Efficient solution technique for solving the Poisson–Boltzmann equation

Sayyed-Ahmad, Abdallah; Tuncay, Kağan; Ortoleva, P.

doi:10.1002/jcc.20039

Cited by 24 publications

(25 citation statements)

References 27 publications

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“…12-14 for recent reviews͒, among which the multigrid methods seem to be the fastest ones [15][16][17] ͑with the exception of some specialized solvers͒. 18,19 It is noteworthy that most of these programs have been tailored to the PB equation and cannot be used to solve other nonlinear second order PDEs. The successes of the applications of these techniques in biology 14,20 boosted the interest for PoissonBoltzmann in biology in general, including its potential use in molecular simulations.…”

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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“…The detailed physical formulation of the PB model adopts integrated notions from our earlier work 11 and that of others. 43 Mobile ion concentrations are assumed to be at equilibrium.…”

Section: Detailed Physical Formulationmentioning

confidence: 99%

“…52 Modules were written to use b ៝ to construct the coarse-grained dielectric constant tensor ៝ ៝ ‫ء‬ and set up the nonlinear coarse-grained PB equation, solved with the alternating direction implicit ͑ADI͒ method. 11 A list of parameters used in the course of the simulation is shown in Table I. There are universal parameters that are applicable over a large array of nanosystems.…”

Section: Multiscale Computationsmentioning

confidence: 99%

“…IV͒ used for the comparison. 11 The model, and hence the comparison, therefore, does not account for the counter-ions in the Stern ͑tight bound͒ layer, i.e., the Stern Layer ͑of thickness equal to the bare ion radius 53 1.81 Å for Cl − ion͒ is considered ion excluded with ␥ l =0. 53,54 In fact, as discussed later, under conditions where counter-ion concentration in Stern layer becomes important the perturbation series tend to diverge.…”

Section: ͑61͒mentioning

confidence: 99%

“…10 An indirect way of solving the equation is by reformulating the elliptic PB equation as the long-time solution of an advection-diffusion equation. 11 Adaptive finite element techniques have been used to solve the PB equation for large systems. Potentials in specific regions within a system are calculated using boundary conditions derived from a coarse-grained PB solution over the entire system.…”

Section: Introductionmentioning

confidence: 99%

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Multiscale analytic continuation approach to nanosystem simulation: Applications to virus electrostatics

Singharoy

Yesnik

Ortoleva

2010

The Journal of Chemical Physics

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Electrostatic effects in nanosystems are understood via a physical picture built on their multiscale character and the distinct behavior of mobile ions versus charge groups fixed to the nanostructure. The Poisson-Boltzmann equation is nondimensionalized to introduce a factor that measures the density of mobile ion charge versus that due to fixed charges; the diffusive smearing and volume exclusion effects of the former tend to diminish its value relative to that from the fixed charges. We introduce the ratio of the average nearest-neighbor atom distance to the characteristic size of the features of the nanostructure of interest ͑e.g., a viral capsomer͒. We show that a unified treatment ͑i.e., ϰ ͒ and a perturbation expansion around = 0 yields, through analytic continuation, an approximation to the electrostatic potential of high accuracy and computational efficiency. The approach was analyzed via Padé approximants and demonstrated on viral system electrostatics; it can be generalized to accommodate extended Poisson-Boltzmann models, and has wider applicability to nonequilibrium electrodiffusion and many-particle quantum systems.

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