Efficient Algorithms for a Nonlocal Dielectric Model for Protein in Ionic Solvent

Abstract

The nonlocal dielectric approach can significantly enhance the classical Poisson dielectric model by including polarization correlations among water molecules. In this paper, a nonlocal dielectric model for protein in ionic solvent is proposed and analyzed, alongside a new efficient numerical algorithm and program package for solving the model. In particular, by using solution splitting and reformulation techniques, it is shown that the solution of the nonlocal dielectric model is unique, and can be found from solving two well-defined partial differential systems and one Poisson-like boundary value problem. Consequently, the singular and computational difficulties caused by Dirac delta distributions and convolution terms are overcome. Furthermore, a nonlocal linearized Poisson-Boltzmann equation with uniform ionic size effect is proposed and numerically tested on three protein molecules with up to 6062 atoms by using a fast finite element solver from the FEniCS project. A nonlocal point charge Born model with a known analytical solution is also tested to validate the new algorithm and program package. Numerical results demonstrate the high performance of the program package, and confirm one advantage of the new algorithm in retaining a high order of accuracy of finite element approximations. Introduction.Calculation of electrostatic potential energy for proteins in ionic solvent is a fundamental task in the simulation study of the structure and biological function of proteins, catalytic activity, and ligand association [17,25,31]. One commonly used mathematical model for estimating the electrostatic potential function in an ionic solvent is the Poisson-Boltzmann equation (PBE) [2,13,16,22,23,27,37]. But, due to the polarization correlations among water molecules and ionic size effect [21], the PBE model may not work well in some important bioengineering applications (such as ion channel studies and rational drug design). To reflect the polarization correlations among water molecules, several nonlocal dielectric models have been developed for a wide range of dielectric materials and dipolar liquids in the last thirty years [4,5,6,7,8,10,19,20,28,30]. Recent progress in the development of fast numerical algorithms has sharply reduced the complexity of solving a nonlocal dielectric model [15,33,35,36], making it possible for a nonlocal dielectric model to be applied to large scale biomolecular simulations.However, the study of a nonlocal model has been limited to the case of pure water solvent so far due to modeling and algorithmic complications. Most significantly, none of the current ionic models incorporate nonlocal dielectric effects. As the first step toward the direction of changing this situation, in this paper, we propose a nonlocal dielectric model for protein in ionic solvent (see (2.2)) by assuming that

Analytical solutions of nonlocal Poisson dielectric models with multiple point charges inside a dielectric sphere

“…A PDB file can be downloaded for free from the Protein Data Bank (http://www.rcsb.org/). As shown in our previous work [16,20], Ψ can be twice continuously differentiable within and outside D p and continuous across the interface Γ for local and nonlocal Poisson models. Thus, G contains all the singularity points of Φ.…”

“…Hence, these MATLAB surface plots reflect the convergent behaviors of these interpolation functions. [16,20]. Figure 3 shows that the linear interpolation functions of Φ became more and more spiky as h was reduced from 0.1 to 0.01, due to the solution singularity caused by the Dirac-delta distributions.…”

“…Motivated by Hildebrandt et al 's novel work, a fast finite element algorithm for solving a Lorentz nonlocal model for water was developed in [15]. This Lorentz nonlocal model was then extended into a nonlocal Poisson dielectric model for a protein in an ionic solvent [16]. Recently, a nonlocal modified Poisson-Boltzmann equation was proposed as the first nonlinear nonlocal dielectric continuum model for computing electrostatics of ionic solvated biomolecules [17].…”

“…With this software package, we also can calculate the convolution of solution for Model 1 and the reaction parts of the analytical solutions, which can be applied to the development of numerical algorithms for solving local and nonlocal dielectric models [16,17,20]. We further added a special data input option to input the required atomic charge numbers and positions directly from a PQR file of protein, which can be produced from a protein data bank (PDB) file by using the program tool PDB2PQR (http://www.poissonboltzmann.org/pdb2pqr/).…”

SDPBS Web Server for Calculation of Electrostatics of Ionic Solvated Biomolecules

“…Furthermore, SDPBS is an ongoing software development project. It will be frequently updated to improve its performance, and will be expanded to include our new nonlocal continuum dielectric software packages reported in [29,60,61]. By way of the SDPBS web server, the end users always has access to the most up-to-date version of SDPBS for their calculations.…”