Equilibrium location for spherical DNA and toroidal cyclodextrin

Abstract

Cyclodextrin comprises a ring structure composed of glucose molecules with an ability to form complexes of certain substances within its central cavity. The compound can be utilised for various applications including food, textiles, cosmetics, pharmaceutics, and gene delivery. In gene transfer, the possibility of forming complexes depends upon the interaction energy between cyclodextrin and DNA molecules which here are modelled as a torus and a sphere, respectively. Our proposed model is derived using the continuum approximation together with the Lennard-Jones potential, and the total interaction energy is obtained by integrating over both the spherical and toroidal surfaces. The results suggest that the DNA prefers to be symmetrically situated about 1.2 Å above the centre of the cyclodextrin to minimise its energy. Furthermore, an optimal configuration can be determined for any given size of torus and sphere.

Appendices

Appendix A: Derivation of the interaction energy

On substitution with $\delta$ into (1), the interaction energy can be obtained using the toroidal surface integration,

$$\begin{aligned} I_n^{\text {(sp)}}&=\int _{-\pi }^\pi \int _{-\pi }^\pi \frac{4\pi a^2r(R+r\cos \phi )}{(n-1)\big [R^2+r^2+\epsilon ^2+\nu ^2+2r(R\cos \phi -\nu \sin \phi )-2\epsilon (R+r\cos \phi )\cos \theta -a^2 \big ]^n} \\&\quad \times U_{n-2}\left( \frac{R^2+r^2+\epsilon ^2+\nu ^2+2r(R\cos \phi -\nu \sin \phi )-2\epsilon (R+r\cos \phi )\cos \theta +a^2}{R^2+r^2+\epsilon ^2+\nu ^2+2r(R\cos \phi -\nu \sin \phi )-2\epsilon (R+r\cos \phi )\cos \theta -a^2}\right) \> \mathrm{d}\theta \mathrm{d}\phi . \end{aligned}$$

On rewriting the Chebyshev polynomial in the series form,

$$\begin{aligned} U_n(x)=\sum _{m=0}^{\lfloor n/2-1\rfloor } (-1)^m \left( {\begin{array}{c}n-m\\ m\end{array}}\right) (2x)^{n-2m}, \end{aligned}$$

we obtain

$$\begin{aligned} I_n^{\text {(st)}}=\frac{\pi a^2}{n-1}\sum _{m=0}^{\lfloor n/2-1\rfloor }(-1)^m\left( {\begin{array}{c}n-m\\ m\end{array}}\right) 2^{n-2m}\int _{-\pi }^\pi r(R+r\cos \phi )Q_{n,m} \> \mathrm{d}\phi , \end{aligned}$$

where $Q_{n,m}$, for $n=3, 6$ and $m=0,1,2,\ldots ,\lfloor n/2-1\rfloor$, are given by

$$\begin{aligned} Q_{n,m}=\int _{-\pi }^\pi \frac{[R^2+r^2+\epsilon ^2+\nu ^2+2r(R\cos \phi -\nu \sin \phi )-2\epsilon (R+r\cos \phi )\cos \theta +a^2\big ]^{n-2m-2}}{\big [R^2+r^2+\epsilon ^2+\nu ^2+2r(R\cos \phi -\nu \sin \phi )-2\epsilon (R+r\cos \phi )\cos \theta -a^2\big ]^{2n-2m-2}} \> \mathrm{d}\theta . \end{aligned}$$

We define $\alpha = R^2+r^2+\epsilon ^2+\nu ^2+2r(R\cos \phi -\nu \sin \phi )+a^2$, $\beta =R^2+r^2+\epsilon ^2+\nu ^2+2r(R\cos \phi -\nu \sin \phi )-a^2$, and $\kappa =2\epsilon (R+r\cos \phi )$, so we have

$$\begin{aligned} Q_{n,m}=\int _{-\pi }^\pi \frac{(\alpha -\kappa \cos \theta )^{n-2m-2}}{(\beta -\kappa \cos \theta )^{2n-2m-2}}\> \mathrm{d}\theta . \end{aligned}$$

By making the half-angle tangent substitution, $t=\tan \theta /2$, the equation above becomes

$$\begin{aligned} Q_{n,m}=2\int _{-\infty }^\infty \frac{(1+t^2)^{n-1}\left[ (\alpha +\kappa )t^2+\alpha -\kappa \right] ^{n-2m-2}}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^{2n-2m-2}}\> \mathrm{d}t. \end{aligned}$$

Converting the integrand into the partial fraction form yields

$$\begin{aligned} Q_{n,m}=\int _{-\infty }^\infty \sum _{j=1}^{2n-2m-2} \frac{\Psi _{n,m,j}(\alpha ,\beta ,\kappa )}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^j} \> \mathrm{d}t=\sum _{j=1}^{2n-2m-2} \Psi _{n,m,j}(\alpha ,\beta ,\kappa )D_{n,m,j}, \end{aligned}$$

where the exact functions $\Psi _{n,m,j}(\alpha ,\beta ,\kappa )$ are as given in the Appendix B, and the $D_{n,m,j}$ for $j=1,2,3,\ldots ,(2n-2m-2)$ are defined as

$$\begin{aligned} D_{n,m,j}=\int _{-\infty }^\infty \frac{1}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^j} \> \mathrm{d}t. \end{aligned}$$

Further, making the substitution $\displaystyle t=\sqrt{\frac{\beta -\kappa }{\beta +\kappa }}\tan \lambda$ leads $D_{n,m,j}$ to

$$\begin{aligned} D_{n,m,j}=\frac{(\beta -\kappa )^{1/2-n}}{(\beta +\kappa )^{1/2}}B(1/2,n-1/2), \end{aligned}$$

where B(x, y) is the beta function.

Appendix B: Details of $\Psi _{n,m,j}(\alpha ,\beta ,\kappa )$

The full expressions of $Q_{n,m}$, for $n=3, 6$ and $m=0,1,2,\ldots ,\lfloor n/2-1\rfloor$ are given as follows

$$\begin{aligned} Q_{3,0}&=2\int _{-\infty }^\infty \left\{ \frac{\alpha +\kappa }{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] (\beta +\kappa )^3}+\frac{2\kappa (3\alpha -\beta +2\kappa )}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^2(\beta +\kappa )^3}\right. \\&\quad \left. +\frac{4\kappa ^2(3\alpha -2\beta +\kappa )}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^3(\beta +\kappa )^3}+\frac{8\kappa ^3(\alpha -\beta )}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^4(\beta +\kappa )^3}\right\} \>\text {d}t, \end{aligned}$$

$$\begin{aligned} Q_{6,0}&=2\int _{-\infty }^\infty \left\{ \frac{(\alpha +\kappa )^4}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] (\beta +\kappa )^9}+\frac{2\kappa (\alpha +\kappa )^3(9\alpha -4\beta +5\kappa )}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^2(\beta +\kappa )^9}\right. \\&\quad +\frac{8\kappa ^2(\alpha +\kappa )^2(18\alpha ^2-16\alpha \beta +20\alpha \kappa +3\beta ^2-10\beta \kappa +5\kappa ^2)}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^3(\beta +\kappa )^9} \\&\quad + \frac{1}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^4(\beta +\kappa )^9} [16\kappa ^3(\alpha +\kappa )(42\alpha ^3-56\alpha ^2\beta +70\alpha ^2\kappa \\&\quad +21\alpha \beta ^2-70\alpha \beta \kappa +35\alpha \kappa ^2-2\beta ^3+15\beta ^2\kappa -20\beta \kappa ^2+5\kappa ^3)]\\&\quad +\frac{1}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^5(\beta +\kappa )^9}[16\kappa ^4(126\alpha ^4-224\alpha ^3\beta +280\alpha ^3\kappa \\&\quad +126\alpha ^2\beta ^2-420\alpha ^2\beta \kappa +210\alpha ^2\kappa ^2-24\alpha \beta ^3+180\alpha \beta ^2\kappa -240\alpha \beta \kappa ^2\\&\quad +60\alpha \beta ^3+\beta ^4-20\beta ^3\kappa +60\beta ^2\kappa ^2-40\beta \kappa ^3+5\kappa ^4)]\\&\quad +\frac{1}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^6(\beta +\kappa )^9}[32\kappa ^5(126\alpha ^4-280\alpha ^3\beta +224\alpha ^3\kappa \\&\quad +210\alpha ^2\beta ^2-420\alpha ^2\beta \kappa +126\alpha ^2\kappa ^2-60\alpha \beta ^3+240\alpha \beta ^2\kappa -180\alpha \beta \kappa ^2\\&\quad +24\alpha \kappa ^3+5\beta ^4-40\beta ^3\kappa +60\beta ^2\kappa ^2-20\beta \kappa ^3+\kappa ^4)] \\&\quad +\frac{1}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^7(\beta +\kappa )^9}[128\kappa ^6(\alpha -\beta )(42\alpha ^3-70\alpha ^2\beta \\&\quad +56\alpha ^2\kappa +35\alpha \beta ^2-70\alpha \beta \kappa +21\alpha \kappa ^2-5\beta ^3+20\beta ^2\kappa -15\beta \kappa ^2+2\kappa ^3)]\\&\quad +\frac{256\kappa ^7(\alpha -\beta )^2(18\alpha ^2-20\alpha \beta +16\alpha \kappa +5\beta ^2-10\beta \kappa +3\kappa ^2)}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^8(\beta +\kappa )^9}\\&\quad +\left. \frac{256\kappa ^8(\alpha -\beta )^3(9\alpha -5\beta +4\kappa )}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^9(\beta +\kappa )^9}+\frac{512\kappa ^9(\alpha -\beta )^4}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^{10}(\beta +\kappa )^9}\right\} \>\text {d}t, \end{aligned}$$

$$\begin{aligned} Q_{6,1}&=2\int _{-\infty }^\infty \left\{ \frac{(\alpha +\kappa )^2}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] (\beta +\kappa )^7}+\frac{2\kappa (\alpha +\kappa )(7\alpha -2\beta +5\kappa )}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^2(\beta +\kappa )^7 }\right. \\&\quad +\frac{4\kappa ^2(21\alpha ^2-12\alpha \beta +30\alpha \kappa +\beta ^2-10\beta \kappa +10\kappa ^2)}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^3(\beta +\kappa )^7}\\&\quad +\frac{40\kappa ^3(7\alpha ^2-6\alpha \beta +8\alpha \kappa +\beta ^2-4\beta \kappa +2\kappa ^2)}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^4(\beta +\kappa )^7}\\&\quad +\frac{80\kappa ^4(7\alpha ^2-8\alpha \beta +6\alpha \kappa +2\beta ^2-4\beta \kappa +\kappa ^2)}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^5(\beta +\kappa )^7}\\&\quad +\frac{32\kappa ^5(21\alpha ^2-30\alpha \beta +12\alpha \kappa +10\beta ^2-10\beta \kappa +\kappa ^2)}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^6(\beta +\kappa )^7}\\&\quad \left. +\frac{64\kappa ^6(\alpha -\beta )(7\alpha -5\beta +2\kappa )}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^7(\beta +\kappa )^7}+\frac{128\kappa ^7(\alpha -\beta )^2}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^8(\beta +\kappa )^7}\right\} \>\text {d}t, \end{aligned}$$

and

$$\begin{aligned} Q_{6,2}&=2\int _{-\infty }^\infty \left\{ \frac{1}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] (\beta +\kappa )^5}+\frac{10\kappa }{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^2(\beta +\kappa )^5}\right. \\&\quad +\frac{40\kappa ^2}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^3(\beta +\kappa )^5}+\frac{80\kappa ^3}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^4(\beta +\kappa )^5}\\&\quad \left. +\frac{80\kappa ^4}{\left[ (\beta +\kappa )t^2+\beta -\kappa \right] ^5(\beta +\kappa )^5} \right\} \>\text {d}t. \end{aligned}$$

where $\alpha ,\beta$, and $\kappa$ are defined, respectively, by $\alpha = R^2+r^2+\epsilon ^2+\nu ^2+2r(R\cos \phi -\nu \sin \phi )+a^2$, $\beta =R^2+r^2+\epsilon ^2+\nu ^2+2r(R\cos \phi -\nu \sin \phi )-a^2$, and $\kappa =2\epsilon (R+r\cos \phi )$.

Appendix C: Energy different between non-functionalised cyclodextrin and its derivatives

Figure 6 shows the energy different between the system of non-functionalised cyclodextrin and its derivatives.

Fig. 6

Energy different of DNA and functionalised cyclodextrins from energy of non-functionalised cyclodextrin where spherical DNA located on a x-axis and b z-axis