A Modified Nonlocal Continuum Electrostatic Model for Protein in Water and Its Analytical Solutions for Ionic Born Models

Xie, Dexuan; Volkmer, Hans

doi:10.4208/cicp.170811.211011s

Cited by 11 publications

(22 citation statements)

References 28 publications

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“…For the protein case, we have shown in our previous work [18] that ε can be modified as a function of two variables r and r ′ in the form…”

Section: Two Nonlocal Poisson Dielectric Test Modelsmentioning

confidence: 99%

“…We set u = Φ * Q λ , Φ p = Φ and u p = u in D p , and Φ s = Φ and u s = u in D s . As shown in [18], we can reformulate the model (15) as the PDE system…”

Section: Analytical Solution Of Modelmentioning

confidence: 99%

“…Hence, the series expressions of Φ j and Φ j * Q λ can be produced from (28), (29), (30), and (31) by substituting r and cos φ to |r| and cos r j , r , respectively. Since cos r j , r can be calculated by cos r j , r = r j · r |r j ||r| , we obtain the series expressions (18) and (19) of Φ j and convolution Φ j * Q λ . This completes the proof.…”

Section: Analytical Solution Of Modelmentioning

confidence: 99%

“…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]. Meanwhile, to validate our finite element algorithms and program packages for solving these nonlocal models, the analytical solutions of three nonlocal Born ball test models (including the Lorentz nonlocal model) were obtained in [18].…”

Section: Introductionmentioning

confidence: 99%

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

2016

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The nonlocal dielectric approach has led to new models and solvers for predicting electrostatics of proteins (or other biomolecules), but how to validate and compare them remains a challenge. To promote such a study, in this paper, two typical nonlocal dielectric models are revisited. Their analytical solutions are then found in the expressions of simple series for a dielectric sphere containing any number of point charges. As a special case, the analytical solution of the corresponding Poisson dielectric model is also derived in simple series, which significantly improves the well known Kirkwood's double series expansion. Furthermore, a convolution of one nonlocal dielectric solution with a commonly-used nonlocal kernel function is obtained, along with the reaction parts of these local and nonlocal solutions. To turn these new series solutions into a valuable research tool, they are programmed as a free Fortran software package, which can input point charge data directly from a protein data bank file. Consequently, different validation tests can be quickly done on different proteins. Finally, a test example for a protein with 488 atomic charges is reported to demonstrate the differences between the local and nonlocal models as well as the importance of using the reaction parts to develop local and nonlocal dielectric solvers.

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“…For the protein case, we have shown in our previous work [18] that ε can be modified as a function of two variables r and r ′ in the form…”

Section: Two Nonlocal Poisson Dielectric Test Modelsmentioning

confidence: 99%

“…We set u = Φ * Q λ , Φ p = Φ and u p = u in D p , and Φ s = Φ and u s = u in D s . As shown in [18], we can reformulate the model (15) as the PDE system…”

Section: Analytical Solution Of Modelmentioning

confidence: 99%

Section: Analytical Solution Of Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

2016

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“…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.…”

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confidence: 99%

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

Xie¹,

Jiang²,

Scott³

2013

SIAM J. Sci. Comput.

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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

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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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A Modified Nonlocal Continuum Electrostatic Model for Protein in Water and Its Analytical Solutions for Ionic Born Models

Cited by 11 publications

References 28 publications

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

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

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

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

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