## Appendix A  
We assume that there are \(M\) species in the sample solution, and the fluorescence intensity emitted from _the_\(m\)-th species can be expressed as:  
\[I_{m}(t)=\delta I_{m}(t)+\overline{I}_{m}, \tag{1}\]  
\[\delta I_{m}(t)=\varepsilon_{m}\delta n_{m}(t), \tag{2}\]  
\[\overline{I}_{m}=\varepsilon_{m}\overline{n}_{m}, \tag{3}\]  
where the subscript \(m\) denotes the \(m\)-th species. \(\delta I\) and \(\overline{I}\) denote the fluctuation component and the mean fluorescence intensity, respectively. \(\varepsilon\) denotes the brightness per molecule, which is called the CPM. \(\delta n\) and \(\overline{n}\) are the fluctuation component and mean value of the number of molecules inside the confocal volume of the FCS, respectively. Therefore, the total fluorescence intensity is  
\[I(t)=\sum_{m=1}^{M}I_{m}(t). \tag{4}\]  
The ACF of FCS is expressed as follows [5]:  
\[G(\tau)=\frac{I(t)\cdot I(t+\tau)}{\overline{I}^{2}}=1+\frac{\langle\delta I( t)\cdot\delta I(t+\tau)\rangle}{\overline{I}^{2}}, \tag{5}\]\[=1+\sum_{m=1}^{M}\frac{\varepsilon_{m}^{2}\overline{n}_{m}}{\overline{I}^{2}}G_{m }(\tau), \tag{10}\]  
\[A_{m}=\frac{\varepsilon_{m}^{2}\overline{n}_{m}}{\overline{I}^{2}}, \tag{11}\]  
where \(A_{m}\) denotes the amplitude of the ACF for _the m_-th species. \(G_{m}\) denotes the decaying ACF with an amplitude of unity. The angle brackets \(\langle\ \rangle\) represent the ensemble average. For three-dimensional (3D) free diffusion, \(G_{m}\) can be expressed as follows [3]:  
\[G_{m}(\tau)=\bigg{(}1+\frac{\tau}{\tau_{D,m}}\bigg{)}^{-1}\bigg{(}1+\frac{1}{ s^{2}}\frac{\tau}{\tau_{D,m}}\bigg{)}^{-\frac{1}{2}}, \tag{12}\]  
where \(\tau_{D,m}\) denotes the diffusion time of the \(m\)-th species. In FCS measurements, the experimentally obtained ACFs are fitted by the following 3D diffusion model equation based on the non-linear least squares:  
\[G(\tau)=1+\frac{1}{N_{\text{ap}}}\sum_{m=1}^{M}F_{\text{ap},m}G_{m}(\tau), \tag{13}\]  
\[\sum_{m=1}^{M}F_{\text{ap},m}=1, \tag{14}\]  
where \(N_{\text{ap}}\) denotes the apparent total number of all the species and \(F_{\text{ap},m}\) denotes the apparent fraction of the \(m\)-th species. If the CPM of each species is the same, \(N_{\text{ap}}\) and \(F_{\text{ap},m}\) are equal to their true values. The apparent number of \(m\)-th species is given by  
\[\delta I_{m}(t)=\varepsilon_{m}\delta n_{m}(t), \tag{2}\]0  
The apparent CPM is defined by  
\[\delta I_{m}(t)=\varepsilon_{m}\delta n_{m}(t), \tag{2}\]1  
Note that the apparent values \(\varepsilon_{ap}\), \(N_{\text{ap}}\), and \(F_{\text{ap},m}\) are incorrect if the sample contain multiple species with different brightness and should not be used. However, those can be easily obtained by most of available FCS instruments and their software using the most common model equation in FCS shown in Equation (13). The calculation procedure to obtain correct values from such the apparent values is useful to most of FCS users. We introduce here the procedure. For simplicity, we assumed that there are two species, a fluorescently labeled monomer protein and its aggregates, and that the CPM of the monomer protein can be measured by a purified sample. The ACF obtained by an FCS measurement on the sample containing a second species with unknown CPM is fitted by Equation (13), and \(N_{\text{ap}}\) and \(F_{\text{ap}}\) are obtained.