Stochastic binding of Ca2+ ions in the dyadic cleft; continuous versus random walk description of diffusion.
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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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Affiliation: Simula Research Laboratory, Lysaker, Norway. hake@simula.no
ABSTRACT
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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. Related in: MedlinePlus |
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Mentions: The binding probability for a Ca2+ ion near to a mobile receptor, i.e., a mobile buffer, was modeled in the same way as for the stationary receptor, with one exception. A mobile buffer moves during a time-step, which leads to a difference in the expected concentration experienced by the buffer from a nearby Ca2+ ion. Instead of evaluating cE at a single point, as for the stationary receptor, we evaluated it for all possible positions, cE(r,t), and weighted these with the probability, pm(r, t), that the buffer was present. For an arbitrary spatial point r, this quantity is(25)\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{Em}}}}^{{\mathrm{p}}}({\mathbf{r}},t)=c_{_{{\mathrm{E}}}}({\mathbf{r}},t){\times}p_{{\mathrm{m}}}({\mathbf{r}},t)=\frac{f_{{\mathrm{c}}}({\mathbf{r}},t)}{Na}f_{{\mathrm{m}}}({\mathbf{r}},t){\Delta}V({\mathbf{r}}),\end{equation*}\end{document}where fc and fm are the values of the probability density for the Ca2+ and the mobile buffer molecule, respectively. The superscript, p, denotes the concentration at a single spatial position. Using angular symmetry, a cylindrical coordinate system was chosen to integrate over all spatial points. The Cartesian coordinate line, z, was placed in line with the two particles (Fig. 2), and the position of the Ca2+ ion defines the origin. The distance between the two particles is ΔS. The result of the integration was the expected Ca2+ concentration experienced by a nearby mobile receptor, at time t, separated by a distance ΔS,(26)\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{gather*}c_{_{{\mathrm{Em}}}}({\Delta}S,t)=\frac{1}{4{\pi}^{2}Na(4\hspace{.167em}D_{{\mathrm{c}}}\hspace{.167em}D_{{\mathrm{b}}}\hspace{.167em}t^{2})^{\displaystyle\frac{3}{2}}} \\{\times}{\int _{-{\infty}}^{{\infty}}}{\int _{0}^{{\infty}}}re^{-\frac{e^{2}+{\mathrm{z}}^{2}}{4{\mathrm{D}}_{{\mathrm{c}}}{\mathrm{t}}}}e^{-\frac{{\mathrm{r}}^{2}+({\Delta}{\mathrm{S}}-{\mathrm{z}})^{2}}{4{\mathrm{D}}_{{\mathrm{b}}}{\mathrm{t}}}}dr\hspace{.167em}dz, \\=\frac{1}{Na(4{\pi}(D_{{\mathrm{c}}}+D_{{\mathrm{b}}})t)^{\displaystyle\frac{3}{2}}}e^{-\frac{{\Delta}{\mathrm{S}}^{2}}{4({\mathrm{D}}_{{\mathrm{c}}}+{\mathrm{D}}_{{\mathrm{b}}}){\mathrm{t}}}}.\end{gather*}\end{document}Here, Dc and Db are the diffusion constants of the Ca2+ ion and the mobile buffer. Notice that this expression is identical to the expected concentration experienced by a stationary receptor, i.e., Eq. 15, with D = Dc + Db. This result made it possible to use Eq. 24 to calculate the binding probability of a Ca2+ ion to a nearby mobile receptor, merely by setting the diffusion constant, D, to the sum of the diffusion constants of the two particles. In Fig. 1 (left panel, dashed line), the expected concentration of a Ca2+ ion experienced by a nearby tentative mobile buffer is plotted. Also in Fig. 1 (right panel, dashed line), the calculated probability of a nearby Ca2+ ion to bind to the same mobile buffer, during a time-step of Δt = 45 ns, with Dc = 105 nm2 ms−1, Db = Dc/2, and k+ = 30 μM s−1, is plotted. |
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Affiliation: Simula Research Laboratory, Lysaker, Norway. hake@simula.no