## 4 Discussion  
The fluorescence stability of _Spirulina_-originated C-Pc gradually decayed as the number of times of exposure increased. Therefore, preservatives for protecting its stability had been actively researched [12, 27, 28], but modeling studies were a prerequisite for a robust use of C-Pc for several applications. However, C-Pc naturally existed complex with APC, PE, and linker polypeptides, and thus isolated C-Pc complex size and formation  
Figure 6: Mathematical modeling of normalized fluorescence decay. Normalized fluorescence intensities were excited at 609 nm and emitted at 643 nm for 1000 times (100 ms/measure). Figure 7: Mathematical modeling of fluorescence decay of phycobiliproteins. depends on the extraction and purification conditions [29; 30; 31]. This work has focused on statistical analysis and mathematical modeling of fluorescence decay of the fractionated C-Pc from _S. maxima_ so that we could discern a difference in stability among each differently aggregated C-Pc. For this purpose, gel chromatography was used to separate C-Pc complex according to its size, using a principle that larger-sized protein was fractioned earlier through the column (Figure 1). Three different fractions were harvested and confirmed to contain C-Pc, by measuring the specified fluorescence excitation/emission light and UV absorbance (Figure 2). As shown in Figure 2, a slightly different spectrum was observed from fraction to fraction, which was caused by each different aggregate composed of C-Pc, APC, PE, or linker polypeptide. Fraction 1 was thought to exist in the largest phycobilisome in the APC fraction that was the highest; Fraction 2 was in the most similar composition with the control C-Pc, but Fraction 3 was presumed to be C-Pc complex with polypeptides other APC or PE, since the fluorescence intensity was the lowest. Owing to doubts about the significance of the difference in fluorescence decay among the three fractions, statistical correlation analysis was performed to discern whether similarity exists among fractions. As illustrated in the results, the decay trend according to exposure times showed a high similarity (Figure 4 and Table 1). The similar decay trend indicated that the aggregate level did not have a critical effect on the fluorescence of C-Pc, implying that the trend of fluorescence decay of C-Pc was not influenced by the existence of APC, PE, and linker polypeptide, or the number of C-Pc in the aggregate. Mass-balance-based modeling was a traditional but powerful method since the principle is based on the \"mass conservation rule [32; 33]\". However, to construct a model for fluorescence intensity in view of mass balance, both values must first be linearly correlated. Then, fluorescence could be considered as a constant times C-Pc mass. In this sense, the correlation study was first performed, and both were found to have a linear correlation (Figure 5). Then, as illustrated in the derivation processes of Equation (2) through (5), a mathematical model was successfully derived but did not represent the experimental results with low accuracy. This is due to the mathematical model focusing only on the exponentially decaying fluorescence (early period), and not considering the linearly decreasing fluorescence (later period). This intuition is proven to be reasonable since Equation (7) successfully fits all the experimental data in its intact form. This work was meaningful in two points. The fluorescence decay of C-Pc was found to be rarely influenced by phycobiliprotein composition and aggregation level, and a mathematical model was developed to successfully represent the decaying pattern. The C-Pc aggregate level or stability depended on the process environment. The results in this study were not necessarily applicable to all operating conditions, but the modeling and statistical approaches were valuable since they could show us quantitative trends and the comparison of C-Pc stability among fractions.