Quantification of fluctuations from fluorescence correlation spectroscopy experiments in reaction-diffusion systems

Fluorescence correlation spectroscopy (FCS) is commonly used to estimate diffusion and reaction rates. In FCS the fluorescence coming from a small volume is recorded and the autocorrelation function (ACF) of the fluorescence fluctuations is computed. Scaling out the fluctuations due to the emission process, this ACF can be related to the ACF of the fluctuations in the number of observed fluorescent molecules. In this paper the ACF of the molecule number fluctuations is studied theoretically, with no approximations, for a reaction-diffusion system in which the fluorescence changes with binding and unbinding. Theoretical ACFs are usually derived assuming that fluctuations in the number of molecules of one species are instantaneously uncorrelated to those of the others and obey Poisson statistics. Under these assumptions, the ACF derived in this paper is characterized only by the diffusive timescale of the fluorescent species and its total weight is the inverse of the mean number of observed fluorescent molecules. The theory is then scrutinized in view of previous experimental results which, for a similar system, gave a different total weight and correct estimates of other diffusive timescales. The total weight mismatch is corrected by assuming that the variance of the number of fluorescent molecules depends on the variance of the particle numbers of the other species, as in the variance decomposition formula. Including the finite acquisition time in its computation, it is shown that the ACF depends on various timescales of the system and that its total weight coincides with the one obtained with the variance decomposition formula. This calculation implies that diffusion coefficients of nonobservable species can be estimated with FCS experiments performed in reaction-diffusion systems. Ways to proceed in future experiments are also discussed.

Figure 1

Three nontrivial eigenvalue branches as functions of the squared wave number, $q^2$, of the linearized reaction-diffusion equations that correspond to a system with ${\mathrm{Ca}}^{2+}$, dye, and Fluo4 with the corresponding parameters of Table 1, $D_{\mathrm{E}}=405\mu {\mathrm{m}}^2/\text{s}$ and the total ${\mathrm{Ca}}^{2+}$ concentrations: ${\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{tot}}=9300\mu \text{M}$ in (a), ${\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{tot}}=9600\mu \text{M}$ in (b), and ${\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{tot}}=10400\mu \text{M}$ in (c). In the three subfigures ${\lambda }_1$ is shown with solid lines, ${\lambda }_2$ with dashed lines, and ${\lambda }_3$ with dashed-dotted lines. The effective eigenvalues, ${\lambda }_{ef1}, {\lambda }_{ef2}$, and ${\lambda }_{ef3}$, are also plotted with thin blue solid lines.

Figure 2

(a) Effective ($D_{ef1}$: dashed line; $D_{ef2}$: dotted line; $D_{ef3}$: dashed-dotted line) and free (solid lines, $D_{{\mathrm{Ca}}^{2+}}>D_{\mathrm{E}}>D_{\mathrm{F}}$) diffusion coefficients as functions of ${\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{tot}}$ computed for a reaction-diffusion system with ${\mathrm{Ca}}^{2+}$, EGTA, and Fluo4 and the corresponding parameters of Table 1 with $D_{\mathrm{E}}=405\mu {\mathrm{m}}^2/\text{s}$. (b) Similar to (a) but for the effective rates ${\nu }_{ef2}$ (dotted line) and ${\nu }_{ef3}$ (dashed-dotted line). (c) Similar to (a) but for the total weight ($G_{\mathrm{tot}}$: gray solid line) and the weights of the four components of the ACF ($A_1$: solid line; $A_2$: dashed line; $A_3$: dashed-dotted line; $A_4$: dotted line).

Figure 3

Weight densities, $g_i(q,\phi =\pi /2)$, as functions of the squared wave number, $q^2$. The ones that correspond to the three branches of nontrivial eigenvalues of Fig. 1 are shown in (a)–(c) for the same ${\mathrm{Ca}}^{2+}$ concentrations and symbols as in Fig. 1. The density that corresponds to ${\lambda }_4$, the eigenvalue associated to the free diffusion coefficient of the dye, is shown in (d) for the case with ${\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{tot}}=10.4\text{mM}$.

Figure 4

Stochastic simulations of the reaction-diffusion system of ${\mathrm{Ca}}^{2+}$, dye, and EGTA. Deviation from the mean of the number of ${\mathrm{Ca}}^{2+}$-bound dye molecules as a function of the deviation in the number of free dye molecules in the observation volume. (a) All the data points and (b) average over the observation time interval.

Figure 5

Cumulative distribution function (CDF) of the ${\mathrm{Ca}}^{2+}$-bound dye molecules in the observation volume obtained with the stochastic simulation of the full reaction-diffusion system (solid line) and when the probability of binding is assumed to be fixed at $p_b=\left[{\mathrm{Ca}}^{2+}\right]/(\left[{\mathrm{Ca}}^{2+}\right]+K_{d\mathrm{F}})$ (dashed line).

Figure 6

Weight densities of the three components of $G_{{\mathrm{Ca}}^{2+}}$ computed as functions of $q^2$ at $\phi =\pi /2$. Symbols and ${\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{tot}}$ in (a)–(c) are the same as in Fig. 1.

Figure 7

Theoretical and experimental results (with lines and symbols, respectively) for the case of experiments performed in aqueous solutions containing ${\mathrm{Ca}}^{2+}$, EGTA, and Fluo8. The theoretical results were obtained using the corresponding parameters of Table 1 with $D_{\mathrm{E}}=352\mu {\text{m}}^2$. (a) Ratio between the weight associated to the free diffusion coefficient of the dye and the total weight of the ACF derived from the experiments $[G{o}_F/G_{\mathrm{tot}}=A_{1,\mathrm{fit}}/(A_{1,\mathrm{fit}}+A_{1,\mathrm{fit}})$: rhombuses with error bars] and computed as in Fig. 2 ($G{o}_F/G_{\mathrm{tot}}=A_4/G_{\mathrm{tot}}$: circles). (b) Effective diffusion coefficients (dashed line: $D_{ef1}$; dash-dotted line: $D_{ef2}$; dotted line: $D_{ef3}$) and coefficients derived by fitting the experimental ACFs with expressions of the form Eq. (31) for each ${\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{tot}}$ probed, circles for the diffusion coefficient derived from ${\tau }_1$, and triangles for the one derived from ${\tau }_2$. The experimental points correspond to averages over three experiments for almost all values of ${\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{tot}}$, two for ${\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{tot}}=9.37$ and 9.57, and six for ${\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{tot}}=9.66\text{mM}$.