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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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(Left panel) Expected [Ca2+], as given by Eq. 15, experienced by a receptor situated a distance ΔS from it, at t = 45 ns. The diffusion constant of Ca2+ is Dc = 105nm2 ms−1. The solid line represents the [Ca2+] experienced by a stationary receptor and the dashed line represents the [Ca2+] experienced by a mobile receptor, with Db = Dc/2, as given by Eq. 26. The right panel shows the corresponding probabilities that a Ca2+ ion will bind to a stationary receptor (solid line) and to a mobile receptor (dashed line), as given by Eq. 24, where Δt = 45 ns and k+ = 30 μM−1 s−1. The probabilities are plotted against the distance between the Ca2+ ion and the receptor. Note the logarithmic scale used for ΔS in the right panel.
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fig1: (Left panel) Expected [Ca2+], as given by Eq. 15, experienced by a receptor situated a distance ΔS from it, at t = 45 ns. The diffusion constant of Ca2+ is Dc = 105nm2 ms−1. The solid line represents the [Ca2+] experienced by a stationary receptor and the dashed line represents the [Ca2+] experienced by a mobile receptor, with Db = Dc/2, as given by Eq. 26. The right panel shows the corresponding probabilities that a Ca2+ ion will bind to a stationary receptor (solid line) and to a mobile receptor (dashed line), as given by Eq. 24, where Δt = 45 ns and k+ = 30 μM−1 s−1. The probabilities are plotted against the distance between the Ca2+ ion and the receptor. Note the logarithmic scale used for ΔS in the right panel.

Mentions: We derived the concept of expected concentration by dividing the entire spatial domain that surrounds the diffusive ligand into N equally spaced shells. Each shell had a volume of where ΔSi = i δs, δs ∝ 1/N, and i = 1…N. Fixing the time to t < Δt, we sampled the position of the diffusive ligand K times. Let Ni be the number of times the ligand occurred in the shell at ΔSi. Dividing this by K, we obtained the averaged number of times the ligand occurred in the ith shell. Then, the average number density of the particle in the same shell is given by(13)\documentclass[10pt]{article}\usepackage{amsmath}\usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{pmc}\usepackage[Euler]{upgreek}\pagestyle{empty}\oddsidemargin -1.0in\begin{document}\begin{equation*}{\bar {n}}_{{\mathrm{i}}}=\frac{N_{{\mathrm{i}}}}{K\hspace{.167em}{\Delta}V_{{\mathrm{i}}}}.\end{equation*}\end{document}Dividing this by Avogadro's number, Na, we arrived at the average concentration given in Molar. Given that we were sampling a deterministic probability distribution K times, we used this information to express the expected number of times a particle occurred in the ith shell, after time t:(14)\documentclass[10pt]{article}\usepackage{amsmath}\usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{pmc}\usepackage[Euler]{upgreek}\pagestyle{empty}\oddsidemargin -1.0in\begin{document}\begin{equation*}N_{_{{\mathrm{E}}}{\mathrm{i}}}=K{\times}P({\Delta}S_{{\mathrm{i}}},t)=K{\times}f({\Delta}S,t){\times}{\Delta}V_{{\mathrm{i}}}.\end{equation*}\end{document}Substituting Ni in Eq. 13 with this value, and letting N → ∞, we obtain the expected concentration that this ligand exerts after t ms at distance ΔS,(15)\documentclass[10pt]{article}\usepackage{amsmath}\usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{pmc}\usepackage[Euler]{upgreek}\pagestyle{empty}\oddsidemargin -1.0in\begin{document}\begin{equation*}c_{_{{\mathrm{E}}}}({\Delta}S,t,D)=\frac{1}{Na}f({\Delta}S,t)=\frac{1}{Na(4{\pi}Dt)^{\displaystyle\frac{3}{2}}}e^{-\frac{{\Delta}{\mathrm{S}}^{2}}{4\hspace{.167em}{\mathrm{Dt}}}}.\end{equation*}\end{document}Here we have divided by Avogadro's constant to obtain the concentration in Molar. We see that the cE is directly proportional to the probability distribution in Eq. 11, which makes sense. The expected concentration of a single Ca2+ ion after t = 45 ns, with D = Dc = 105 nm2 ms−1, is plotted against ΔS in Fig. 1 (left panel, solid line).


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

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

(Left panel) Expected [Ca2+], as given by Eq. 15, experienced by a receptor situated a distance ΔS from it, at t = 45 ns. The diffusion constant of Ca2+ is Dc = 105nm2 ms−1. The solid line represents the [Ca2+] experienced by a stationary receptor and the dashed line represents the [Ca2+] experienced by a mobile receptor, with Db = Dc/2, as given by Eq. 26. The right panel shows the corresponding probabilities that a Ca2+ ion will bind to a stationary receptor (solid line) and to a mobile receptor (dashed line), as given by Eq. 24, where Δt = 45 ns and k+ = 30 μM−1 s−1. The probabilities are plotted against the distance between the Ca2+ ion and the receptor. Note the logarithmic scale used for ΔS in the right panel.
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Related In: Results  -  Collection

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

fig1: (Left panel) Expected [Ca2+], as given by Eq. 15, experienced by a receptor situated a distance ΔS from it, at t = 45 ns. The diffusion constant of Ca2+ is Dc = 105nm2 ms−1. The solid line represents the [Ca2+] experienced by a stationary receptor and the dashed line represents the [Ca2+] experienced by a mobile receptor, with Db = Dc/2, as given by Eq. 26. The right panel shows the corresponding probabilities that a Ca2+ ion will bind to a stationary receptor (solid line) and to a mobile receptor (dashed line), as given by Eq. 24, where Δt = 45 ns and k+ = 30 μM−1 s−1. The probabilities are plotted against the distance between the Ca2+ ion and the receptor. Note the logarithmic scale used for ΔS in the right panel.
Mentions: We derived the concept of expected concentration by dividing the entire spatial domain that surrounds the diffusive ligand into N equally spaced shells. Each shell had a volume of where ΔSi = i δs, δs ∝ 1/N, and i = 1…N. Fixing the time to t < Δt, we sampled the position of the diffusive ligand K times. Let Ni be the number of times the ligand occurred in the shell at ΔSi. Dividing this by K, we obtained the averaged number of times the ligand occurred in the ith shell. Then, the average number density of the particle in the same shell is given by(13)\documentclass[10pt]{article}\usepackage{amsmath}\usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{pmc}\usepackage[Euler]{upgreek}\pagestyle{empty}\oddsidemargin -1.0in\begin{document}\begin{equation*}{\bar {n}}_{{\mathrm{i}}}=\frac{N_{{\mathrm{i}}}}{K\hspace{.167em}{\Delta}V_{{\mathrm{i}}}}.\end{equation*}\end{document}Dividing this by Avogadro's number, Na, we arrived at the average concentration given in Molar. Given that we were sampling a deterministic probability distribution K times, we used this information to express the expected number of times a particle occurred in the ith shell, after time t:(14)\documentclass[10pt]{article}\usepackage{amsmath}\usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{pmc}\usepackage[Euler]{upgreek}\pagestyle{empty}\oddsidemargin -1.0in\begin{document}\begin{equation*}N_{_{{\mathrm{E}}}{\mathrm{i}}}=K{\times}P({\Delta}S_{{\mathrm{i}}},t)=K{\times}f({\Delta}S,t){\times}{\Delta}V_{{\mathrm{i}}}.\end{equation*}\end{document}Substituting Ni in Eq. 13 with this value, and letting N → ∞, we obtain the expected concentration that this ligand exerts after t ms at distance ΔS,(15)\documentclass[10pt]{article}\usepackage{amsmath}\usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{pmc}\usepackage[Euler]{upgreek}\pagestyle{empty}\oddsidemargin -1.0in\begin{document}\begin{equation*}c_{_{{\mathrm{E}}}}({\Delta}S,t,D)=\frac{1}{Na}f({\Delta}S,t)=\frac{1}{Na(4{\pi}Dt)^{\displaystyle\frac{3}{2}}}e^{-\frac{{\Delta}{\mathrm{S}}^{2}}{4\hspace{.167em}{\mathrm{Dt}}}}.\end{equation*}\end{document}Here we have divided by Avogadro's constant to obtain the concentration in Molar. We see that the cE is directly proportional to the probability distribution in Eq. 11, which makes sense. The expected concentration of a single Ca2+ ion after t = 45 ns, with D = Dc = 105 nm2 ms−1, is plotted against ΔS in Fig. 1 (left panel, solid line).

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