# Fluorescence Correlation Spectroscopy Measurement Based on Fiber Optics for Biological Materials
## 3 Results and Discussion  
### Concentration Dependence  
The concentration dependence of FB-FCS was first confirmed using an organic fluorescent dye, the NIST-traceable fluorescein standard. The fluorescein dilution series was prepared using a dilution buffer (100 mM borate buffer, pH 9.5). The results are shown in Figure 2. The measurement duration was 60 s. The excitation laser power was 1.1 \(\mu\)W at the focal plane of the objective lens. The background count rate was found to be 163.7 kHz using a dilution buffer as the sample. Figure 2a shows the average ACFs measured for each concentration. The ACFs at low concentrations are not shown here to avoid the plot becoming too busy. The excluded plots are shown in Figure A1 in Appendix A. Because the amplitude of the ACF in FCS is inversely proportional to the number of molecules, the amplitudes of the results increased as the concentration decreased. Figure 2b shows the normalized ACFs. The amplitudes of the fluorescent components were normalized to unity based on the number of molecules obtained by the fitting analysis. The decay of each normalized ACF did not change significantly. This was because the diffusion speed was not changed by the concentration because the effect of the concentration of fluorescein on the viscosity can be considered negligible at low concentrations. Figure 2c shows the linear relationship between the fluorescence count rate and the concentration. The red broken line is a regression line for concentrations ranging from 63 nM to 2000 nM (\(y=1.10x-0.72\), \(R^{2}=0.9992\)). At low concentrations, the count rate was too low compared to the value predicted by the linear fitting line. This was likely due to the background correction error because of the long-term (time scale of several ten minutes) fluctuations in the laser power. Figure 2d shows the linear relationship between the number of molecules and concentration. The red broken line is a regression line for concentrations ranging from 63 nM to 2000 nM (\(y=1.19x-1.26\), \(R^{2}=0.9997\)). Good linearity was maintained above 63 nM, at least under these measurement conditions. The non-zero intercept of the regression line likely indicates that the system underestimated the number of fluorescein by 1.26. To measure sparser samples, a more stable excitation light source and elongation of the measurement duration are needed [16]. Figure 2e,f shows the results of the diffusion time and counts per molecule (CPM). The CPM is the brightness per single fluorescent molecule, defined by  
\[\varepsilon=\frac{\overline{I}-\overline{I}_{\text{bg}}}{N}. \tag{3}\]  
Both the diffusion time and CPM must be independent of the concentration. The concentration of samples needed to be more than 63 nM for diffusion time measurement and 250 nM for CPM measurement under these measurement conditions. The lateral radius of measurement volume \(w_{0}\) can be estimated as follows:  
\[w_{0}=\sqrt{4D\tau_{\text{D}}}, \tag{4}\]  
Figure 2: FB-FCS measurements on fluorescein solution. (**a**) Average ACF at different concentrations (\(n=5\)). The symbols show the measurement results. Solid lines are the fitting curves. (**b**) Average normalized ACFs (\(n=5\)). (**c**) Linear relationship between fluorescence count rate and concentration. The background count rate was subtracted. (**d**) Linear relationship between the measured number of molecules and concentration. (**e**) Measured diffusion time at each concentration. (**f**) Measured CPM at each concentration. (**c**,**f**) Error bars show standard error (\(n=5\)). Red broken lines show the results of the linear fitting. where \(D\) is a diffusion coefficient of the sample molecule. Using the known diffusion coefficient of \(D\) = 436 \(\upmu\)m\({}^{2}\)/s [17] and the measured diffusion time of 19.6 \(\upmu\)s for 1000 nM fluorescein, we obtained \(w_{0}\) = 185.0 nm. We can obtain the diffusion coefficient of an unknown sample using \(w_{0}\) and Equation (4).