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The new generations of nano-devices successfully apply with great promise as drug carriers in the treatment of different diseases. The proposed model aims to determine the pharmacological targets and evaluate the bio-safety of usefulness of carbon nanotube conjugated with two different antiviral compounds, Acetylcholine and Ravastigmine, for treating Alzheimer disease. We also obtain the medicinal model mathematically to evaluate the interaction energy arising from encapsulation of each antiviral compound inside the single-walled carbon nanotube. Acetylcholine is modelled as two-connected spheres, while Ravastigmine has two possible structures which are an ellipsoid and cylinder, all interacting with the interior wall of single-walled carbon nanotubes with variant radii
*r _{c}* . Our calculations show that the single-walled carbon nanotube of radius

*r*greater than 3.391 Å that will accept both drugs which are quite closer to the recent findings.

_{c}Research in nanobiotechnology has rapidly increased since the development of molecular dynamic simulations (MDSs) in 1980, X-ray crystallography and scanning tunneling microscope in 1982. Nanobiotechnology is an interdisciplinary area combining the molecular biological approach with the micronanotechnology. This combination aims to design and develop new devices by manipulating at the nanoscale their size, shape, distinct properties. It also presents revolutionary opportunities by implying the creation of nano-materials, against the infection, cancer cells and cardiac disorder, designed to interact with targeted sites in the body at sub-cellular scales with a high degree of certainty. This has led the scientific researchers to find the lightest, strongest and most conductive carbon nano-materials that capable of transporting different biomolecules through their surfaces. Carbon nanodevices are a family of very small tubes which are wholly composed of carbon atoms, having diameter measured by nanometre level (one-billionth scale) which is about to ten-thousand times smaller than the human hair, such as peptides, fullerenes, nano-rods, nano-buds, graphenes and cylindrical carbon nanotubes. Carbon nanotubes (CNTs) are selective nanoparticle because of their huge potential, low solubility and toxicity, outstanding properties, maximum loading capability and extraordinary thermal conductivity [

CNTs have attracted a lot of interesting researches since their discovery [

AD is the most common type of dementia, progressive and irreversible brain disorder that causes problems with behavior and slowly destroys thinking skills and memory. Its symptoms often develops gradually and gets worse overtime and classified into three stages; mild, moderate then becomes severe, which can be noticeably seen in the X-ray image shown in

of losing memory, improve quality of daily tasks and prevent Alzheimer’s symptoms from developing. Yang et al. [

The proposed model is designed to investigate the mechanism of encapsulation of two different antiviral compounds, Acetylcholine (used for the early stages) and Ravastigmine (RAV) (used for the moderate and severe stages) with chemical formulas C 7 NH 16 O 2 + and C 14 H 22 N 2 O 2 as shown in _{c}_{ }. Firstly, we obtain the mathematical geometry for each antiviral compound and also evaluate the total energy arising form each antiviral compound encapsulated inside a SWCNT of radius r_{c}_{ }. ACh structure comprised as two-connected sphere with different radii r_{s}_{ }, while RAV modelled as two possible structres; an ellipsoid and cylinder, each interacting with the interior wall of SWCNTs with variant radii r_{c}_{ }. This paper is structured as, in the first section, we briefly outline the significance of usefulness of SWCNTs in many modern applications at nano-scales. Next, we apply the van der Waals force, Lennard-Jones potential and a discrete-continuum approximation to investigate the medical application which describes the possibility of conjugation between a SWCNT and antiviral compounds specialized to treat AD by inhibiting the growth of MTCDH (infected site in the brain). We also determine the magnitude of the potential energy arising from the interaction between antiviral compounds and nanotubes with various radii r_{c}_{ }. Followed by a discussion and analysis our results in section 3. Finally, conclusions and remarks are given in the last section.

Here, we obtain the bio-physical model which describes the adsorption of two different drugs into SWCNTs with variant radii r_{c} as a mathematical model by using van der Waals forces. We use the Lennard-Jones potential and the continuum approach together to model the encapsulation of these drugs inside a SWCNT. Next, we use the Cartesian coordinate ( x , y , z ) as a reference system to model the two interacting molecules, the certain biomolecule and the cylindrical nanotube. We assume that the point at atom P has coordinates ( δ ,0,0 ) and the SWCNT defined as a cylindrical tube parameterized by ( r c c o s θ , r c s i n θ , z ) , where 0 ≤ δ ≤ r c , 0 ≤ r c ≤ 1 , − π < θ ≤ π and − L < z ≤ L as shown in

ρ 2 = ( r c cos θ − δ ) 2 + r c 2 sin 2 θ + z 2 = ( r c − δ ) 2 − 4 δ r c sin 2 ( θ / 2 ) + z 2 . (1)

The Lennard-Jones potential given as

β ( ρ ) = − A ρ 6 + B ρ 12 , (2)

where β ( ρ ) is the potential function, ρ denotes the distance between two molecular structures, and A and B are the attractive and repulsive constants. The physical parameters, A = 4 ε σ 6 and B = 4 ε σ 12 , are calculated by using the empirical combining laws given by given by ε i j = ε i ε j , σ i j = ( σ i + σ j ) / 2 and ζ i j = ζ i ζ j , where ε is the well depth, σ is the van der Waals diameter and ζ is the non-bond energy [

E a = η c ∫ V β ( ρ ) d V = η c ∫ V ( − A I 3 + B I 6 ) d V (3)

where η c is the atomic surface densities of atoms on the nanotube and d V is a typical surface element located on the interacting molecule. The integral I n ( n = 3 , 6 ) is defined by

I n = r c ∫ − ∞ ∞ ∫ − π π 1 [ ( r c − δ ) 2 − 4 δ r sin 2 ( θ / 2 ) + z 2 ] d θ d z , (4)

we may re-write the equation 4 by using the hypergeometric function as

I n = 2 π r c 2 n − 2 B ( n − 1 / 2 , 1 / 2 ) ∑ m = 0 ∞ ( ( n − 1 / 2 ) m δ m m ! r c m ) 2 . (5)

Next, we assume that the atom at point P is within the volume element of each biomolecule. Thus, we can determine the molecular interaction arising from the certain drug by performing the volume integral of E d over the volume of the certain drug, namely

E d = η s ∫ V E a ( δ ) d V = η s η c ∫ V ( − A I 3 ( δ ) + B I 6 ( δ ) ) d V = η s η c ( − A K 3 + B K 6 ) , (6)

where δ is the distance from the nanotube axis to a typical point of the certain biomolecule and η s is the mean volume density of the biomolecule, which depends on the assumed configuration of the interacting biomolecule and K n can be given as

K n = ∫ V I n ( δ ) d V (7)

Here, we assume ACh structure modelled as two-connected spheres, the larger sphere centred at the origin point with radius r s 1 and the samller sphere with radius r s 2 located on the left side of the origin point. Each sphere assumed to be as a spherical shell parameterized ( r s cos θ sin ϕ , r s sin θ sin ϕ , r s cos ϕ ) , where − π < θ ≤ π , 0 ≤ ϕ ≤ π , 0 ≤ r s ≤ 1 and r s is the radius of the spherical shell as shown in

E Sphc-CNT = η c η s ( − A D 3 + B D 6 ) = η c η s ∫ V ( − A J 3 + B J 6 ) d V , (8)

where η c and η s are the atomic volume densities of the cylindrical nanotube and spheroidal molecule, respectively. So, the integral J n ( n = 3 , 6 ) is given by

J n = ∫ − π π ∫ 0 π ∫ 0 1 r s 2 n + 2 r c 2 n + 2 sin 2 n + 2 ϕ d r d ϕ d θ , (9)

by using the relation between the beta and hypergeometric functions, D n can be expressed in terms of

D n = 8 π 2 r s 3 3 r c 2 n − 2 B ( n − 1 / 2 , 1 / 2 ) ∑ m = 0 ∞ ( n − 1 / 2 ) m ( n − 1 / 2 ) m ( 5 / 2 ) m m ! ( r s 2 r c 2 ) m . (10)

To evaluate the total energy arising from the RAV drug interaction with SWCNT of radius r_{c}_{ }, we consider two possible structures as models for RAV molecule which are an ellipsoid and cylinder as shown in

The RAV molecule assumed to be as a spheroidal structure, parameterized by ( a r sin ϕ cos θ , a r sin ϕ sin θ , b r cos ϕ ) , where 0 ≤ r ≤ 1 , − π < θ ≤ π , 0 ≤ ϕ ≤ π , and a and b are the equatorial semi-axis length and polar semi-axis length (along the z-axis) of spheroidal structure, respectively, as shown in

E Sphd-CNT = η c η l ( − A T 3 + B T 6 ) = η c η l ∫ V ( − A W 3 + B W 6 ) d V , (11)

where η l is the mean volume density of the spheroidal molecule, respectively. So, the integral W n ( n = 3 , 6 ) is given by

W n = ∫ − π π ∫ 0 π ∫ 0 1 a 2 n + 2 b r 2 n + 2 sin 2 n + 2 ϕ d r d ϕ d θ . (12)

The Integral T n can be expressed in terms of

T n = 8 π 2 a 2 b 3 r c 2 n − 2 B ( n − 1 / 2 , 1 / 2 ) ∑ m = 0 ∞ ( n − 1 / 2 ) m ( n − 1 / 2 ) m ( 5 / 2 ) m m ! ( a 2 r c 2 ) m . (13)

To obtain and evaluate the interaction energy for each configuration as shown in

Here, we model the RAV molecule modelled as a perfect cylinder located at the origin (centered) with radius a and length L = 2 b as shown in

E Cyld-CNT = η c η d ( − A Y 3 + B Y 6 ) = η c η d ∫ V ( − A G 3 + B G 6 ) d V , (14)

where η d is the mean volume density of the cylindrical molecule. So, the integral G n ( n = 3 , 6 ) is given by

G n = ∫ − L L ∫ − π π ∫ 0 1 a 2 n + 2 r 2 n + 2 d r d θ d z . (15)

So, Y n is given as

Y n = 4 π 2 a 2 L r c 2 n − 2 B ( n − 1 / 2 , 1 / 2 ) ∑ m = 0 ∞ ( n − 1 / 2 ) m ( n − 1 / 2 ) m ( 2 ) m m ! ( a 2 r c 2 ) m . (16)

In this section, we apply Lennard-Jones potential and the discrete-continuum approach to evaluate the interaction energy of each drug interacting inside SWCNTs with variant radii r_{c}_{ }. The non-bond energy, well-depth ε and van der Waals diameter σ are shown in _{c} of CNTs are given in _{c} along the range of z-axis. In this model, we observe the encapsulation of ACh and RAV inside the nanotubes with radius in the range 3.204 Å < r_{c} < 7.551 Å and the minimum energies obtained for both configurations when r_{c} greater than 3.391 Å. The lowest interaction energy for ACh-SWCNT and RAV-SWCNT interactions is obtained when the radius of nanotube in the range 3.86 Å < r_{c} < 4.07 Å as shown in Figures 4-6.

For the three proposed configurations, we note that the both antiviral compounds, ACh-SWCNT and RAV-SWCNT, are repulsive and unstable when r c < 3.325 and r c < 3.391 Å, respectively, and the (9, 2) SWCNT of radius r c = 3.973 Å is the most favorable nanotube followed by the condition where r c = 3.861 , 3.775, 4.615, 3.590, 3.523, 5.523, 6.102, 7.551 and 3.391 Å, respectively. For all interactions, ACh-SWCNT (connected-spheres), RAV-SWCNT (spheroidal) and RAV-SWCNT (cylindrical), are with minimum energies of approximately −0.664, −1.059 and 1.204 kcal/mol, respectively. Furthermore, we can noticeably see that the magnitude of the minimum energy for RAV-SWCNT (an ellipsoid structure) interaction is slightly smaller than that of RAV-SWCNT (perfect cylinder). We also note that the perfect cylinder (RAV) has the

Interaction | ε (Å) | σ (Å) | ζ (Kcal/mol) | Interaction | ε (Å) | σ (Å) | ζ (Kcal/mol) |
---|---|---|---|---|---|---|---|

H-H | 0.74 | 2.886 | 0.044 | O-H | 0.96 | 3.193 | 0.051 |

O-O (sb) | 1.48 | 3.500 | 0.060 | O-O (db) | 1.21 | 3.500 | 0.060 |

N-N | 1.45 | 3.660 | 0.069 | N-H | 1.00 | 3.273 | 0.055 |

C-C (sb) | 1.54 | 3.851 | 0.105 | C-H | 1.09 | 3.368 | 0.068 |

C-C (db) | 1.34 | 3.851 | 0.105 | C-O (sb) | 1.43 | 3.675 | 0.079 |

C-O (db) | 1.20 | 3.675 | 0.079 | C-N | 1.47 | 3.755 | 0.085 |

Radius of CNT (7, 2) | 3.204 Å [ | Radius of CNT (8, 1) | 3.325 Å [ |
---|---|---|---|

Radius of CNT (5, 5) | 3.390 Å [ | Radius of CNT (9, 0) | 3.523 Å [ |

Radius of CNT (8, 2) | 3.591 Å [ | Radius of CNT (7, 4) | 3.775 Å [ |

Radius of CNT (8, 3) | 3.861 Å [ | Radius of CNT (9, 2) | 3.973 Å [ |

Radius of CNT (10, 3) | 4.615 Å [ | Radius of CNT (13, 2) | 5.523 Å [ |

Radius of CNT (9, 9) | 6.102 Å [ | Radius of CNT (14, 8) | 7.551 Å [ |

Radius of the smaller sphere | r s 1 = 2.03 Å | Radius of the larger sphere | r s 2 = 2.52 Å |

Equatorial-axes of the spheroidal molecule | a = 2.54 Å | Polar semi-axes of the spheroidal molecule | b = 6.205 Å |

Length of cylinder molecule | L = 2 b = 12.41 Å | Surface density for the SWCNT | η c = 0.381 Å^{−2} |

Volume density for the larger sphere | η s 1 = 0.2388 Å^{−3} | Volume density for the smaller sphere | η s 2 = 0.2856 Å^{−3} |

Volume density for the spheroidal molecule | η l = 0.1849 Å^{−3} | Volume density for the cylindrical molecule | η d = 0.0617 Å^{−3} |

Interaction | Attractive | Value (Å^{6} kcal/mol) | Repulsive | Value (Å^{12} × 10^{3} kcal/mol) |
---|---|---|---|---|

ACh | A ACh | 23.38 | B ACh | 54.386 |

RAV | A RAV | 23.43 | B RAV | 56.598 |

CNT | A CNT | 17.40 | B CNT | 29.000 |

Spherical shell ( r s 1 = 2.52 ) | A s 1 | 23.52 | B s 1 | 54.825 |

Spherical shell ( r s 2 = 2.03 ) | A s 2 | 23.24 | B s 2 | 53.946 |

An ellipsoid configuration (RAV) | A l | 23.43 | B l | 56.598 |

Cylindrical configuration (RAV) | A c | 23.43 | B c | 56.598 |

maximum binding energy because of its volume 502.804 Å^{3} being larger than that of an ellipsoid structure (RAV) which is 167.601 Å^{3}. This means that smaller size of an ellipsoid ends requires smaller size of nanotube to accommodate the spheroidal shell (RAV) compared with that of cylindrical structure (RAV) despite having similar dimensions. Moreover, we observe that our results consistently agree with the most recent research findings, for example, Dresselhaus et al. [_{c} ≤ 8 Å (8 Å < diameter = 2r_{c} < 16 Å) delivered to the target and infected cells [

In this study, the Lennard-Jones potential and continuum approach are adopted to evaluate the minimum energy for each configuration. The proposed model obtained mathematically by representing each molecule using the rectangular coordinate ( x , y , z ) as a reference system. Through investigation, we find that the SWCNT plays a significant role by increasing the effectiveness of the antiviral compounds against the growth and symptoms of the AD. The SWCNT is a selective tool because of its distinct properties, such as high conductivity and low solubility in aqueous media. It can be concluded that the RAV antiviral compound is more effective against the AD growth, and both antiviral compounds, ACh and RAV, would not be accepted when r c < 3.391 Å. For all possible configurations, we note that the lowest minimum energy obtained when r c = 3.973 Å. Our results are in very good agreement with Yang’s work who has shown that the ACh antiviral compound is successfully carried out and conjugated with SWCNTs with variant radii r c [

The author acknowledges financial support from the Research and Consultation Centre (RCC) at the University of Business and Technology.

This project funded by the Research and Consultation Centre (RCC) at the University of Business and Technology.

Has no conflict of Interest.

This article does not contain any studies with animals performed by any of authors.

Al Garalleh, H. (2018) Modelling of the Usefulness of Carbon Nanotubes as Antiviral Compounds for Treating Alzheimer Disease. Advances in Alzheimer’s Disease, 7, 79-92. https://doi.org/10.4236/aad.2018.73006