# Kinetics of Drug Release from Clay Using Enhanced Sampling Methods

^{*}

## Abstract

**:**

## 1. Introduction

_{f}) [23]. The OPES

_{f}method has been carefully benchmarked in a ligand–protein system, for which there are accurate experimental data directly comparable with the computations [24]. In that study, a calculated residence time of 6.87 × 10

^{2}s

^{−1}was computed, which agrees with the experimental value of 6.00 ± 3.00 × 10

^{2}s

^{−1}. This methodology, with such a level of accuracy, is of high relevance in the pharmaceutical field and it is therefore our objective to use it in the present work. We shall interpret our previous experiments on the praziquantel release from montmorillonite [19,20], quantifying the kinetics of the process.

## 2. Methods

_{f}) [23], which ensure in a relatively simple way that no bias is added to the transition state. We outline here these two methods that differ in the way the bias is constructed. The description of the two methods is considerably simplified if we limit ourselves to describing how to use them to compute the escape times from the bound state, instead of reconstructing the full free energy landscape. We refer to the original literature for a full description of the two methods.

- GAMBES

- OPES

_{f}) that is a modification of OPES designed to avoid depositing bias in the transition region [23,36]. As in GAMBES, the $\mathsf{\u03f5}$ is used to avoid overflowing the basin. Furthermore, the parameter EXCLUDED_REGION can be used to prevent OPES

_{f}from depositing bias in the preassigned region of the configurational space, corresponding to the transition state.

#### 2.1. The Model

#### 2.1.1. The Surface Model System

_{24}(Al

_{76}Mg

_{20})(Si

_{188}Al

_{4})O

_{480}(OH)

_{96}. The drug was positioned on the layer surface, and it was immersed in a bath of 1300 water molecules. The periodicity along the direction z perpendicular to the surface was 45.90 Å (Figure 1).

#### 2.1.2. Model System for the Case of the Interlayer Adsorbed Drug

_{48}(Al

_{152}Mg

_{40})(Si

_{376}Al

_{8})O

_{900}(OH)

_{312}·96H

_{2}O. Both layers are identical, and each interlayer space has 48 waters, that is, 2 waters per sodium according to experiments [19]. In one of the interlayer spaces, the praziquantel molecule was adsorbed and 2881 water molecules were placed outside the clay, as shown in Figure 2. Periodic boundary conditions were applied. The box size was L

_{x}= 30.96, L

_{y}= 128.06, L

_{z}= 30.00, α = 90.00, β = 100.46 and γ = 90.00 (distances in Å and angles in °).

#### 2.2. Computational Details

^{0}). The biased simulations started with the drug at X

^{0}, from which it diffuses before escaping. The static bias $V\left(\mathit{d}\right)$ was constructed to act only on this known state and to drive the drug release process. To limit the bias deposition, the energy cutoff related to the ϵ parameter was 7 kcal/mol. This value allowed the drug release.

_{f}, we used as CV $\mathbf{s}$ the same variable as in GAMBES. We prevented depositing bias in the region y > 8.5 or < $-$8.5 Å (EXCLUDED_REGION). To calculate $\tau $ of different structures and then the diffusion coefficient, the cutoff was 7 kcal/mol when the starting point of the drug molecule was at X

^{0}(Figure 3A), and also when it was between X

^{0}and the edge of the clay (Figure 3B). No bias was required when it was in the edge of the interlayer space (Figure 3C).

_{f}methods give very close results. However, in this application, OPES

_{f}appears to be more efficient. The results presented in the main text are based on OPES

_{f}.

## 3. Results and Discussion

#### 3.1. Drug Adsorbed on the Clay’s Surface

#### 3.2. Drug Adsorbed in the Clay’s Interlayer Space

_{f}was needed when the molecule was inside the clay and therefore we ran two sets of 25 biased simulations from the structures of Figure 3A, B. In the case of the drug positioned at the edge (Figure 3C), observing the release did not require enhanced sampling and we carried out 25 unbiased simulations.

^{0}(Figure 3A), $\tau $ = 200 µs. This τ decreases to 54.4 µs when the molecule starts at a position closer to the edge (Figure 3B). Finally, at the edge, we obtained a τ value of only 5.47 ns (Figure 3C).

^{−11}cm

^{2}s

^{−1}. This value is consistent with previous experimental results on a similar organic molecule (tryptophan) trapped in a clay-based material ($D~$5 × 10

^{−11}cm

^{2}s

^{−1}) [46]. It is five orders of magnitude smaller than the diffusion coefficient of praziquantel dissolved in water [47,48].

^{−11}cm

^{2}s

^{−1}is compatible with such a scenario. The use of a methodology in the present work, that has been demonstrated in protein–ligand studies [24] to be highly accurate, allows us to quantify the diffusion coefficient of the praziquantel release from the interlayer space of montmorillonite as 1.10 × 10

^{−11}cm

^{2}s

^{−1}. It agrees with the fast drug release profiles observed in the experiments and complements the experimental perspective.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Model of praziquantel adsorbed in the interlayer space of montmorillonite in aqueous solution. Here, only some of the waters in y direction are shown. The geometrical descriptor used in the GAMBES and OPES

_{f}simulations is highlighted, corresponding to the y-component of the vector, connecting the center of mass of the drug molecule and a fixed point in the middle of the clay interlayer space (X

^{0}) (see text).

**Figure 3.**Different initial structures with the praziquantel placed in the center (

**A**), between the center and the edge (

**B**), and in the edge (

**C**), of the montmorillonite interlayer space.

**Figure 4.**Desorption of praziquantel from montmorillonite surface in aqueous solution, obtained from standard molecular dynamics simulations.

**Figure 5.**Praziquantel release mechanism from the interlayer space of the montmorillonite in aqueous solution.

**Table 1.**Drug release time ($\tau $ ) and rate (k = 1/$\tau $ ). p-value measures the quality of the fit using the Kolmogorov–Smirnov analysis. We also present the average release time µ and its standard deviation σ.

τ (10^{−12} s)
| k (10^{9} s^{−1}) | p-Value | µ ± σ (10^{−12} s) | |
---|---|---|---|---|

Surface, MD | 363 | 2.76 | 0.76 | 344 ± 218 |

**Table 2.**Drug release time ($\tau $ ) and rate (k = 1/$\tau $ ) for structures A, B and C of Figure 3. p-value measures the quality of the fit using the Kolmogorov–Smirnov analysis. We also present the average release time µ and its standard deviation σ.

$\mathit{\tau}({10}^{-6}\mathbf{s})$ | k (10^{6} s^{−1}) | p-Value | µ ± σ (10^{−6} s) | |
---|---|---|---|---|

A, OPES_{f} | 200.0 | 0.005 | 0.76 | 198.0 ± 176.0 |

B, OPES_{f} | 54.4 | 0.018 | 0.41 | 55.4 ± 64.5 |

C, MD | 0.00547 | 182.8 | 0.42 | 0.00536 ± 0.00393 |

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**MDPI and ACS Style**

Borrego-Sánchez, A.; Debnath, J.; Parrinello, M.
Kinetics of Drug Release from Clay Using Enhanced Sampling Methods. *Pharmaceutics* **2022**, *14*, 2586.
https://doi.org/10.3390/pharmaceutics14122586

**AMA Style**

Borrego-Sánchez A, Debnath J, Parrinello M.
Kinetics of Drug Release from Clay Using Enhanced Sampling Methods. *Pharmaceutics*. 2022; 14(12):2586.
https://doi.org/10.3390/pharmaceutics14122586

**Chicago/Turabian Style**

Borrego-Sánchez, Ana, Jayashrita Debnath, and Michele Parrinello.
2022. "Kinetics of Drug Release from Clay Using Enhanced Sampling Methods" *Pharmaceutics* 14, no. 12: 2586.
https://doi.org/10.3390/pharmaceutics14122586