A hybrid solver of size modified Poisson–Boltzmann equation by domain decomposition, finite element, and finite difference

Ying, Jinyong; Xie, Dexuan

doi:10.1016/j.apm.2017.09.026

Cited by 12 publications

(15 citation statements)

References 39 publications

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“…For a complex C consisting of molecules A and B , the binding free energy E b ( I s ) is defined by

E_{b} (I_{s}) = E (C, I_{s}) - E (A, I_{s}) - E (B, I_{s}),

where E ( X , I s ) denotes an electrostatic solvation free energy of molecule X in an ionic solvent with ionic strength I s , which is defined in . Following the description in Ying and Xie, E b can be transformed to a linear function of ξ by the variable change

ξ = \ln I_{s}

as follows

E_{b} = m ξ + b,

where m and b are two constants to be determined. Then a scaled slope m s , which is called the binding affinity to measure the salt dependence, is defined by

m_{s} = - m / (N_{A} k_{B} T) .

…”

Section: Numerical Resultsmentioning

confidence: 99%

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An accelerated nonlocal Poisson‐Boltzmann equation solver for electrostatics of biomolecule

Ying

Xie

2018

Numer Methods Biomed Eng

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The nonlocal modified Poisson-Boltzmann equation (NMPBE) is one important variant of a commonly used dielectric continuum model, the Poisson-Boltzmann equation (PBE), for computing electrostatics of biomolecules. In this paper, an accelerated NMPBE solver is constructed by finite element and finite difference hybrid techniques. It is then programmed as a software package for computing electrostatic solvation and binding free energies for a protein in a symmetric 1:1 ionic solvent. Numerical results validate the new solver and its numerical stability. They also demonstrate that the new solver has much better performance than the corresponding finite element solver in terms of computer CPU time. Furthermore, they show that the binding free energies produced by NMPBE can match chemical experiment data better than those by PBE.

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“…For a complex C consisting of molecules A and B , the binding free energy E b ( I s ) is defined by

E_{b} (I_{s}) = E (C, I_{s}) - E (A, I_{s}) - E (B, I_{s}),

ξ = \ln I_{s}

as follows

E_{b} = m ξ + b,

where m and b are two constants to be determined. Then a scaled slope m s , which is called the binding affinity to measure the salt dependence, is defined by

m_{s} = - m / (N_{A} k_{B} T) .

…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Using our new solver as a simulation tool, we plan to perform more studies on NMPBE and its applications. Since ionic size effects may significantly affect the calculation of ionic concentrations, we will follow what we did for the size‐modified PBE to develop size‐modified NMPBE models and their efficient numerical solvers in the future.…”

Section: Resultsmentioning

confidence: 99%

An accelerated nonlocal Poisson‐Boltzmann equation solver for electrostatics of biomolecule

Ying

Xie

2018

Numer Methods Biomed Eng

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“…Moreover, as shown in Refs. [ ], SMPBE can yield more physically reasonable ionic concentrations than PBE.…”

Section: Smpbe Definition and Applicationsmentioning

confidence: 99%

“…[ ] and a finite element and finite difference hybrid solver reported in Ref. [ ] have been adopted to solve SMPBE numerically. To avoid the singularity difficulties caused by the atomic point charges from the protein, these two solvers have been developed using the following solution decomposition:

u = G + Ψ + \overset{Φ}{true},

where G is given by the analytical expression

G (r) = \frac{10^{10} e_{c}^{2}}{4 π ɛ_{p} ɛ_{0} k_{B} T} \sum_{j = 1}^{n_{p}} \frac{z_{j}}{| r - r_{j} |},

…”

Section: Smpbe Definition and Applicationsmentioning

confidence: 99%

“…Recently, we developed two efficient SMPBE solvers [21,22] as the extensions of our fast PBE finite element solver [23] and our PBE finite element and finite difference hybrid solver, [24] respectively. Our PBE and SMPBE solvers were constructed using advanced numerical techniques such as finite element, solution decomposition, domain decomposition, and multigrid preconditioning.…”

Section: Introductionmentioning

confidence: 99%

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SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson‐Boltzmann equation

2017

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SMPBS (Size Modified Poisson-Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson-Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson-Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile-friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware-accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc.

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A stabilized finite volume element method for solving Poisson–Nernst–Planck equations

Ying

2021

Numer Methods Biomed Eng

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One difficulty in solving the Poisson-Nernst-Planck (PNP) equations used for studying the ion transport in channel proteins is the possible convectiondominant problem in the Nernst-Planck equations. In this paper, to overcome this issue, considering the general mixed boundary conditions of concentration functions on the interface, a novel stabilized finite volume element method based on the standard weak formulation to solve the steady-state PNP equations is proposed and analyzed. Numerical tests on four ion-channel proteins served as benchmark with varying boundary conditions in a certain range show that the new stabilized technique not only improves the robustness of the new PNP solver, but also makes the computed (especially the maximal) concentration values much more reasonable.

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A hybrid solver of size modified Poisson–Boltzmann equation by domain decomposition, finite element, and finite difference

Cited by 12 publications

References 39 publications

An accelerated nonlocal Poisson‐Boltzmann equation solver for electrostatics of biomolecule

An accelerated nonlocal Poisson‐Boltzmann equation solver for electrostatics of biomolecule

SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson‐Boltzmann equation

A stabilized finite volume element method for solving Poisson–Nernst–Planck equations

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