Domain Decomposition Method for Poisson--Boltzmann Equations based on Solvent Excluded Surface

TL;DR

This work tackles the nonlinear Poisson-Boltzmann equation for implicit solvation by placing the solute boundary on the solvent-excluded surface and incorporating steric effects through a Stern layer. It develops a Schwarz-domain-decomposition framework that reduces the problem to two coupled equations in a bounded cavity and employs a hybrid linear-nonlinear solver with local spectral discretizations (unit-ball HSP and GSP solvers) to solve the subproblems. A continuous SES-based dielectric and ion-exclusion function, together with a single-layer potential coupling, enables accurate boundary communication between the solute cavity and the bulk solvent. Numerical results on small molecules demonstrate the method’s convergence, sensitivity to discretization, and the necessity of nonlinear modeling near the molecular surface, highlighting potential for scalable simulations of larger systems.

Abstract

In this paper, we develop a domain decomposition method for the nonlinear Poisson-Boltzmann equation based on a solvent-excluded surface widely used in computational chemistry. The model relies on a nonlinear equation defined in $\mathbb{R}^3$ with a space-dependent dielectric permittivity and an ion-exclusion function that accounts for steric effects. Potential theory arguments transform the nonlinear equation into two coupled equations defined in a bounded domain. Then, the Schwarz decomposition method is used to formulate local problems by decomposing the cavity into overlapping balls and only solving a set of coupled sub-equations in each ball. The main novelty of the proposed method is the introduction of a hybrid linear-nonlinear solver used to solve the equation. A series of numerical experiments are presented to test the method and show the importance of the nonlinear model.

Figures (15)

• Figure 1: SES surface with a probe of radius $r_p$ (left) and with $r_p+a$ (right) .
• Figure 2: Solute probes and solute-solvent boundary for a diatmoic molecule.
• Figure 3: Schematic diagram of the dielectric permittivity, $\varepsilon(\boldsymbol{x})$ (left $y$-axis) and the ion-exclusion function, $\lambda(\boldsymbol{x})$ (right $y$-axis) with respect to $f_{\mathrm{SAS}}$.
• Figure 4: PDEs defined in the solute cavity $\Omega_0=\Omega_{\mathrm{SES}}\cup \mathcal{L}$, and the bulk solvent region $\Omega_{\infty}$.
• Figure 5: 2-D schematic diagram of $\Gamma_j^{\mathtt{i}}$ and $\Gamma_j^{\mathtt{e}}$.

Theorems & Definitions (12)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Remark 8
• Remark 9
• Remark 10