Fluorescence correlation spectroscopy experiments to quantify free diffusion coefficients in reaction-diffusion systems: The case of ${\mathrm{Ca}}^{2+}$ and its dyes

Many cell signaling pathways involve the diffusion of messengers that bind and unbind to and from intracellular components. Quantifying their net transport rate under different conditions then requires having separate estimates of their free diffusion coefficient and binding or unbinding rates. In this paper, we show how performing sets of fluorescence correlation spectroscopy (FCS) experiments under different conditions, it is possible to quantify free diffusion coefficients and on and off rates of reaction-diffusion systems. We develop the theory and present a practical implementation for the case of the universal second messenger, calcium (${\mathrm{Ca}}^{2+}$) and single-wavelength dyes that increase their fluorescence upon ${\mathrm{Ca}}^{2+}$ binding. We validate the approach with experiments performed in aqueous solutions containing ${\mathrm{Ca}}^{2+}$ and Fluo4 dextran (both in its high and low affinity versions). Performing FCS experiments with tetramethylrhodamine-dextran in Xenopus laevis oocytes, we infer the corresponding free diffusion coefficients in the cytosol of these cells. Our approach can be extended to other physiologically relevant reaction-diffusion systems to quantify biophysical parameters that determine the dynamics of various variables of interest.

Figure 1

Average ACF. (a) Typical stationary time-course of fluorescence fluctuations obtained from FCS measurements performed in aqueous solutions with ${\mathrm{Ca}}^{2+}$ and Fluo4 dextran. (b) Prototypical example of an average ACF.

Figure 2

Diffusion coefficients obtained from the fitting of the experimental data using Eq. (5), $D_0$ (open squares), $D_1$ (circles), $D_2$ (triangles), and the sum, $D_1+D_2$ (solid squares), as functions of the total calcium dye concentration of the aqueous solutions, ${\left[F4\right]}_{\text{tot}}$. In solid line, $D_F=85\mu {\text{m}}^2/\text{s}$, and in dash lines expected effective diffusion coefficients, $D_{\text{ef}1}$ (bold) and $D_{\text{ef}2}$ (light) given by Eqs. (10) and (11), respectively, with the calcium and dye concentrations employed in the aqueous solutions, $D_{\text{Ca}}=760\mu {\text{m}}^2/\text{s}, D_F=85\mu {\text{m}}^2/\text{s}$ and the dissociation constant given by the manufacturer, $K_d=772$ nM and 2600 nM for high (a) and low (b) affinity Fluo4.

Figure 3

Parameters derived from the fitting of the experimental ACFs (with symbols) and theoretical expected values (solid curves) computed using the concentrations of Table 1 and the dissociation constants of Table 2. (a, d) Inverse of $g_F$ as function of the total dye concentration used in the solutions, ${\left[F4\right]}_{\text{tot}}$. (b, e) Inverse of the sum of all the weights, $g_{\text{tot}}$, as function of the ${\mathrm{Ca}}^{2+}$-bound dye concentration, $\left[CF4\right]$, computed theoretically setting $\left[CF4\right]={\left[CF\right]}_{\text{eq}}$ and $F_{\text{tot}}={\left[F4\right]}_{\text{tot}}$ in Eq. (4) with the constant, $K_d$, given by the manufacturer (Table 2) and $C_{\text{tot}}^{\left(s\right)}=\left[CF4\right]+C_{\text{eq}}$ with $C_{\text{tot}}^{\left(s\right)}$ and ${\left[F4\right]}_{\text{tot}}$, the total concentrations of Table 1. (c, f) $\nu$ as function of the total dye concentration used in the solutions, ${\left[F4\right]}_{\text{tot}}$. (a, b, c) correspond to experiments with Fluo4 high affinity and (d, e, f) with Fluo4 low affinity.

Figure 4

Parameters of the underlying biophysical model derived from the fitting parameters for each aqueous solution, $D_C, D_F$, and $k_{\text{off}}$ (mean and SEM over two or three experiments with one or two fits) and average of the values obtained for all experiment types in the case of $D_F$ and $k_{\text{off}}$ and only those for which the sum $D_1+D_2$ remains invariant in the case of $D_C$ (solid line), see Table 3. (a, b, c) Fluo4 high affinity and (d, e, f) Fluo4 low affinity. In all cases we include the expected values (dashed line) based on the total concentrations used in the solutions and on the previous estimates of Table 2.

Figure 5

(a) Inverse of the total weight of the ACF computed theoretically using the dissociation constant provided by the vendor, the effective volume of the calibration, $V_{\text{ef}}=0.59\mu {\text{m}}^3$, and the total ${\mathrm{Ca}}^{2+}$ and dye concentrations of the experimental solutions $(V_{\text{ef}}\left[CF4\right]$, solid curve$)$ or the fractions, ${\zeta }_C=0.2$ and ${\zeta }_F=0.5$, respectively, of those total concentrations $(V_{\text{ef}}{\left[CF\right]}_{\text{eq}}$, dashed curve$)$ as functions of $\left[CF4\right]$ for the high affinity dye. (b) Similar to (a) but for the low affinity dye. (c) Ratio between the total weights, $g_{\text{tot}}$ (circles), obtained in experiments with high and low affinity Fluo4 and ratios of the theoretical values, ${\left[CF\right]}_{\text{eq}}$(low)/ ${\left[CF\right]}_{\text{eq}}$(high) (dashed line) and $\left[CF4\right]$(low)/ $\left[CF4\right]$(high) (solid line), displayed in (a) and (b) as functions of the total dye concentration used to prepare the solutions, ${\left[F4\right]}_{\text{tot}}$.

Figure 6

(a) Difference between the full and approximated ACFs obtained using the parameters of Table 2 for Fluo4 high (solid line) and low (dashed line) affinity. (b) Ratios ${\tau }_0/{\tau }_F$ (solid line), ${\tau }_1/{\tau }_{\text{ef}1}$ (dashed line), ${\tau }_2/{\tau }_{\text{ef}2}$ (dotted line), and ${\nu }_F/\nu$ (dashed-dotted line) between fitting and fast reaction approximation times as functions of ${\left[F4\right]}_{\text{tot}}$ for high affinity Fluo4 when all parameters are free to fit in Eq. (5). (c) Similar to (b) but for the ratios between the fitted and the fast reaction weights, $g/g_F$ (solid line), $g_1/g_{\text{ef}1}$ (dashed line), $g_2/g_{\text{ef}2}$ (dotted line), as functions of ${\left[F4\right]}_{\text{tot}}$ for Fluo4 high affinity when the timescales in Eq. (5) are fixed at the fast reaction approximation values.

Figure 7

(a) ACF obtained from FCS experiments performed in X. laevis oocytes microinjected with 37 $\text{nl}$ of TMR-D = $30\mu \text{M}$ (dashed line) fitted by Eq. (2) (solid line). (b) As in (a) for solution of TMR-D = 50 nM. (c) ACFs from the fits performed in (a) and (b) (solid and dashed line, respectively), normalized.