A Mathematical Modeling and Statistical Analysis of Phycobiliprotein Fluorescence Decay under Exposure to Excitation Light

Abstract

Photosynthetic phycobiliprotein complexes from Spirulina maxima were purified and fractioned by gel chromatography. A mathematical model was developed for the fractionated phycobiliprotein complexes to successfully represent the fluorescence decay rate under exposure to excitation light. Each fractionated complex had a different ratio of phycobiliproteins, such as allophycocyanin, phycocyanin, or phycoerythrin, but their fluorescence decay trends were determined to statistically have a high similarity. The mathematical model was derived based on mass balance in the sense that the fluorescence of phycobiliprotein complex was linearly dependent on its mass. The model considered both exponentially decreasing (early light-exposure period) and linearly decreasing (later period), and successfully fit the whole period of fluorescence decay trend.

1. Introduction

C-phycocyanin (C-Pc), a photosynthetic pigment protein, has been widely used as a food ingredient [1,2], as well as in cosmetics [3], pharmaceuticals [2,4], and sensors [3], and it has recently been highlighted as a light-harvesting molecule for photocurrent applications [5,6,7]. However, C-Pc loses its original optical properties under the influence of temperature, pH, and exposure to light [8,9,10,11,12]. Some previous reports have also shown that its optical properties are influenced by its aggregation level [13]. In scenarios where the optical characteristics need to be conserved, preservatives can be effective. In these scenarios, basic physicochemical knowledge of C-Pc stability is a prerequisite for successfully working on C-Pc.

The typical optical characteristic of C-Pc is its fluorescence with a specified excitation and emission frequency [14,15,16]. This fluorescence is also destined to be photobleached under multiple exposures to excitation light [17,18]. Therefore, to allow for proper use of the fluorescent signals of C-Pc in an appropriate field, the stability of protein structure and fluorescence signals of C-Pc must be assessed under the exposure to excitation impulses. However, C-Pc exists as a hexamer, trimer, or monomer, and is in complex with allophycocyanin (APC), phycoerythrin (PE), and linker polypeptides, depending on isolation, purification, or solvent conditions, and thus the fluorescence stability must be investigated with respect to C-Pc aggregate [13,19,20].

Mathematical modeling is a powerful approach to ascertain the relationship between fluorescence signals and excitation impulses and elucidates physical properties such as the half-life of decay and reaction rates [21,22,23]. Some reports have employed mathematical modeling to characterize the optical signals of C-Pc, however, they dealt mainly with the absorbance spectrum, not with fluorescence [24,25]. The objective of this work is to develop a mathematical model to represent fluorescence decay of C-Pc. For this work, C-Pc was separated with respect to aggregate level via gel chromatography, and then a mathematical model was developed to understand the fluorescence decay rate with respect to exposure times of the excitation impulse, and the effect of C-Pc aggregate on its fluorescence decay was evaluated using a statistical analysis of the correlation of signals.

2. Materials and Methods

2.1. Materials

Piperazine, sodium chloride, and C-Pc from Spirulina sp. were supplied by Sigma Aldrich, St. Louis, MO, USA. Tris(hydroxymethyl)aminomethane (Tris) and ethylenediaminetetraacetic acid (EDTA) disomium salt dihydrate, and sucrose were purchased from USB corporation, Cleveland, OH, USA. Ammonium sulfate and hydrochloric acid were bought from Daejung Chemical Co., and Samchun Chemical Co., Pyeongtaek-si, South Korea, respectively.

2.2. Extraction and Purification of C-Pc Complex

C-Pc was obtained from Spirulina maxima. To break the cell wall, lysis buffer and 50 mM Piperazine buffer (pH 6) were used for 1 h at 4 °C. The lysis buffer contained 1 M Tris (pH 8), 0.5 M EDTA, and 20% sucrose. Cell debris was removed by centrifugation (Supra 25K, Hanil, South Korea) at 10,000× g, for 15 min, at 4 °C. The precipitate of ammonium sulfate (0 to 30% saturation) was discarded, and the supernatant was precipitated in ammonium sulfate (30 to 50% saturation), following a previous report [26]. Then, the precipitated protein underwent two steps of chromatography, anion exchange, and then size exclusion after dialysis (MEMBRA-CEL MC18 X 100 CLR (SERVA, Heidelberg, Germany). Each chromatography was HiTrap Q HP and HiPrep Sepharcyl S-100 HR (GE Healthcare, Chicago, IL, USA), respectively. All chromatography used on AKTA start purification system and Frac30 fraction collector (GE Healthcare). For anion exchange chromatography, the column was equilibrated with 0.005 M Na-phosphate butter (pH 7) and then washed with the same buffer. Elution was performed by increasing the concentration from 0 to 0.5 M NaCl with a flow rate of 1 mL/min. For size exclusion chromatography, 0.005 M Na-phosphate butter (pH 7) was used as a mobile phase with flow rate 0.5 mL/min. Samples were fractioned to three conical tubes, and called Fraction 1, Fraction 2, and Fraction 3, following several the tubes. The amount of protein in each fraction was measured via the Pierce bicinchoninic acid (BCA) protein kit (Thermo Scientific, Waltham, MA, USA).

2.3. Fluorescence and Absorbance Spectroscopy

Fluorescence intensity was measured using fluorescence chromatography for excitation at 609 nm and emission at 643 nm. The fluorescence spectrum was also measured at 609 nm for excitation with a range of emission from 627 to 750 nm. The absorbance spectrum was obtained with a range from 500 to 700 nm. Fluorescence decay samples were exposed 1000 times, for 100 ms each measure. All measured spectroscopy was used via a microplate reader, that was equipped with a 50 W xenon flash lamp (Varioskan LUX, Thermo Scientific, Waltham, MA, USA).

2.4. Statistical Analyses

All experiments were repeated three times and fluorescence values were averaged with standard deviation. Correlation analysis of fluorescence among fractions was performed with the Pearson correlation analysis function in Minitab 18. The Pearson correlation analysis was calculated by Equation (1).

$$\rho =\frac{{{\displaystyle \sum }}_{i=1}^n\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}{\left(n-1\right)S_x{S}_y}$$

(1)

$\rho$, $S_x, \overline{x}$, $S_y,$$\overline{y}$, and n are the Pearson correlation coefficient, sample mean for the first variable, standard deviation for the first variable, sample mean for the second variable, standard deviation for the second variable, and column length, respectively. The variables were fluorescence intensity of gel-chromatography fractions. p-value for Pearson correlation coefficient was calculated by the t-distribution.

3. Results

3.1. Purification of C-Pc from S. maxima

C-Pc was successfully purified through a serial step of ammonium sulfate precipitation, ion exchange, and then gel chromatography. These three fractions according to the main peaks, as indicated in the chromatogram, were harvested and identified to contain C-Pc as a main component (Figure 1) and the same number of proteins were compared under absorbance and fluorescence spectroscopy (Figure 2a,b). C-Pc and APC have a maximal absorbance at 620 and 650 nm in absorbance spectrum, respectively [16,25,26]. In conformity with this fact, the control C-Pc showed a maximal peak at 619 nm and a shoulder peak at 658 nm in absorbance spectroscopy. Fractions 1, 2, and 3 spectra expressed a shoulder peak at around 659 nm, as well as a maximal peak around 620 nm, similarly with the control C-Pc. The absorbance at 609 nm was less than 0.1, which guaranteed avoiding the internal filter error in the fluorescence measurements (Figure 2a). For fluorescence spectra (Figure 2b), Fraction 1, control C-pc, Fraction 2, and 3 had a maximal emission at 647, 657, 647, and 641 nm, respectively. However, shift peak towards a higher wavenumber was observed in fraction 1. The fluorescence and absorbance spectra verified that the purified protein was C-Pc, but their intensities were different with respect to the ratio of C-Pc and APC, even though the same protein amount was measured for the four samples.

3.2. Time Serial Measurement of Fluorescence Intensity of C-Pc

Fluorescence decay was measured as excitation exposure times increased for the fractionated samples. All the fractionated C-Pc expressed a similarly nonlinear decay trend as exposure times increased (Figure 3), but the fluorescence intensities were each different at each exposure time. The order of fluorescence intensity was matched with those of absorbance and fluorescence spectra as seen in Figure 2.

3.3. Statistical Analysis of the Similarity of the Decay Trend among Fractions

Fluorescence values were discretized per each exposure time. Correlation analysis was performed to determine whether the fluorescence decays of three fractions were similar or not. As seen in Figure 4 and

Table 1. Correlation analysis of fluorescence decay of phycobiliproteins with each different aggregate.

Fraction	Pearson Correlation Coefficient	p-Value
1 vs. 2	0.982	p < 0.001
1 vs. 3	0.993	p < 0.001
2 vs. 3	0.993	p < 0.001

, Fraction 1 showed 0.982 and 0.993 Pearson correlation coefficient with Fraction 2 and 3, respectively, and the value was 0.993 between Fraction 2 and 3. As a result, a high similarity in the decay trend was ascertained among fractions.

3.4. 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:

$$\frac{\mathrm{d}M}{\mathrm{d}t}=-k{M}^n$$

(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

$$\frac{\mathrm{d}I}{\mathrm{d}t}=-k{I}^n$$

(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, 1st, and 2nd 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 1st order reaction, Equation (3) can be transformed to Equation (4), and after applying boundary conditions, a mathematical model was established (Equation (5)).

$$\frac{\mathrm{d}{I}_{relative}}{\mathrm{d}{\tau }_{relative}}=\frac{\mathrm{d}\left[\frac{I-I_{min}}{I_{max}-I_{min}}\right]}{\mathrm{d}\left(\tau \right)}=-k{\tau }_{max}\left[\frac{I-I_{min}}{I_{max}-I_{min}}\right]$$

(4)

$$I_{relative}=\frac{I_{max}}{I_{max}-I_{min}}{e}^{-k{t}_{max}\tau }-\frac{I_{min}}{I_{max}-I_{min}}, k{t}_{max}=\ln \left(\frac{I_{max}}{I_{min}}\right)$$

(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.

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)).

$$\frac{\mathrm{d}I}{\mathrm{d}t}=-kI+a constant a\ge 0$$

(6)

$$I=\left(I_{max}-\frac{a}{k}\right)e^{-kt}+\frac{a}{k}$$

(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).

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 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. Several fluorescence preservatives have been developed, but their efficacy could be correctly evaluated under the condition that C-Pc fluorescence is measured exactly. The modeling and statistical approaches were meaningful as they provided a guideline for this purpose.

5. Conclusions

The fluorescence decay rate of C-Pc complex under serial exposure times of fluorescence light has been measured and modeled according to the aggregate level. In this work, C-Pc complex from S. maxima was purified and fractionized by using a serial chromatography of ion and gel. The fluorescence decay followed a nonlinear shape in all fractions, but a general mathematical model was developed to represent their decay trend.