# Synergy of Small Antiviral Molecules on a Black-Phosphorus Nanocarrier: Machine Learning and Quantum Chemical Simulation Insights

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## Abstract

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## 1. Introduction

## 2. Outcomes and Discussion

#### 2.1. Data Description

#### 2.2. Data Analysis

#### 2.3. Prediction Results

#### 2.3.1. Hamiltonian Energy Prediction

#### 2.3.2. Non-Bond Energy Prediction

## 3. Pharmachemical Implications

#### 3.1. Drug Release

#### 3.1.1. The pH Sensitivity

#### 3.1.2. Thermotherapy Properties

## 4. Materials and Methods

#### 4.1. Molecular Dynamics Simulations

#### 4.2. Gaussian Process Regressor

**x**) is considered as a random variable. It is worth pointing out that the uncertainty on f could decrease significantly by the observation of the function’s output at different input points.

#### 4.3. Bayesian Optimization Procedure

#### 4.4. SVR Models

#### 4.5. Bagged Tree Model

#### 4.6. Evaluation Metrics

- RMSE: Root Mean Squared Error measures the differences between predicted and actual values. It is calculated by taking the square root of the average of the squared differences between the predicted and actual values.$$RMSE=\sqrt{\frac{1}{n}\sum _{t=1}^{n}{({y}_{t}-{\widehat{y}}_{t})}^{2}},$$
- MAPE: Mean Absolute Percentage Error is a measure of the accuracy of a model in predicting values. It is calculated as the average of the absolute percentage difference between predicted and actual values.$$MAPE=\frac{100}{n}\sum _{t=1}^{n}\left|\frac{{y}_{t}-{\widehat{y}}_{t}}{{y}_{t}}\right|\%,$$
- MAE: Mean Absolute Error is a measure of the average magnitude of the errors in a set of predictions, without considering their direction. It is calculated as the average of the absolute differences between predicted and actual values.$$MAE=\frac{{\sum}_{t=1}^{n}\left(\right)open="|"\; close="|">{y}_{t}-{\widehat{y}}_{t}}{}n$$
- R-squared: ${R}^{2}$ is a statistical measure that represents the proportion of the variance for a dependent variable that is explained by an independent variable or variables in a regression model. It is also known as the coefficient of determination, and its value ranges between 0 and 1.$${R}^{2}=\frac{{\sum}_{t=1}^{n}{[({y}_{t}-\overline{y})\xb7({\widehat{y}}_{t}-\overline{y})]}^{2}}{\sqrt{{\sum}_{t=1}^{n}{({y}_{t}-\overline{y})}^{2}}\xb7\sqrt{{\sum}_{t=1}^{n}{({\widehat{y}}_{t}-\overline{y})}^{2}}},$$

#### 4.7. Prediction Framework

#### 4.8. Ab Initio Calculations

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Prediction Distribution of prediction errors of the four models based on testing datasets: (

**a**) MLFF-EB, (

**b**) MLFF-FP, and (

**c**) MLFF-FP_EB.

**Figure 4.**Predicted Hamiltonian energy from the GPR model based on test data: (

**Top**) MLFF-EB, (

**Middle**) MLFF-FP, and (

**Bottom**) MLFF-FP_EB.

**Figure 5.**Prediction errors of the prediction models from the testing datasets: (

**a**) MLFF-EB, (

**b**) MLFF-FP, and (

**c**) MLFF-FP_EB.

**Figure 7.**The calculated adsorption energy process as a function of net charge at the solvent phase of the BP sheet with favipiravir, ebselen, and favipiravir + ebselen hybrid drugs. The minus red and plus blue sign colors represent the difference between pH < 0 and pH > 0 values, respectively.

**Figure 8.**The free Gibbs adsorption energy process as a function of temperature at the vacuum and solvent phases of the BP sheet with favipiravir, ebselen, and favipiravir+ebselen hybrid drugs.

**Figure 10.**Illustration of the used prediction framework and general steps in antiviral drug discovery for small molecules on black phosphorus, developing quantitative structure–activity relationship models to optimize lead structures for pharmaceutical implication.

MLFF-FP | ||||
---|---|---|---|---|

Methods | ${R}^{2}$ | RMSE | MAE | MAPE |

BT | 0.9743 | 0.0266 | 0.0217 | 0.0049 |

GPR | 0.9663 | 0.0304 | 0.0238 | 0.0053 |

SVR | 0.9767 | 0.0253 | 0.0208 | 0.0047 |

RT | 0.9674 | 0.0299 | 0.0238 | 0.0053 |

MLFF-EB | ||||

Methods | ${R}^{2}$ | RMSE | MAE | MAPE |

BT | 0.9513 | 0.1001 | 0.0655 | 0.0115 |

GPR | 0.9614 | 0.0891 | 0.0606 | 0.0106 |

SVR | 0.9447 | 0.1067 | 0.0809 | 0.0142 |

RT | 0.9501 | 0.1014 | 0.0654 | 0.0114 |

MLFF-FP_EB | ||||

Methods | ${R}^{2}$ | RMSE | MAE | MAPE |

BT | 0.9489 | 0.1459 | 0.1011 | 0.0179 |

GPR | 0.9669 | 0.1173 | 0.0718 | 0.0127 |

SVR | 0.9665 | 0.1182 | 0.0710 | 0.0125 |

RT | 0.9259 | 0.1756 | 0.1109 | 0.0196 |

${\mathit{R}}^{2}$ | RMSE | MAE | MAPE | |
---|---|---|---|---|

BT | 0.9582 | 0.0909 | 0.0628 | 0.0114 |

GPR | 0.9649 | 0.0789 | 0.0521 | 0.0095 |

SVR | 0.9626 | 0.0834 | 0.0576 | 0.0105 |

RT | 0.9478 | 0.1023 | 0.0667 | 0.0121 |

MLFF-FP | ||||
---|---|---|---|---|

${R}^{2}$ | RMSE | MAE | MAPE | |

BT | 0.9772 | 0.0279 | 0.0218 | 0.2466 |

GPR | 0.9792 | 0.0266 | 0.0208 | 0.2348 |

SVR | 0.9772 | 0.0279 | 0.0211 | 0.2382 |

RT | 0.9775 | 0.0277 | 0.0221 | 0.2502 |

MLFF-EB | ||||

${R}^{2}$ | RMSE | MAE | MAPE | |

BT | 0.9685 | 0.0758 | 0.0493 | 0.4496 |

GPR | 0.9735 | 0.0695 | 0.0468 | 0.4277 |

SVR | 0.9705 | 0.0734 | 0.0492 | 0.4503 |

RT | 0.9729 | 0.0703 | 0.0486 | 0.4448 |

MLFF-FP_EB | ||||

${R}^{2}$ | RMSE | MAE | MAPE | |

BT | 0.9648 | 0.1253 | 0.0843 | 2.5297 |

GPR | 0.9725 | 0.1107 | 0.0732 | 2.2111 |

SVR | 0.9710 | 0.1138 | 0.0771 | 2.3273 |

RT | 0.9689 | 0.1178 | 0.0795 | 2.3512 |

${\mathit{R}}^{2}$ | RMSE | MAE | MAPE | |
---|---|---|---|---|

BT | 0.9701 | 0.0763 | 0.0518 | 1.0753 |

GPR | 0.9751 | 0.0689 | 0.0469 | 0.9579 |

SVR | 0.9729 | 0.0717 | 0.0491 | 1.0053 |

RT | 0.9731 | 0.0719 | 0.0501 | 1.0154 |

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## Share and Cite

**MDPI and ACS Style**

Laref, S.; Harrou, F.; Wang, B.; Sun, Y.; Laref, A.; Laleg-Kirati, T.-M.; Gojobori, T.; Gao, X.
Synergy of Small Antiviral Molecules on a Black-Phosphorus Nanocarrier: Machine Learning and Quantum Chemical Simulation Insights. *Molecules* **2023**, *28*, 3521.
https://doi.org/10.3390/molecules28083521

**AMA Style**

Laref S, Harrou F, Wang B, Sun Y, Laref A, Laleg-Kirati T-M, Gojobori T, Gao X.
Synergy of Small Antiviral Molecules on a Black-Phosphorus Nanocarrier: Machine Learning and Quantum Chemical Simulation Insights. *Molecules*. 2023; 28(8):3521.
https://doi.org/10.3390/molecules28083521

**Chicago/Turabian Style**

Laref, Slimane, Fouzi Harrou, Bin Wang, Ying Sun, Amel Laref, Taous-Meriem Laleg-Kirati, Takashi Gojobori, and Xin Gao.
2023. "Synergy of Small Antiviral Molecules on a Black-Phosphorus Nanocarrier: Machine Learning and Quantum Chemical Simulation Insights" *Molecules* 28, no. 8: 3521.
https://doi.org/10.3390/molecules28083521