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This study employs a combination of mathematical derivation and optimization technique to investigate the adsorption of drug molecules on nanocarriers. Specifically, the chemotherapy drugs, fluorouracil, proflavine, and methylene blue, are non-covalently bonded with either a flat graphene sheet or a spherical C 60 fullerene. Mathematical expressions for the interaction energy between an atom and graphene, as well as between an atom and C 60 fullerene, are derived. Subsequently, a discrete summation is evaluated for all atoms on the drug molecule utilizing the U-NSGA-III algorithm. The stable configurations' three-dimensional architectures are presented, accompanied by numerical values for crucial parameters. The results indicate that the nanocarrier's structure effectively accommodates the atoms on the drug's carbon planes. The three drug types' molecules disperse across the graphene surface, whereas only fluorouracil spreads on the C 60 surface; proflavine and methylene blue stack vertically to form a layer. Furthermore, all atomic positions of equilibrium configurations for all systems are obtained. This hybrid method, integrating analytical expressions and an optimization process, significantly reduces computational time, representing an initial step in studying the binding of drug molecules on nanocarriers., (© 2024. The Author(s).)

The catalytic effect of graphene on the corannulene bowl-to-bowl inversion is confirmed in this paper using a pair-wise dispersion interaction model. In particular, a continuum approach together with the Lennard-Jones potential are adopted to determine the interaction energy between corannulene and graphene. These results are consistent with previous quantum chemical studies, which showed that a graphene sheet reduces the barrier height for the bowl-to-bowl inversion in corannulene. However, the results presented here demonstrate, for the first time, that the catalytic activity of graphene can be reproduced using pair-wise dispersion interactions alone. This demonstrates the major role that pair-wise dispersion interactions play in the catalytic activity of graphene., Competing Interests: There are no conflicts of interest to declare., (This journal is © The Royal Society of Chemistry.)

Carbon nanotubes can be used as ultrafast liquid transporters for water purification and drug delivery applications. In this study, we mathematically model the interaction between water clusters and carbon nanotubes using a continuum approach with the Lennard-Jones potential. Since the structure of water clusters depends on the confining material, this paper models the cluster as a cylindrical column of water molecules located inside a carbon nanotube. By assuming the system of two concentric cylinders, we derive analytical expressions for the interaction energy and force, which are used to determine the mechanics and physical parameters that optimize water transport in the nanotubes. Additionally, we adopt Verlet algorithm to investigate the ultrahigh-speed dynamics of water clusters inside carbon nanotubes. For a given carbon nanotube, we find that the cluster's length and the surface's wettability are important factors in controlling the dynamics of water transport. Our findings here demonstrate the possibility of using carbon nanotubes as effective nanopumps in water purification and nanomedical devices., Competing Interests: The authors declare no competing financial interest., (© 2023 The Authors. Published by American Chemical Society.)

Nanoparticles in drug delivery have been widely studied and have become a potential technique for cancer treatment. Doxorubicin (DOX) and carbon graphene are candidates as a drug and a nanocarrier, respectively, and they can be modified or decorated by other molecular functions to obtain more controllable and stable systems. A number of researchers focus on investigating the energy, atomic distance, bond length, system formation and their properties using density function theory and molecular dynamic simulation. In this study, we propose metaheuristic optimization algorithms, NSGA-II and U-NSGA-III, to find the interaction energy between DOXH molecules and pristine graphene in three systems: (i) interacting between two DOXHs, (ii) one DOXH interacting with graphene and (iii) two DOXHs interacting with graphene. The result shows that the position of the carbon ring plane of DOXH is noticeably a key factor of stability. In the first system, there are three possible, stable configurations where their carbon ring planes are oppositely parallel, overlapping and perpendicular. In the second system, the most stable configuration is the parallel form between the DOXH carbon ring plane and graphene, and the spacing distance from the closest atom on the DOXH to the graphene is 2.57 Å. In the last system, two stable configurations are formed, where carbon ring planes from the two DOXHs lie either in the opposite direction or in the same direction and are parallel to the graphene sheet. All numerical results show good agreement with other studies.

Targeted drug delivery provides a possible method for the transfer of drug molecules into cancer cells. Liposomes together with a drug, such as Doxorubicin (DOX) inside the liposomes, can be formed as a nano-capsule. In this study, we are interested in finding a favorable size of liposome and an appropriate shape of DOX cluster: sphere, cylinder or ellipsoid. Using mathematical modeling, the interaction energy of the system is obtained from the Lennard-Jones potential and the continuum assumption which assumes that discrete atomic structure can be replaced by an average atomic density spread over a surface. The numerical results show that the spherical shape gives the lowest energy at the equilibrium configuration amongst the three shapes. In the case of equivalent surface areas, the spherical shape gives the energy lower than -4,000 kJ/mol at the equilibrium while the energies for the other cases do not come close to this level. Further in the case of a liposome of 50 nm in radius, the sphere of radius 49.726 nm, equivalent to 31,072 nm(2) surface area, gives the minimum energy at -6,642 kJ/mol. However, an equivalent cylindrical shape is not possible due to geometric constraints. The lowest minimum energy for the ellipsoid occurs for equal major and minor axes, namely for the spherical case. The results presented here are a first step in the design and implementation of a drug molecule for a targeted drug delivery system.

Nanoparticles may be taken up into cells via endocytotic processes whereby the foreign particles are encapsulated in vesicles formed by lipid bilayers. After uptake into these endocytic vesicles, intracellular targeting processes and vesicle fusion might cause transfer of the vesicle cargo into other vesicle types, e.g., early or late endosomes, lysosomes, or others. In addition, nanoparticles might be taken up as single particles or larger agglomerates and the agglomeration state of the particles might change during vesicle processing. In this study, liposomes are regarded as simple models for intracellular vesicles. We compared the energetic balance between two liposomes encapsulating each a single silica nanoparticle and a large liposome containing two silica nanoparticles. Analytical expressions were derived that show how the energy of the system depends on the particle size and the distance between the particles. We found that the electrostatic contributions to the total energy of the system are negligibly small. In contrast, the van der Waals term strongly favors arrangements where the liposome snugly fits around the nanoparticle(s). Thus the two separated small liposomes have a more favorable energy than a larger liposome encapsulating two nanoparticles.

Due to the large number of possible applications of nanoparticles in cosmetic and medical products, the possible hazards of nanoparticles in the human body are a major concern. A worst-case scenario is that nanoparticles might cause health issues such as skin damage or even induce cancer. As a first step to study the toxicity of nanoparticles, we investigate the energy behaviour of a C(60) fullerene interacting with a lipid bilayer. Using the 6-12 Lennard-Jones potential function and the continuous approximation, the equilibrium spacing between the two layers of a bilayer is predicted to be 3.36 Å. On assuming that there is a circular hole in the lipid bilayer, a relation for the molecular interaction energy is determined, involving the circular radius b of the hole and the perpendicular distance Z of the spherical fullerene from the hole. A graph of the minimum energy location Z ( min ) verses the hole radius b shows that a C(60) fullerene first penetrates through a lipid bilayer when b > 6.81 Å, and shows a simple circular relation [Formula: see text] for Z ( min ) positive and b ≤ 6.81 Å. For b > 6.81, the fullerene relocates from the surface of the bilayer to the interior, and as the hole radius increases further it moves to the centre of the bilayer and remains there for increasing hole radii. Accordingly, our modelling indicates that at least for the system with no external forces, the C(60) fullerene will not penetrate through the lipid bilayer but rather remains encased between the two layers at the mid-plane location.

Experimental and predicted flow rates through carbon nanotubes vary considerably but generally are reported to be well in excess of that predicted by the conventional Poiseuille flow, and therefore nanotubes embedded in a matrix might provide membranes with exceptional mass transport properties. In this paper, applied mathematical modelling is undertaken to estimate the three forces acting on a nanotube bundle, namely the molecular interaction force, the viscous force, and the static pressure force. In deducing estimates of these forces we introduce a modification of the notion of the effective dead area for a carbon nanotube membrane, and we calculate the total forces necessary to push one or more of the nanotubes out of the bundle, thus creating a channel through which further enhancement of flow may take place. However, careful analysis shows that the nett dislodgement force is entirely independent on the useable flow area, but rather depends only on the total cross-sectional area perpendicular to the flow. This rather surprising result is a consequence of the flow being steady and a balance of the viscous and pressure forces.

For future nanoelectromechanical signalling devices, it is vital to understand how to connect various nanostructures. Since boron nitride nanostructures are believed to be good electronic materials, in this paper we elucidate the classification of defect geometries for combining boron nitride structures. Specifically, we determine possible joining structures between a boron nitride nanotube and a flat sheet of hexagonal boron nitride. Firstly, we determine the appropriate defect configurations on which the tube can be connected, given that the energetically favourable rings for boron nitride structures are rings with an even number of sides. A new formula E = 6+2J relating the number of edges E and the number of joining positions J is established for each defect, and the number of possible distinct defects is related to the so-called necklace and bracelet problems of combinatorial theory. Two least squares approaches, which involve variation in bond length and variation in bond angle, are employed to determine the perpendicular connection of both zigzag and armchair boron nitride nanotubes with a boron nitride sheet. Here, three boron nitride tubes, which are (3, 3), (6, 0) and (9, 0) tubes, are joined with the sheet, and Euler's theorem is used to verify geometrically that the connected structures are sound, and their relationship with the bonded potential energy function approach is discussed. For zigzag tubes (n,0), it is proved that such connections investigated here are possible only for n divisible by 3.