A super-Gaussian Poisson–Boltzmann model for electrostatic free energy calculation: smooth dielectric distribution for protein cavities and in both water and vacuum states

Abstract

Calculations of electrostatic potential and solvation free energy of macromolecules are essential for understanding the mechanism of many biological processes. In the classical implicit solvent Poisson–Boltzmann (PB) model, the macromolecule and water are modeled as two-dielectric media with a sharp border. However, the dielectric property of interior cavities and ion-channels is difficult to model realistically in a two-dielectric setting. In fact, the detection of water molecules in a protein cavity remains to be an experimental challenge. This introduces an uncertainty, which affects the subsequent solvation free energy calculation. In order to compensate this uncertainty, a novel super-Gaussian dielectric PB model is introduced in this work, which devices an inhomogeneous dielectric distribution to represent the compactness of atoms and characterizes empty cavities via a gap dielectric value. Moreover, the minimal molecular surface level set function is adopted so that the dielectric profile remains to be smooth when the protein is transferred from water phase to vacuum. An important feature of this new model is that as the order of super-Gaussian function approaches the infinity, the dielectric distribution reduces to a piecewise constant of the two-dielectric model. Mathematically, an effective dielectric constant analysis is introduced in this work to benchmark the dielectric model and select optimal parameter values. Computationally, a pseudo-time alternative direction implicit (ADI) algorithm is utilized for solving the super-Gaussian PB equation, which is found to be unconditionally stable in a smooth dielectric setting. Solvation free energy calculation of a Kirkwood sphere and various proteins is carried out to validate the super-Gaussian model and ADI algorithm. One macromolecule with both water filled and empty cavities is employed to demonstrate how the cavity uncertainty in protein structure can be bypassed through dielectric modeling in biomolecular electrostatic analysis.

Appendix

Theorem

The density function for the $i^{th}$ atom is defined by

$$\begin{aligned} g_i^s(\vec {r})=\exp \left[ -\left( \frac{|\vec {r}-r_i|^2}{\sigma ^2 R_i^2}\right) ^m\right] \end{aligned}$$

where $r_i$ and $R_i$ are the center and radius of the $i\mathrm{th}$ atom, respectively. Also, here $\vec {r}$ is the position vector, $\sigma$ is the relative variance and m is the power of super-Gaussian function. Suppose $\sigma =1$ for simplicity. Next, the total density function of a biomolecular system is defined as $g_0^s=1-\prod (1-g_i^s(\vec {r}))$ and the dielectric function of that system is modeled as

$$\begin{aligned} \displaystyle {\epsilon _G^s=\epsilon _m g_0^s+\epsilon _s (1-g_0^s)}. \end{aligned}$$

Here $\epsilon _m$ and $\epsilon _s$ are the dielectric constants of the solute and solvent respectively. Then we have that $\displaystyle {\lim _{m\rightarrow \infty }\epsilon _G^s=\epsilon _2}$ at the solute and solvent regions where $\epsilon _2$ is the dielectric function of the classical two-dielectric model.

Proof

Let us consider three cases where the position vector is either inside or outside the solute, or on the Van der Walls (VDW) molecular surface.

• Case I: There exists an atom (say ith atom) such that $|\vec {r}-r_i|< R_i$ or, $\displaystyle {\frac{|\vec {r}-r_i|}{ R_i}}<1$. In this case $\displaystyle {\lim _{m\rightarrow \infty }\Big (\frac{|\vec {r}-r_i|}{ R_i}\Big )^{2m}}=0$. Hence $\displaystyle {\lim _{m\rightarrow \infty }\exp \Big [-\Big (\frac{|\vec {r}-r_i|}{ R_i}\Big )^{2m}\Big ]}=1$, which means $g^s_i(\vec {r}) =1$ and $g^s_0(\vec {r}) =1$. Therefore, if $|\vec {r}-r_i|< R_i$ for some i (inside the VDW surface), $\epsilon _G^s=\epsilon _m$.

• Case II: For all atoms, we have $|\vec {r}-r_i|> R_i$ or $\displaystyle {\frac{|\vec {r}-r_i|}{ R_i}}>1$ for any i. In this case $\displaystyle {\lim _{m\rightarrow \infty }\Big (\frac{|\vec {r}-r_i|}{ R_i}\Big )^{2m}}=\infty$. So, $\displaystyle {\lim _{m\rightarrow \infty }\exp \Big [-\Big (\frac{|\vec {r}-r_i|}{ R_i}\Big )^{2m}\Big ]}=0$, which means that $g^s_i(\vec {r}) =0$ for all i. Hence $g^s_0(\vec {r}) =0$. Therefore, if $|\vec {r}-r_i|> R_i$ for all i (outside the VDW surface), $\epsilon _G^s=\epsilon _s$.

• Case III: In the last case, the position vector $\vec {r}$ has to be located on the VDW surface. Without the loss of generality, we assume that $\vec {r}$ is on the sphere boundary of the $i^{th}$ atom and does not locate inside any other atoms. So, we have $|\vec {r}-r_i|= R_i$ or $\displaystyle {\frac{|\vec {r}-r_i|}{ R_i}}=1$. And, for any $j \ne i$, $|\vec {r}-r_j|> R_j$. In this case $\displaystyle {\lim _{m\rightarrow \infty }\exp \Big [-\Big (\frac{|\vec {r}-r_i|}{ R_i}\Big )^{2m}\Big ]}=\frac{1}{e}$, which means $\displaystyle g^s_i(\vec {r}) =\frac{1}{e}$. For any $j \ne i$, $g^s_j(\vec {r}) =0$. Hence, $\displaystyle g^s_0(\vec {r}) =\frac{1}{e}$. Therefore, on the VDW surface, we have $\displaystyle {\epsilon _G^s=\epsilon _m g_0^s+\epsilon _s (1-g_0^s)=\epsilon _m \frac{1}{e}+\epsilon _s (1-\frac{1}{e})}=50.9375$ for $\epsilon _m=1$ and $\epsilon _s=80$. In all cases, the new dielectric model converges to a two-dielectric model based on the VDW surface

  $$\begin{aligned} \lim _{m\rightarrow \infty }\epsilon _G^s(\vec {r})=\epsilon _2(\vec {r}) = \left\{ \begin{array}{ll} \epsilon _m, &{}\quad \vec {r}~~\text {is inside the VDW surface}\\ \epsilon _m/e + \epsilon _s(e-1)/e, &{}\quad \vec {r}~~\text {is on the VDW surface}\\ \epsilon _s, &{}\quad \vec {r}~~\text {is outside the VDW surface}. \\ \end{array} \right. \end{aligned}$$

  (35)