Second-order Poisson–Nernst–Planck solver for ion transport

Zheng, Qiong; Chen, Duan; Wei, Guo-Wei

doi:10.1016/j.jcp.2011.03.020

Cited by 113 publications

(144 citation statements)

References 72 publications

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“…(11) and (17). The PBNP system differs from the original PNP system in the sense that only the ions of interest are described by the Nernst-Planck equation while others are described by the Boltzmann distribution.…”

Section: B Poisson-boltzmann-nernst-planck (Pbnp) Modelmentioning

confidence: 99%

“…For the closely related PNP equations, a second order accurate scheme has not been developed until very recently for realistic biomolecular systems. 17 Here, "second order accurate" means that when the mesh size is halved, the accuracy increases by a factor of 2 2 times. Although achieving this factor may appear easy in the solution of many other partial differential equations, it is a severe challenge in solving the PBNP equations for realistic biomolecular systems.…”

Section: Numerical Algorithmsmentioning

confidence: 99%

“…Therefore, PNP theory has gained much attention in the sense that it provides a general framework to successfully describe the essential ionic transport phenomenon in electrochemistry. 17 It offers reasonable predictions for many applications and its limitations in some specific applications, such as ion channel systems, can be improved by simply adjusting the diffusion coefficients, or by considering additional force contributions in the ion flux term of the Nernst-Planck equation. Despite many limitations, the PNP theory is still one of the most wellestablished models in this field.…”

Section: Introductionmentioning

confidence: 99%

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Poisson–Boltzmann–Nernst–Planck model

Zheng

Wei

2011

The Journal of Chemical Physics

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141

111

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The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external voltages. Extensive numerical experiments show that there is an excellent consistency between the results predicted from the present PBNP model and those obtained from the PNP model in terms of the electrostatic potentials, ion concentration profiles, and current-voltage (I-V) curves. The present PBNP model is further validated by a comparison with experimental measurements of I-V curves under various ion bulk concentrations. Numerical experiments indicate that the proposed PBNP model is more efficient than the original PNP model in terms of simulation time.

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Section: B Poisson-boltzmann-nernst-planck (Pbnp) Modelmentioning

confidence: 99%

Section: Numerical Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Poisson–Boltzmann–Nernst–Planck model

Zheng

Wei

2011

The Journal of Chemical Physics

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141

111

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“…The proposed high-order fractional PDEs are applied to the surface construction of macromolecular surfaces, which are crucial components in the implicit solvent models [24], charge transport models [74] and variational multiscale models [60, 62]. We consider three types of initial values to study the proposed high-order fractional PDEs.…”

Section: Discussionmentioning

confidence: 99%

“…Surface generation time is a bottleneck for many practical applications such as the Poisson-Boltzmann based molecular dynamics [23] and the solution of coupled nonlinear Poisson-Nernst-Planck equations [74, 75]. Here we examine the computational efficiency of the fractional PDE transform based biomolecular surface generation.…”

Section: Numerical Test and Validationmentioning

confidence: 99%

High-order fractional partial differential equation transform for molecular surface construction

Chen

Wei

2012

Computational and Mathematical Biophysics

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Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

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Quantum dynamics in continuum for proton transport II: Variational solvent–solute interface

Chen

Guo

2011

Numer Methods Biomed Eng

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Proton transport plays an important role in biological energy transduction and sensory systems. Therefore it has attracted much attention in biological science and biomedical engineering in the past few decades. The present work proposes a multiscale/multiphysics model for the understanding of the molecular mechanism of proton transport in transmembrane proteins involving continuum, atomic and quantum descriptions, assisted with the evolution, formation and visualization of membrane channel surfaces. We describe proton dynamics quantum mechanically via a new density functional theory based on the Boltzmann statistics, while implicitly model numerous solvent molecules as a dielectric continuum to reduce the number of degrees of freedom. The density of all other ions in the solvent is assumed to obey the Boltzmann distribution in a dynamic manner. The impact of protein molecular structure and its charge polarization on the proton transport is considered explicitly at the atomic scale. A variational solute-solvent interface is designed to separate the explicit molecule and implicit solvent regions. We formulate a total free energy functional to put proton kinetic and potential energies, the free energy of all other ions, the polar and nonpolar energies of the whole system on an equal footing. The variational principle is employed to derive coupled governing equations for the proton transport system. Generalized Laplace-Beltrami equation, generalized Poisson-Boltzmann equation and generalized Kohn-Sham equation are obtained from the present variational framework. The variational solvent-solute interface is generated and visualized to facilitate the multiscale discrete/continuum/quantum descriptions. Theoretical formulations for the proton density and conductance are constructed based on fundamental laws of physics. A number of mathematical algorithms, including the Dirichlet to Neumann mapping (DNM), matched interface and boundary (MIB) method, Gummel iteration, and Krylov space techniques are utilized to implement the proposed model in a computationally efficient manner. The Gramicidin A (GA) channel is used to validate the performance of the proposed proton transport model and demonstrate the efficiency of the proposed mathematical algorithms. The proton channel conductances are studied over a number of applied voltages and reference concentrations. A comparison with experimental data verifies the present model predictions and confirms the proposed model.

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Second-order Poisson–Nernst–Planck solver for ion transport

Cited by 113 publications

References 72 publications

Poisson–Boltzmann–Nernst–Planck model

Poisson–Boltzmann–Nernst–Planck model

High-order fractional partial differential equation transform for molecular surface construction

Quantum dynamics in continuum for proton transport II: Variational solvent–solute interface

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