mi> r c , $0\le {r}_{c}\le 1$ , $-\text{π}<\theta \le \text{π}$ and $-L as shown in Figure 2(c). Therefore, the distance $\rho$ from point P to typical surface element of the cylindrical tube given by

${\rho }^{2}={\left({r}_{c}\text{ }\mathrm{cos}\theta -\delta \right)}^{2}+{r}_{c}^{2}{\mathrm{sin}}^{2}\theta +{z}^{2}={\left({r}_{c}-\delta \right)}^{2}-4\delta {r}_{c}{\mathrm{sin}}^{2}\left(\theta /2\right)+{z}^{2}.$ (1)

The Lennard-Jones potential given as

$\beta \left(\rho \right)=\frac{-A}{{\rho }^{6}}+\frac{B}{{\rho }^{12}},$ (2)

where $\beta \left(\rho \right)$ is the potential function, $\rho$ denotes the distance between two molecular structures, and A and B are the attractive and repulsive constants. The physical parameters, $A=4\epsilon {\sigma }^{6}$ and $B=4\epsilon {\sigma }^{12}$ , are calculated by using the empirical combining laws given by given by ${\epsilon }_{ij}=\sqrt{{\epsilon }_{i}{\epsilon }_{j}}$ , ${\sigma }_{ij}=\left({\sigma }_{i}+{\sigma }_{j}\right)/2$ and ${\zeta }_{ij}=\sqrt{{\zeta }_{i}{\zeta }_{j}}$ , where $\epsilon$ is the well depth, $\sigma$ is the van der Waals diameter and $\zeta$ is the non-bond energy   . Here, we apply continuum approximation, atoms are assumed to be uniformly distributed over the surfaces of the two interacting molecules, to evaluate the interaction energy between two well-defined molecules by performing double integral over the surface of each molecule. From the work of Thamwattana et al.  , the interaction energy is given by

${E}_{a}={\eta }_{c}{\int }_{V}\beta \left(\rho \right)\text{d}V={\eta }_{c}{\int }_{V}\left(-A{I}_{3}+B{I}_{6}\right)\text{d}V$ (3)

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

${I}_{n}={r}_{c}{\int }_{-\infty }^{\infty }{\int }_{-\text{π}}^{\text{π}}\frac{1}{\left[{\left({r}_{c}-\delta \right)}^{2}-4\delta r{\mathrm{sin}}^{2}\left(\theta /2\right)+{z}^{2}\right]}\text{d}\theta \text{ }\text{d}z\text{ },$ (4)

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

${I}_{n}=\frac{2\text{π}}{{r}_{c}^{2n-2}}B\left(n-1/2,1/2\right)\underset{m=0}{\overset{\infty }{\sum }}{\left(\frac{{\left(n-1/2\right)}_{m}{\delta }^{m}}{m!\text{ }{r}_{c}^{m}}\right)}^{2}.$ (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

$\begin{array}{c}{E}_{d}={\eta }_{s}{\int }_{V}{E}_{a}\left(\delta \right)\text{d}V\\ ={\eta }_{s}{\eta }_{c}{\int }_{V}\left(-A{I}_{3}\left(\delta \right)+B{I}_{6}\left(\delta \right)\right)\text{d}V\\ ={\eta }_{s}{\eta }_{c}\left(-A{K}_{3}+B{K}_{6}\right),\end{array}$ (6)

where $\delta$ is the distance from the nanotube axis to a typical point of the certain biomolecule and ${\eta }_{s}$ 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

${K}_{n}={\int }_{V}{I}_{n}\left(\delta \right)\text{d}V$ (7)

2.1. Insertion of ACh as Two-Connected Spheres into SWCNT

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 $\left({r}_{s}\mathrm{cos}\theta \mathrm{sin}\varphi ,{r}_{s}\mathrm{sin}\theta \mathrm{sin}\varphi ,{r}_{s}\mathrm{cos}\varphi \right)$ , where $-\text{π}<\theta \le \text{π}$ , $0\le \varphi \le \text{π}$ , $0\le {r}_{s}\le 1$ and ${r}_{s}$ is the radius of the spherical shell as shown in Figure 3(a). Further, the distance is given by ${\rho }^{2}={r}_{s}^{2}{r}_{c}^{2}{\mathrm{sin}}^{2}\varphi$ and the spherical volume element is $\text{d}V={r}_{s}\text{ }{r}_{c}^{2}\mathrm{sin}\varphi \text{ }\text{d}r\text{ }\text{d}\varphi \text{ }\text{d}\theta$ . From the work of Thamwattana et al.  , the interaction energy between a spherical molecule and a cylindrical nanotube is given as

${E}_{\text{Sphc-CNT}}={\eta }_{c}{\eta }_{s}\left(-A{D}_{3}+B{D}_{6}\right)={\eta }_{c}{\eta }_{s}{\int }_{V}\left(-A{J}_{3}+B{J}_{6}\right)\text{d}V,$ (8)

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

${J}_{n}={\int }_{-\text{π}}^{\text{π}}{\int }_{0}^{\text{π}}{\int }_{0}^{1}\text{ }\text{ }{r}_{s}^{2n+2}{r}_{c}^{2n+2}{\mathrm{sin}}^{2n+2}\varphi \text{d}r\text{d}\varphi \text{d}\theta ,$ (9)

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

${D}_{n}=\frac{8{\text{π}}^{2}{r}_{s}^{3}}{3{r}_{c}^{2n-2}}B\left(n-1/2,1/2\right)\underset{m=0}{\overset{\infty }{\sum }}\frac{{\left(n-1/2\right)}_{m}{\left(n-1/2\right)}_{m}}{{\left(5/2\right)}_{m}m!}{\left(\frac{{r}_{s}^{2}}{{r}_{c}^{2}}\right)}^{m}.$ (10)

2.2. Insertion of RAV into SWCNT

To evaluate the total energy arising from the RAV drug interaction with SWCNT of radius rc , we consider two possible structures as models for RAV molecule which are an ellipsoid and cylinder as shown in Figure 3(b) and Figure 3(c), respectively.

2.2.1. An Ellipsoid Model

The RAV molecule assumed to be as a spheroidal structure, parameterized by $\left(a\text{ }r\mathrm{sin}\varphi \mathrm{cos}\theta ,a\text{ }r\mathrm{sin}\varphi \mathrm{sin}\theta ,b\text{ }r\mathrm{cos}\varphi \right)$ , where $0\le r\le 1$ , $-\text{π}<\theta \le \text{π}$ , $0\le \varphi \le \text{π}$ , 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 Figure 3(b). Further, the distance is given by ${\rho }^{2}={a}^{2}{r}^{2}{\mathrm{sin}}^{2}\varphi$ and the spheroidal volume element is $\text{d}V={a}^{2}b{r}^{2}\mathrm{sin}\varphi \text{ }\text{d}r\text{ }\text{d}\varphi \text{ }\text{d}\theta$ . From the work of Thamwattana et al.  , the interaction energy between a spheroidal molecule and a cylindrical nanotube is given as

${E}_{\text{Sphd-CNT}}={\eta }_{c}{\eta }_{l}\left(-A{T}_{3}+B{T}_{6}\right)={\eta }_{c}{\eta }_{l}{\int }_{V}\left(-A{W}_{3}+B{W}_{6}\right)\text{d}V,$ (11)

where ${\eta }_{l}$ is the mean volume density of the spheroidal molecule, respectively. So, the integral ${W}_{n}$ ( $n=3,6$ ) is given by

${W}_{n}={\int }_{-\text{π}}^{\text{π}}{\int }_{0}^{\text{π}}{\int }_{0}^{1}{a}^{2n+2}b{r}^{2n+2}{\mathrm{sin}}^{2n+2}\varphi \text{ }\text{d}r\text{ }\text{d}\varphi \text{ }\text{d}\theta \text{ }.$ (12)

The Integral ${T}_{n}$ can be expressed in terms of

Figure 3. The possible configuration for each antiviral compound: (a) ACh molecule splitted as two-connected sphere located on the left and right sides of the origin point; (b) RAV molecule as an ellipsoid structure; (c) RAV molecule as a perfect cylinder, each configuration interacting with a SWCNT of radius rc .

${T}_{n}=\frac{8{\text{π}}^{2}{a}^{2}b}{3{r}_{c}^{2n-2}}B\left(n-1/2,1/2\right)\underset{m=0}{\overset{\infty }{\sum }}\frac{{\left(n-1/2\right)}_{m}{\left(n-1/2\right)}_{m}}{{\left(5/2\right)}_{m}m!}{\left(\frac{{a}^{2}}{{r}_{c}^{2}}\right)}^{m}.$ (13)

To obtain and evaluate the interaction energy for each configuration as shown in Figure 3, we need to determine the potential energy arising from the specific atom at point P inside the cylindrical nanotube as shown in Figure 2(c) (this atom is withing the volume of RAV).

2.2.2. Cylindrical Model

Here, we model the RAV molecule modelled as a perfect cylinder located at the origin (centered) with radius a and length $L=2b$ as shown in Figure 3(c). A typical point in the cylinder can be parameterized by $\left(a\text{ }r\text{ }\mathrm{cos}\theta ,a\text{ }r\text{ }\mathrm{sin}\theta ,z\right)$ , where $0\le r\le 1$ , $-\text{π}\le \theta \le \text{π}$ and $-L\le z\le L$ . Therefore, the distance $\delta$ is given by ${\delta }^{2}={a}^{2}{r}^{2}$ and the volume element of cylinder is $\text{d}V={a}^{2}r\text{ }\text{d}r\text{ }\text{d}\theta \text{ }\text{d}z$ . From Thamwattana’s work et al.  , the interaction energy between a cylindrical molecule and a cylindrical nanotube given as

${E}_{\text{Cyld-CNT}}={\eta }_{c}{\eta }_{d}\left(-A{Y}_{3}+B{Y}_{6}\right)={\eta }_{c}{\eta }_{d}{\int }_{V}\left(-A{G}_{3}+B{G}_{6}\right)\text{d}V\text{ },$ (14)

where ${\eta }_{d}$ is the mean volume density of the cylindrical molecule. So, the integral ${G}_{n}$ ( $n=3,6$ ) is given by

${G}_{n}={\int }_{-L}^{L}{\int }_{-\text{π}}^{\text{π}}{\int }_{0}^{1}\text{ }{a}^{2n+2}{r}^{2n+2}\text{d}r\text{ }\text{d}\theta \text{ }\text{d}z.$ (15)

So, ${Y}_{n}$ is given as

${Y}_{n}=\frac{4{\text{π}}^{2}{a}^{2}L}{{r}_{c}^{2n-2}}B\left(n-1/2,1/2\right)\underset{m=0}{\overset{\infty }{\sum }}\frac{{\left(n-1/2\right)}_{m}{\left(n-1/2\right)}_{m}}{{\left(2\right)}_{m}m!}{\left(\frac{{a}^{2}}{{r}_{c}^{2}}\right)}^{m}.$ (16)

3. Results and Discussion

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 rc . The non-bond energy, well-depth $\epsilon$ and van der Waals diameter $\sigma$ are shown in Table 1. The physical parameters and illustrated radii rc of CNTs are given in Table 2. The attractive and repulsive constants are calculated by using the combining laws ( $A=4\epsilon {\sigma }^{6}$ and $B=4\epsilon {\sigma }^{12}$ ) and are given in Table 3. The volume density for each configuration calculated as the total number of atoms that are containing the specific molecule are divided by the volume of the molecule structure, spherical shape ( ${\eta }_{s}$ ), spheroidal structure ( ${\eta }_{l}$ ) and cylindrical shell ( ${\eta }_{d}$ ), which are ${\eta }_{s}=26/\left(4\text{π}{r}_{s}^{3}/3\right)$ , ${\eta }_{l}=31/\left(4\text{π}{a}^{2}b/3\right)$ and ${\eta }_{d}=31/\left(2\text{π}{a}^{2}L\right)$ , respectively. Next, we evaluate and plot the minimum energies (for all configurations) which are arising from the ACh-SWCNT and RAV-SWCNT interactions. We also deduce the critical radius of SWCNT that will accept both antiviral compounds (ACh-SWCNT and RAV-SWCNT). The minimum energies for all configurations are obtained based on the equilibrium position of each molecule being away from the interior wall of CNT and its radius rc 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 < rc < 7.551 Å and the minimum energies obtained for both configurations when rc 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 < rc < 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

Table 1. The Lennard-Jones constants (ε: Bond length, σ: Non-bond distance and ζ: Non-bond energy) (single bond: sb, double bond: db)    .

Table 2. Parameters for carbon nanotubes, ACh and RAV molecules.

Table 3. Numerical values of the significant constants (A and B) involved in this model.

Figure 4. Interaction energy (E) arising from the encapsulation of two antiviral compounds (ACh and RAV) inside SWCNTs with varian radii ${r}_{c}$ a) ${r}_{c}=3.204$ b) ${r}_{c}=3.325$ c) ${r}_{c}=3.391$ d) ${r}_{c}=3.523$ Å.

Figure 5. Interaction energy (E) arising from the encapsulation of two antiviral compounds (ACh and RAV) inside SWCNTs with varian radii ${r}_{c}$ a) ${r}_{c}=3.590$ b) ${r}_{c}=3.775$ c) ${r}_{c}=3.861$ d) ${r}_{c}=3.973$ Å.

Figure 6. Interaction energy (E) arising from the encapsulation of two antiviral compounds (ACh and RAV) inside SWCNTs with varian radii ${r}_{c}$ a) ${r}_{c}=4.615$ b) ${r}_{c}=5.523$ c) ${r}_{c}=6.102$ d) ${r}_{c}=7.551$ Å.

maximum binding energy because of its volume 502.804 Å3 being larger than that of an ellipsoid structure (RAV) which is 167.601 Å3. 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.  predict that the (5, 5) CNT of ${r}_{c}=3.391$ could be the most significant and smallest effective physical nanotube, and ACh drug can be carried with SWCNT of radius in the range of 4 Å ≤ rc ≤ 8 Å (8 Å < diameter = 2rc < 16 Å) delivered to the target and infected cells  .

4. Conclusion

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 $\left(x,y,z\right)$ 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 ${r}_{c}$  .

Acknowledgements

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

Funding

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

Ethical Approval

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

Cite this paper
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. doi: 10.4236/aad.2018.73006.
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