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

Ying, Jinyong; Xie, Dexuan

doi:10.1002/cnm.3129

Cited by 5 publications

(4 citation statements)

References 23 publications

(87 reference statements)

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“…Same as the classic implicit continuum models, 10,15,19,20,22,24,36 in order to handle the singularities caused by the Dirac delta distributions, it is a common sense now to use the solution decomposition schemes to isolate the singularities. For the presentation purpose in this paper, we follow 10,15 to decompose the electrostatic potential u as follows…”

Section: The Dimensionless Pnp Equations and Its Solution Decompositionmentioning

confidence: 99%

u

as follows

u ((), r) goodbreak= ({, \begin{array}{c} G goodbreak+ u_{c} goodbreak+ normalΨ & if boldr \in Ω_{p}, \\ normalΨ & if boldr \in Ω_{s}, \end{array})

where

G

is given by

G ((), r) goodbreak= \frac{α}{4 π ɛ_{p}} \sum_{j = 1}^{n_{p}} \frac{z_{j}}{∥ boldr - r_{j} ∥},

u_{c}

is a solution of the following Laplace equation

({, \begin{array}{c} goodbreak- ɛ_{p} normalΔ u_{c} goodbreak= 0, in Ω_{p}, \\ u_{c} goodbreak= goodbreak- G, on normalΓ, \end{array})

and

normalΨ

together with

C_{k}

satisfies the following coupled equations

({, \begin{array}{c} goodbreak- ɛ_{p} ΔΨ ((), r) goodbreak= 0, & boldr \in \end{array})

…”

Section: The Dimensionless Pnp Equations and Its Solution Decompositionmentioning

confidence: 99%

See 1 more Smart Citation

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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Section: The Dimensionless Pnp Equations and Its Solution Decompositionmentioning

confidence: 99%

u

as follows

u ((), r) goodbreak= ({, \begin{array}{c} G goodbreak+ u_{c} goodbreak+ normalΨ & if boldr \in Ω_{p}, \\ normalΨ & if boldr \in Ω_{s}, \end{array})

where

G

is given by

G ((), r) goodbreak= \frac{α}{4 π ɛ_{p}} \sum_{j = 1}^{n_{p}} \frac{z_{j}}{∥ boldr - r_{j} ∥},

u_{c}

is a solution of the following Laplace equation

({, \begin{array}{c} goodbreak- ɛ_{p} normalΔ u_{c} goodbreak= 0, in Ω_{p}, \\ u_{c} goodbreak= goodbreak- G, on normalΓ, \end{array})

and

normalΨ

together with

C_{k}

satisfies the following coupled equations

({, \begin{array}{c} goodbreak- ɛ_{p} ΔΨ ((), r) goodbreak= 0, & boldr \in \end{array})

…”

Section: The Dimensionless Pnp Equations and Its Solution Decompositionmentioning

confidence: 99%

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

Ying

2021

Numer Methods Biomed Eng

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“…Yet the localization method, proposed originally by Hildebrandt [34,69] and later developed, enhanced and implemented by others [70][71][72], does rely on the standard differentiation rules for full-space convolutions (Section 2.3).…”

Section: The Domain Of Convolution Makes a Significant Differencementioning

confidence: 99%

“…An ingenious idea proposed by Hildebrandt et al [34,69] in the early 2000s and later enhanced and efficiently implemented by by Xie et al [70][71][72], allows one to convert nonlocal electrostatic problems to coupled local ones. This conversion is valid under the simplification assumptions noted below.…”

Section: The Hildebrandt Localizationmentioning

confidence: 99%

Trefftz Functions for Nonlocal Electrostatics

Tsukerman¹

2020

Preprint

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Electrostatic interactions in solvents play a major role in biophysical systems. There is a consensus in the literature that the dielectric response of aqueous solutions is nonlocal: polarization depends on the electric field not only at a given point but in the vicinity of that point as well. This is typically modeled via a convolution of the electric field with an appropriate integral kernel.A primary problem with nonlocal models is high computational cost. A secondary problem is restriction of convolution integrals to the solvent, as opposed to their evaluation over the whole space.The paper develops a computational tool alleviating the "curse of nonlocality" and helping to handle the integration correctly. This tool is Trefftz approximations, which tend to furnish much higher accuracy than traditional polynomial ones. In the paper, Trefftz approximations are developed for problems of nonlocal electrostatics, with the goal of numerically "localizing" the original nonlocal problem. This approach can be extended to nonlocal problems in other areas of computational mathematics, physics and engineering.

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A flux-jump preserved gradient recovery technique for accurately predicting the electrostatic field of an immersed biomolecule

Ying

2019

Journal of Computational Physics

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

Cited by 5 publications

References 23 publications

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

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

Trefftz Functions for Nonlocal Electrostatics

A flux-jump preserved gradient recovery technique for accurately predicting the electrostatic field of an immersed biomolecule

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