# Modeling the Kinetics of the Singlet Oxygen Effect in Aqueous Solutions of Proteins Exposed to Thermal and Laser Radiation

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Algorithm

#### 2.2. Physical Experiment Protocol

## 3. Results

#### 3.1. ROS Reactions in Aqueous Protein Solutions

_{2}+ hν →

^{1}O

_{2},

^{1}Δ

_{g}or

^{1}Σ

_{g}

^{+}, whose position corresponds to wavelengths of 1269 nm and 763 nm [37]. The state

^{1}Σ

_{g}

^{+}is short-lived and quickly relaxes in

^{1}Δ

_{g}. It is important that a transition in collision complexes (dimoles) is possible,

^{3}Σ

_{g}

^{−}+

^{3}Σ

_{g}

^{−}+ hν →

^{1}Δ

_{g}+

^{1}Δ

_{g}, with light absorption at a wavelength of ~630 nm (bandwidth of 18 nm), which allows the use of helium–neon laser radiation with a wavelength of 632.8 nm to generate singlet oxygen [10].

^{−}+

^{1}O

_{2}+ H

^{+}→

^{•}OH + HO

_{2}

^{•},

_{2}

^{•}+ HO

_{2}

^{•}→ H

_{2}O

_{2}+

^{1}O

_{2},

^{•}OH +

^{•}OH → H

_{2}O

_{2},

^{•}OH with a hydrogen atom at the α-carbon atom of the polypeptide chain in the protein molecule P leads to the detachment of the hydrogen atom and the formation of the α-carbon radical [38]:

^{•}OH → P

_{cα}

^{•}+ H

^{•},

_{cα}

^{•}+

^{1}O

_{2}→ P

_{cα}OO

^{•},

_{cα}OO

^{•}+H

^{•}→ 2 P

_{e}+ HO

_{2}

^{•},

^{•}OH +

^{1}O

_{2}→ P

_{cα}OO

^{•}+ H

^{•},

_{cα}OO

^{•}+ P → 2P

_{e}+ H

_{2}O

_{2}+

^{1}O

_{2},

_{e}is the final product of the conversion of LRPS from the original protein.

_{2}O

_{2}+ P → 2 H

_{2}O +

^{1}O

_{2},

#### 3.2. Kinetic Model

_{2}

^{•}radical, which, in turn, leads to the formation of H

_{2}O

_{2}in accordance with (15). In the proposed model, two channels of peroxide decomposition are considered: decomposition due to the catalase activity of the protein (16) and self-decomposition (17). For the decay reactions, a mismatch of the experimentally determined reaction orders with the formal stoichiometric coefficients is typical; these orders can be both integer and fractional [41]. Therefore, the kinetic equations for the concentrations of ROS (HO

_{2}

^{•}and H

_{2}O

_{2}), corresponding to the above system of chemical reactions, can be written in the general form:

#### 3.3. Stationary Dependences of the Hydrogen Peroxide Concentration on the Protein Concentration in Aqueous Solutions of BSA and BGG

_{2}O

_{2}and HO

_{2}

^{•}, we obtain

_{2}

^{•}:

_{2}O

_{2}:

_{2}O

_{2}was experimentally studied under the influence of moderate thermal radiation at 40 °C for 2 h on the BGG and BSA solutions and upon exposure to He–Ne laser radiation for 30 min. The measured dependences of the H

_{2}O

_{2}concentration on the concentration of BGG and BSA are shown in Figure 2 and Figure 3, respectively.

_{2}O

_{2}concentration from the Equation (22), we set α and β equal to 1, which is typical for the decomposition of peroxide under normal conditions. Then, the dependence of the peroxide concentration on the protein concentration takes the form:

#### 3.4. Dynamics of Hydrogen Peroxide Formation in Aqueous Solutions of BSA and BGG

_{2}

^{•}], $y\left(t\right)$ ≡ [H

_{2}O

_{2}]) with free coefficients (A, B, C):

#### 3.5. Chemical Constants

## 4. Discussion

_{2}O

_{2}concentration in BSA and BGG solutions subjected to thermal and laser radiation, we determined the reaction order of the protein-induced decomposition of hydrogen peroxide with respect to the protein and the kinetic constants describing the ROS conversions.

_{2}O

_{2}and HO

_{2}

^{•}concentrations derived in an analytical form containing the kinetic parameters is applicable for describing the ROS dynamics in solutions of various proteins under both types of external exposure.

_{2}O

_{2}concentration on the protein content and time in solutions of the different proteins, both under the action of thermal radiation and the radiation of the He–Ne laser, allowed us to conclude that the fundamental mechanisms of the formation of H

_{2}O

_{2}are identical in these cases. Thus, similar mathematical approaches can be applied to modeling the ROS conversions in various protein systems, including those that are substantially close to living systems.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Scheme of the setup for measuring chemoluminescence of protein solutions: (1) Biotox-7A chemoluminometer; (2) focusing mirror; (3) cell with protein solution; (4) photo shutter; (5) optical filter; (6) photomultiplier tube; (7) personal computer.

**Figure 2.**Variation of H

_{2}O

_{2}concentration with the protein concentration in aqueous solutions exposed to moderate thermal radiation at 40 °C for 2 h: (

**a**) BGG; (

**b**) BSA. The circles are experimentally measured H

_{2}O

_{2}concentrations. The solid lines display theoretical approximation of the experimental points by Levenberg–Marquardt method.

**Figure 3.**Variation of H

_{2}O

_{2}concentration with the protein concentration in aqueous solutions exposed to He–Ne laser radiation (3 mW) for 30 min: (

**a**) BGG; (

**b**) BSA. The circles are experimentally measured H

_{2}O

_{2}concentrations. The solid lines display theoretical approximation of the experimental points by Levenberg–Marquardt method.

**Figure 4.**Dynamics of hydrogen peroxide generation in solutions of proteins exposed to heat (45 °C, 2 h) in phosphate buffer (10 mM Na

_{2}HPO

_{4}and 150 mM NaCl, pH 7.4): (

**a**) BGG (2 μM); (

**b**) BSA (10 μM). The circles are experimentally measured H

_{2}O

_{2}concentrations. The solid lines display theoretical approximation of the experimental points by Levenberg–Marquardt method.

**Figure 5.**Dynamics of hydrogen peroxide generation in solutions of proteins exposed to laser radiation He–Ne laser (3 mW) for 15 min in phosphate buffer (10 mM Na

_{2}HPO

_{4}and 150 mM NaCl, pH 7.4): (

**a**) BGG (2 μM); (

**b**) BSA (10 μM). The circles are experimentally measured H

_{2}O

_{2}concentrations. The solid lines display theoretical approximation of the experimental points by Levenberg–Marquardt method.

Protein | Exposure | a (L^{2}·Micromole^{−2}) | b | γ |
---|---|---|---|---|

BGG | Thermal (Figure 2a) | 5.17 | 23.33 | 2.45 |

Laser (Figure 3a) | 36.45 | 315.14 | 2.23 | |

BSA | Thermal (Figure 2b) | 3.32 | 76.52 | 1.81 |

Laser (Figure 3b) | 69.84 | 125.78 | 1.43 |

Protein | Exposure | $\mathit{\xi}$ (h ^{−1}) | ${\mathit{\eta}}_{0}$ | $\mathit{C}$ (h ^{−1}) | ${\mathit{y}}_{0}$ (nmole·L ^{−1}) | $\mathit{A}$ (nmole·L ^{−1}·h^{−1}) |
---|---|---|---|---|---|---|

BGG | Thermal (Figure 4a) | 0.066 | 0.136 | 0.426 | 0.028 | 1.38·10^{−3} |

Laser (Figure 5a) | 0.096 | 0.148 | 0.939 | 0.00023 | 4.86·10^{−4} | |

BSA | Thermal (Figure 4b) | 0.029 | 0.163 | 0.354 | 0.033 | 1.34·10^{−3} |

Laser (Figure 5b) | 0.044 | 0.295 | 0.396 | 0.00041 | 9.77·10^{−4} |

Protein | Exposure | ${\mathit{k}}_{1}^{\mathbf{\left(}\mathit{e}\mathit{f}\mathit{f}\mathbf{\right)}}\phantom{\rule{0ex}{0ex}}\mathbf{\left(}{\mathbf{s}}^{-1}\mathbf{\right)}$ | k_{2}$(\mathbf{L}\cdot {\mathbf{mole}}^{-1}\cdot {\mathbf{s}}^{-1})$ | k_{3}$({\mathbf{L}}^{2}\cdot {\mathbf{mole}}^{-2}\cdot {\mathbf{s}}^{-1})$ | k_{4}$\left({\mathbf{s}}^{-1}\right)$ |
---|---|---|---|---|---|

BGG | Thermal | 4.6⋅10^{−6} | 4.4⋅10^{5} | 5.9⋅10^{6} | 2.7⋅10^{−5} |

Laser | 1.07⋅10^{−6} | 2.6⋅10^{6} | 9.8⋅10^{6} | 8.5⋅10^{−5} | |

BSA | Thermal | 6.8⋅10^{−7} | 0.87⋅10^{5} | 5.6⋅10^{5} | 1.3⋅10^{−5} |

Laser | 1.1⋅10^{−7} | 2.8⋅10^{5} | 1.9⋅10^{6} | 3.5⋅10^{−6} |

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## Share and Cite

**MDPI and ACS Style**

Shkirin, A.V.; Chirikov, S.N.; Suyazov, N.V.; Reut, V.E.; Grigorieva, D.V.; Gorudko, I.V.; Bruskov, V.I.; Gudkov, S.V.
Modeling the Kinetics of the Singlet Oxygen Effect in Aqueous Solutions of Proteins Exposed to Thermal and Laser Radiation. *Mathematics* **2022**, *10*, 4295.
https://doi.org/10.3390/math10224295

**AMA Style**

Shkirin AV, Chirikov SN, Suyazov NV, Reut VE, Grigorieva DV, Gorudko IV, Bruskov VI, Gudkov SV.
Modeling the Kinetics of the Singlet Oxygen Effect in Aqueous Solutions of Proteins Exposed to Thermal and Laser Radiation. *Mathematics*. 2022; 10(22):4295.
https://doi.org/10.3390/math10224295

**Chicago/Turabian Style**

Shkirin, Alexey V., Sergey N. Chirikov, Nikolai V. Suyazov, Veronika E. Reut, Daria V. Grigorieva, Irina V. Gorudko, Vadim I. Bruskov, and Sergey V. Gudkov.
2022. "Modeling the Kinetics of the Singlet Oxygen Effect in Aqueous Solutions of Proteins Exposed to Thermal and Laser Radiation" *Mathematics* 10, no. 22: 4295.
https://doi.org/10.3390/math10224295