## 3 Results
### Mathematical Modeling of the Fluorescence Intensity of C-Pc  
Looking at the correlation results, three fluorescence decays expressed a liner correlation among them with high similarity. However, the decay reaction according to exposure times could be slightly different. Mathematical modeling was performed for the fluorescence of C-Pc. In a closed system exposed to fluorescent light, if C-Pc lost its fluorescence, the mass of the C-Pc could be lost, since it is no longer C-Pc in its intact form. Based on this viewpoint, C-Pc mass follows the \"mass conservation rule\" when its fluorescence decays as follows:  
\[\frac{\mathrm{d}M}{\mathrm{d}t}=-kM^{n} \tag{2}\]  
where \(M\), \(t\), \(k\), and \(n\) represent the mass, exposure time, decay constant, and reaction order of C-Pc, respectively. However, as can be seen in Figure 5, the fluorescence intensity had a linear relationship with C-Pc mass, and thus Equation (2) can be transformed into  
\[\frac{\mathrm{d}I}{\mathrm{d}t}=-kI^{n} \tag{3}\]  
where \(I\) is the fluorescence intensity of C-Pc exposed to light. Applying this model to the experimental results, no successful model was constructed for 0, 1 st, and 2 nd order of reaction (data not shown). In order to try another regression model to fit the experimental data, the fluorescence experimental data were normalized. In case of the 1 st order reaction, Equation (3) can be transformed to Equation (4), and after applying boundary conditions, a mathematical model was established (Equation (5)).  
\[\frac{\mathrm{d}I_{relative}}{\mathrm{d}\tau_{relative}}=\frac{\mathrm{d} \left[\frac{I-I_{min}}{I_{max}-I_{min}}\right]}{\mathrm{d}(\tau)}=-k\tau_{max} \left[\frac{I-I_{min}}{I_{max}-I_{min}}\right] \tag{4}\]  
\[I_{relative}=\frac{I_{max}}{I_{max}-I_{min}}e^{-k_{max}\tau}-\frac{I_{min}}{I_{ max}-I_{min}},\;k_{max}=\ln\left(\frac{I_{max}}{I_{min}}\right) \tag{5}\]  
where \(k_{relative}\), \(I_{max}\), and \(I_{min}\) are decay constant (s\({}^{-1}\)), maximal and minimal fluorescence of each fraction, respectively. Normalized fluorescence of control C-Pc was fitted with Equation (5), leaving a regression model as seen in Figure 6. Even though the regression model seems to be successfully fit the normalized fluorescence data, the theoretical and expected \(kt_{max}\) are 0.2400 and 3.0855, respectively. Therefore, the regression model Equation (5) was determined not to be appropriate for representing the decay pattern of fluorescence of C-Pc.  
Figure 5: Fluorescence of phycobiliproteins with respect to their concentration correlation of fluorescence intensity with concentration of phycobiliproteins; (a) Control C-Pc; (b) Fraction 1; (c) Fraction 2; (d) Fraction 3.  
Another differential model was suggested to represent both the exponentially decreasing (early period) and linearly decreasing (later period) pattern of fluorescence (Equation (6)), and its algebraic model was also formulated (Equation (7)).  
\[\frac{\mathrm{d}I}{\mathrm{d}t}=-kI+a\ constant\ a\geq 0 \tag{6}\]  
\[I=\Big{(}I_{max}-\frac{a}{k}\Big{)}e^{-kt}+\frac{a}{k} \tag{7}\]  
Original fluorescence intensities of all four samples were examined against Equation (7). As a result, the regression model successfully fit all the four experimental data not in normalized but in intact form (Figure 7).