Poisson-Boltzmann Calculations: van der Waals or Molecular Surface?

AbstractThe Poisson-Boltzmann equation is widely used for modeling the electro-statics of biomolecules, but the calculation results are sensitive to the choice of the boundary between the low solute dielectric and the high solvent dielectric. The default choice for the dielectric boundary has been the molecular surface, but the use of the van der Waals surface has also been advocated. Here we review recent studies in which the two choices are tested against experimental results and explicit-solvent calculations. The assignment of the solvent high dielectric constant to interstitial voids in the solute is often used as a criticism against the van der Waals surface. However, this assignment may not be as unrealistic as previously thought, since hydrogen exchange and other NMR experiments have firmly established that all interior parts of proteins are transiently accessible to the solvent.

Download Full-text

Computing Protein pKas Using the TABI Poisson–Boltzmann Solver

Journal of Computational Biophysics and Chemistry ◽

10.1142/s2737416520420065 ◽

2020 ◽

pp. 2042006

Author(s):

Jiahui Chen ◽

Jingzhen Hu ◽

Yongjia Xu ◽

Robert Krasny ◽

Weihua Geng

Keyword(s):

Boltzmann Distribution ◽

Dielectric Medium ◽

Boundary Integral ◽

Acid Dissociation ◽

Low Dielectric ◽

Poisson Boltzmann ◽

Computing Procedure ◽

Poisson Boltzmann Equation ◽

High Dielectric ◽

Electrostatic Calculations

A common approach to computing protein pKas uses a continuum dielectric model in which the protein is a low dielectric medium with embedded atomic point charges, the solvent is a high dielectric medium with a Boltzmann distribution of ionic charges, and the pKa is related to the electrostatic free energy which is obtained by solving the Poisson–Boltzmann equation. Starting from the model pKa for a titrating residue, the method obtains the intrinsic pKa and then computes the protonation probability for a given pH including site–site interactions. This approach assumes that acid dissociation does not affect protein conformation aside from adding or deleting charges at titratable sites. In this work, we demonstrate our treecode-accelerated boundary integral (TABI) solver for the relevant electrostatic calculations. The pKa computing procedure is enclosed in a convenient Python wrapper which is publicly available at the corresponding author’s website. Predicted results are compared with experimental pKas for several proteins. Among ongoing efforts to improve protein pKa calculations, the advantage of TABI is that it reduces the numerical errors in the electrostatic calculations so that attention can be focused on modeling assumptions.

Download Full-text

Modified Poisson–Boltzmann equations for characterizing biomolecular solvation

Journal of Theoretical and Computational Chemistry ◽

10.1142/s021963361440001x ◽

2014 ◽

Vol 13 (03) ◽

pp. 1440001 ◽

Cited By ~ 8

Author(s):

Patrice Koehl ◽

Frederic Poitevin ◽

Henri Orland ◽

Marc Delarue

Keyword(s):

Electrostatic Interactions ◽

Degrees Of Freedom ◽

Lattice Gas ◽

Homogeneous Medium ◽

Variable Density ◽

Solvent Density ◽

Poisson Boltzmann ◽

Poisson Boltzmann Equation ◽

High Dielectric ◽

Solvation Free Energies

Methods for computing electrostatic interactions often account implicitly for the solvent, due to the much smaller number of degrees of freedom involved. In the Poisson–Boltzmann (PB) approach the electrostatic potential is obtained by solving the Poisson–Boltzmann equation (PBE), where the solvent region is modeled as a homogeneous medium with a high dielectric constant. PB however is not exempt of problems. It does not take into account for example the sizes of the ions in the atmosphere surrounding the solute, nor does it take into account the inhomogeneous dielectric response of water due to the presence of a highly charged surface. In this paper we review two major modifications of PB that circumvent these problems, namely the size-modified PB (SMPB) equation and the Dipolar Poisson–Boltzmann Langevin (DPBL) model. In SMPB, steric effects between ions are accounted for with a lattice gas model. In DPBL, the solvent region is no longer modeled as a homogeneous dielectric media but rather as an assembly of self-orienting interacting dipoles of variable density. This model results in a dielectric profile that transits smoothly from the solute to the solvent region as well as in a variable solvent density that depends on the charges of the solute. We show successful applications of the DPBL formalism to computing the solvation free energies of isolated ions in water. Further developments of more accurately modified PB models are discussed.

Download Full-text

Accurate estimation of electrostatic binding energy with Poisson-Boltzmann equation solver DelPhi program

Journal of Theoretical and Computational Chemistry ◽

10.1142/s0219633616500711 ◽

2016 ◽

Vol 15 (08) ◽

pp. 1650071

Author(s):

Anbang Li ◽

Kaifu Gao

Keyword(s):

Boltzmann Equation ◽

Binding Energy ◽

Binding Free Energy ◽

Molecular Surface ◽

Accurate Estimation ◽

Free Energies ◽

Grid Spacing ◽

Electrostatic Binding ◽

Poisson Boltzmann ◽

Poisson Boltzmann Equation

Poisson–Boltzmann (PB) model is a widely used implicit solvent approximation in biophysical modeling because of its ability to provide accurate and reliable PB electrostatic salvation free energies ([Formula: see text] as well as electrostatic binding free energy ([Formula: see text] estimations. However, a recent study has warned that the 0.5[Formula: see text]Å grid spacing which is normally adopted can produce unacceptable errors in [Formula: see text] estimation with the solvent excluded surface (SES) (Harris RC, Boschitsch AH and Fenley MO, Influence of grid spacing in Poisson–Boltzmann equation binding energy estimation, J Chem Theory Comput 19: 3677–3685, 2013). In this work, we investigate the grid dependence of the widely used PB solver DelPhi v6.2 with molecular surface (MS) for estimating both electrostatic solvation free energies and electrostatic binding free energies. Our results indicate that, for the molecular complex and components the absolute errors of [Formula: see text] are smaller than that of [Formula: see text], and grid spacing of 0.8[Formula: see text]Å with DelPhi program ensures the accuracy and reliability of [Formula: see text]; however, the accuracy of [Formula: see text] largely relies on the order of magnitude of [Formula: see text] itself rather than that of [Formula: see text] or [Formula: see text]. Our findings suggest that grid spacing of 0.5[Formula: see text]Å is enough to produce accurate [Formula: see text] for molecules whose [Formula: see text] are large, but finer grids are needed when [Formula: see text] is very small.

Download Full-text

Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation

Communications in Computational Physics ◽

10.4208/cicp.081009.130611a ◽

2012 ◽

Vol 11 (1) ◽

pp. 179-214 ◽

Cited By ~ 25

Author(s):

M. Holst ◽

J.A. McCammon ◽

Z. Yu ◽

Y.C. Zhou ◽

Y. Zhu

Keyword(s):

Finite Element ◽

Boltzmann Equation ◽

Geometric Model ◽

A Priori ◽

Molecular Surface ◽

Adaptive Finite Element Method ◽

Adaptive Finite Element ◽

Regularization Technique ◽

Poisson Boltzmann ◽

Poisson Boltzmann Equation

AbstractWe consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a prioriL∞ estimates. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEMalgorithmis also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.

Download Full-text

Between Algorithm and Model: Different Molecular Surface Definitions for the Poisson-Boltzmann Based Electrostatic Characterization of Biomolecules in Solution

Communications in Computational Physics ◽

10.4208/cicp.050711.111111s ◽

2013 ◽

Vol 13 (1) ◽

pp. 61-89 ◽

Cited By ~ 30

Author(s):

Sergio Decherchi ◽

Jose Colmenares ◽

Chiara Eva Catalano ◽

Michela Spagnuolo ◽

Emil Alexov ◽

...

Keyword(s):

Surface Area ◽

Geometric Analysis ◽

Point Of View ◽

Molecular Surface ◽

Implicit Solvent ◽

Computationally Efficient ◽

Molecular Graphics ◽

Biomolecular Systems ◽

Explicit Solvent ◽

Poisson Boltzmann

AbstractThe definition of a molecular surface which is physically sound and computationally efficient is a very interesting and long standing problem in the implicit solvent continuum modeling of biomolecular systems as well as in the molecular graphics field. In this work, two molecular surfaces are evaluated with respect to their suitability for electrostatic computation as alternatives to the widely used Connolly-Richards surface: the blobby surface, an implicit Gaussian atom centered surface, and the skin surface. As figures of merit, we considered surface differentiability and surface area continuity with respect to atom positions, and the agreement with explicit solvent simulations. Geometric analysis seems to privilege the skin to the blobby surface, and points to an unexpected relationship between the non connectedness of the surface, caused by interstices in the solute volume, and the surface area dependence on atomic centers. In order to assess the ability to reproduce explicit solvent results, specific software tools have been developed to enable the use of the skin surface in Poisson-Boltzmann calculations with the DelPhi solver. The results indicate that the skin and Connolly surfaces have a comparable performance from this last point of view.

Download Full-text

Fully implicit ADI schemes for solving the nonlinear Poisson-Boltzmann equation

Computational and Mathematical Biophysics ◽

10.2478/mlbmb-2013-0006 ◽

2013 ◽

Vol 1 ◽

pp. 109-123 ◽

Cited By ~ 6

Author(s):

Weihua Geng ◽

Shan Zhao

Keyword(s):

Blow Up ◽

Operator Splitting ◽

Dielectric Constants ◽

Molecular Surface ◽

Diffusion Term ◽

Tridiagonal Matrices ◽

Fully Implicit ◽

Poisson Boltzmann ◽

Adi Schemes ◽

Poisson Boltzmann Equation

AbstractThe Poisson-Boltzmann (PB) model is an effective approach for the electrostatics analysis of solvated biomolecules. The nonlinearity associated with the PB equation is critical when the underlying electrostatic potential is strong, but is extremely difficult to solve numerically. In this paper, we construct two operator splitting alternating direction implicit (ADI) schemes to efficiently and stably solve the nonlinear PB equation in a pseudo-transient continuation approach. The operator splitting framework enables an analytical integration of the nonlinear term that suppresses the nonlinear instability. A standard finite difference scheme weighted by piecewise dielectric constants varying across the molecular surface is employed to discretize the nonhomogeneous diffusion term of the nonlinear PB equation, and yields tridiagonal matrices in the Douglas and Douglas-Rachford type ADI schemes. The proposed time splitting ADI schemes are different from all existing pseudo-transient continuation approaches for solving the classical nonlinear PB equation in the sense that they are fully implicit. In a numerical benchmark example, the steady state solutions of the fully-implicit ADI schemes based on different initial values all converge to the time invariant analytical solution, while those of the explicit Euler and semi-implicit ADI schemes blow up when the magnitude of the initial solution is large. For the solvation analysis in applications to real biomolecules with various sizes, the time stability of the proposed ADI schemes can be maintained even using very large time increments, demonstrating the efficiency and stability of the present methods for biomolecular simulation.

Download Full-text

Enhanced performance of supercapacitors by constructing a “mini parallel-plate capacitor” in an electrode with high dielectric constant materials

Journal of Materials Chemistry A ◽

10.1039/d0ta04364h ◽

2020 ◽

Vol 8 (32) ◽

pp. 16661-16668

Author(s):

Huayao Tu ◽

Shouzhi Wang ◽

Hehe Jiang ◽

Zhenyan Liang ◽

Dong Shi ◽

...

Keyword(s):

Dielectric Constant ◽

Carbon Fiber ◽

Parallel Plate ◽

Sandwich Structure ◽

High Dielectric Constant ◽

Parallel Plate Capacitor ◽

Plate Capacitor ◽

Fiber Metal ◽

High Dielectric ◽

Mini Plate

The carbon fiber/metal oxide/metal oxynitride layer sandwich structure is constructed in the electrode to form a mini-plate capacitor. High dielectric constant metal oxides act as dielectric to increase their capacitance.

Download Full-text