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A Nonlocal size modified Poisson-Boltzmann Model and Its Finite Element Solver for Protein in Multi-Species Ionic Solution

Dexuan Xie, Liam Jemison, Yi Jiang

TL;DR

The paper tackles the limitation of existing Poisson-Boltzmann models by integrating ionic-size effects with nonlocal dielectric correlations in a single NSMPB framework for proteins in multi-species ionic solutions. It introduces a solution decomposition $u=G+\Psi+\tilde{Φ}$ to manage singularities and develops a finite element solver built around a damped modified Newton method, with a damping scheme and multiple initial-iterate strategies to ensure robust convergence. The methodology includes a linear NSMPB model, a convolution-free reformulation of the Newton step, and a FE variational formulation, all underpinned by theoretical justification (Theorems 3.2–3.4), and is implemented in a Python/Fortran package that integrates PDB/PQR processing and mesh generation. Numerical experiments on proteins with thousands of atoms and multi-species ions demonstrate fast convergence, robustness to initialization, and scalable performance on large irregular meshes, positioning NSMPB as a practical tool for accurate electrostatics and solvation energy calculations in biomolecular simulations.

Abstract

The Poisson-Boltzmann (PB) model is a widely used implicit solvent model in protein simulations. Although variants, such as the size modified PB and nonlocal modified PB models, have been developed to account for ionic size effects and nonlocal dielectric correlations, no existing PB variants simultaneously incorporate both, due to significant modeling and computational challenges. To address this gap, in this paper, a nonlocal size modified PB (NSMPB) model is introduced and solved using a finite element method for a protein with a three-dimensional molecular structure and an ionic solution containing multiple ion species. In particular, a novel solution decomposition is proposed to overcome the difficulties caused by the increased nonlinearity, nonlocality, and solution singularities of the model. It is then applied to the development of the NSMPB finite element solver, which includes an efficient modified Newton iterative method, an effective damping parameter selection strategy, and good selections of initial iterations. Moreover, the construction of the modified Newton iterative method is mathematically justified. Furthermore, an NSMPB finite element package is developed by integrating a mesh generation tool, a protein data bank file retrieval program, and the PDB2PQR package to simplify and accelerate its usage and application. Finally, numerical experiments are conducted on an ionic solution with four species, proteins with up to 11439 atoms, and irregular interface-fitted tetrahedral box meshes with up to 1188840 vertices. The numerical results confirm the fast convergence and strong robustness of the modified Newton iterative method, demonstrate the high performance of the package, and highlight the crucial roles played by the damping parameter and initial iteration selections in enhancing the method's convergence. The package will be a valuable tool in protein simulations.

A Nonlocal size modified Poisson-Boltzmann Model and Its Finite Element Solver for Protein in Multi-Species Ionic Solution

TL;DR

The paper tackles the limitation of existing Poisson-Boltzmann models by integrating ionic-size effects with nonlocal dielectric correlations in a single NSMPB framework for proteins in multi-species ionic solutions. It introduces a solution decomposition to manage singularities and develops a finite element solver built around a damped modified Newton method, with a damping scheme and multiple initial-iterate strategies to ensure robust convergence. The methodology includes a linear NSMPB model, a convolution-free reformulation of the Newton step, and a FE variational formulation, all underpinned by theoretical justification (Theorems 3.2–3.4), and is implemented in a Python/Fortran package that integrates PDB/PQR processing and mesh generation. Numerical experiments on proteins with thousands of atoms and multi-species ions demonstrate fast convergence, robustness to initialization, and scalable performance on large irregular meshes, positioning NSMPB as a practical tool for accurate electrostatics and solvation energy calculations in biomolecular simulations.

Abstract

The Poisson-Boltzmann (PB) model is a widely used implicit solvent model in protein simulations. Although variants, such as the size modified PB and nonlocal modified PB models, have been developed to account for ionic size effects and nonlocal dielectric correlations, no existing PB variants simultaneously incorporate both, due to significant modeling and computational challenges. To address this gap, in this paper, a nonlocal size modified PB (NSMPB) model is introduced and solved using a finite element method for a protein with a three-dimensional molecular structure and an ionic solution containing multiple ion species. In particular, a novel solution decomposition is proposed to overcome the difficulties caused by the increased nonlinearity, nonlocality, and solution singularities of the model. It is then applied to the development of the NSMPB finite element solver, which includes an efficient modified Newton iterative method, an effective damping parameter selection strategy, and good selections of initial iterations. Moreover, the construction of the modified Newton iterative method is mathematically justified. Furthermore, an NSMPB finite element package is developed by integrating a mesh generation tool, a protein data bank file retrieval program, and the PDB2PQR package to simplify and accelerate its usage and application. Finally, numerical experiments are conducted on an ionic solution with four species, proteins with up to 11439 atoms, and irregular interface-fitted tetrahedral box meshes with up to 1188840 vertices. The numerical results confirm the fast convergence and strong robustness of the modified Newton iterative method, demonstrate the high performance of the package, and highlight the crucial roles played by the damping parameter and initial iteration selections in enhancing the method's convergence. The package will be a valuable tool in protein simulations.
Paper Structure (14 sections, 4 theorems, 69 equations, 7 figures, 3 tables)

This paper contains 14 sections, 4 theorems, 69 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A nonlocal size modified Poisson-Boltzmann model
  3. An NSMPB finite element iterative method
  4. Formulation of a nonlinear finite element variational problem
  5. A modified Newton iterative method
  6. Reformulation of the modified Newton iterative scheme to avoid direct convolution calculations
  7. A linear NSMPB model
  8. Initial iterate selections
  9. The NSMPB finite element program package and numerical results
  10. Construction of six box meshes per protein case
  11. Performance of the NSMPB finite element program package
  12. Tests on our damping parameter selection scheme
  13. Tests on finite element solution sequence convergence
  14. Conclusions

Key Result

Theorem 3.1

Let $b(\tilde{\Phi},v)$ be defined in b-def and $\| \cdot \|_{H^{1}(\Omega)}$ denote the norm of $H^{1}(\Omega)$. Then a linear expansion of $b(\tilde{\Phi}+p,v)$ is given as follows: where $b^{\prime}(\tilde{\Phi},v;p)$ is the Fréchet derivative of the nonlinear functional $b(\tilde{\Phi},v)$ at $p$, which is given in the expression Here $A_1$, $A_2$, and $A_3$ are defined by and

Figures (7)

  • Figure 1: An illustration of the box domain $\Omega$ partitioned into the protein region $D_p$ (green), the solvent region $D_s$ (gray), and the interface $\Gamma$ between $D_p$ and $D_s$. In this figure, $\Gamma$ is set as a molecular surface of the protein with the protein data bank identifier (PDB ID) 1C4K.
  • Figure 2: Three proteins' crystallographic molecular structures depicted in cartoon and sphere representations wrapped by the three protein regions $D_p$ (in gray color) generated by our NSMPB software package. Here 4PTI, 1CID, and 1C4K are the Protein Data Bank identifications (PDB ID).
  • Figure 3: (a, b, c) One view of the box mesh $\Omega_{h_j}$ (i.e., Mesh $j$ for $j=1,2,3$) generated by our NSMPB software package in the protein 1CID case. (d, e, f) One clipped view (with $x=0$) of the solvent region mesh $D_{s,h_j}$, along with the triangular boundary surface mesh of the protein region mesh $D_{p, h_j}$, shown in green, highlighting the mesh irregularity between $D_{s,h_j}$ and $D_{p, h_j}$.
  • Figure 4: Convergence processes of our modified Newton iterative method \ref{['Newton_scheme2']} on Meshes 1 to 6 in the case of protein 1CID.
  • Figure 5: A comparison of the iterative process of the modified Newton method \ref{['Newton_scheme2']} using Selection 3 of initial iteration \ref{['NMPBE-initial3']} with that using Selection 4 of \ref{['NMPBE-initial4']} for the protein 1C4K case on Mesh 3 in terms of the absolute residual error $\|F(\tilde{\Phi}^{(k)})\|$ and the damping parameter $\omega_k$ of the modified Newton iterative method.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4