Treatment of electrostatic effects in proteins: Multigrid‐based newton iterative method for solution of the full nonlinear poisson–boltzmann equation

Holst, Michael; Kozack, Richard E.; Saied, Faisal; Subramaniam, Shankar

doi:10.1002/prot.340180304

Cited by 82 publications

(72 citation statements)

References 63 publications

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“…21 These methods rely on the linearization of the DebyeHückel term that accounts for counterions around the solute; it is known however that this approximation is not valid for highly charged systems. 22 Boschitsch and Fenley 23 proposed a correction to the BME methods to account for the nonlinearity in the PB equation. Clearly, we need a more general framework for solving PB equations, especially in light of the recent theoretical modifications.…”

Section: Introductionmentioning

confidence: 99%

AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

Koehl

Delarue

2010

The Journal of Chemical Physics

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The Poisson-Boltzmann ͑PB͒ formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin ͑DPBL͒ formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation ͑PDE͒. Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered ͑i.e., inside and outside of the molecules considered͒. While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied ͓J. Comput. Chem. 16, 337 ͑1995͔͒ to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available.

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Section: Introductionmentioning

confidence: 99%

AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

Koehl

Delarue

2010

The Journal of Chemical Physics

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“…For comparison with the FFT-based solution of the Poisson equation, one finite difference multigrid (FDMG) calculation was done using the PMG multigrid library [75][76][77][78] (http://www.fetk.org/) via the APBS software package [37] (http://apbs.sf.net/) to obtain the potential due to the solvent charge distribution around the uncharged protein. The charge density obtained from our simulations was padded on all sides to accommodate a 321 × 289 × 289 grid suitable for the multi-grid algorithm in APBS, but grid spacings were otherwise kept the same.…”

Section: Implicit Solvent Electrostatic Calculations Using the Finitementioning

confidence: 99%

Solvent reaction field potential inside an uncharged globular protein: A bridge between implicit and explicit solvent models?

Cerutti

Baker

McCammon

2007

The Journal of Chemical Physics

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The solvent reaction field potential of an uncharged protein immersed in Simple Point Charge/ Extended (SPC/E) explicit solvent was computed over a series of molecular dynamics trajectories, intotal 1560 ns of simulation time. A finite, positive potential of 13 to 24 k b Te c −1 (where T = 300K), dependent on the geometry of the solvent-accessible surface, was observed inside the biomolecule. The primary contribution to this potential arose from a layer of positive charge density 1.0 Å from the solute surface, on average 0.008 e c /Å 3 , which we found to be the product of a highly ordered first solvation shell. Significant second solvation shell effects, including additional layers of charge density and a slight decrease in the short-range solvent-solvent interaction strength, were also observed. The impact of these findings on implicit solvent models was assessed by running similar explicit-solvent simulations on the fully charged protein system. When the energy due to the solvent reaction field in the uncharged system is accounted for, correlation between per-atom electrostatic energies for the explicit solvent model and a simple implicit (Poisson) calculation is 0.97, and correlation between per-atom energies for the explicit solvent model and a previously published, optimized Poisson model is 0.99.

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“…We remark that initial numerical experiments with larger mesh sizes show that the improvement of the damped-inexact-Newton-multilevel approach over methods grows with the problem size [49].…”

Section: Discussionmentioning

confidence: 85%

Numerical solution of the nonlinear Poisson–Boltzmann equation: Developing more robust and efficient methods

1995

Self Cite

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We present a robust and efficient numerical method for solution of the nonlinear Poisson-Boltzmann equation arising in molecular biophysics. The equation is discretized with the box method, and solution of the discrete equations is accomplished with a global inexact-Newton method, combined with linear multilevel techniques we have described in a paper appearing previously in this journal. A detailed analysis of the resulting method is presented, with comparisons to other methods that have been proposed in the literature, including the classical nonlinear multigrid method, the nonlinear conjugate gradient method, and nonlinear relaxation methods such as successive over-relaxation. Both theoretical and numerical evidence suggests that this method will converge in the case of molecules for which many of the existing methods will not. In addition, for problems which the other methods are able to solve, numerical experiments show that the new method is substantially more efficient, and the superiority of this method grows with the problem size. The method is easy to implement once a linear multilevel solver is available, and can also easily be used in conjunction with linear methods other than multigrid.

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Treatment of electrostatic effects in proteins: Multigrid‐based newton iterative method for solution of the full nonlinear poisson–boltzmann equation

Cited by 82 publications

References 63 publications

AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

Solvent reaction field potential inside an uncharged globular protein: A bridge between implicit and explicit solvent models?

Numerical solution of the nonlinear Poisson–Boltzmann equation: Developing more robust and efficient methods

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