the results from one simulation of the continuous model. Each line, solid, dashed, and dash-dotted, represents the concentration of Ca2+, mobile buffer, and stationary buffer, respectively. The right y axis shows the scale for the stationary buffer. The colored lines are the mean concentrations from 40 runs of the RW model (red lines), and the concentrations from a single RW simulation (green lines).
© Copyright Policy
Related In: Results  -  Collection

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

fig7: Simulation results from the continuous model (black lines) and from the Random Walk (RW) model (colored lines). The results represent the average concentrations from the whole cleft. The same simulation setup was used for the two models, including buffers from Table 1. One LCC is open from the start. After 2 ms, one more opens. Then, after 4 ms, both close. The black lines are the results from one simulation of the continuous model. Each line, solid, dashed, and dash-dotted, represents the concentration of Ca2+, mobile buffer, and stationary buffer, respectively. The right y axis shows the scale for the stationary buffer. The colored lines are the mean concentrations from 40 runs of the RW model (red lines), and the concentrations from a single RW simulation (green lines).
Mentions: To confirm that the solution from the continuous model coincided with the mean concentrations from the RW model, we did one run with the continuous model and 40 runs of the RW model, using the same parameters (Fig. 7 for result). The black lines are the concentration in the cleft given by the continuous model of, respectively, Ca2+ (solid line), mobile buffer (dashed line), and stationary buffer (dash-dotted line). The colored lines, partly covered by the black lines, are 1), the concentration results from a single RW simulation (green lines); and 2), the average results from 40 RW runs (red lines). Note that the scale for the stationary buffer traces is given in the right y axis. One LCC was opened at t = 0, to act as a Ca2+ source in the cleft. After ∼1 ms, the steady state, in which most of the stationary buffers were bound to Ca2+, was achieved. After 2 ms, a second LCC was opened. This time, the steady state occurred more quickly, due to the fact that less stationary buffer was available. We see that the [Ca2+] in the single RW run fluctuates a great deal in the steady-state period, but the mean concentration does not. After 4 ms, both LCCs were turned off and the Ca2+ left the cleft quickly. Some Ca2+ remained, due to the unbinding of Ca2+ from the stationary buffer.

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