Three-dielectric-layer hybrid solvation model with spheroidal cavities in biomolecular simulations

This paper extends the three-dielectric-layer hybrid solvation model for treating electrostatic interactions in biomolecular simulations from the spherical geometry [Comm. Comp. Phys. 6, 955 (2009)] to the prolate/oblate spheroidal geometries. In the resulting model, the inner spheroidal cavity of a low dielectric constant ${\epsilon }_i$ contains an irregular-shaped solute and some explicit solvent molecules, while the unbounded outer layer is used to implicitly model the bulk solvent as a dissimilar continuum medium of a high dielectric constant ${\epsilon }_o$. The thin intermediate translation layer, which also consists of explicit solvent molecules, assumes a continuous variation of the dielectric permittivity $\epsilon \left(\mathbf{r}\right)$ changing smoothly from ${\epsilon }_i$ to ${\epsilon }_o$. In particular, the so-called quasiharmonic dielectric permittivity profile is introduced based on a harmonic interpolation, thus allowing analytical series solutions of the generalized Coulomb potential and the self-polarization energy in terms of the associated Legendre functions. A key advantage of the proposed quasiharmonic dielectric model lies in the fact that it overcomes the inherent mathematical divergence in the self-polarization energy that exists in the simple and widely used steplike dielectric model. Numerical examples are included to show some simulated behaviors of the quasiharmonic dielectric model and the corresponding analytical solution.

Figure 1

Schematic illustration of the steplike dielectric model: the dielectric constants of a prolate spheroid and the surrounding medium are ${\epsilon }_i$ and ${\epsilon }_o$, respectively. The prolate spheroid, defined by the equation $\xi ={\xi }_b$, has an interfocal distance $a$. A point charge $e_s$ is located at the point ${\mathbf{r}}_s=\left({\xi }_s,{\eta }_s,{\varphi }_s=0\right)$ in the $xz$ plane.

Figure 2

Schematic illustration of a three-layer dielectric model: The inner layer $\left(\xi \le {\xi }_b-\delta \right)$ has a dielectric constant of ${\epsilon }_i$, while the outer layer $\left(\xi \ge {\xi }_b+\delta \right)$ has a dielectric constant of ${\epsilon }_o$. The intermediate translation layer $\left({\xi }_b-\delta <\xi <{\xi }_b+\delta \right)$ assumes a continuous dielectric permittivity profile $\epsilon \left(\xi \right)$ that connects ${\epsilon }_i$ and ${\epsilon }_o$.

Figure 3

(Color online) Illustration of several dielectric models for the translation layer between ${\xi }_b-\delta$ and ${\xi }_b+\delta$, assuming ${\epsilon }_i<{\epsilon }_o$. Dot-dashed line, the steplike model; dotted line, the linear model; dashed line, the cosinelike model; and solid line, the quasiharmonic model.

Figure 4

(Color online) Self-polarization energy $V_s$ along a minor axis of the prolate spheroid $\left(x^2+y^2\right)/{10}^2+z^2/{20}^2=1$ corresponding to the quasiharmonic model with $\delta =0.1$, 0.05, 0.02, and 0.01. (a) ${\epsilon }_o=10$, and (b) ${\epsilon }_o=80$.

Figure 5

(Color online) Self-polarization energy $V_s$ along the ray extending from the center to the point (10,0,20) corresponding to the quasiharmonic model with $\delta =0.1$, 0.05, 0.02, and 0.01. (a) ${\epsilon }_o=10$, and (b) ${\epsilon }_o=80$.

Figure 6

(Color online) Self-polarization energy $V_s$ of a prolate spheroidal quantum dot as a function of $r$ corresponding to the quasiharmonic model with $\delta =0.1$, 0.05, 0.02, and 0.01. (a) Along a minor axis of the prolate spheroid, and (b) along the ray extending from the center to the point (10,0,20).

Figure 7

(Color online) The relative error $E$ in the self-polarization energy defined in Eq. (19) for the case of Fig. 4a with $\delta =0.1$.