Towards optimal boundary integral formulations of the Poisson–Boltzmann equation for molecular electrostatics

Abstract

The Poisson‐Boltzmann equation offers an efficient way to study electrostatics in molecular settings. Its numerical solution with the boundary element method is widely used, as the complicated molecular surface is accurately represented by the mesh, and the point charges are accounted for explicitly. In fact, there are several well‐known boundary integral formulations available in the literature. This work presents a generalized expression of the boundary integral representation of the implicit solvent model, giving rise to new forms to compute the electrostatic potential. Moreover, it proposes a strategy to build efficient preconditioners for any of the resulting systems, improving the convergence of the linear solver. We perform systematic benchmarking of a set of formulations and preconditioners, focusing on the time to solution, matrix conditioning, and eigenvalue spectrum. We see that the eigenvalue clustering is a good indicator of the matrix conditioning, and show that they can be easily manipulated by scaling the preconditioner. Our results suggest that the optimal choice is problem‐size dependent, where a simpler direct formulation is the fastest for small molecules, but more involved second‐kind equations are better for larger problems. We also present a fast Calderón preconditioner for first‐kind formulations, which shows promising behavior for future analysis. This work sets the basis towards choosing the most convenient boundary integral formulation of the Poisson–Boltzmann equation for a given problem.

Accurate Boundary Integral Formulations for the Calculation of Electrostatic Forces with an Implicit-Solvent Model

“…Here we use the simplest form, termed direct formulation , which is implemented in the code . The direct formulation simply takes the limit of the expressions in eq as x → Γ, leaving ϕ 1 , normalΓ 2 + K normalΓ , normalL normalΓ ( ϕ 1 , Γ ) − V normalΓ , normalL normalΓ ( ∂ ϕ 1 , Γ ∂ n ) = 1 ϵ 1 ∑ k = 1 N q q k 4 π | x Γ − x k | ϕ 1 , normalΓ 2 − K normalΓ , normalY normalΓ ( ϕ 1 , Γ ) + italicϵ 1 <...…”

Accurate boundary-integral formulations for the calculation of electrostatic forces with an implicit-solvent model

“…Combinations of the expressions in Eq. ( 2) yield different boundary integral formulations, 27 that vary in complexity and the conditioning of the resulting matrix. Here we use the simplest form, termed direct formulation, 29 which is implemented in the PBJ code.…”

“…Here we use the simplest form, termed direct formulation, 29 which is implemented in the PBJ code. 27 The direct formulation simply takes the limit of the expressions in Eq. ( 2) as x → Γ, leaving…”

“…Second, we perform a thorough assessment of the accuracy of the force computed with different methods, implemented in the Poisson-Boltzmann & Jupyter (PBJ) code. 27 In the next section we present the implicit solvent model, and how the Poisson-Boltzmann equation is formulated with a boundary integral approach. This section also gives details on the calculation of the energy and force in a Poisson-Boltzmann continuum.…”

Arbitrary-Shape Dielectric Particles Interacting in the Linearized Poisson–Boltzmann Framework: An Analytical Treatment

“…This, however, does not diminish the importance of studying the behavior of the LPBE also in the case of highly charged objects: their electrostatics may still be correctly described at sufficiently long distances (as compared to the Debye length) by the usual DH approximation provided that the sources of the electric field are properly renormalized. − (See also the recent ref for additional comments concerning the ranges of applicability of the DH theory.) This once again emphasizes the importance of a thorough study of the DH approximations, both theoretically and numerically, and justifies the constant stream of works related to the LPBE (see recent refs , , and − , and references therein).…”