The Experts below are selected from a list of 264 Experts worldwide ranked by ideXlab platform
Marcia O. Fenley - One of the best experts on this subject based on the ideXlab platform.
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Sensitivities to parameterization in the size-modified Poisson-Boltzmann Equation
The Journal of chemical physics, 2014Co-Authors: Robert C. Harris, Alexander H. Boschitsch, Marcia O. FenleyAbstract:Experimental results have demonstrated that the numbers of counterions surrounding nucleic acids differ from those predicted by the nonlinear Poisson-Boltzmann Equation, NLPBE. Some studies have fit these data against the ion size in the size-modified Poisson-Boltzmann Equation, SMPBE, but the present study demonstrates that other parameters, such as the Stern layer thickness and the molecular surface definition, can change the number of bound ions by amounts comparable to varying the ion size. These parameters will therefore have to be fit simultaneously against experimental data. In addition, the data presented here demonstrate that the derivative, SK, of the electrostatic binding free energy, ΔGel, with respect to the logarithm of the salt concentration is sensitive to these parameters, and experimental measurements of SK could be used to parameterize the model. However, although better values for the Stern layer thickness and ion size and better molecular surface definitions could improve the model's predictions of the numbers of ions around biomolecules and SK, ΔGel itself is more sensitive to parameters, such as the interior dielectric constant, which in turn do not significantly affect the distributions of ions around biomolecules. Therefore, improved estimates of the ion size and Stern layer thickness to use in the SMPBE will not necessarily improve the model's predictions of ΔGel.
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Comparing the Predictions of the Nonlinear Poisson-Boltzmann Equation and the Ion Size-Modified Poisson-Boltzmann Equation for a Low-Dielectric Charged Spherical Cavity in an Aqueous Salt Solution.
Journal of chemical theory and computation, 2010Co-Authors: Alexander R.j. Silalahi, Alexander H. Boschitsch, Robert C. Harris, Marcia O. FenleyAbstract:The ion size-modified Poisson−Boltzmann Equation (SMPBE) is applied to the simple model problem of a low-dielectric spherical cavity containing a central charge in an aqueous salt solution to investigate the finite ion size effect upon the electrostatic free energy and its sensitivity to changes in salt concentration. The SMPBE is shown to predict a very different electrostatic free energy than the nonlinear Poisson−Boltzmann Equation (NLPBE) due to the additional entropic cost of placing ions in solution. Although the energy predictions of the SMPBE can be reproduced by fitting an appropriately sized Stern layer, or ion-exclusion layer to the NLPBE calculations, the size of the Stern layer is difficult to estimate a priori. The SMPBE also produces a saturation layer when the central charge becomes sufficiently large. Ion competition effects on various integrated quantities, such the total number of ions predicted by the SMPBE, are qualitatively similar to those given by the NLPBE and those found in avail...
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Fast Boundary Element Method for the Linear Poisson-Boltzmann Equation
The Journal of Physical Chemistry B, 2002Co-Authors: Alexander H. Boschitsch, Marcia O. Fenley, Huan-xiang ZhouAbstract:This article summarizes the development of a fast boundary element method for the linear Poisson-Boltzmann Equation governing biomolecular electrostatics. Unlike previous fast boundary element implementations, the present treatment accommodates finite salt concentrations thus enabling the study of biomolecular electrostatics under realistic physiological conditions. This is achieved by using multipole expansions specifically designed for the exponentially decaying Green’s function of the linear Poisson -Boltzmann Equation. The particular formulation adopted in the boundary element treatment directly affects the numerical conditioning and thus convergence behavior of the method. Therefore, the formulation and reasons for its choice are first presented. Next, the multipole approximation and its use in the context of a fast boundary element method are described together with the iteration method employed to extract the surface distributions. The method is then subjected to a series of computational tests involving a sphere with interior charges. The purpose of these tests is to assess accuracy and verify the anticipated computational performance trends. Finally, the salt dependence of electrostatic properties of several biomolecular systems (alanine dipeptide, barnase, barstar, and coiled coil tetramer) is examined with the method and the results are compared with finite difference Poisson-Boltzmann codes.
Dexuan Xie - One of the best experts on this subject based on the ideXlab platform.
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On the analysis and application of an ion size-modified Poisson–Boltzmann Equation
Nonlinear Analysis: Real World Applications, 2019Co-Authors: Jinyong Ying, Dexuan XieAbstract:Abstract In this paper, an improved electrostatic free energy functional is presented as an extension of the one proposed in Xie and Li (2015) to reflect ion size effects. It is then shown to have a unique minimizer, resulting in the solution existence and uniqueness of one commonly-used ion size-modified Poisson–Boltzmann Equation (SMPBE). As for applications, SMPBE is used to calculate the electrostatic solvation free energy with the new derived well-defined formula and simulate an electric double layer numerically to demonstrate the advantage of SMPBE over the classic Poisson–Boltzmann Equation in the prediction of ionic concentrations.
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An accelerated nonlocal Poisson-Boltzmann Equation solver for electrostatics of biomolecule.
International journal for numerical methods in biomedical engineering, 2018Co-Authors: Jinyong Ying, Dexuan XieAbstract: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 nonlocal modified Poisson–Boltzmann Equation and finite element solver for computing electrostatics of biomolecules
Journal of Computational Physics, 2016Co-Authors: Dexuan Xie, Yi JiangAbstract:Abstract The nonlocal dielectric approach has been studied for more than forty years but only limited to water solvent until the recent work of Xie et al. (2013) [20] . As the development of this recent work, in this paper, a nonlocal modified Poisson–Boltzmann Equation (NMPBE) is proposed to incorporate nonlocal dielectric effects into the classic Poisson–Boltzmann Equation (PBE) for protein in ionic solvent. The focus of this paper is to present an efficient finite element algorithm and a related software package for solving NMPBE. Numerical results are reported to validate this new software package and demonstrate its high performance for protein molecules. They also show the potential of NMPBE as a better predictor of electrostatic solvation and binding free energies than PBE.
Michael Holst - One of the best experts on this subject based on the ideXlab platform.
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Goal-Oriented Adaptivity and Multilevel Preconditioning for the Poisson-Boltzmann Equation
arXiv: Numerical Analysis, 2011Co-Authors: Burak Aksoylu, Stephen D. Bond, Eric C. Cyr, Michael HolstAbstract:In this article, we develop goal-oriented error indicators to drive adaptive refinement algorithms for the Poisson-Boltzmann Equation. Empirical results for the solvation free energy linear functional demonstrate that goal-oriented indicators are not sufficient on their own to lead to a superior refinement algorithm. To remedy this, we propose a problem-specific marking strategy using the solvation free energy computed from the solution of the linear regularized Poisson-Boltzmann Equation. The convergence of the solvation free energy using this marking strategy, combined with goal-oriented refinement, compares favorably to adaptive methods using an energy-based error indicator. Due to the use of adaptive mesh refinement, it is critical to use multilevel preconditioning in order to maintain optimal computational complexity. We use variants of the classical multigrid method, which can be viewed as generalizations of the hierarchical basis multigrid and Bramble-Pasciak-Xu (BPX) preconditioners.
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Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation
arXiv: Numerical Analysis, 2010Co-Authors: Michael Holst, James Andrew Mccammon, Y. C. Zhou, Yunrong ZhuAbstract:We develop an efficient and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann Equation (PBE). We first examine the regularization technique of Chen, Holst, and Xu; this technique made possible the first a priori pointwise estimates and the first complete solution and approximation theory for the Poisson-Boltzmann Equation. It also made possible the first provably convergent discretization of the PBE, and allowed for the development of a provably convergent AFEM for the PBE. However, in practice the regularization turns out to be numerically ill-conditioned. In this article, we examine a second regularization, and establish a number of basic results to ensure that the new approach produces the same mathematical advantages of the original regularization, without the ill-conditioning property. We then design an AFEM scheme based on 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 priori pointwise estimates to establish quasi-orthogonality. 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. The stability advantages of the new regularization are demonstrated using an FETK-based implementation, through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.
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The Finite Element Approximation of the Nonlinear Poisson-Boltzmann Equation
arXiv: Numerical Analysis, 2010Co-Authors: Long Chen, Michael Holst, Jinchao XuAbstract:A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson-Boltzmann Equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson-Boltzmann Equation is introduced as an auxiliary problem, making it possible to study the original nonlinear Equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson-Boltzmann Equation based on certain quasi-uniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson-Boltzmann Equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson-Boltzmann Equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson-Boltzmann Equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the Equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.
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protein electrostatics rapid multigrid based newton algorithm for solution of the full nonlinear poisson boltzmann Equation
Journal of Biomolecular Structure & Dynamics, 1994Co-Authors: Michael Holst, Richard E Kozack, Faisal Saied, Shankar SubramaniamAbstract:Abstract A new method for solving the full nonlinear Poisson-Boltzmann Equation is outlined. This method is robust and efficient, and uses a combination of the multigrid and inexact Newton algorithms. The novelty of this approach lies in the appropriate combination of the two methods, neither of which by themselves are capable of solving the nonlinear problem accurately. Features of the Poisson-Boltzmann Equation are fully exploited by each component of the hybrid algorithm to provide robustness and speed. The advantages inherent in this method increase with the size of the problem. The efficacy of the method is illustrated by calculations of the electrostatic potential around the enzyme Superoxide Dismutase. The CPU time required to solve the full nonlinear Equation is less than half that needed for a conjugate gradient solution of the corresponding linearized Poisson-Boltzmann Equation. The solutions reveal that the field around the active sites is significantly reduced as compared to that obtained by s...
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Multigrid solution of the Poisson-Boltzmann Equation
Journal of Computational Chemistry, 1993Co-Authors: Michael Holst, Faisal SaiedAbstract:A multigrid method is presented for the numerical solution of the linearized Poisson–Boltzmann Equation arising in molecular biophysics. The Equation is discretized with the finite volume method, and the numerical solution of the discrete Equations is accomplished with multiple grid techniques originally developed for twodimensional interface problems occurring in reactor physics. A detailed analysis of the resulting method is presented for several computer architectures, including comparisons to diagonally scaled CG, ICCG, vectorized ICCG and MICCG, and to SOR provided with an optimal relaxation parameter. Our results indicate that the multigrid method is superior to the preconditioned CG methods and SOR and that the advantage of multigrid grows with the problem size. © 1993 John Wiley & Sons, Inc.
Jehanzeb H. Chaudhry - One of the best experts on this subject based on the ideXlab platform.
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A Posteriori Analysis and Efficient Refinement Strategies for the Poisson--Boltzmann Equation
SIAM Journal on Scientific Computing, 2018Co-Authors: Jehanzeb H. ChaudhryAbstract:The Poisson--Boltzmann Equation (PBE) models the electrostatic interactions of charged bodies such as molecules and proteins in an electrolyte solvent. The PBE is a challenging Equation to solve nu...
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A Posteriori Analysis and Efficient Refinement Strategies for the Poisson-Boltzmann Equation
arXiv: Numerical Analysis, 2017Co-Authors: Jehanzeb H. ChaudhryAbstract:The Poisson-Boltzmann Equation (PBE) models the electrostatic interactions of charged bodies such as molecules and proteins in an electrolyte solvent. The PBE is a challenging Equation to solve numerically due to the presence of singularities, discontinuous coefficients and boundary conditions. Hence, there is often large error in the numerical solution of the PBE that needs to be quantified. In this work, we use adjoint based a posteriori analysis to accurately quantify the error in an important quantity of interest, the solvation free energy, for the finite element solution of the PBE. We identify various sources of error and propose novel refinement strategies based on a posteriori error estimates.
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Finite Element Approximation to a Finite-Size Modified Poisson-Boltzmann Equation
Journal of Scientific Computing, 2010Co-Authors: Jehanzeb H. Chaudhry, Stephen D. Bond, Luke N. OlsonAbstract:The inclusion of steric effects is important when determining the electrostatic potential near a solute surface. We consider a modified form of the Poisson-Boltzmann Equation, often called the Poisson-Bikerman Equation, in order to model these effects. The modifications lead to bounded ionic concentration profiles and are consistent with the Poisson-Boltzmann Equation in the limit of zero-size ions. Moreover, the modified Equation fits well into existing finite element frameworks for the Poisson-Boltzmann Equation. In this paper, we advocate a wider use of the modified Equation and establish well-posedness of the weak problem along with convergence of an associated finite element formulation. We also examine several practical considerations such as conditioning of the linearized form of the nonlinear modified Poisson-Boltzmann Equation, implications in numerical evaluation of the modified form, and utility of the modified Equation in the context of the classical Poisson-Boltzmann Equation.
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A first-order system least-squares finite element method for the Poisson-Boltzmann Equation.
Journal of computational chemistry, 2010Co-Authors: Stephen D. Bond, Jehanzeb H. Chaudhry, Eric C. Cyr, Luke N. OlsonAbstract:The Poisson-Boltzmann Equation is an important tool in modeling solvent in biomolecular systems. In this article, we focus on numerical approximations to the electrostatic potential expressed in the regularized linear Poisson-Boltzmann Equation. We expose the flux directly through a first-order system form of the Equation. Using this formulation, we propose a system that yields a tractable least-squares finite element formulation and establish theory to support this approach. The least-squares finite element approximation naturally provides an a posteriori error estimator and we present numerical evidence in support of the method. The computational results highlight optimality in the case of adaptive mesh refinement for a variety of molecular configurations. In particular, we show promising performance for the Born ion, Fasciculin 1, methanol, and a dipole, which highlights robustness of our approach. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010
Alexander H. Boschitsch - One of the best experts on this subject based on the ideXlab platform.
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Sensitivities to parameterization in the size-modified Poisson-Boltzmann Equation
The Journal of chemical physics, 2014Co-Authors: Robert C. Harris, Alexander H. Boschitsch, Marcia O. FenleyAbstract:Experimental results have demonstrated that the numbers of counterions surrounding nucleic acids differ from those predicted by the nonlinear Poisson-Boltzmann Equation, NLPBE. Some studies have fit these data against the ion size in the size-modified Poisson-Boltzmann Equation, SMPBE, but the present study demonstrates that other parameters, such as the Stern layer thickness and the molecular surface definition, can change the number of bound ions by amounts comparable to varying the ion size. These parameters will therefore have to be fit simultaneously against experimental data. In addition, the data presented here demonstrate that the derivative, SK, of the electrostatic binding free energy, ΔGel, with respect to the logarithm of the salt concentration is sensitive to these parameters, and experimental measurements of SK could be used to parameterize the model. However, although better values for the Stern layer thickness and ion size and better molecular surface definitions could improve the model's predictions of the numbers of ions around biomolecules and SK, ΔGel itself is more sensitive to parameters, such as the interior dielectric constant, which in turn do not significantly affect the distributions of ions around biomolecules. Therefore, improved estimates of the ion size and Stern layer thickness to use in the SMPBE will not necessarily improve the model's predictions of ΔGel.
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Comparing the Predictions of the Nonlinear Poisson-Boltzmann Equation and the Ion Size-Modified Poisson-Boltzmann Equation for a Low-Dielectric Charged Spherical Cavity in an Aqueous Salt Solution.
Journal of chemical theory and computation, 2010Co-Authors: Alexander R.j. Silalahi, Alexander H. Boschitsch, Robert C. Harris, Marcia O. FenleyAbstract:The ion size-modified Poisson−Boltzmann Equation (SMPBE) is applied to the simple model problem of a low-dielectric spherical cavity containing a central charge in an aqueous salt solution to investigate the finite ion size effect upon the electrostatic free energy and its sensitivity to changes in salt concentration. The SMPBE is shown to predict a very different electrostatic free energy than the nonlinear Poisson−Boltzmann Equation (NLPBE) due to the additional entropic cost of placing ions in solution. Although the energy predictions of the SMPBE can be reproduced by fitting an appropriately sized Stern layer, or ion-exclusion layer to the NLPBE calculations, the size of the Stern layer is difficult to estimate a priori. The SMPBE also produces a saturation layer when the central charge becomes sufficiently large. Ion competition effects on various integrated quantities, such the total number of ions predicted by the SMPBE, are qualitatively similar to those given by the NLPBE and those found in avail...
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Fast Boundary Element Method for the Linear Poisson-Boltzmann Equation
The Journal of Physical Chemistry B, 2002Co-Authors: Alexander H. Boschitsch, Marcia O. Fenley, Huan-xiang ZhouAbstract:This article summarizes the development of a fast boundary element method for the linear Poisson-Boltzmann Equation governing biomolecular electrostatics. Unlike previous fast boundary element implementations, the present treatment accommodates finite salt concentrations thus enabling the study of biomolecular electrostatics under realistic physiological conditions. This is achieved by using multipole expansions specifically designed for the exponentially decaying Green’s function of the linear Poisson -Boltzmann Equation. The particular formulation adopted in the boundary element treatment directly affects the numerical conditioning and thus convergence behavior of the method. Therefore, the formulation and reasons for its choice are first presented. Next, the multipole approximation and its use in the context of a fast boundary element method are described together with the iteration method employed to extract the surface distributions. The method is then subjected to a series of computational tests involving a sphere with interior charges. The purpose of these tests is to assess accuracy and verify the anticipated computational performance trends. Finally, the salt dependence of electrostatic properties of several biomolecular systems (alanine dipeptide, barnase, barstar, and coiled coil tetramer) is examined with the method and the results are compared with finite difference Poisson-Boltzmann codes.
